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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Rougerie%2C+N&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Rougerie%2C+N&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Rougerie%2C+N&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.13597">arXiv:2412.13597</a> <span> [<a href="https://arxiv.org/pdf/2412.13597">pdf</a>, <a href="https://arxiv.org/format/2412.13597">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> From bosonic canonical ensembles to non-linear Gibbs measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Dinh%2C+v+D">van Duong Dinh</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.13597v1-abstract-short" style="display: inline;"> We study the mean-field limit of the 1D bosonic canonical ensemble in a superharmonic trap. This is the regime with temperature proportional to particle number, both diverging to infinity, and correspondingly scaled interactions. We prove that the limit model is a classical field theory based on a non-linear Schr{枚}dinger-Gibbs measure conditioned on the L2 mass, thereby obtaining a canonical anal… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.13597v1-abstract-full').style.display = 'inline'; document.getElementById('2412.13597v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.13597v1-abstract-full" style="display: none;"> We study the mean-field limit of the 1D bosonic canonical ensemble in a superharmonic trap. This is the regime with temperature proportional to particle number, both diverging to infinity, and correspondingly scaled interactions. We prove that the limit model is a classical field theory based on a non-linear Schr{枚}dinger-Gibbs measure conditioned on the L2 mass, thereby obtaining a canonical analogue of previous results for the grand-canonical ensemble. We take advantage of this set-up with fixed mass to include focusing/attractive interactions/non-linearities in our study. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.13597v1-abstract-full').style.display = 'none'; document.getElementById('2412.13597v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.06670">arXiv:2305.06670</a> <span> [<a href="https://arxiv.org/pdf/2305.06670">pdf</a>, <a href="https://arxiv.org/ps/2305.06670">ps</a>, <a href="https://arxiv.org/format/2305.06670">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> </div> <p class="title is-5 mathjax"> Dimensional reduction for a system of 2D anyons </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Yang%2C+Q">Qiyun Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.06670v2-abstract-short" style="display: inline;"> Anyons with a statistical phase parameter $伪\in(0,2)$ are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave-guide. More specifically, we study the 2D magnetic-gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the $y$-direction… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.06670v2-abstract-full').style.display = 'inline'; document.getElementById('2305.06670v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.06670v2-abstract-full" style="display: none;"> Anyons with a statistical phase parameter $伪\in(0,2)$ are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave-guide. More specifically, we study the 2D magnetic-gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the $y$-direction than in the $x$-direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the $x$-direction is given by the impenetrable Tonks-Girardeau Bose gas, which has no dependency on $伪$, and no trace left of the long-range interactions of the 2D model. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.06670v2-abstract-full').style.display = 'none'; document.getElementById('2305.06670v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Annales Henri Poincar{茅}, In press</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.05220">arXiv:2302.05220</a> <span> [<a href="https://arxiv.org/pdf/2302.05220">pdf</a>, <a href="https://arxiv.org/ps/2302.05220">ps</a>, <a href="https://arxiv.org/format/2302.05220">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Anyons in a tight wave-guide and the Tonks-Girardeau gas </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Yang%2C+Q">Qiyun Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.05220v3-abstract-short" style="display: inline;"> We consider a many-body system of 2D anyons, free quantum particles with general statistics parameter 伪\in ]0,2[. In the magnetic gauge picture they are described as bosons attached to Aharonov-Bohm fluxes of intensity 2 蟺伪generating long-range magnetic forces. A dimensional reduction to 1D is obtained by imposing a strongly anisotropic trapping potential. This freezes the motion in the direction… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.05220v3-abstract-full').style.display = 'inline'; document.getElementById('2302.05220v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.05220v3-abstract-full" style="display: none;"> We consider a many-body system of 2D anyons, free quantum particles with general statistics parameter 伪\in ]0,2[. In the magnetic gauge picture they are described as bosons attached to Aharonov-Bohm fluxes of intensity 2 蟺伪generating long-range magnetic forces. A dimensional reduction to 1D is obtained by imposing a strongly anisotropic trapping potential. This freezes the motion in the direction of strong trapping, leading to 1D physics along the weak direction. The latter is governed to leading order by the Tonks-Girardeau model of impenetrable bosons, independently of 伪. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.05220v3-abstract-full').style.display = 'none'; document.getElementById('2302.05220v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.14215">arXiv:2209.14215</a> <span> [<a href="https://arxiv.org/pdf/2209.14215">pdf</a>, <a href="https://arxiv.org/format/2209.14215">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> </div> </div> <p class="title is-5 mathjax"> Quantum Hall Phases of Cold Bose Gases </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Yngvason%2C+J">Jakob Yngvason</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.14215v1-abstract-short" style="display: inline;"> Cold atomic gases of interacting bosons subject to rapid rotation and confined in anharmonic traps can theoretically exhibit analogues of the fractional quantum Hall effect for electrons in strong magnetic fields. In this setting the Coriolis force due to the rotation mimics the Lorentz force on charged particles but artificial gauge fields can also be obtained by coupling the internal structure o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.14215v1-abstract-full').style.display = 'inline'; document.getElementById('2209.14215v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.14215v1-abstract-full" style="display: none;"> Cold atomic gases of interacting bosons subject to rapid rotation and confined in anharmonic traps can theoretically exhibit analogues of the fractional quantum Hall effect for electrons in strong magnetic fields. In this setting the Coriolis force due to the rotation mimics the Lorentz force on charged particles but artificial gauge fields can also be obtained by coupling the internal structure of the atoms to light fields. The chapter discusses mathematical aspects of transitions to different strongly correlated phases that appear when the parameters of a model Hamiltonian are varied. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.14215v1-abstract-full').style.display = 'none'; document.getElementById('2209.14215v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, chapter for Encyclopedia of Condensed Matter Physics, 2nd edition (Elsevier). arXiv admin note: substantial text overlap with arXiv:1402.0706</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81V70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.02543">arXiv:2209.02543</a> <span> [<a href="https://arxiv.org/pdf/2209.02543">pdf</a>, <a href="https://arxiv.org/ps/2209.02543">ps</a>, <a href="https://arxiv.org/format/2209.02543">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11005-022-01627-x">10.1007/s11005-022-01627-x <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Lieb-Thirring inequality for extended anyons </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Girardot%2C+T">Th茅otime Girardot</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.02543v3-abstract-short" style="display: inline;"> We derive a Pauli exclusion principle for extended fermion-based anyons of any positive radius and any non-trivial statistics parameter. That is, we consider 2D fermionic particles coupled to magnetic flux tubes of non-zero radius, and prove a Lieb-Thirring inequality for the corresponding many-body kinetic energy operator. The implied constant is independent of the radius of the flux tubes, and p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.02543v3-abstract-full').style.display = 'inline'; document.getElementById('2209.02543v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.02543v3-abstract-full" style="display: none;"> We derive a Pauli exclusion principle for extended fermion-based anyons of any positive radius and any non-trivial statistics parameter. That is, we consider 2D fermionic particles coupled to magnetic flux tubes of non-zero radius, and prove a Lieb-Thirring inequality for the corresponding many-body kinetic energy operator. The implied constant is independent of the radius of the flux tubes, and proportional to the statistics parameter. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.02543v3-abstract-full').style.display = 'none'; document.getElementById('2209.02543v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.08317">arXiv:2208.08317</a> <span> [<a href="https://arxiv.org/pdf/2208.08317">pdf</a>, <a href="https://arxiv.org/ps/2208.08317">ps</a>, <a href="https://arxiv.org/format/2208.08317">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Blow-up of 2D attractive Bose-Einstein condensates at the crittical rotational speed </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Dinh%2C+V+D">Van Duong Dinh</a>, <a href="/search/math-ph?searchtype=author&query=Nguyen%2C+D">Dinh-Thi Nguyen</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.08317v1-abstract-short" style="display: inline;"> We study the ground states of a 2D focusing non-linear Schr枚dinger equation with rotation and harmonic trapping. When the strength of the interaction approaches a critical value from below, the system collapses to a profile obtained from the optimizer of a Gagliardo--Nirenberg interpolation inequality. This was established before in the case of fixed rotation frequency. We extend the result to rot… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.08317v1-abstract-full').style.display = 'inline'; document.getElementById('2208.08317v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.08317v1-abstract-full" style="display: none;"> We study the ground states of a 2D focusing non-linear Schr枚dinger equation with rotation and harmonic trapping. When the strength of the interaction approaches a critical value from below, the system collapses to a profile obtained from the optimizer of a Gagliardo--Nirenberg interpolation inequality. This was established before in the case of fixed rotation frequency. We extend the result to rotation frequencies approaching, or even equal to, the critical frequency at which the centrifugal force compensates the trap. We prove that the blow-up scenario is to leading order unaffected by such a strong deconfinement mechanism. In particular the blow-up profile remains independent of the rotation frequency. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.08317v1-abstract-full').style.display = 'none'; document.getElementById('2208.08317v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q40; 35Q51; 35Q55 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.02619">arXiv:2208.02619</a> <span> [<a href="https://arxiv.org/pdf/2208.02619">pdf</a>, <a href="https://arxiv.org/format/2208.02619">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Strongly Correlated Electrons">cond-mat.str-el</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> On the stability of Laughlin's fractional quantum Hall phase </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.02619v2-abstract-short" style="display: inline;"> The fractional quantum Hall effect in 2D electron gases submitted to large magnetic fields remains one of the most striking phenomena in condensed matter physics. Historically, the first observed signature is a Hall resistance quantized to the value (2m+1) when the filling factor (electron density divided by magnetic flux quantum density) of a 2D electron gas is in the vicinity of an inverse odd i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.02619v2-abstract-full').style.display = 'inline'; document.getElementById('2208.02619v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.02619v2-abstract-full" style="display: none;"> The fractional quantum Hall effect in 2D electron gases submitted to large magnetic fields remains one of the most striking phenomena in condensed matter physics. Historically, the first observed signature is a Hall resistance quantized to the value (2m+1) when the filling factor (electron density divided by magnetic flux quantum density) of a 2D electron gas is in the vicinity of an inverse odd integer 1/(2m +1). This was one of the first observation of fractional quantum numbers. A large part of our basic theoretical understanding of this effect (and descendants) originates from Laughlin's theory of 1983, reviewed here from a mathematical physics perspective. We explain in which sense Laughlin's proposed ground and excited states for the system are rigid/incompressible liquids, and why this is crucial for the explanation of the effect. