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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Approximate model used to describe the transition between conducting and insulating systems</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2D-Hubbard-model.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/2D-Hubbard-model.png/220px-2D-Hubbard-model.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/2D-Hubbard-model.png/330px-2D-Hubbard-model.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a8/2D-Hubbard-model.png 2x" data-file-width="355" data-file-height="258" /></a><figcaption>2-dimensional Hubbard model.</figcaption></figure> <p>The <b>Hubbard model</b> is an <a href="/wiki/Approximation" title="Approximation">approximate model</a> used to describe the transition between <a href="/wiki/Conductor_(material)" class="mw-redirect" title="Conductor (material)">conducting</a> and <a href="/wiki/Electrical_insulation" class="mw-redirect" title="Electrical insulation">insulating systems</a>.<sup id="cite_ref-Altland_1-0" class="reference"><a href="#cite_note-Altland-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is particularly useful in <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a>. The model is named for <a href="/wiki/John_Hubbard_(physicist)" title="John Hubbard (physicist)">John Hubbard</a>. </p><p>The Hubbard model states that each electron experiences competing forces: one pushes it to tunnel to neighboring atoms, while the other pushes it away from its neighbors.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Its <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> thus has two terms: a kinetic term allowing for <a href="/wiki/Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">tunneling</a> ("hopping") of particles between lattice sites and a potential term reflecting on-site interaction. The particles can either be <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a>, as in Hubbard's original work, or <a href="/wiki/Bosons" class="mw-redirect" title="Bosons">bosons</a>, in which case the model is referred to as the "<a href="/wiki/Bose%E2%80%93Hubbard_model" title="Bose–Hubbard model">Bose–Hubbard model</a>". </p><p>The Hubbard model is a useful approximation for particles in a periodic potential at sufficiently low temperatures, where all the particles may be assumed to be in the lowest <a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch band</a>, and long-range interactions between the particles can be ignored. If interactions between particles at different sites of the lattice are included, the model is often referred to as the "extended Hubbard model". In particular, the Hubbard term, most commonly denoted by <i>U</i>, is applied in first principles based simulations using <a href="/wiki/Density_Functional_Theory" class="mw-redirect" title="Density Functional Theory">Density Functional Theory</a>, DFT. The inclusion of the Hubbard term in DFT simulations is important as this improves the prediction of electron localisation and thus it prevents the incorrect prediction of metallic conduction in insulating systems.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The Hubbard model introduces short-range interactions between electrons to the <a href="/wiki/Tight_binding" title="Tight binding">tight-binding model</a>, which only includes kinetic energy (a "hopping" term) and interactions with the atoms of the lattice (an "atomic" potential). When the interaction between electrons is strong, the behavior of the Hubbard model can be qualitatively different from a tight-binding model. For example, the Hubbard model correctly predicts the existence of <a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulators</a>: materials that are insulating due to the strong repulsion between electrons, even though they satisfy the usual criteria for conductors, such as having an odd number of electrons per unit cell. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hubbard_model&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The model was originally proposed in 1963 to describe electrons in solids.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Hubbard, <a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Martin Gutzwiller</a> and Junjiro Kanamori each independently proposed it.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Since then, it has been applied to the study of <a href="/wiki/High-temperature_superconductivity" title="High-temperature superconductivity">high-temperature superconductivity</a>, quantum magnetism, and charge density waves.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Narrow_energy_band_theory">Narrow energy band theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hubbard_model&action=edit&section=2" title="Edit section: Narrow energy band theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Hubbard model is based on the <a href="/wiki/Tight_binding" title="Tight binding">tight-binding</a> approximation from <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a>, which describes particles moving in a periodic potential, typically referred to as a <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a>. For real materials, each lattice site might correspond with an ionic core, and the particles would be the valence electrons of these ions. In the tight-binding approximation, the Hamiltonian is written in terms of <a href="/wiki/Wannier_function" title="Wannier function">Wannier states</a>, which are localized states centered on each lattice site. Wannier states on neighboring lattice sites are coupled, allowing particles on one site to "hop" to another. Mathematically, the strength of this coupling is given by a "hopping integral", or "transfer integral", between nearby sites. The system is said to be in the tight-binding limit when the strength of the hopping integrals falls off rapidly with distance. This coupling allows states associated with each lattice site to hybridize, and the <a href="/wiki/Eigenstates" class="mw-redirect" title="Eigenstates">eigenstates</a> of such a <a href="/wiki/Crystal" title="Crystal">crystalline</a> system are <a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch's functions</a>, with the energy levels divided into separated <a href="/wiki/Electronic_band_structure" title="Electronic band structure">energy bands</a>. The width of the bands depends upon the value of the hopping integral. </p><p>The Hubbard model introduces a contact interaction between particles of opposite spin on each site of the lattice. When the Hubbard model is used to describe electron systems, these interactions are expected to be repulsive, stemming from the <a href="/wiki/Electric-field_screening" title="Electric-field screening">screened Coulomb interaction</a>. However, attractive interactions have also been frequently considered. The physics of the Hubbard model is determined by competition between the strength of the hopping integral, which characterizes the system's <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>, and the strength of the interaction term. The Hubbard model can therefore explain the transition from metal to insulator in certain interacting systems. For example, it has been used to describe <a href="/wiki/Metal" title="Metal">metal</a> oxides as they are heated, where the corresponding increase in nearest-neighbor spacing reduces the hopping integral to the point where the on-site potential is dominant. Similarly, the Hubbard model can explain the transition from conductor to insulator in systems such as <a href="/wiki/Rare-earth" class="mw-redirect" title="Rare-earth">rare-earth</a> <a href="/wiki/Pyrochlore" title="Pyrochlore">pyrochlores</a> as the <a href="/wiki/Atomic_number" title="Atomic number">atomic number</a> of the rare-earth metal increases, because the <a href="/wiki/Lattice_parameter" class="mw-redirect" title="Lattice parameter">lattice parameter</a> increases (or the angle between atoms can also change) as the rare-earth element atomic number increases, thus changing the relative importance of the hopping integral compared to the on-site repulsion. </p> <div class="mw-heading mw-heading2"><h2 id="Example:_one_dimensional_hydrogen_atom_chain">Example: one dimensional hydrogen atom chain</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hubbard_model&action=edit&section=3" title="Edit section: Example: one dimensional hydrogen atom chain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a> has one electron, in the so-called <i>s</i> orbital, which can either be spin up (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span>) or spin down (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span>). This orbital can be occupied by at most two electrons, one with <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> up and one down (see <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>). </p><p>Under <a href="/wiki/Band_theory" class="mw-redirect" title="Band theory">band theory</a>, for a 1D chain of hydrogen atoms, the 1s orbital forms a continuous band, which would be exactly half-full. The 1D chain of hydrogen atoms is thus predicted to be a conductor under conventional band theory. This 1D string is the only configuration simple enough to be solved directly.<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>But in the case where the spacing between the hydrogen atoms is gradually increased, at some point the chain must become an insulator. </p><p>Expressed using the Hubbard model, the Hamiltonian is made up of two terms. The first term describes the kinetic energy of the system, parameterized by the hopping integral, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. The second term is the on-site interaction of strength <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> that represents the electron repulsion. Written out in <a href="/wiki/Second_quantization" title="Second quantization">second quantization</a> notation, the Hubbard <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> then takes the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}=-t\sum _{i,\sigma }\left({\hat {c}}_{i,\sigma }^{\dagger }{\hat {c}}_{i+1,\sigma }+{\hat {c}}_{i+1,\sigma }^{\dagger }{\hat {c}}_{i,\sigma }\right)+U\sum _{i}{\hat {n}}_{i\uparrow }{\hat {n}}_{i\downarrow },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>t</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>σ</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>σ</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>σ</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>U</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">↑</mo> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">↓</mo> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-t\sum _{i,\sigma }\left({\hat {c}}_{i,\sigma }^{\dagger }{\hat {c}}_{i+1,\sigma }+{\hat {c}}_{i+1,\sigma }^{\dagger }{\hat {c}}_{i,\sigma }\right)+U\sum _{i}{\hat {n}}_{i\uparrow }{\hat {n}}_{i\downarrow },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa6146e256ef045a454b3d93b4f600f900d6b2a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.143ex; height:6.343ex;" alt="{\displaystyle {\hat {H}}=-t\sum _{i,\sigma }\left({\hat {c}}_{i,\sigma }^{\dagger }{\hat {c}}_{i+1,\sigma }+{\hat {c}}_{i+1,\sigma }^{\dagger }{\hat {c}}_{i,\sigma }\right)+U\sum _{i}{\hat {n}}_{i\uparrow }{\hat {n}}_{i\downarrow },}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}_{i\sigma }={\hat {c}}_{i\sigma }^{\dagger }{\hat {c}}_{i\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>σ</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {n}}_{i\sigma }={\hat {c}}_{i\sigma }^{\dagger }{\hat {c}}_{i\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f07faabb41988d097c561cb58313a961814626e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.296ex; height:3.509ex;" alt="{\displaystyle {\hat {n}}_{i\sigma }={\hat {c}}_{i\sigma }^{\dagger }{\hat {c}}_{i\sigma }}"></span> is the spin-density operator for spin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> on the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th site. The density operator is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {n}}_{i}={\hat {n}}_{i\uparrow }+{\hat {n}}_{i\downarrow }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">↑</mo> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">↓</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {n}}_{i}={\hat {n}}_{i\uparrow }+{\hat {n}}_{i\downarrow }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/207b040a5b9ee0932306141e4d7e09942799edfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.166ex; height:2.843ex;" alt="{\displaystyle {\hat {n}}_{i}={\hat {n}}_{i\uparrow }+{\hat {n}}_{i\downarrow }}"></span> and occupation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th site for the wavefunction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}=\langle \Phi \vert {\hat {n}}_{i}\vert \Phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Φ</mi> <mo fence="false" stretchy="false">|</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> <mi mathvariant="normal">Φ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}=\langle \Phi \vert {\hat {n}}_{i}\vert \Phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e9d4f38b1390149a52b2e56c1cd8175b2d93b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.946ex; height:2.843ex;" alt="{\displaystyle n_{i}=\langle \Phi \vert {\hat {n}}_{i}\vert \Phi \rangle }"></span>. Typically <i>t</i> is taken to be positive, and <i>U</i> may be either positive or negative, but is assumed to be positive when considering electronic systems. </p><p>Without the contribution of the second term, the Hamiltonian resolves to the <a href="/wiki/Tight_binding" title="Tight binding">tight binding</a> formula from regular band theory. </p><p>Including the second term yields a realistic model that also predicts a transition from conductor to insulator as the ratio of interaction to hopping, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U/t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U/t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cba4d9f04656b82ee4ae5238ab736d9bc1b0e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.785ex; height:2.843ex;" alt="{\displaystyle U/t}"></span>, is varied. This ratio can be modified by, for example, increasing the inter-atomic spacing, which would decrease the magnitude of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> without affecting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>. In the limit where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U/t\gg 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> <mo>≫</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U/t\gg 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd3d3ab8207a348bd31f316e88684fc9d4b2941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.561ex; height:2.843ex;" alt="{\displaystyle U/t\gg 1}"></span>, the chain simply resolves into a set of isolated <a href="/wiki/Magnetic_moment" title="Magnetic moment">magnetic moments</a>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U/t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U/t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cba4d9f04656b82ee4ae5238ab736d9bc1b0e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.785ex; height:2.843ex;" alt="{\displaystyle U/t}"></span> is not too large, the overlap integral provides for <a href="/wiki/Superexchange" title="Superexchange">superexchange</a> interactions between neighboring magnetic moments, which may lead to a variety of interesting magnetic correlations, such as ferromagnetic, antiferromagnetic, etc. depending on the model parameters. The one-dimensional Hubbard model was solved by <a href="/wiki/Elliott_H._Lieb" title="Elliott H. Lieb">Lieb</a> and Wu using the <a href="/wiki/Bethe_ansatz" title="Bethe ansatz">Bethe ansatz</a>. Essential progress was achieved in the 1990s: a <a href="/wiki/Yangian" title="Yangian">hidden symmetry</a> was discovered, and the <a href="/wiki/S-matrix" title="S-matrix">scattering matrix</a>, <a href="/wiki/Correlation_function_(statistical_mechanics)" title="Correlation function (statistical mechanics)">correlation functions</a>, <a href="/wiki/Thermodynamic" class="mw-redirect" title="Thermodynamic">thermodynamic</a> and <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">quantum entanglement</a> were evaluated.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="More_complex_systems">More complex systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hubbard_model&action=edit&section=4" title="Edit section: More complex systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although Hubbard is useful in describing systems such as a 1D chain of hydrogen atoms, it is important to note that more complex systems may experience other effects that the Hubbard model does not consider. In general, insulators can be divided into <a href="/wiki/Metal%E2%80%93insulator_transition" title="Metal–insulator transition">Mott–Hubbard</a> insulators and <a href="/wiki/Charge-transfer_insulators" title="Charge-transfer insulators">charge-transfer insulators</a>. </p><p>A Mott–Hubbard insulator can be described as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathrm {Ni} ^{2+}\mathrm {O} ^{2-})_{2}\longrightarrow \mathrm {Ni} ^{3+}\mathrm {O} ^{2-}+\mathrm {Ni} ^{1+}\mathrm {O} ^{2-}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">⟶</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {Ni} ^{2+}\mathrm {O} ^{2-})_{2}\longrightarrow \mathrm {Ni} ^{3+}\mathrm {O} ^{2-}+\mathrm {Ni} ^{1+}\mathrm {O} ^{2-}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53d345fe37cde22bbf92405bf0e889430e3f975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.038ex; height:3.176ex;" alt="{\displaystyle (\mathrm {Ni} ^{2+}\mathrm {O} ^{2-})_{2}\longrightarrow \mathrm {Ni} ^{3+}\mathrm {O} ^{2-}+\mathrm {Ni} ^{1+}\mathrm {O} ^{2-}.