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Special emphasize is given to the long-distance decay of the impurity-impurity pair correlation function. It is shown that for the lattice sizes considered (L=10-96) and for the two different impurity distributions (purely random and mutually avoiding lines) the function is governed by the power law of 1/r a with an universal exponent a≈2. This result supports our findings about the numerical values of the critical exponents governing magnetic phase transition in the 3D Ising model with long-range-correlated disorder.","publication_date":"2007,,","grobid_abstract_attachment_id":"41677341"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"IMPURITY-IMPURITY PAIR CORRELATION FUNCTION AND PARAMAGNETIC-TO-FERROMAGNETIC TRANSITION IN THE RANDOM ISING MODEL","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [42254679]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":41677341,"attachmentType":"pdf"}"><img alt="First page of “IMPURITY-IMPURITY PAIR CORRELATION FUNCTION AND PARAMAGNETIC-TO-FERROMAGNETIC TRANSITION IN THE RANDOM ISING MODEL”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/41677341/mini_magick20190218-3265-640cp7.png?1550551490" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">IMPURITY-IMPURITY PAIR CORRELATION FUNCTION AND PARAMAGNETIC-TO-FERROMAGNETIC TRANSITION IN THE RANDOM ISING MODEL</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="42254679" href="https://independent.academia.edu/JaroslavIlnytskyi"><img alt="Profile image of Jaroslav Ilnytskyi" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Jaroslav Ilnytskyi</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2007</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">4 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 21026817; 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">In this Monte Carlo study we concentrated on the influence of non-magnetic impurities arranged as the lines with random orientation on paramagnetic-to-ferromagnetic phase transition in the 3D Ising model. Special emphasize is given to the long-distance decay of the impurity-impurity pair correlation function. It is shown that for the lattice sizes considered (L=10-96) and for the two different impurity distributions (purely random and mutually avoiding lines) the function is governed by the power law of 1/r a with an universal exponent a≈2. This result supports our findings about the numerical values of the critical exponents governing magnetic phase transition in the 3D Ising model with long-range-correlated disorder.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":41677341,"attachmentType":"pdf","workUrl":"https://www.academia.edu/21026817/IMPURITY_IMPURITY_PAIR_CORRELATION_FUNCTION_AND_PARAMAGNETIC_TO_FERROMAGNETIC_TRANSITION_IN_THE_RANDOM_ISING_MODEL"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":41677341,"attachmentType":"pdf","workUrl":"https://www.academia.edu/21026817/IMPURITY_IMPURITY_PAIR_CORRELATION_FUNCTION_AND_PARAMAGNETIC_TO_FERROMAGNETIC_TRANSITION_IN_THE_RANDOM_ISING_MODEL"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas" data-impression-entity-id="21026817" data-impression-entity-type="2" data-impression-source="signup-banner"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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Whereas both theoretical and numerical studies agree on the validity of extended Harris criterion (A. Weinrib, B.I. Halperin, Phys. Rev. B 27 (1983) 413) and indicate the existence of a new universality class, the numerical values of the critical exponents found so far differ essentially. To resolve this discrepancy we perform extensive Monte Carlo simulations of a 3d Ising model with non-magnetic impurities being arranged in a form of lines along randomly chosen axes of a lattice. The Swendsen-Wang algorithm is used alongside with a histogram reweighting technique and the finite-size scaling analysis to evaluate the values of critical exponents governing the magnetic phase transition. Our estimates for these exponents differ from both previous numerical simulations and are in favour of a non-trivial dependency of the critical exponents on the peculiarities of long-range correlations decay.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the universality class of the 3d Ising model with long-range-correlated disorder","attachmentId":41677343,"attachmentType":"pdf","work_url":"https://www.academia.edu/21026836/On_the_universality_class_of_the_3d_Ising_model_with_long_range_correlated_disorder","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/21026836/On_the_universality_class_of_the_3d_Ising_model_with_long_range_correlated_disorder"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="28790438" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/28790438/3_DIMENSIONAL_Ferromagnetic_Ising_Models_with_Quenched_Random_Nonmagnetic_Impurities">3-DIMENSIONAL Ferromagnetic Ising-Models with Quenched, Random Nonmagnetic Impurities</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="50940" href="https://granada.academia.edu/JoaquinMarro">Joaquin Marro</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physica B & C, 1986</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate the thermal and magnetic properties of three-dimensional, ferromagnetic Ising models with quenched and random non-magnetic (site) impurities in the case of simple cubic lattices. The reported thermodynamic phase diagrams reveal in particular a sharp phase transition (e.g. in the magnetic susceptibility) for the (relatively small) impurity concentrations</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"3-DIMENSIONAL Ferromagnetic Ising-Models with Quenched, Random Nonmagnetic Impurities","attachmentId":49208732,"attachmentType":"pdf","work_url":"https://www.academia.edu/28790438/3_DIMENSIONAL_Ferromagnetic_Ising_Models_with_Quenched_Random_Nonmagnetic_Impurities","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/28790438/3_DIMENSIONAL_Ferromagnetic_Ising_Models_with_Quenched_Random_Nonmagnetic_Impurities"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="109145067" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/109145067/Critical_disorder_and_critical_magnetic_field_of_the_nonequilibrium_athermal_random_field_Ising_model_in_thin_systems">Critical disorder and critical magnetic field of the nonequilibrium athermal random-field Ising model in thin systems</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="40901382" href="https://bg.academia.edu/DjordjeSpasojevic">Djordje Spasojevic</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical review, 2019</p><p class="ds-related-work--abstract ds2-5-body-sm">In the present study of the nonequilibrium athermal random-field Ising model we focus on the behavior of the critical disorder R c (l) and the critical magnetic field H c (l) under different boundary conditions when the system thickness l varies. We propose expressions for R c (l) and H c (l) as well as for the effective critical disorder R eff c (l, L) and effective critical magnetic field H eff c (l, L) playing the role of the effective critical parameters for the L × L × l lattices of finite lateral size L. We support these expressions by the scaling collapses of the magnetization and susceptibility curves obtained in extensive simulations. The collapses are achieved with the two-dimensional (2D) exponents for l below some characteristic value, providing thus a numerical evidence that the thin systems exhibit a 2D-like criticality which should be relevant for the experimental analyses of thin ferromagnetic samples.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Critical disorder and critical magnetic field of the nonequilibrium athermal random-field Ising model in thin systems","attachmentId":107354894,"attachmentType":"pdf","work_url":"https://www.academia.edu/109145067/Critical_disorder_and_critical_magnetic_field_of_the_nonequilibrium_athermal_random_field_Ising_model_in_thin_systems","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/109145067/Critical_disorder_and_critical_magnetic_field_of_the_nonequilibrium_athermal_random_field_Ising_model_in_thin_systems"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="30873646" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/30873646/Universality_and_logarithmic_corrections_in_two_dimensional_random_Ising_ferromagnets">Universality and logarithmic corrections in two-dimensional random Ising ferromagnets</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="58764247" href="https://ufrj.academia.edu/RaimundodosSantos">Raimundo dos Santos</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="58804473" href="https://independent.academia.edu/QueirozSde">S. de Queiroz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review B, 1997</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Universality and logarithmic corrections in two-dimensional random Ising ferromagnets","attachmentId":51300131,"attachmentType":"pdf","work_url":"https://www.academia.edu/30873646/Universality_and_logarithmic_corrections_in_two_dimensional_random_Ising_ferromagnets","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/30873646/Universality_and_logarithmic_corrections_in_two_dimensional_random_Ising_ferromagnets"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="73784927" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/73784927/Critical_behavior_of_the_three_dimensional_random_field_Ising_model_Two_exponent_scaling_and_discontinuous_transition">Critical behavior of the three-dimensional random-field Ising model: Two-exponent scaling and discontinuous transition</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="212183904" href="https://independent.academia.edu/WeilunYuan">Weilun Yuan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review B, 1995</p><p class="ds-related-work--abstract ds2-5-body-sm">In extensive Monte Carlo simulations the phase transition of the random field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite size scaling. For a Gaussian distribution of the random fields it is found that the correlation length ξ diverges with an exponent ν = 1.1 ± 0.2 at the critical temperature and that χ ∼ ξ 2−η with η = 0.50 ± 0.05 for the connected susceptibility and χ dis ∼ ξ 4−η with η = 1.03 ± 0.05 for the disconnected susceptibility. Together with the amplitude ratio A = limT →Tc χ dis /χ 2 (hr/T) 2 being close to one this gives further support for a two exponent scaling scenario implying η = 2η. The magnetization behaves discontinuously at the transition, i.e. β = 0, indicating a first order transition. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multi-peak structure that is characteristic for the phase-coexistence at first order phase transition points.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Critical behavior of the three-dimensional random-field Ising model: Two-exponent scaling and discontinuous transition","attachmentId":82171043,"attachmentType":"pdf","work_url":"https://www.academia.edu/73784927/Critical_behavior_of_the_three_dimensional_random_field_Ising_model_Two_exponent_scaling_and_discontinuous_transition","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/73784927/Critical_behavior_of_the_three_dimensional_random_field_Ising_model_Two_exponent_scaling_and_discontinuous_transition"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="65875123" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/65875123/Temperature_dependent_criticality_in_random_2D_Ising_models">Temperature-dependent criticality in random 2D Ising models</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="28693089" href="https://independent.academia.edu/LucZorrilla">Luc Zorrilla</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The European Physical Journal Plus, 2021</p><p class="ds-related-work--abstract ds2-5-body-sm">We consider 2D random Ising ferromagnetic models, where quenched disorder is represented either by random local magnetic fields (random-field Ising model) or by a random distribution of interaction couplings (random-bond Ising model). In both cases, we first perform zero- and finite-temperature Monte Carlo simulations to determine how the critical temperature depends on the disorder parameter. We then focus on the reversal transition triggered by an external field and study the associated Barkhausen noise. Our main result is that the critical exponents characterizing the power law associated with the Barkhausen noise exhibit a temperature dependence in line with existing experimental observations.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Temperature-dependent criticality in random 2D Ising models","attachmentId":77282868,"attachmentType":"pdf","work_url":"https://www.academia.edu/65875123/Temperature_dependent_criticality_in_random_2D_Ising_models","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/65875123/Temperature_dependent_criticality_in_random_2D_Ising_models"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="85681433" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/85681433/Critical_scaling_of_the_mutual_information_in_two_dimensional_disordered_Ising_models">Critical scaling of the mutual information in two-dimensional disordered Ising models</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="2667348" href="https://snu-in.academia.edu/IpsitaMandal">Ipsita Mandal</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Mechanics: Theory and Experiment, 2018</p><p class="ds-related-work--abstract ds2-5-body-sm">Rényi Mutual information (RMI), computed from second Rényi entropies, can identify classical phase transitions from their finite-size scaling at the critical points. We apply this technique to examine the presence or absence of finite temperature phase transitions in various two-dimensional models on a square lattice, which are extensions of the conventional Ising model by adding a quenched disorder. When the quenched disorder causes the nearest neighbor bonds to be both ferromagnetic and antiferromagnetic, (a) a spin glass phase exists only at zero temperature, and (b) a ferromagnetic phase exists at a finite temperature when the antiferromagnetic bond distributions are sufficiently dilute. Furthermore, finite temperature paramagnetic-ferromagnetic transitions can also occur when the disordered bonds involve only ferromagnetic couplings of random strengths. In our numerical simulations, the "zero temperature only" phase transitions are identified when there is no consistent finite-size scaling of the RMI curves, while for finite temperature critical points, the curves can identify the critical temperature Tc by their crossings at Tc and 2 Tc.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Critical scaling of the mutual information in two-dimensional disordered Ising models","attachmentId":90305507,"attachmentType":"pdf","work_url":"https://www.academia.edu/85681433/Critical_scaling_of_the_mutual_information_in_two_dimensional_disordered_Ising_models","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/85681433/Critical_scaling_of_the_mutual_information_in_two_dimensional_disordered_Ising_models"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="48600954" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/48600954/Self_averaging_in_the_random_two_dimensional_Ising_ferromagnet">Self-averaging in the random two-dimensional Ising ferromagnet</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3474085" href="https://nas.academia.edu/MaxymDudka">Maxym Dudka</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E</p><p class="ds-related-work--abstract ds2-5-body-sm">We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size L like ∼ L ln ln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ-peak in the thermodynamic limit L → ∞. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Self-averaging in the random two-dimensional Ising ferromagnet","attachmentId":67128407,"attachmentType":"pdf","work_url":"https://www.academia.edu/48600954/Self_averaging_in_the_random_two_dimensional_Ising_ferromagnet","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/48600954/Self_averaging_in_the_random_two_dimensional_Ising_ferromagnet"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="106634769" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/106634769/Monte_Carlo_simulation_of_phase_transition_in_2D_and_3D_ising_model">Monte-Carlo simulation of phase transition in 2D and 3D ising model</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="218122458" href="https://independent.academia.edu/KaranGiri7">Karan Giri</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Scientific World</p><p class="ds-related-work--abstract ds2-5-body-sm">In this work, Markov Chain-Monte Carlo technique was used to study the phase transition in two and three dimensional Ising Model (IM) in a square and cubic lattice. The study of temperature dependence of average magnetization and specific heat in different magnetic fields has been carried out in the 3x3 and 3x3x3 lattice with periodic boundary. Critical temperature point kBTc / J for 2D and 3D Ising Model has been observed at around 2.2 and 4.3 respectively at zero field. Our work satisfies Onsager’s critical value in 2D IM. The simulation suggests bifurcation in average magnetization below critical temperature Tc. Temperature plays the role of increasing randomness of spins. We found that Ising Model in small lattice size still retains interesting features like spontaneous magnetization and symmetry breaking below Tc at B = 0. At a non-zero field, the likelihood of spins to prefer certain alignment depends on the direction of the external field and magnitude of magnetization depend...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Monte-Carlo simulation of phase transition in 2D and 3D ising model","attachmentId":105735256,"attachmentType":"pdf","work_url":"https://www.academia.edu/106634769/Monte_Carlo_simulation_of_phase_transition_in_2D_and_3D_ising_model","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/106634769/Monte_Carlo_simulation_of_phase_transition_in_2D_and_3D_ising_model"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="84522679" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/84522679/Theory_of_the_Random_Field_Ising_Model">Theory of the Random Field Ising Model</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="41603281" href="https://independent.academia.edu/TNattermann">T. Nattermann</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Series on Directions in Condensed Matter Physics, 1997</p><p class="ds-related-work--abstract ds2-5-body-sm">A review is given on some recent developments in the theory of the Ising model in a random field. This model is a good representation of a large number of impure materials. After a short repetition of earlier arguments, which prove the absence of ferromagnetic order in d ≤ 2 space dimensions for uncorrelated random fields, we consider different random field correlations and in particular the generation of uncorrelated from anti-correlated random fields by thermal fluctuations. In discussing the phase transition, we consider the transition to be characterized by a divergent correlation length and compare the critical exponents obtained from various methods (real space RNG, Monte Carlo calculations, weighted mean field theory etc.). The ferromagnetic transition is believed to be preceded by a spin glass transition which manifests itself by replica symmetry breaking. In the discussion of dynamical properties, we concentrate mainly on the zero temperature depinning transition of a domain wall, which represents a critical point far from equilibrium with new scaling relations and critical exponents.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Theory of the Random Field Ising Model","attachmentId":89515559,"attachmentType":"pdf","work_url":"https://www.academia.edu/84522679/Theory_of_the_Random_Field_Ising_Model","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/84522679/Theory_of_the_Random_Field_Ising_Model"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":41677341,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":41677341,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_41677341" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="5447394" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/5447394/Low_Temperature_Impurity_Pairing_in_the_Frustrated_2d_Ising_Model">Low-Temperature Impurity Pairing in the Frustrated 2d Ising Model</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="7630487" href="https://bu.academia.edu/EugeneStanley">Eugene Stanley</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Low-Temperature Impurity Pairing in the Frustrated 2d Ising Model","attachmentId":32570659,"attachmentType":"pdf","work_url":"https://www.academia.edu/5447394/Low_Temperature_Impurity_Pairing_in_the_Frustrated_2d_Ising_Model","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/5447394/Low_Temperature_Impurity_Pairing_in_the_Frustrated_2d_Ising_Model"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="1" data-entity-id="100814125" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/100814125/Short_Time_Correlations_in_a_Two_Dimensional_Ising_Model_with_a_Line_of_Defects">Short Time Correlations in a Two-Dimensional Ising Model with a Line of Defects</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="43424912" href="https://usp-br.academia.edu/TTome">T. 