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.02619v2-abstract-full').style.display = 'none'; document.getElementById('2208.02619v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Contribution to the second edition of the Encyclopedia of condensed matter physics. Partially based on two previous review texts by the same author. arXiv admin note: text overlap with arXiv:2203.06952</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.06952">arXiv:2203.06952</a> <span> [<a href="https://arxiv.org/pdf/2203.06952">pdf</a>, <a href="https://arxiv.org/format/2203.06952">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> The classical Jellium and the Laughlin phase </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.06952v1-abstract-short" style="display: inline;"> I discuss results bearing on a variational problem of a new type, inspired by fractional quantum Hall physics. In the latter context, the main result reviewed herein can be spelled as "the phase of independent quasi-holes generated from Laughlin's wave-function is stable against external potentials and weak long-range interactions". The main ingredient of the proof is a connection between fraction… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.06952v1-abstract-full').style.display = 'inline'; document.getElementById('2203.06952v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.06952v1-abstract-full" style="display: none;"> I discuss results bearing on a variational problem of a new type, inspired by fractional quantum Hall physics. In the latter context, the main result reviewed herein can be spelled as "the phase of independent quasi-holes generated from Laughlin's wave-function is stable against external potentials and weak long-range interactions". The main ingredient of the proof is a connection between fractional quantum Hall wave-functions and statistical mechanics problems that generalize the 2D one-component plasma (jellium model). Universal bounds on the density of such systems, coined "Incompressibility estimates" are obtained via the construction of screening regions for any configuration of points with positive electric charges. The latter regions are patches of constant, negative electric charge density, whose shape is optimized for the total system (points plus patch) not to generate any electric potential in its exterior. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.06952v1-abstract-full').style.display = 'none'; document.getElementById('2203.06952v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.04418">arXiv:2201.04418</a> <span> [<a href="https://arxiv.org/pdf/2201.04418">pdf</a>, <a href="https://arxiv.org/ps/2201.04418">ps</a>, <a href="https://arxiv.org/format/2201.04418">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/paa.2022.4.535">10.2140/paa.2022.4.535 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Thomas-Fermi profile of a fast rotating Bose-Einstein condensate </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Nguyen%2C+D">Dinh-Thi Nguyen</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.04418v3-abstract-short" style="display: inline;"> We study the minimizers of a magnetic 2D non-linear Schr枚dinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effectiv… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.04418v3-abstract-full').style.display = 'inline'; document.getElementById('2201.04418v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.04418v3-abstract-full" style="display: none;"> We study the minimizers of a magnetic 2D non-linear Schr枚dinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effective Thomas-Fermi functional is linked to the emergence of vortex lattices (the Abrikosov problem). We define it via a low density expansion of the energy of the corresponding homogeneous gas in the thermodynamic limit. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.04418v3-abstract-full').style.display = 'none'; document.getElementById('2201.04418v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">The previous version, submitted in August 2022, had an unphysical restriction on the allowed scaling of physical parameters of the problem. We bypass this by a kind of bootstrap argument. We also correct a mistake in the proof of the density convergence in the LLL case</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q40; 81V70; 81S05; 46N50 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Pure Appl. Analysis 4 (2022) 535-569 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.03518">arXiv:2201.03518</a> <span> [<a href="https://arxiv.org/pdf/2201.03518">pdf</a>, <a href="https://arxiv.org/ps/2201.03518">ps</a>, <a href="https://arxiv.org/format/2201.03518">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Strongly Correlated Electrons">cond-mat.str-el</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/pmp.2023.4.803">10.2140/pmp.2023.4.803 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum statistics transmutation via magnetic flux attachment </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lambert%2C+G">Gaultier Lambert</a>, <a href="/search/math-ph?searchtype=author&query=Lundholm%2C+D">Douglas Lundholm</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.03518v3-abstract-short" style="display: inline;"> We consider a model for two types (bath and tracers) of 2D quantum particles in a perpendicular magnetic field. Interactions are short range and inter-species, and we assume that the bath particles are fermions, all lying in the lowest Landau level of the magnetic field. Heuristic arguments then indicate that, if the tracers are strongly coupled to the bath, they effectively change their quantum s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.03518v3-abstract-full').style.display = 'inline'; document.getElementById('2201.03518v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.03518v3-abstract-full" style="display: none;"> We consider a model for two types (bath and tracers) of 2D quantum particles in a perpendicular magnetic field. Interactions are short range and inter-species, and we assume that the bath particles are fermions, all lying in the lowest Landau level of the magnetic field. Heuristic arguments then indicate that, if the tracers are strongly coupled to the bath, they effectively change their quantum statistics, from bosonic to fermionic or vice-versa. We rigorously compute the energy of a natural trial state, indeed exhibiting this phenomenon of statistics transmutation. The proof involves estimates for the characteristic polynomial of the Ginibre ensemble of random matrices. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.03518v3-abstract-full').style.display = 'none'; document.getElementById('2201.03518v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">49 pages. Minor corrections, clarifications and updated references</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 81V70; 81V27; 32A70 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Prob. Math. Phys. 4 (2023) 803-848 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.08690">arXiv:2101.08690</a> <span> [<a href="https://arxiv.org/pdf/2101.08690">pdf</a>, <a href="https://arxiv.org/ps/2101.08690">ps</a>, <a href="https://arxiv.org/format/2101.08690">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/apde.2023.16.1885">10.2140/apde.2023.16.1885 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Bosons in a double well: two-mode approximation and fluctuations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Olgiati%2C+A">Alessandro Olgiati</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Spehner%2C+D">Dominique Spehner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.08690v3-abstract-short" style="display: inline;"> We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity. Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose-Hubbard Hamiltonian. We prove (i) an energy expansion, includi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.08690v3-abstract-full').style.display = 'inline'; document.getElementById('2101.08690v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.08690v3-abstract-full" style="display: none;"> We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity. Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose-Hubbard Hamiltonian. We prove (i) an energy expansion, including the two-modes Bose-Hubbard energy and two independent Bogoliubov corrections (one for each potential well), (ii) a variance bound for the number of particles falling inside each potential well. The latter is a signature of a correlated ground state in that it violates the central limit theorem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.08690v3-abstract-full').style.display = 'none'; document.getElementById('2101.08690v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Analysis & PDE 16 (2023) 1885-1954 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.04457">arXiv:2101.04457</a> <span> [<a href="https://arxiv.org/pdf/2101.04457">pdf</a>, <a href="https://arxiv.org/ps/2101.04457">ps</a>, <a href="https://arxiv.org/format/2101.04457">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00220-021-04164-1">10.1007/s00220-021-04164-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Semiclassical limit for almost fermionic anyons </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Girardot%2C+T">Th茅otime Girardot</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.04457v4-abstract-short" style="display: inline;"> In two-dimensional space there are possibilities for quantum statistics continuously interpolating between the bosonic and the fermionic one. Quasi-particles obeying such statistics can be described as ordinary bosons and fermions with magnetic interactions. We study a limit situation where the statistics/magnetic interaction is seen as a "perturbation from the fermionic end". We vindicate a mean-… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.04457v4-abstract-full').style.display = 'inline'; document.getElementById('2101.04457v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.04457v4-abstract-full" style="display: none;"> In two-dimensional space there are possibilities for quantum statistics continuously interpolating between the bosonic and the fermionic one. Quasi-particles obeying such statistics can be described as ordinary bosons and fermions with magnetic interactions. We study a limit situation where the statistics/magnetic interaction is seen as a "perturbation from the fermionic end". We vindicate a mean-field approximation, proving that the ground state of a gas of anyons is described to leading order by a semi-classical, Vlasov-like, energy functional. The ground state of the latter displays anyonic behavior in its momentum distribution. Our proof is based on coherent states, Husimi functions, the Diaconis-Freedman theorem and a quantitative version of a semi-classical Pauli pinciple. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.04457v4-abstract-full').style.display = 'none'; document.getElementById('2101.04457v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2004.14729">arXiv:2004.14729</a> <span> [<a href="https://arxiv.org/pdf/2004.14729">pdf</a>, <a href="https://arxiv.org/ps/2004.14729">ps</a>, <a href="https://arxiv.org/format/2004.14729">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> The Hartree functional in a double well </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Olgiati%2C+A">Alessandro Olgiati</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2004.14729v4-abstract-short" style="display: inline;"> We consider a non-linear Hartree energy for bosonic particles in a symmetric double-well potential. In the limit where the wells are fare apart and the potential barrier is high, we prove that the ground state and first excited state are given to leading order by an even, respectively odd, superposition of ground states in single wells. We evaluate the resulting tunneling term splitting the corres… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.14729v4-abstract-full').style.display = 'inline'; document.getElementById('2004.14729v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2004.14729v4-abstract-full" style="display: none;"> We consider a non-linear Hartree energy for bosonic particles in a symmetric double-well potential. In the limit where the wells are fare apart and the potential barrier is high, we prove that the ground state and first excited state are given to leading order by an even, respectively odd, superposition of ground states in single wells. We evaluate the resulting tunneling term splitting the corresponding energies precisely. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.14729v4-abstract-full').style.display = 'none'; document.getElementById('2004.14729v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">The results are inputs in a forthcoming joint work with Dominique Spehner</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.02678">arXiv:2002.02678</a> <span> [<a href="https://arxiv.org/pdf/2002.02678">pdf</a>, <a href="https://arxiv.org/ps/2002.02678">ps</a>, <a href="https://arxiv.org/format/2002.02678">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Scaling limits of bosonic ground states, from many-body to nonlinear Schr{枚}dinger </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2002.02678v2-abstract-short" style="display: inline;"> How and why may an interacting system of many particles be described assuming that all particles are independent and identically distributed ? This question is at least as old as statistical mechanics itself. Its quantum version has been rejuvenated by the birth of cold atoms physics. In particular the experimental creation of Bose-Einstein condensates directly asks the following variant: why and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.02678v2-abstract-full').style.display = 'inline'; document.getElementById('2002.02678v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.02678v2-abstract-full" style="display: none;"> How and why may an interacting system of many particles be described assuming that all particles are independent and identically distributed ? This question is at least as old as statistical mechanics itself. Its quantum version has been rejuvenated by the birth of cold atoms physics. In particular the experimental creation of Bose-Einstein condensates directly asks the following variant: why and how can a large assembly of very cold interacting bosons (quantum particles deprived of the Pauli exclusion principle) all populate the same quantum state ? In this text I review the various mathematical techniques allowing to prove that the lowest energy state of a bosonic system forms, in a reasonable macroscopic limit of large particle number, a Bose-Einstein condensate. This means that indeed in the relevant limit all particles approximately behave as if independent and identically distributed, according to a law determined by minimizing a non-linear Schr{枚}dinger energy functional. This is a particular instance of the justification of the mean-field approximation in statistical mechanics, starting from the basic many-body Schr{枚}dinger Hamiltonian. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.02678v2-abstract-full').style.display = 'none'; document.getElementById('2002.02678v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> EMS Surveys in Mathematical Sciences, EMS, In press </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1912.10904">arXiv:1912.10904</a> <span> [<a href="https://arxiv.org/pdf/1912.10904">pdf</a>, <a href="https://arxiv.org/ps/1912.10904">ps</a>, <a href="https://arxiv.org/format/1912.10904">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Strongly Correlated Electrons">cond-mat.str-el</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/5.0004111">10.1063/5.0004111 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Holomorphic quantum Hall states in higher Landau levels </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Yngvason%2C+J">Jakob Yngvason</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1912.10904v3-abstract-short" style="display: inline;"> Eigenstates of the planar magnetic Laplacian with homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems, in particular we represent effective Hamiltonians and particle densiti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.10904v3-abstract-full').style.display = 'inline'; document.getElementById('1912.10904v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1912.10904v3-abstract-full" style="display: none;"> Eigenstates of the planar magnetic Laplacian with homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems, in particular we represent effective Hamiltonians and particle densities in higher Landau levels by corresponding quantities in the lowest Landau level. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.10904v3-abstract-full').style.display = 'none'; document.getElementById('1912.10904v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 December, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version. To be published in Journal of Mathematical Physics</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1912.07890">arXiv:1912.07890</a> <span> [<a href="https://arxiv.org/pdf/1912.07890">pdf</a>, <a href="https://arxiv.org/format/1912.07890">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Strongly Correlated Electrons">cond-mat.str-el</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1103/PhysRevB.102.144109">10.1103/PhysRevB.102.144109 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Quantum Impurity Model for Anyons </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Yakaboylu%2C+E">Enderalp Yakaboylu</a>, <a href="/search/math-ph?searchtype=author&query=Ghazaryan%2C+A">Areg Ghazaryan</a>, <a href="/search/math-ph?searchtype=author&query=Lundholm%2C+D">Douglas Lundholm</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Lemeshko%2C+M">Mikhail Lemeshko</a>, <a href="/search/math-ph?searchtype=author&query=Seiringer%2C+R">Robert Seiringer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1912.07890v2-abstract-short" style="display: inline;"> One of the hallmarks of quantum statistics, tightly entwined with the concept of topological phases of matter, is the prediction of anyons. Although anyons are predicted to be realized in certain fractional quantum Hall systems, they have not yet been unambiguously detected in experiment. Here we introduce a simple quantum impurity model, where bosonic or fermionic impurities turn into anyons as a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.07890v2-abstract-full').style.display = 'inline'; document.getElementById('1912.07890v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1912.07890v2-abstract-full" style="display: none;"> One of the hallmarks of quantum statistics, tightly entwined with the concept of topological phases of matter, is the prediction of anyons. Although anyons are predicted to be realized in certain fractional quantum Hall systems, they have not yet been unambiguously detected in experiment. Here we introduce a simple quantum impurity model, where bosonic or fermionic impurities turn into anyons as a consequence of their interaction with the surrounding many-particle bath. A cloud of phonons dresses each impurity in such a way that it effectively attaches fluxes/vortices to it and thereby converts it into an Abelian anyon. The corresponding quantum impurity model, first, provides a new approach to the numerical solution of the many-anyon problem, along with a new concrete perspective of anyons as emergent quasiparticles built from composite bosons or fermions. More importantly, the model paves the way towards realizing anyons using impurities in crystal lattices as well as ultracold gases. In particular, we consider two heavy electrons interacting with a two-dimensional lattice crystal in a magnetic field, and show that when the impurity-bath system is rotated at the cyclotron frequency, impurities behave as anyons as a consequence of the angular momentum exchange between the impurities and the bath. A possible experimental realization is proposed by identifying the statistics parameter in terms of the mean square distance of the impurities and the magnetization of the impurity-bath system, both of which are accessible to experiment. Another proposed application are impurities immersed in a two-dimensional weakly interacting Bose gas. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.07890v2-abstract-full').style.display = 'none'; document.getElementById('1912.07890v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 December, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys. Rev. B 102, 144109 (2020) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1912.03150">arXiv:1912.03150</a> <span> [<a href="https://arxiv.org/pdf/1912.03150">pdf</a>, <a href="https://arxiv.org/ps/1912.03150">ps</a>, <a href="https://arxiv.org/format/1912.03150">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> On two properties of the Fisher information </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1912.03150v3-abstract-short" style="display: inline;"> Alternative proofs for the superadditivity and the affinity (in the large system limit) of the usual and some fractional Fisher informations of a probability density of many variables are provided. They are consequences of the fact that such informations can be interpreted as quantum kinetic energies. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1912.03150v3-abstract-full" style="display: none;"> Alternative proofs for the superadditivity and the affinity (in the large system limit) of the usual and some fractional Fisher informations of a probability density of many variables are provided. They are consequences of the fact that such informations can be interpreted as quantum kinetic energies. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.03150v3-abstract-full').style.display = 'none'; document.getElementById('1912.03150v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 December, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Revised version</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.08902">arXiv:1909.08902</a> <span> [<a href="https://arxiv.org/pdf/1909.08902">pdf</a>, <a href="https://arxiv.org/ps/1909.08902">ps</a>, <a href="https://arxiv.org/format/1909.08902">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Improved stability for 2D attractive Bose gases </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1909.08902v2-abstract-short" style="display: inline;"> We study the ground-state energy of N attractive bosons in the plane. The interaction is scaled for the gas to be dilute, so that the corresponding mean-field problem is a local non-linear Schr{枚}dinger (NLS) equation. We improve the conditions under which one can prove that the many-body problem is stable (of the second kind). This implies, using previous results, that the many-body ground states… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.08902v2-abstract-full').style.display = 'inline'; document.getElementById('1909.08902v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.08902v2-abstract-full" style="display: none;"> We study the ground-state energy of N attractive bosons in the plane. The interaction is scaled for the gas to be dilute, so that the corresponding mean-field problem is a local non-linear Schr{枚}dinger (NLS) equation. We improve the conditions under which one can prove that the many-body problem is stable (of the second kind). This implies, using previous results, that the many-body ground states and dynamics converge to the NLS ones for an extended range of diluteness parameters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.08902v2-abstract-full').style.display = 'none'; document.getElementById('1909.08902v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1906.11656">arXiv:1906.11656</a> <span> [<a href="https://arxiv.org/pdf/1906.11656">pdf</a>, <a href="https://arxiv.org/format/1906.11656">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Strongly Correlated Electrons">cond-mat.str-el</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> On the Laughlin function and its perturbations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1906.11656v2-abstract-short" style="display: inline;"> The Laughlin state is an ansatz for the ground state of a system of 2D quantum particles submitted to a strong magnetic field and strong interactions. The two effects conspire to generate strong and very specific correlations between the particles. I present a mathematical approach to the rigidity these correlations display in their response to perturbations. This is an important ingredient in the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.11656v2-abstract-full').style.display = 'inline'; document.getElementById('1906.11656v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.11656v2-abstract-full" style="display: none;"> The Laughlin state is an ansatz for the ground state of a system of 2D quantum particles submitted to a strong magnetic field and strong interactions. The two effects conspire to generate strong and very specific correlations between the particles. I present a mathematical approach to the rigidity these correlations display in their response to perturbations. This is an important ingredient in the theory of the fractional quantum Hall effect. The main message is that potentials generated by impurities and residual interactions can be taken into account by generating uncorrelated quasi-holes on top of Laughlin's wave-function. An appendix contains a conjecture (not due to me) that should be regarded as a major open mathematical problem of the field, relating to the spectral gap of a certain zero-range interaction. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.11656v2-abstract-full').style.display = 'none'; document.getElementById('1906.11656v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Expository text based on joint works with Elliott H. Lieb, Alessandro Olgiati, Sylvia Serfaty and Jakob Yngvason</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1906.05564">arXiv:1906.05564</a> <span> [<a href="https://arxiv.org/pdf/1906.05564">pdf</a>, <a href="https://arxiv.org/ps/1906.05564">ps</a>, <a href="https://arxiv.org/format/1906.05564">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Strongly Correlated Electrons">cond-mat.str-el</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00205-020-01536-0">10.1007/s00205-020-01536-0 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Stability of the Laughlin phase against long-range interactions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Olgiati%2C+A">Alessandro Olgiati</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1906.05564v2-abstract-short" style="display: inline;"> A natural, "perturbative", problem in the modelization of the fractional quantum Hall effect is to minimize a classical energy functional within a variational set based on Laughlin's wave-function. We prove that, for small enough pair interactions, and asymptotically for large particle numbers, a minimizer can always be looked for in the particular form of uncorrelated quasi-holes superimposed to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.05564v2-abstract-full').style.display = 'inline'; document.getElementById('1906.05564v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.05564v2-abstract-full" style="display: none;"> A natural, "perturbative", problem in the modelization of the fractional quantum Hall effect is to minimize a classical energy functional within a variational set based on Laughlin's wave-function. We prove that, for small enough pair interactions, and asymptotically for large particle numbers, a minimizer can always be looked for in the particular form of uncorrelated quasi-holes superimposed to Laughlin's wave-function. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.05564v2-abstract-full').style.display = 'none'; document.getElementById('1906.05564v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.01271">arXiv:1903.01271</a> <span> [<a href="https://arxiv.org/pdf/1903.01271">pdf</a>, <a href="https://arxiv.org/ps/1903.01271">ps</a>, <a href="https://arxiv.org/format/1903.01271">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.5094331">10.1063/1.5094331 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lewin%2C+M">Mathieu Lewin</a>, <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1903.01271v2-abstract-short" style="display: inline;"> We review our recent result on the rigorous derivation of the renormalized Gibbs measure from the many-body Gibbs state in 1D and 2D. The many-body renormalization is accomplished by simply tuning the chemical potential in the grand-canonical ensemble, which is analogous to the Wick ordering in the classical field theory. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.01271v2-abstract-full" style="display: none;"> We review our recent result on the rigorous derivation of the renormalized Gibbs measure from the many-body Gibbs state in 1D and 2D. The many-body renormalization is accomplished by simply tuning the chemical potential in the grand-canonical ensemble, which is analogous to the Wick ordering in the classical field theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.01271v2-abstract-full').style.display = 'none'; document.getElementById('1903.01271v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 May, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Contribution to Proceedings of the International Congress of Mathematical Physics, Montreal, Canada, July 23-28, 2018</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1901.10739">arXiv:1901.10739</a> <span> [<a href="https://arxiv.org/pdf/1901.10739">pdf</a>, <a href="https://arxiv.org/format/1901.10739">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1209/0295-5075/126/20005">10.1209/0295-5075/126/20005 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Vortex patterns in the almost-bosonic anyon gas </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Correggi%2C+M">Michele Correggi</a>, <a href="/search/math-ph?searchtype=author&query=Duboscq%2C+R">Romain Duboscq</a>, <a href="/search/math-ph?searchtype=author&query=Lundholm%2C+D">Douglas Lundholm</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1901.10739v2-abstract-short" style="display: inline;"> We study theoretically and numerically the ground state of a gas of 2D abelian anyons in an external trapping potential. We treat anyon statistics in the magnetic gauge picture, perturbatively around the bosonic end. This leads to a mean-field energy functional, whose ground state displays vortex lattices similar to those found in rotating Bose-Einstein condensates. A crucial difference is however… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1901.10739v2-abstract-full').style.display = 'inline'; document.getElementById('1901.10739v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1901.10739v2-abstract-full" style="display: none;"> We study theoretically and numerically the ground state of a gas of 2D abelian anyons in an external trapping potential. We treat anyon statistics in the magnetic gauge picture, perturbatively around the bosonic end. This leads to a mean-field energy functional, whose ground state displays vortex lattices similar to those found in rotating Bose-Einstein condensates. A crucial difference is however that the vortex density is proportional to the underlying matter density of the gas. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1901.10739v2-abstract-full').style.display = 'none'; document.getElementById('1901.10739v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 May, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 January, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Europhys. Lett. 126 (2019), 20005 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1901.09561">arXiv:1901.09561</a> <span> [<a href="https://arxiv.org/pdf/1901.09561">pdf</a>, <a href="https://arxiv.org/ps/1901.09561">ps</a>, <a href="https://arxiv.org/format/1901.09561">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Non linear Schr{枚}dinger limit of bosonic ground states, again </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1901.09561v2-abstract-short" style="display: inline;"> I review an information-theoretic variant of the quantum de Finetti theorem due to Brand{茫}o and Harrow and discuss its applications to the topic of bosonic mean-field limits. This leads to slightly improved methods for the derivation of the local non-linear Schr{枚}dinger energy functional from many-body quantum mechanics. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1901.09561v2-abstract-full" style="display: none;"> I review an information-theoretic variant of the quantum de Finetti theorem due to Brand{茫}o and Harrow and discuss its applications to the topic of bosonic mean-field limits. This leads to slightly improved methods for the derivation of the local non-linear Schr{枚}dinger energy functional from many-body quantum mechanics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1901.09561v2-abstract-full').style.display = 'none'; document.getElementById('1901.09561v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 January, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.06755">arXiv:1811.06755</a> <span> [<a href="https://arxiv.org/pdf/1811.06755">pdf</a>, <a href="https://arxiv.org/ps/1811.06755">ps</a>, <a href="https://arxiv.org/format/1811.06755">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Limites de champ moyen bosoniques {脿} temp{茅}rature positive </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.06755v3-abstract-short" style="display: inline;"> I present recent results in quantum statistical mechanics, obtained in joint works with Mathieu Lewin and Phan Th{脿}nh Nam. We consider a certain mean-field limit of the grand-canonical ensemble for a Bose gas at positive temperature. In this limit, the reduced density matrices of the quantum theory converge to their analogues in classical field theory, given by a non-linear Gibbs measure. In part… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.06755v3-abstract-full').style.display = 'inline'; document.getElementById('1811.06755v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.06755v3-abstract-full" style="display: none;"> I present recent results in quantum statistical mechanics, obtained in joint works with Mathieu Lewin and Phan Th{脿}nh Nam. We consider a certain mean-field limit of the grand-canonical ensemble for a Bose gas at positive temperature. In this limit, the reduced density matrices of the quantum theory converge to their analogues in classical field theory, given by a non-linear Gibbs measure. In particular, we deal with cases where the latter must be defined via a renormalization procedure. The corresponding renormalization at the level of the original quantum grand-canonical model, with non-commuting fields, is one of the important difficulties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.06755v3-abstract-full').style.display = 'none'; document.getElementById('1811.06755v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 May, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Text in french, intended for the proceedings of the second congress of the french mathematical society (Lille, June 2018)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.08370">arXiv:1810.08370</a> <span> [<a href="https://arxiv.org/pdf/1810.08370">pdf</a>, <a href="https://arxiv.org/ps/1810.08370">ps</a>, <a href="https://arxiv.org/format/1810.08370">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Classical field theory limit of many-body quantum Gibbs states in 2D and 3D </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lewin%2C+M">Mathieu Lewin</a>, <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.08370v5-abstract-short" style="display: inline;"> We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schr{枚}dinger-type classical field theory, in terms of partition functions and reduced density mat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.08370v5-abstract-full').style.display = 'inline'; document.getElementById('1810.08370v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.08370v5-abstract-full" style="display: none;"> We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schr{枚}dinger-type classical field theory, in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose-Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.08370v5-abstract-full').style.display = 'none'; document.getElementById('1810.08370v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This revised version (to appear in Inventiones Mathematicae) covers both the 2D and 3D cases. It replaces an older 2018 work that was limited to 2D. The older, non-refereed, version is accessible as v1-v2 of the preprint</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1809.07085">arXiv:1809.07085</a> <span> [<a href="https://arxiv.org/pdf/1809.07085">pdf</a>, <a href="https://arxiv.org/ps/1809.07085">ps</a>, <a href="https://arxiv.org/format/1809.07085">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> On the stability of 2D dipolar Bose-Einstein condensates </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Eychenne%2C+A">Arnaud Eychenne</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1809.07085v2-abstract-short" style="display: inline;"> We study the existence of energy minimizers for a Bose-Einstein condensate with dipole-dipole interactions, tightly confined to a plane. The problem is critical in that the kinetic energy and the (partially attractive) interaction energy behave the same under mass-preserving scalings of the wave-function. We obtain a sharp criterion for the existence of ground states, involving the optimal constan… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.07085v2-abstract-full').style.display = 'inline'; document.getElementById('1809.07085v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1809.07085v2-abstract-full" style="display: none;"> We study the existence of energy minimizers for a Bose-Einstein condensate with dipole-dipole interactions, tightly confined to a plane. The problem is critical in that the kinetic energy and the (partially attractive) interaction energy behave the same under mass-preserving scalings of the wave-function. We obtain a sharp criterion for the existence of ground states, involving the optimal constant of a certain generalized Gagliardo-Nirenberg inequality. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.07085v2-abstract-full').style.display = 'none'; document.getElementById('1809.07085v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 September, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 September, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1805.03506">arXiv:1805.03506</a> <span> [<a href="https://arxiv.org/pdf/1805.03506">pdf</a>, <a href="https://arxiv.org/ps/1805.03506">ps</a>, <a href="https://arxiv.org/format/1805.03506">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Physics">physics.comp-ph</span> </div> </div> <p class="title is-5 mathjax"> The interacting 2D Bose gas and nonlinear Gibbs measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lewin%2C+M">Mathieu Lewin</a>, <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1805.03506v1-abstract-short" style="display: inline;"> We announce a new theorem bearing on high-temperature 2D Bose gases. In a certain mean-field-like regime, the grand-canonical quantum Gibbs state reduces to a nonlinear Gibbs measure constructed from a renormalized mean-field energy functional. This establishes a rigorous connection between quantum fields and (singular) classical fields in the regime of interest. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1805.03506v1-abstract-full" style="display: none;"> We announce a new theorem bearing on high-temperature 2D Bose gases. In a certain mean-field-like regime, the grand-canonical quantum Gibbs state reduces to a nonlinear Gibbs measure constructed from a renormalized mean-field energy functional. This establishes a rigorous connection between quantum fields and (singular) classical fields in the regime of interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.03506v1-abstract-full').style.display = 'none'; document.getElementById('1805.03506v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">The result was announced at the Oberwolfach mini-workshop "Gibbs measures for nonlinear dispersive equations" organized by Giuseppe Genovese, Benjamin Schlein and Vedran Sohinger</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1802.01854">arXiv:1802.01854</a> <span> [<a href="https://arxiv.org/pdf/1802.01854">pdf</a>, <a href="https://arxiv.org/ps/1802.01854">ps</a>, <a href="https://arxiv.org/format/1802.01854">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Blow-up profile of rotating 2D focusing Bose gases </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lewin%2C+M">Mathieu Lewin</a>, <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1802.01854v2-abstract-short" style="display: inline;"> We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation $惟$. First we study the behavior of the ground state when the coupling constant approaches $a\_*$ , the critical strength of the cubic nonlinearity for the focusing nonlinear Schr{枚}dinger equation. We prove… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.01854v2-abstract-full').style.display = 'inline'; document.getElementById('1802.01854v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1802.01854v2-abstract-full" style="display: none;"> We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation $惟$. First we study the behavior of the ground state when the coupling constant approaches $a\_*$ , the critical strength of the cubic nonlinearity for the focusing nonlinear Schr{枚}dinger equation. We prove that blow-up always happens at the center of the trap, with the blow-up profile given by the Gagliardo-Nirenberg solution. In particular, the blow-up scenario is independent of $惟$, to leading order. This generalizes results obtained by Guo and Seiringer (Lett. Math. Phys., 2014, vol. 104, p. 141--156) in the non-rotating case. In a second part we consider the many-particle Hamiltonian for $N$ bosons, interacting with a potential rescaled in the mean-field manner $--a\_N N^{2尾--1} w(N^尾 x), with $w$ a positive function such that $\int\_{\mathbb{R}^2} w(x) dx = 1$. Assuming that $尾< 1/2$ and that $a\_N \to a\_*$ sufficiently slowly, we prove that the many-body system is fully condensed on the Gross-Pitaevskii ground state in the limit $N \to \infty$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.01854v2-abstract-full').style.display = 'none'; document.getElementById('1802.01854v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 February, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1707.05059">arXiv:1707.05059</a> <span> [<a href="https://arxiv.org/pdf/1707.05059">pdf</a>, <a href="https://arxiv.org/ps/1707.05059">ps</a>, <a href="https://arxiv.org/format/1707.05059">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11005-017-1020-5">10.1007/s11005-017-1020-5 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Laughlin liquid in an external potential </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Yngvason%2C+J">Jakob Yngvason</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1707.05059v2-abstract-short" style="display: inline;"> We study natural perturbations of the Laughlin state arising from the effects of trapping and disorder. These are N-particle wave functions that have the form of a product of Laughlin states and analytic functions of the N variables. We derive an upper bound to the ground state energy in a confining external potential, matching exactly a recently derived lower bound in the large N limit. Irrespect… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.05059v2-abstract-full').style.display = 'inline'; document.getElementById('1707.05059v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1707.05059v2-abstract-full" style="display: none;"> We study natural perturbations of the Laughlin state arising from the effects of trapping and disorder. These are N-particle wave functions that have the form of a product of Laughlin states and analytic functions of the N variables. We derive an upper bound to the ground state energy in a confining external potential, matching exactly a recently derived lower bound in the large N limit. Irrespective of the shape of the confining potential, this sharp upper bound can be achieved through a modification of the Laughlin function by suitably arranged quasi-holes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.05059v2-abstract-full').style.display = 'none'; document.getElementById('1707.05059v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Typos corrected and one remark added. To be published in Letters in Mathematical Physics</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1706.00654">arXiv:1706.00654</a> <span> [<a href="https://arxiv.org/pdf/1706.00654">pdf</a>, <a href="https://arxiv.org/format/1706.00654">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Superconductivity">cond-mat.supr-con</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1140/epjb/e2017-80498-3">10.1140/epjb/e2017-80498-3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Universal and shape dependent features of surface superconductivity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Correggi%2C+M">Michele Correggi</a>, <a href="/search/math-ph?searchtype=author&query=Devanarayanan%2C+B">Bharathiganesh Devanarayanan</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1706.00654v2-abstract-short" style="display: inline;"> We analyze the response of a type II superconducting wire to an external magnetic field parallel to it in the framework of Ginzburg-Landau theory. We focus on the surface superconductivity regime of applied field between the second and third critical values, where the superconducting state survives only close to the sample's boundary. Our first finding is that, in first approximation, the shape of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.00654v2-abstract-full').style.display = 'inline'; document.getElementById('1706.00654v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1706.00654v2-abstract-full" style="display: none;"> We analyze the response of a type II superconducting wire to an external magnetic field parallel to it in the framework of Ginzburg-Landau theory. We focus on the surface superconductivity regime of applied field between the second and third critical values, where the superconducting state survives only close to the sample's boundary. Our first finding is that, in first approximation, the shape of the boundary plays no role in determining the density of superconducting electrons. A second order term is however isolated, directly proportional to the mean curvature of the boundary. This demonstrates that points of higher boundary curvature (counted inwards) attract superconducting electrons. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.00654v2-abstract-full').style.display = 'none'; document.getElementById('1706.00654v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> Roma01.Math.MP </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1705.03203">arXiv:1705.03203</a> <span> [<a href="https://arxiv.org/pdf/1705.03203">pdf</a>, <a href="https://arxiv.org/ps/1705.03203">ps</a>, <a href="https://arxiv.org/format/1705.03203">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1090/conm/717">10.1090/conm/717 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Local Density Approximation for Almost-Bosonic Anyons </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Correggi%2C+M">M. Correggi</a>, <a href="/search/math-ph?searchtype=author&query=Lundholm%2C+D">D. Lundholm</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">N. Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1705.03203v2-abstract-short" style="display: inline;"> We discuss the average-field approximation for a trapped gas of non-interacting anyons in the quasi-bosonic regime. In the homogeneous case, i.e., for a confinement to a bounded region, we prove that the energy in the regime of large statistics parameter, i.e., for "less-bosonic" anyons, is independent of boundary conditions and of the shape of the domain. When a non-trivial trapping potential is… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.03203v2-abstract-full').style.display = 'inline'; document.getElementById('1705.03203v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1705.03203v2-abstract-full" style="display: none;"> We discuss the average-field approximation for a trapped gas of non-interacting anyons in the quasi-bosonic regime. In the homogeneous case, i.e., for a confinement to a bounded region, we prove that the energy in the regime of large statistics parameter, i.e., for "less-bosonic" anyons, is independent of boundary conditions and of the shape of the domain. When a non-trivial trapping potential is present, we derive a local density approximation in terms of a Thomas-Fermi-like model. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.03203v2-abstract-full').style.display = 'none'; document.getElementById('1705.03203v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 May, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Contribution to the proceedings of QMath13: Mathematical Results in Quantum Physics, 8-11 October 2016, Atlanta, US</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> Roma01.Math.MP </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> in "Mathematical Problems in Quantum Physics", F. Bonetto, D. Borthwick, E. Harrell, M. Loss edts., Contemp. Math. 717, 77-92, AMS, 2018 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1703.09422">arXiv:1703.09422</a> <span> [<a href="https://arxiv.org/pdf/1703.09422">pdf</a>, <a href="https://arxiv.org/ps/1703.09422">ps</a>, <a href="https://arxiv.org/format/1703.09422">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lewin%2C+M">Mathieu Lewin</a>, <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1703.09422v3-abstract-short" style="display: inline;"> We prove that Gibbs measures based on 1D defocusing nonlinear Schr{枚}dinger functionals with sub-harmonic trapping can be obtained as the mean-field/large temperature limit of the corresponding grand-canonical ensemble for many bosons. The limit measure is supported on Sobolev spaces of negative regularity and the corresponding density matrices are not trace-class. The general proof strategy is th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.09422v3-abstract-full').style.display = 'inline'; document.getElementById('1703.09422v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1703.09422v3-abstract-full" style="display: none;"> We prove that Gibbs measures based on 1D defocusing nonlinear Schr{枚}dinger functionals with sub-harmonic trapping can be obtained as the mean-field/large temperature limit of the corresponding grand-canonical ensemble for many bosons. The limit measure is supported on Sobolev spaces of negative regularity and the corresponding density matrices are not trace-class. The general proof strategy is that of a previous paper of ours, but we have to complement it with Hilbert-Schmidt estimates on reduced density matrices. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.09422v3-abstract-full').style.display = 'none'; document.getElementById('1703.09422v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 March, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Minor changes and precisions</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1701.09064">arXiv:1701.09064</a> <span> [<a href="https://arxiv.org/pdf/1701.09064">pdf</a>, <a href="https://arxiv.org/ps/1701.09064">ps</a>, <a href="https://arxiv.org/format/1701.09064">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00220-018-3181-1">10.1007/s00220-018-3181-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Local incompressibility estimates for the Laughlin phase </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lieb%2C+E">Elliott Lieb</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Yngvason%2C+J">Jakob Yngvason</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1701.09064v3-abstract-short" style="display: inline;"> We prove sharp density upper bounds on optimal length-scales for the ground states of classical 2D Coulomb systems and generalizations thereof. Our method is new, based on an auxiliary Thomas-Fermi-like variational model. Moreover, we deduce density upper bounds for the related low-temperature Gibbs states. Our motivation comes from fractional quantum Hall physics, more precisely, the perturbation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.09064v3-abstract-full').style.display = 'inline'; document.getElementById('1701.09064v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1701.09064v3-abstract-full" style="display: none;"> We prove sharp density upper bounds on optimal length-scales for the ground states of classical 2D Coulomb systems and generalizations thereof. Our method is new, based on an auxiliary Thomas-Fermi-like variational model. Moreover, we deduce density upper bounds for the related low-temperature Gibbs states. Our motivation comes from fractional quantum Hall physics, more precisely, the perturbation of the Laughlin state by external potentials or impurities. These give rise to a class of many-body wave-functions that have the form of a product of the Laughlin state and an analytic function of many variables. This class is related via Laughlin's plasma analogy to Gibbs states of the generalized classical Coulomb systems we consider. Our main result shows that the perturbation of the Laughlin state cannot increase the particle density anywhere, with implications for the response of FQHE systems to external perturbations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.09064v3-abstract-full').style.display = 'none'; document.getElementById('1701.09064v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 January, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Version 3. Some details clarified, recent results of arXiv:1707.05059 taken into account</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.05758">arXiv:1612.05758</a> <span> [<a href="https://arxiv.org/pdf/1612.05758">pdf</a>, <a href="https://arxiv.org/ps/1612.05758">ps</a>, <a href="https://arxiv.org/format/1612.05758">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Interacting bosons in a double-well potential : localization regime </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Spehner%2C+D">Dominique Spehner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1612.05758v4-abstract-short" style="display: inline;"> We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.05758v4-abstract-full').style.display = 'inline'; document.getElementById('1612.05758v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.05758v4-abstract-full" style="display: none;"> We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of them live in each well). We start from the full many-body Schr{枚}dinger Hamiltonian in a large-filling situation where the on-site interaction and kinetic energies are comparable. When tunneling is negligible against interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly suppressed. The modes in which the particles condense are minimizers of nonlinear Schr{枚}dinger-type functionals. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.05758v4-abstract-full').style.display = 'none'; document.getElementById('1612.05758v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Comments on what is meant by localization added. The conjecture on the localization/delocalization transition has been modified based on more precise heuristics</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1611.00942">arXiv:1611.00942</a> <span> [<a href="https://arxiv.org/pdf/1611.00942">pdf</a>, <a href="https://arxiv.org/ps/1611.00942">ps</a>, <a href="https://arxiv.org/format/1611.00942">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/apde.2017.10.1169">10.2140/apde.2017.10.1169 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Local density approximation for the almost-bosonic anyon gas </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Correggi%2C+M">Michele Correggi</a>, <a href="/search/math-ph?searchtype=author&query=Lundholm%2C+D">Douglas Lundholm</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1611.00942v3-abstract-short" style="display: inline;"> We study the minimizers of an energy functional with a self-consistent magnetic field, which describes a quantum gas of almost-bosonic anyons in the average-field approximation. For the homogeneous gas we prove the existence of the thermodynamic limit of the energy at fixed effective statistics parameter, and the independence of such a limit from the shape of the domain. This result is then used i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.00942v3-abstract-full').style.display = 'inline'; document.getElementById('1611.00942v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1611.00942v3-abstract-full" style="display: none;"> We study the minimizers of an energy functional with a self-consistent magnetic field, which describes a quantum gas of almost-bosonic anyons in the average-field approximation. For the homogeneous gas we prove the existence of the thermodynamic limit of the energy at fixed effective statistics parameter, and the independence of such a limit from the shape of the domain. This result is then used in a local density approximation to derive an effective Thomas--Fermi-like model for the trapped anyon gas in the limit of a large effective statistics parameter (i.e., "less-bosonic" anyons). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.00942v3-abstract-full').style.display = 'none'; document.getElementById('1611.00942v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 April, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 November, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Minor typo corrected, version accepted in Analysis and PDEs</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> Roma01.Math.MP </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Analysis & PDE 10 (2017) 1169-1200 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1609.03818">arXiv:1609.03818</a> <span> [<a href="https://arxiv.org/pdf/1609.03818">pdf</a>, <a href="https://arxiv.org/ps/1609.03818">ps</a>, <a href="https://arxiv.org/format/1609.03818">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> </div> </div> <p class="title is-5 mathjax"> Rigidity of the Laughlin liquid </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lieb%2C+E">Elliott Lieb</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Yngvason%2C+J">Jakob Yngvason</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1609.03818v5-abstract-short" style="display: inline;"> We consider general N-particle wave functions that have the form of a product of the Laughlin state with filling factor $1/\ell$ and an analytic function of the N variables. This is the most general form of a wave function that can arise through a perturbation of the Laughlin state by external potentials or impurities, while staying in the lowest Landau level and maintaining the strong correlation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.03818v5-abstract-full').style.display = 'inline'; document.getElementById('1609.03818v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1609.03818v5-abstract-full" style="display: none;"> We consider general N-particle wave functions that have the form of a product of the Laughlin state with filling factor $1/\ell$ and an analytic function of the N variables. This is the most general form of a wave function that can arise through a perturbation of the Laughlin state by external potentials or impurities, while staying in the lowest Landau level and maintaining the strong correlations of the original state. We show that the perturbation can only shift or lower the 1-particle density but nowhere increase it above a maximum value. Consequences of this bound for the response of the Laughlin state to external fields are discussed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.03818v5-abstract-full').style.display = 'none'; document.getElementById('1609.03818v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Revised version, to appear in Journal of Statistical Physics</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1607.03833">arXiv:1607.03833</a> <span> [<a href="https://arxiv.org/pdf/1607.03833">pdf</a>, <a href="https://arxiv.org/ps/1607.03833">ps</a>, <a href="https://arxiv.org/format/1607.03833">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> </div> </div> <p class="title is-5 mathjax"> Some contributions to many-body quantum mathematics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1607.03833v2-abstract-short" style="display: inline;"> The results summarized here are intended as rigorous mathematical statements on various physical models coming from condensed matter physics, statistical mechanics (classical and quantum), quantum field theory and cold atoms physics. The main tools are mostly those of the mathematical analyst: partial differential equations, functional analysis, spectral theory, calculus of variations, with some i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1607.03833v2-abstract-full').style.display = 'inline'; document.getElementById('1607.03833v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1607.03833v2-abstract-full" style="display: none;"> The results summarized here are intended as rigorous mathematical statements on various physical models coming from condensed matter physics, statistical mechanics (classical and quantum), quantum field theory and cold atoms physics. The main tools are mostly those of the mathematical analyst: partial differential equations, functional analysis, spectral theory, calculus of variations, with some incursions into probability theory. A running thread is the construction, by rigorous asymptotic analysis most of the time, of bridges between different levels of mathematical modeling of physical phenomena. This is the manuscript for the author's habilitation thesis, summarizing results obtained between 2011 and 2016, in collaboration with: Michele Correggi, Mathieu Lewin, Douglas Lundholm, Phan Th脿nh Nam, Robert Seiringer, Sylvia Serfaty and Jakob Yngvason. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1607.03833v2-abstract-full').style.display = 'none'; document.getElementById('1607.03833v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 July, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">118 pages, "habilitation 脿 diriger des recherches" thesis. The defense took place in Grenoble on November 8th, 2016. The composition of the committee has been added on the front page</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1602.05166">arXiv:1602.05166</a> <span> [<a href="https://arxiv.org/pdf/1602.05166">pdf</a>, <a href="https://arxiv.org/ps/1602.05166">ps</a>, <a href="https://arxiv.org/format/1602.05166">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Bose Gases at Positive Temperature and Non-Linear Gibbs Measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lewin%2C+M">Mathieu Lewin</a>, <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1602.05166v1-abstract-short" style="display: inline;"> We summarize recent results on positive temperature equilibrium states of large bosonic systems. The emphasis will be on the connection between bosonic grand-canonical thermal states and the (semi-) classical Gibbs measures on one-body quantum states built using the corresponding mean-field energy functionals. An illustrative comparison with the case of "distinguishable" particles (boltzons) is pr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.05166v1-abstract-full').style.display = 'inline'; document.getElementById('1602.05166v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1602.05166v1-abstract-full" style="display: none;"> We summarize recent results on positive temperature equilibrium states of large bosonic systems. The emphasis will be on the connection between bosonic grand-canonical thermal states and the (semi-) classical Gibbs measures on one-body quantum states built using the corresponding mean-field energy functionals. An illustrative comparison with the case of "distinguishable" particles (boltzons) is provided. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.05166v1-abstract-full').style.display = 'none'; document.getElementById('1602.05166v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Proceedings of the 18th ICMP, Santiago de Chile, July 2015</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1601.02508">arXiv:1601.02508</a> <span> [<a href="https://arxiv.org/pdf/1601.02508">pdf</a>, <a href="https://arxiv.org/ps/1601.02508">ps</a>, <a href="https://arxiv.org/format/1601.02508">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Strongly Correlated Electrons">cond-mat.str-el</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mesoscale and Nanoscale Physics">cond-mat.mes-hall</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1103/PhysRevLett.116.170401">10.1103/PhysRevLett.116.170401 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Emergence of fractional statistics for tracer particles in a Laughlin liquid </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lundholm%2C+D">Douglas Lundholm</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1601.02508v3-abstract-short" style="display: inline;"> We consider a thought experiment where two distinct species of 2D particles in a perpendicular magnetic field interact via repulsive potentials. If the magnetic field and the interactions are strong enough, one type of particles forms a Laughlin state and the other ones couple to Laughlin quasi-holes. We show that in this situation, the motion of the second type of particles is described by an eff… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.02508v3-abstract-full').style.display = 'inline'; document.getElementById('1601.02508v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1601.02508v3-abstract-full" style="display: none;"> We consider a thought experiment where two distinct species of 2D particles in a perpendicular magnetic field interact via repulsive potentials. If the magnetic field and the interactions are strong enough, one type of particles forms a Laughlin state and the other ones couple to Laughlin quasi-holes. We show that in this situation, the motion of the second type of particles is described by an effective Hamiltonian, corresponding to the magnetic gauge picture for non-interacting anyons. The argument is in accord with, but distinct from, the Berry phase calculation of Arovas-Schrieffer-Wilczek. It suggests possibilities to observe the influence of effective anyon statistics in fractional quantum Hall systems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.02508v3-abstract-full').style.display = 'none'; document.getElementById('1601.02508v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Several revisions taking into account remarks from referees. To appear in Phys Rev Lett</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys. Rev. Lett. 116, 170401 (2016) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1511.01278">arXiv:1511.01278</a> <span> [<a href="https://arxiv.org/pdf/1511.01278">pdf</a>, <a href="https://arxiv.org/format/1511.01278">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Superconductivity">cond-mat.supr-con</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11005-016-0824-z">10.1007/s11005-016-0824-z <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Effects of boundary curvature on surface superconductivity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Correggi%2C+M">Michele Correggi</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1511.01278v2-abstract-short" style="display: inline;"> We investigate, within 2D Ginzburg-Landau theory, the ground state of a type-II superconducting cylinder in a parallel magnetic field varying between the second and third critical values. In this regime, superconductivity is restricted to a thin shell along the boundary of the sample and is to leading order constant in the direction tangential to the boundary. We exhibit a correction to this effec… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.01278v2-abstract-full').style.display = 'inline'; document.getElementById('1511.01278v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1511.01278v2-abstract-full" style="display: none;"> We investigate, within 2D Ginzburg-Landau theory, the ground state of a type-II superconducting cylinder in a parallel magnetic field varying between the second and third critical values. In this regime, superconductivity is restricted to a thin shell along the boundary of the sample and is to leading order constant in the direction tangential to the boundary. We exhibit a correction to this effect, showing that the curvature of the sample affects the distribution of superconductivity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.01278v2-abstract-full').style.display = 'none'; document.getElementById('1511.01278v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 November, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Lett. Math. Phys. 106 (2016), 445-467 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1509.09045">arXiv:1509.09045</a> <span> [<a href="https://arxiv.org/pdf/1509.09045">pdf</a>, <a href="https://arxiv.org/ps/1509.09045">ps</a>, <a href="https://arxiv.org/format/1509.09045">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> A note on 2D focusing many-boson systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lewin%2C+M">Mathieu Lewin</a>, <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1509.09045v2-abstract-short" style="display: inline;"> We consider a 2D quantum system of $N$ bosons in a trapping potential $|x|^s$, interacting via a pair potential of the form $N^{2尾-1} w(N^尾x)$. We show that for all $0 \textless{} 尾\textless{} (s+1)/(s+2)$, the leading order behavior of ground states of the many-body system is described in the large $N$ limit by the corresponding cubic nonlinear Schr{枚}dinger energy functional. Our result covers t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.09045v2-abstract-full').style.display = 'inline'; document.getElementById('1509.09045v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1509.09045v2-abstract-full" style="display: none;"> We consider a 2D quantum system of $N$ bosons in a trapping potential $|x|^s$, interacting via a pair potential of the form $N^{2尾-1} w(N^尾x)$. We show that for all $0 \textless{} 尾\textless{} (s+1)/(s+2)$, the leading order behavior of ground states of the many-body system is described in the large $N$ limit by the corresponding cubic nonlinear Schr{枚}dinger energy functional. Our result covers the focusing case ($w \leq 0$) where even the stability of the many-body system is not obvious. This answers an open question mentioned by X.~Chen and J.~Holmer for harmonic traps ($s=2$). Together with the BBGKY hierarchy approach used by these authors, our result implies the convergence of the many-body quantum dynamics to the focusing NLS equation with harmonic trap for all $0 \textless{} 尾\textless{} 3/4$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.09045v2-abstract-full').style.display = 'none'; document.getElementById('1509.09045v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 October, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1507.01440">arXiv:1507.01440</a> <span> [<a href="https://arxiv.org/pdf/1507.01440">pdf</a>, <a href="https://arxiv.org/ps/1507.01440">ps</a>, <a href="https://arxiv.org/format/1507.01440">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> From Bosonic Grand-Canonical Ensembles to Nonlinear Gibbs Measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1507.01440v1-abstract-short" style="display: inline;"> In a recent paper, in collaboration with Mathieu Lewin and Phan Th{脿}nh Nam, we showed that nonlinear Gibbs measures based on Gross-Pitaevskii like functionals could be derived from many-body quantum mechanics, in a mean-field limit. This text summarizes these findings. It focuses on the simplest, but most physically relevant, case we could treat so far, namely that of the defocusing cubic NLS fun… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1507.01440v1-abstract-full').style.display = 'inline'; document.getElementById('1507.01440v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1507.01440v1-abstract-full" style="display: none;"> In a recent paper, in collaboration with Mathieu Lewin and Phan Th{脿}nh Nam, we showed that nonlinear Gibbs measures based on Gross-Pitaevskii like functionals could be derived from many-body quantum mechanics, in a mean-field limit. This text summarizes these findings. It focuses on the simplest, but most physically relevant, case we could treat so far, namely that of the defocusing cubic NLS functional on a 1D interval. The measure obtained in the limit, which (almost) lives over H^{1/2} , has been previously shown to be invariant under the NLS flow by Bourgain. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1507.01440v1-abstract-full').style.display = 'none'; document.getElementById('1507.01440v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 July, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This is the text of a Laurent Schwartz X-EDP seminar I gave in November 2014. It summarizes some of the results of arXiv:1410.0335</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1506.05263">arXiv:1506.05263</a> <span> [<a href="https://arxiv.org/pdf/1506.05263">pdf</a>, <a href="https://arxiv.org/ps/1506.05263">ps</a>, <a href="https://arxiv.org/format/1506.05263">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> De Finetti theorems, mean-field limits and Bose-Einstein condensation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1506.05263v2-abstract-short" style="display: inline;"> These notes deal with the mean-field approximation for equilibrium states of N-body systems in classical and quantum statistical mechanics. A general strategy for the justification of effective models based on statistical independence assumptions is presented in details. The main tools are structure theorems {脿} la de Finetti, describing the large N limits of admissible states for these systems. T… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.05263v2-abstract-full').style.display = 'inline'; document.getElementById('1506.05263v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1506.05263v2-abstract-full" style="display: none;"> These notes deal with the mean-field approximation for equilibrium states of N-body systems in classical and quantum statistical mechanics. A general strategy for the justification of effective models based on statistical independence assumptions is presented in details. The main tools are structure theorems {脿} la de Finetti, describing the large N limits of admissible states for these systems. These rely on the symmetry under exchange of particles, due to their indiscernability. Emphasis is put on quantum aspects, in particular the mean-field approximation for the ground states of large bosonic systems, in relation with the Bose-Einstein condensation phenomenon. Topics covered in details include: the structure of reduced density matrices for large bosonic systems, Fock-space localization methods, derivation of effective energy functionals of Hartree or non-linear Schr{枚}dinger type, starting from the many-body Schr{枚}dinger Hamiltonian. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.05263v2-abstract-full').style.display = 'none'; document.getElementById('1506.05263v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Lectures notes from a course at the LMU, Munich. Translated and slightly expanded version of my cours Peccot, hal-01060125v4, arXiv:1409.1182. A wrong proof has been removed from Appendix A</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.05982">arXiv:1505.05982</a> <span> [<a href="https://arxiv.org/pdf/1505.05982">pdf</a>, <a href="https://arxiv.org/ps/1505.05982">ps</a>, <a href="https://arxiv.org/format/1505.05982">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> The Average Field Approximation for Almost Bosonix Extended Anyons </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lundholm%2C+D">Douglas Lundholm</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1505.05982v3-abstract-short" style="display: inline;"> Anyons are 2D or 1D quantum particles with intermediate statistics, interpolating between bosons and fermions. We study the ground state of a large number N of 2D anyons, in a scaling limit where the statistics parameter is proportional to the inverse of N. This means that the statistics is seen as a "perturbation from the bosonic end." We model this situation in the magnetic gauge picture by boso… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.05982v3-abstract-full').style.display = 'inline'; document.getElementById('1505.05982v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.05982v3-abstract-full" style="display: none;"> Anyons are 2D or 1D quantum particles with intermediate statistics, interpolating between bosons and fermions. We study the ground state of a large number N of 2D anyons, in a scaling limit where the statistics parameter is proportional to the inverse of N. This means that the statistics is seen as a "perturbation from the bosonic end." We model this situation in the magnetic gauge picture by bosons interacting through long-range magnetic potentials. We assume that these effective statistical gauge potentials are generated by magnetic charges carried by each particle, smeared over discs of radius R (extended anyons). Our method allows to take R to 0 not too fast at the same time as N to infinity. In this limit we rigorously justify the so-called "average field approximation": the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.05982v3-abstract-full').style.display = 'none'; document.getElementById('1505.05982v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1503.07061">arXiv:1503.07061</a> <span> [<a