}"></span></dd></dl> <p>This can be seen as analogous to the Hubbard model for hydrogen chains, where conduction between unit cells can be described by a transfer integral. </p><p>However, it is possible for the electrons to exhibit another kind of behavior: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Ni} ^{2+}\mathrm {O} ^{2-}\longrightarrow \mathrm {Ni} ^{1+}\mathrm {O} ^{1-}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> </mrow> </msup> <mo stretchy="false">⟶</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Ni} ^{2+}\mathrm {O} ^{2-}\longrightarrow \mathrm {Ni} ^{1+}\mathrm {O} ^{1-}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4b6395e5aeaad6a01f098f0dcb08e028c6ac43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.47ex; height:2.676ex;" alt="{\displaystyle \mathrm {Ni} ^{2+}\mathrm {O} ^{2-}\longrightarrow \mathrm {Ni} ^{1+}\mathrm {O} ^{1-}.}"></span></dd></dl> <p>This is known as charge transfer and results in <a href="/wiki/Charge-transfer_insulators" title="Charge-transfer insulators">charge-transfer insulators</a>. Unlike Mott–Hubbard insulators electron transfer happens only within a unit cell. </p><p>Both of these effects may be present and compete in complex ionic systems. </p> <div class="mw-heading mw-heading2"><h2 id="Numerical_treatment">Numerical treatment</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hubbard_model&action=edit&section=5" title="Edit section: Numerical treatment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fact that the Hubbard model has not been solved analytically in arbitrary dimensions has led to intense research into numerical methods for these strongly correlated electron systems.<sup id="cite_ref-ScalapinoNumerical_StudiesOfThe2DHubbardModel_7-0" class="reference"><a href="#cite_note-ScalapinoNumerical_StudiesOfThe2DHubbardModel-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SimonsBenchmark_8-0" class="reference"><a href="#cite_note-SimonsBenchmark-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> One major goal of this research is to determine the low-temperature phase diagram of this model, particularly in two-dimensions. Approximate numerical treatment of the Hubbard model on finite systems is possible via various methods. </p><p>One such method, the <a href="/wiki/Lanczos_algorithm" title="Lanczos algorithm">Lanczos algorithm</a>, can produce static and dynamic properties of the system. Ground state calculations using this method require the storage of three vectors of the size of the number of states. The number of states scales exponentially with the size of the system, which limits the number of sites in the lattice to about 20 on 21st century hardware. With projector and finite-temperature <a href="/wiki/Auxiliary-field_Monte_Carlo" title="Auxiliary-field Monte Carlo">auxiliary-field Monte Carlo</a>, two statistical methods exist that can obtain certain properties of the system. For low temperatures, convergence problems appear that lead to an exponential computational effort with decreasing temperature due to the so-called fermion <a href="/wiki/Sign_problem" class="mw-redirect" title="Sign problem">sign problem</a>. </p><p>The Hubbard model can be studied within <a href="/wiki/Dynamical_mean-field_theory" title="Dynamical mean-field theory">dynamical mean-field theory</a> (DMFT). This scheme maps the Hubbard Hamiltonian onto a <a href="/wiki/Anderson_impurity_model" title="Anderson impurity model">single-site impurity model</a>, a mapping that is formally exact only in infinite dimensions and in finite dimensions corresponds to the exact treatment of all purely local correlations only. DMFT allows one to compute the local <a href="/wiki/Green%27s_function" title="Green's function">Green's function</a> of the Hubbard model for a given <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> and a given temperature. Within DMFT, the evolution of the <a href="/wiki/Spectral_function" class="mw-redirect" title="Spectral function">spectral function</a> can be computed and the appearance of the upper and lower Hubbard bands can be observed as correlations increase. </p> <div class="mw-heading mw-heading2"><h2 id="Simulator">Simulator</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hubbard_model&action=edit&section=6" title="Edit section: Simulator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stacks of heterogeneous 2-dimensional <a href="/wiki/Transition_metal_dichalcogenide_monolayers" title="Transition metal dichalcogenide monolayers">transition metal dichalcogenides (TMD)</a> have been used to simulate geometries in more than one dimension. <a href="/wiki/Tungsten_diselenide" title="Tungsten diselenide">Tungsten diselenide</a> and tungsten sulfide were stacked. This created a <a href="/wiki/Moir%C3%A9" class="mw-redirect" title="Moiré">moiré</a> superlattice consisting of hexagonal <a href="/wiki/Supercell_(crystal)" title="Supercell (crystal)">supercells</a> (repetition units defined by the relationship of the two materials). Each supercell then behaves as though it were a single atom. The distance between supercells is roughly 100x that of the atoms within them. This larger distance drastically reduces electron tunneling across supercells.