href="https://arxiv.org/pdf/1503.07061">pdf</a>, <a href="https://arxiv.org/ps/1503.07061">ps</a>, <a href="https://arxiv.org/format/1503.07061">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/apde.2016.9.459">10.2140/apde.2016.9.459 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Ground states of large bosonic systems: The gross-pitaevskii limit revisited </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Seiringer%2C+R">Robert Seiringer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1503.07061v3-abstract-short" style="display: inline;"> We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schr枚dinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.07061v3-abstract-full').style.display = 'inline'; document.getElementById('1503.07061v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1503.07061v3-abstract-full" style="display: none;"> We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schr枚dinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson's lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.07061v3-abstract-full').style.display = 'none'; document.getElementById('1503.07061v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Anal. PDE 9 (2016) 459-485 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1411.2361">arXiv:1411.2361</a> <span> [<a href="https://arxiv.org/pdf/1411.2361">pdf</a>, <a href="https://arxiv.org/ps/1411.2361">ps</a>, <a href="https://arxiv.org/format/1411.2361">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Strongly Correlated Electrons">cond-mat.str-el</span> </div> </div> <p class="title is-5 mathjax"> Incompressibility estimates for the Laughlin phase, part II </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a>, <a href="/search/math-ph?searchtype=author&query=Yngvason%2C+J">Jakob Yngvason</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1411.2361v4-abstract-short" style="display: inline;"> We consider fractional quantum Hall states built on Laughlin's original N-body wave-functions, i.e., they are of the form holomorphic times gaussian and vanish when two particles come close, with a given polynomial rate. Such states appear naturally when looking for the ground state of 2D particles in strong magnetic fields, interacting via repulsive forces and subject to an external potential due… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.2361v4-abstract-full').style.display = 'inline'; document.getElementById('1411.2361v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1411.2361v4-abstract-full" style="display: none;"> We consider fractional quantum Hall states built on Laughlin's original N-body wave-functions, i.e., they are of the form holomorphic times gaussian and vanish when two particles come close, with a given polynomial rate. Such states appear naturally when looking for the ground state of 2D particles in strong magnetic fields, interacting via repulsive forces and subject to an external potential due to trapping and/or disorder. We prove that all functions in this class satisfy a universal local density upper bound, in a suitable weak sense. Such bounds are useful to investigate the response of fractional quantum Hall phases to variations of the external potential. Contrary to our previous results for a restricted class of wave-functions, the bound we obtain here is not optimal, but it does not require any additional assumptions on the wave-function, besides analyticity and symmetry of the pre-factor modifying the Laughlin function. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.2361v4-abstract-full').style.display = 'none'; document.getElementById('1411.2361v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 November, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2014. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1410.0335">arXiv:1410.0335</a> <span> [<a href="https://arxiv.org/pdf/1410.0335">pdf</a>, <a href="https://arxiv.org/ps/1410.0335">ps</a>, <a href="https://arxiv.org/format/1410.0335">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Derivation of nonlinear Gibbs measures from many-body quantum mechanics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lewin%2C+M">Mathieu Lewin</a>, <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1410.0335v3-abstract-short" style="display: inline;"> We prove that nonlinear Gibbs measures can be obtained from the corresponding many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where the temperature T diverges and the interaction behaves as 1/T. We proceed by characterizing the interacting Gibbs state as minimizing a functional counting the free-energy relatively to the non-interacting case. We then perform an infinite-dime… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1410.0335v3-abstract-full').style.display = 'inline'; document.getElementById('1410.0335v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1410.0335v3-abstract-full" style="display: none;"> We prove that nonlinear Gibbs measures can be obtained from the corresponding many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where the temperature T diverges and the interaction behaves as 1/T. We proceed by characterizing the interacting Gibbs state as minimizing a functional counting the free-energy relatively to the non-interacting case. We then perform an infinite-dimensional analogue of phase-space semiclassical analysis, using fine properties of the quantum relative entropy, the link between quantum de Finetti measures and upper/lower symbols in a coherent state basis, as well as Berezin-Lieb type inequalities. Our results cover the measure built on the defocusing nonlinear Schr{枚}dinger functional on a finite interval, as well as smoother interactions in dimensions d\textgreater{}1. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1410.0335v3-abstract-full').style.display = 'none'; document.getElementById('1410.0335v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 October, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">to appear in Journal de l'Ecole Polytechnique</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1409.1182">arXiv:1409.1182</a> <span> [<a href="https://arxiv.org/pdf/1409.1182">pdf</a>, <a href="https://arxiv.org/ps/1409.1182">ps</a>, <a href="https://arxiv.org/format/1409.1182">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Th茅or猫mes de de Finetti, limites de champ moyen et condensation de Bose-Einstein </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1409.1182v4-abstract-short" style="display: inline;"> These lecture notes treat the mean-field approximation for equilibrium states of N body systems in classical and quantum statistical mechanics. A general strategy to justify effective models based on assumptions of statistical independence of the particles is in presented in detail. The main tools are a structure theorems of de Finetti that describe large N limits of states accessible to the syst… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.1182v4-abstract-full').style.display = 'inline'; document.getElementById('1409.1182v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1409.1182v4-abstract-full" style="display: none;"> These lecture notes treat the mean-field approximation for equilibrium states of N body systems in classical and quantum statistical mechanics. A general strategy to justify effective models based on assumptions of statistical independence of the particles is in presented in detail. The main tools are a structure theorems of de Finetti that describe large N limits of states accessible to the systems in question, exploiting the indistinguishablity of particles. The focus is on quantum aspects, particularly the mean-field approximation for the ground state of a large system of bosons, in connection with Bose-Einstein condensation: structure of reduced density matrices of a large bosonic system, localization methods in Fock space, derivation of Hartree and non-linear Schr枚dinger effective energy functionals. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.1182v4-abstract-full').style.display = 'none'; document.getElementById('1409.1182v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">These are the lectures notes from my cours Peccot at the Coll猫ge de France, given in March-April 2014. They are in french for the moment, some english translation should be available soon(er or later). Two appendices contain each an unpublished result obtained in collaboration with Mathieu Lewin. Further small corrections and a few more references in this version.</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1406.2259">arXiv:1406.2259</a> <span> [<a href="https://arxiv.org/pdf/1406.2259">pdf</a>, <a href="https://arxiv.org/format/1406.2259">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Superconductivity">cond-mat.supr-con</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00205-015-0900-z">10.1007/s00205-015-0900-z <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Boundary Behavior of the Ginzburg-Landau Order Parameter in the Surface Superconductivity Regime </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Correggi%2C+M">M. Correggi</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">N. Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1406.2259v2-abstract-short" style="display: inline;"> We study the 2D Ginzburg-Landau theory for a type-II superconductor in an applied magnetic field varying between the second and third critical value. In this regime the order parameter minimizing the GL energy is concentrated along the boundary of the sample and is well approximated to leading order by a simplified 1D profile in the direction perpendicular to the boundary. Motivated by a conjectur… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1406.2259v2-abstract-full').style.display = 'inline'; document.getElementById('1406.2259v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1406.2259v2-abstract-full" style="display: none;"> We study the 2D Ginzburg-Landau theory for a type-II superconductor in an applied magnetic field varying between the second and third critical value. In this regime the order parameter minimizing the GL energy is concentrated along the boundary of the sample and is well approximated to leading order by a simplified 1D profile in the direction perpendicular to the boundary. Motivated by a conjecture of Xing-Bin Pan, we address the question of whether this approximation can hold uniformly in the boundary region. We prove that this is indeed the case as a corollary of a refined, second order energy expansion including contributions due to the curvature of the sample. Local variations of the GL order parameter are controlled by the second order term of this energy expansion, which allows us to prove the desired uniformity of the surface superconductivity layer. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1406.2259v2-abstract-full').style.display = 'none'; document.getElementById('1406.2259v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 January, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 June, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Arch. Rational Mech. Anal. 219 (2015), 553-606 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1405.3220">arXiv:1405.3220</a> <span> [<a href="https://arxiv.org/pdf/1405.3220">pdf</a>, <a href="https://arxiv.org/ps/1405.3220">ps</a>, <a href="https://arxiv.org/format/1405.3220">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Gases">cond-mat.quant-gas</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> The mean-field approximation and the non-linear Schr枚dinger functional for trapped Bose gases </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math-ph?searchtype=author&query=Lewin%2C+M">Mathieu Lewin</a>, <a href="/search/math-ph?searchtype=author&query=Nam%2C+P+T">Phan Th脿nh Nam</a>, <a href="/search/math-ph?searchtype=author&query=Rougerie%2C+N">Nicolas Rougerie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1405.3220v3-abstract-short" style="display: inline;"> We study the ground state of a trapped Bose gas, starting from the full many-body Schr{枚}dinger Hamiltonian, and derive the nonlinear Schr{枚}dinger energy functional in the limit of large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field ap… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.3220v3-abstract-full').style.display = 'inline'; document.getElementById('1405.3220v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1405.3220v3-abstract-full" style="display: none;"> We study the ground state of a trapped Bose gas, starting from the full many-body Schr{枚}dinger Hamiltonian, and derive the nonlinear Schr{枚}dinger energy functional in the limit of large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive nonlinear Schr{枚}dinger ground state. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.3220v3-abstract-full').style.display = 'none'; document.getElementById('1405.3220v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 May, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2014. </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Rougerie%2C+N&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a 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