<sup id="cite_ref-:0_2-3" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>They can be used to form <a href="/wiki/Wigner_crystal" title="Wigner crystal">Wigner crystals</a>. <a href="/wiki/Electrodes" class="mw-redirect" title="Electrodes">Electrodes</a> can be attached to regulate an <a href="/wiki/Electric_field" title="Electric field">electric field</a>. The electric field controls how many electrons fill each supercell. The number of electrons per supercell effectively determines which "atom" the lattice simulates. One electron/cell behaves like hydrogen, two/cell like helium, etc. As of 2022, supercells with up to eight electrons (<a href="/wiki/Oxygen" title="Oxygen">oxygen</a>) could be simulated. One result of the simulation showed that the difference between metal and insulator is a continuous function of the electric field strength.<sup id="cite_ref-:0_2-4" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>A "backwards" stacking regime allows the creation of a Chern insulator via the <a href="/wiki/Anomalous_quantum_Hall_effect" class="mw-redirect" title="Anomalous quantum Hall effect">anomalous quantum Hall effect</a> (with the edges of the device acting as a conductor while the interior acted as an insulator.) The device functioned at a temperature of 5 <a href="/wiki/Kelvin" title="Kelvin">Kelvins</a>, far above the temperature at which the effect had first been observed.<sup id="cite_ref-:0_2-5" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hubbard_model&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Anderson_impurity_model" title="Anderson impurity model">Anderson impurity model</a></li> <li><a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch's theorem</a></li> <li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">Electronic band structure</a></li> <li><a href="/wiki/Solid-state_physics" title="Solid-state physics">Solid-state physics</a></li> <li><a href="/wiki/Bose%E2%80%93Hubbard_model" title="Bose–Hubbard model">Bose–Hubbard model</a></li> <li><a href="/wiki/T-J_model" title="T-J model">t-J model</a></li> <li><a href="/wiki/Heisenberg_model_(quantum)" class="mw-redirect" title="Heisenberg model (quantum)">Heisenberg model (quantum)</a></li> <li><a href="/wiki/Dynamical_mean-field_theory" title="Dynamical mean-field theory">Dynamical mean-field theory</a></li> <li><a href="/wiki/Stoner_criterion" title="Stoner criterion">Stoner criterion</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hubbard_model&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Altland-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Altland_1-0">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAltlandSimons2006" class="citation book cs1">Altland, A.; Simons, B. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0KMkfAMe3JkC&pg=RA4-PA58">"Interaction effects in the tight-binding system"</a>. <i>Condensed Matter Field Theory</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 58 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-84508-3" title="Special:BookSources/978-0-521-84508-3"><bdi>978-0-521-84508-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Interaction+effects+in+the+tight-binding+system&rft.btitle=Condensed+Matter+Field+Theory&rft.pages=58+%27%27ff%27%27&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-84508-3&rft.aulast=Altland&rft.aufirst=A.&rft.au=Simons%2C+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0KMkfAMe3JkC%26pg%3DRA4-PA58&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:0_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-:0_2-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWood2022" class="citation web cs1">Wood, Charlie (16 August 2022). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/physics-duo-finds-magic-in-two-dimensions-20220816/">"Physics Duo Finds Magic in Two Dimensions"</a>. <i>Quanta Magazine</i><span class="reference-accessdate">. Retrieved <span class="nowrap">21 August</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quanta+Magazine&rft.atitle=Physics+Duo+Finds+Magic+in+Two+Dimensions&rft.date=2022-08-16&rft.aulast=Wood&rft.aufirst=Charlie&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fphysics-duo-finds-magic-in-two-dimensions-20220816%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFronziAssadiHanaor2019" class="citation journal cs1">Fronzi, Marco; Assadi, M. Hussein N.; Hanaor, Dorian A.H. (2019). "Theoretical insights into the hydrophobicity of low index CeO2 surfaces". <i>Applied Surface Science</i>. <b>478</b>: <span class="nowrap">68–</span>74. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1902.02662">1902.02662</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019ApSS..478...68F">2019ApSS..478...68F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.apsusc.2019.01.208">10.1016/j.apsusc.2019.01.208</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118895100">118895100</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Surface+Science&rft.atitle=Theoretical+insights+into+the+hydrophobicity+of+low+index+CeO2+surfaces&rft.volume=478&rft.pages=%3Cspan+class%3D%22nowrap%22%3E68-%3C%2Fspan%3E74&rft.date=2019&rft_id=info%3Aarxiv%2F1902.02662&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118895100%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.apsusc.2019.01.208&rft_id=info%3Abibcode%2F2019ApSS..478...68F&rft.aulast=Fronzi&rft.aufirst=Marco&rft.au=Assadi%2C+M.+Hussein+N.&rft.au=Hanaor%2C+Dorian+A.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHubbard1963" class="citation journal cs1">Hubbard, J. (26 November 1963). <a rel="nofollow" class="external text" href="https://royalsocietypublishing.org/doi/10.1098/rspa.1963.0204">"Electron correlations in narrow energy bands"</a>. <i>Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences</i>. <b>276</b> (1365): <span class="nowrap">238–</span>257. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963RSPSA.276..238H">1963RSPSA.276..238H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1963.0204">10.1098/rspa.1963.0204</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0080-4630">0080-4630</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:35439962">35439962</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London.+Series+A.+Mathematical+and+Physical+Sciences&rft.atitle=Electron+correlations+in+narrow+energy+bands&rft.volume=276&rft.issue=1365&rft.pages=%3Cspan+class%3D%22nowrap%22%3E238-%3C%2Fspan%3E257&rft.date=1963-11-26&rft_id=info%3Adoi%2F10.1098%2Frspa.1963.0204&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A35439962%23id-name%3DS2CID&rft.issn=0080-4630&rft_id=info%3Abibcode%2F1963RSPSA.276..238H&rft.aulast=Hubbard&rft.aufirst=J.&rft_id=https%3A%2F%2Froyalsocietypublishing.org%2Fdoi%2F10.1098%2Frspa.1963.0204&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAuerbach,_Assa.1994" class="citation book cs1">Auerbach, Assa. (1994). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/30028928"><i>Interacting electrons and quantum magnetism</i></a>. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94286-6" title="Special:BookSources/0-387-94286-6"><bdi>0-387-94286-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/30028928">30028928</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Interacting+electrons+and+quantum+magnetism&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1994&rft_id=info%3Aoclcnum%2F30028928&rft.isbn=0-387-94286-6&rft.au=Auerbach%2C+Assa.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F30028928&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEsslerFrahmGöhmannKlümper2005" class="citation book cs1">Essler, F. H. L.; Frahm, H.; Göhmann, F.; Klümper, A.; Korepin, V. E. (2005). <i>The One-Dimensional Hubbard Model</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-80262-8" title="Special:BookSources/978-0-521-80262-8"><bdi>978-0-521-80262-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+One-Dimensional+Hubbard+Model&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0-521-80262-8&rft.aulast=Essler&rft.aufirst=F.+H.+L.&rft.au=Frahm%2C+H.&rft.au=G%C3%B6hmann%2C+F.&rft.au=Kl%C3%BCmper%2C+A.&rft.au=Korepin%2C+V.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></span> </li> <li id="cite_note-ScalapinoNumerical_StudiesOfThe2DHubbardModel-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-ScalapinoNumerical_StudiesOfThe2DHubbardModel_7-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScalapino2006" class="citation journal cs1">Scalapino, D. J. (2006). <a rel="nofollow" class="external text" href="https://archive.org/details/arxiv-cond-mat0610710">"Numerical Studies of the 2D Hubbard Model"</a>: cond–mat/0610710. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0610710">cond-mat/0610710</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006cond.mat.10710S">2006cond.mat.10710S</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Numerical+Studies+of+the+2D+Hubbard+Model&rft.pages=cond-mat%2F0610710&rft.date=2006&rft_id=info%3Aarxiv%2Fcond-mat%2F0610710&rft_id=info%3Abibcode%2F2006cond.mat.10710S&rft.aulast=Scalapino&rft.aufirst=D.+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Farxiv-cond-mat0610710&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-SimonsBenchmark-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-SimonsBenchmark_8-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeBlanc2015" class="citation journal cs1">LeBlanc, J. (2015). <a rel="nofollow" class="external text" href="https://archive.org/details/arxiv-1505.02290">"Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms"</a>. <i>Physical Review X</i>. <b>5</b> (4): 041041. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1505.02290">1505.02290</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PhRvX...5d1041L">2015PhRvX...5d1041L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevX.5.041041">10.1103/PhysRevX.5.041041</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+X&rft.atitle=Solutions+of+the+Two-Dimensional+Hubbard+Model%3A+Benchmarks+and+Results+from+a+Wide+Range+of+Numerical+Algorithms&rft.volume=5&rft.issue=4&rft.pages=041041&rft.date=2015&rft_id=info%3Aarxiv%2F1505.02290&rft_id=info%3Adoi%2F10.1103%2FPhysRevX.5.041041&rft_id=info%3Abibcode%2F2015PhRvX...5d1041L&rft.aulast=LeBlanc&rft.aufirst=J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Farxiv-1505.02290&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hubbard_model&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHubbard1963" class="citation journal cs1">Hubbard, J. (1963). "Electron Correlations in Narrow Energy Bands". <i><a href="/wiki/Proceedings_of_the_Royal_Society_of_London" class="mw-redirect" title="Proceedings of the Royal Society of London">Proceedings of the Royal Society of London</a></i>. <b>276</b> (1365): <span class="nowrap">238–</span>257. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963RSPSA.276..238H">1963RSPSA.276..238H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1963.0204">10.1098/rspa.1963.0204</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2414761">2414761</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:35439962">35439962</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London&rft.atitle=Electron+Correlations+in+Narrow+Energy+Bands&rft.volume=276&rft.issue=1365&rft.pages=%3Cspan+class%3D%22nowrap%22%3E238-%3C%2Fspan%3E257&rft.date=1963&rft_id=info%3Adoi%2F10.1098%2Frspa.1963.0204&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A35439962%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2414761%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1963RSPSA.276..238H&rft.aulast=Hubbard&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBachLiebSolovej1994" class="citation journal cs1">Bach, V.; Lieb, E. H.; Solovej, J. P. (1994). "Generalized Hartree–Fock Theory and the Hubbard Model". <i><a href="/wiki/Journal_of_Statistical_Physics" title="Journal of Statistical Physics">Journal of Statistical Physics</a></i>. <b>76</b> (<span class="nowrap">1–</span>2): 3. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9312044">cond-mat/9312044</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994JSP....76....3B">1994JSP....76....3B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02188656">10.1007/BF02188656</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207143">207143</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Physics&rft.atitle=Generalized+Hartree%E2%80%93Fock+Theory+and+the+Hubbard+Model&rft.volume=76&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=3&rft.date=1994&rft_id=info%3Aarxiv%2Fcond-mat%2F9312044&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A207143%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02188656&rft_id=info%3Abibcode%2F1994JSP....76....3B&rft.aulast=Bach&rft.aufirst=V.&rft.au=Lieb%2C+E.+H.&rft.au=Solovej%2C+J.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLieb1995" class="citation journal cs1">Lieb, E. H. (1995). "The Hubbard Model: Some Rigorous Results and Open Problems". <i>Xi Int. Cong. Mp, Int. Press (?)</i>. <b>1995</b>: cond–mat/9311033. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9311033">cond-mat/9311033</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993cond.mat.11033L">1993cond.mat.11033L</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Xi+Int.+Cong.+Mp%2C+Int.+Press+%28%3F%29&rft.atitle=The+Hubbard+Model%3A+Some+Rigorous+Results+and+Open+Problems&rft.volume=1995&rft.pages=cond-mat%2F9311033&rft.date=1995&rft_id=info%3Aarxiv%2Fcond-mat%2F9311033&rft_id=info%3Abibcode%2F1993cond.mat.11033L&rft.aulast=Lieb&rft.aufirst=E.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGebhard1997" class="citation book cs1">Gebhard, F. (1997). "Metal–Insulator Transition". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uCPgHgEKnwEC&pg=PA1"><i>The Mott Metal–Insulator Transition: Models and Methods</i></a>. Springer Tracts in Modern Physics. Vol. 137. <a href="/wiki/Springer_(publisher)" class="mw-redirect" title="Springer (publisher)">Springer</a>. pp. <span class="nowrap">1–</span>48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540614814" title="Special:BookSources/9783540614814"><bdi>9783540614814</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Metal%E2%80%93Insulator+Transition&rft.btitle=The+Mott+Metal%E2%80%93Insulator+Transition%3A+Models+and+Methods&rft.series=Springer+Tracts+in+Modern+Physics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E48&rft.pub=Springer&rft.date=1997&rft.isbn=9783540614814&rft.aulast=Gebhard&rft.aufirst=F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuCPgHgEKnwEC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiebWu2003" class="citation journal cs1">Lieb, E. H.; Wu, F. Y. (2003). "The one-dimensional Hubbard model: A reminiscence". <i><a href="/wiki/Physica_A" class="mw-redirect" title="Physica A">Physica A</a></i>. <b>321</b> (1): <span class="nowrap">1–</span>27. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0207529">cond-mat/0207529</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003PhyA..321....1L">2003PhyA..321....1L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0378-4371%2802%2901785-5">10.1016/S0378-4371(02)01785-5</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:44758937">44758937</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+A&rft.atitle=The+one-dimensional+Hubbard+model%3A+A+reminiscence&rft.volume=321&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E27&rft.date=2003&rft_id=info%3Aarxiv%2Fcond-mat%2F0207529&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A44758937%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2FS0378-4371%2802%2901785-5&rft_id=info%3Abibcode%2F2003PhyA..321....1L&rft.aulast=Lieb&rft.aufirst=E.+H.&rft.au=Wu%2C+F.+Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArovasBergKivelsonRahgu2022" class="citation journal cs1">Arovas, Daniel P.; Berg, Erez; Kivelson, Steven; Rahgu, Sri (2022). "The Hubbard Model". <i><a href="/wiki/Annual_Review_of_Condensed_Matter_Physics" title="Annual Review of Condensed Matter Physics">Annual Review of Condensed Matter Physics</a></i>. <b>13</b>: <span class="nowrap">239–</span>274. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2103.12097">2103.12097</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022ARCMP..13..239A">2022ARCMP..13..239A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev-conmatphys-031620-102024">10.1146/annurev-conmatphys-031620-102024</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Condensed+Matter+Physics&rft.atitle=The+Hubbard+Model&rft.volume=13&rft.pages=%3Cspan+class%3D%22nowrap%22%3E239-%3C%2Fspan%3E274&rft.date=2022&rft_id=info%3Aarxiv%2F2103.12097&rft_id=info%3Adoi%2F10.1146%2Fannurev-conmatphys-031620-102024&rft_id=info%3Abibcode%2F2022ARCMP..13..239A&rft.aulast=Arovas&rft.aufirst=Daniel+P.&rft.au=Berg%2C+Erez&rft.au=Kivelson%2C+Steven&rft.au=Rahgu%2C+Sri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHubbard+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQinSchäferAndergassenCorboz2022" class="citation journal cs1">Qin, Mingpu; Schäfer, Thomas; Andergassen, Sabine; Corboz, Philippe; Gull, Emanuel (2022). "The Hubbard Model: A Computational Perspective". <i><a href="/wiki/Annual_Review_of_Condensed_Matter_Physics" title="Annual Review of Condensed Matter Physics">Annual Review of Condensed Matter Physics</a></i>. <b>13</b>: <span class="nowrap">275–</span>302. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2104.00064">2104.00064</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022ARCMP..13..275Q">2022ARCMP..13..275Q</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev-conmatphys-090921-033948">10.1146/annurev-conmatphys-090921-033948</a>. <a 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title="Condensed matter physics">Condensed matter physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/State_of_matter" title="State of matter">States of matter</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Solid" title="Solid">Solid</a></li> <li><a href="/wiki/Liquid" title="Liquid">Liquid</a></li> <li><a href="/wiki/Gas" title="Gas">Gas</a></li> <li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li> <li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li> <li><a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a></li> <li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li> <li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluid</a></li> <li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li> <li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:QuantumPhaseTransition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/220px-QuantumPhaseTransition.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/330px-QuantumPhaseTransition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/440px-QuantumPhaseTransition.svg.png 2x" data-file-width="512" data-file-height="369" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phase phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li> <li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Critical_phenomena" title="Critical phenomena">Critical phenomena</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electrons in solids</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a></li> <li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li> <li><a href="/wiki/Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li> <li><a href="/wiki/Quantum_spin_Hall_effect" title="Quantum spin Hall effect">Quantum spin Hall effect</a></li> <li><a href="/wiki/Berry_phase" class="mw-redirect" title="Berry phase">Berry phase</a></li> <li><a href="/wiki/Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a></li> <li><a href="/wiki/Josephson_effect" title="Josephson effect">Josephson effect</a></li> <li><a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Drude_model" title="Drude model">Drude model</a></li> <li><a href="/wiki/Free_electron_model" title="Free electron model">Free electron model</a></li> <li><a href="/wiki/Nearly_free_electron_model" title="Nearly free electron model">Nearly free electron model</a></li> <li><a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch's theorem</a></li> <li><a href="/wiki/Fermi_liquid_theory" title="Fermi liquid theory">Fermi liquid theory</a></li> <li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">electronic band structure</a></li> <li><a href="/wiki/Anderson_localization" title="Anderson localization">Anderson localization</a></li> <li><a href="/wiki/BCS_theory" title="BCS theory">BCS theory</a></li> <li><a href="/wiki/Tight_binding_model" class="mw-redirect" title="Tight binding model">tight binding model</a></li> <li><a class="mw-selflink selflink">Hubbard model</a></li> <li><a href="/wiki/Density_functional_theory" title="Density functional theory">Density functional theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conduction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li> <li><a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a></li> <li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a></li> <li><a href="/wiki/Semimetal" title="Semimetal">Semimetal</a></li> <li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li> <li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductor</a></li> <li><a href="/wiki/Topological_insulator" title="Topological insulator">Topological insulator</a></li> <li><a href="/wiki/Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Couplings</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">Thermoelectricity</a></li> <li><a href="/wiki/Piezoelectricity" title="Piezoelectricity">Piezoelectricity</a></li> <li><a href="/wiki/Ferroelectricity" title="Ferroelectricity">Ferroelectricity</a></li> <li><a href="/wiki/Flexoelectricity" title="Flexoelectricity">Flexoelectricity</a></li> <li><a href="/wiki/Electrostriction" title="Electrostriction">Electrostriction</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Magnetic phases</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_magnet" title="Amorphous magnet">Amorphous magnet</a></li> <li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnet</a></li> <li><a href="/wiki/Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a></li> <li><a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnet</a></li> <li><a href="/wiki/Superparamagnetism" title="Superparamagnetism">Superparamagnet</a></li> <li><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li> <li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a></li> <li><a href="/wiki/Metamagnetism" title="Metamagnetism">Metamagnet</a></li> <li><a href="/wiki/Spin_glass" title="Spin glass">Spin glass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> 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