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magnets","attachmentId":43914072,"attachmentType":"pdf","work_url":"https://www.academia.edu/23476797/Correlations_and_fractality_in_random_Ising_magnets","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/23476797/Correlations_and_fractality_in_random_Ising_magnets"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="5" data-entity-id="76957714" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/76957714/Self_averaging_in_the_random_2D_Ising_ferromagnet">Self-averaging in the random 2D Ising ferromagnet</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3474085" href="https://nas.academia.edu/MaxymDudka">Maxym Dudka</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2017</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Self-averaging in the random 2D Ising ferromagnet","attachmentId":84472412,"attachmentType":"pdf","work_url":"https://www.academia.edu/76957714/Self_averaging_in_the_random_2D_Ising_ferromagnet","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a 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Dekeyser</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Influence of the Pair Correlations on the Phase Transition in an Ising Lattice","attachmentId":45218805,"attachmentType":"pdf","work_url":"https://www.academia.edu/24897553/Influence_of_the_Pair_Correlations_on_the_Phase_Transition_in_an_Ising_Lattice","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/24897553/Influence_of_the_Pair_Correlations_on_the_Phase_Transition_in_an_Ising_Lattice"><span class="ds2-5-text-link__content">View PDF</span><span 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data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Boundary critical behaviour of two-dimensional random Ising models","attachmentId":36667886,"attachmentType":"pdf","work_url":"https://www.academia.edu/10866060/Boundary_critical_behaviour_of_two_dimensional_random_Ising_models","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/10866060/Boundary_critical_behaviour_of_two_dimensional_random_Ising_models"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="8" data-entity-id="17201853" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/17201853/Critical_aspects_of_the_random_field_Ising_model">Critical aspects of the random-field Ising model</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="13731965" href="https://independent.academia.edu/IoannisLelidis">Ioannis Lelidis</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The European Physical Journal B, 2013</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Critical aspects of the random-field Ising model","attachmentId":39386320,"attachmentType":"pdf","work_url":"https://www.academia.edu/17201853/Critical_aspects_of_the_random_field_Ising_model","alternativeTracking":true}"><span 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class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26393887" href="https://independent.academia.edu/FerencSzalma">Ferenc Szalma</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Physics, 1999</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Two-Dimensional Dilute Ising Models: Critical Behavior near Defect Lines","attachmentId":47066718,"attachmentType":"pdf","work_url":"https://www.academia.edu/10866053/Two_Dimensional_Dilute_Ising_Models_Critical_Behavior_near_Defect_Lines","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" 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data-author-id="44196240" href="https://independent.academia.edu/LevShchur">Lev Shchur</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Experimental and Theoretical Physics, 2000</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Universality of the ratio of the critical amplitudes of the magnetic susceptibility in a two-dimensional ising model with nonmagnetic impurities","attachmentId":43221673,"attachmentType":"pdf","work_url":"https://www.academia.edu/22624011/Universality_of_the_ratio_of_the_critical_amplitudes_of_the_magnetic_susceptibility_in_a_two_dimensional_ising_model_with_nonmagnetic_impurities","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a 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ds2-5-body-sm ds2-5-body-link" data-author-id="58764247" href="https://ufrj.academia.edu/RaimundodosSantos">Raimundo dos Santos</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="58804473" href="https://independent.academia.edu/QueirozSde">S. de Queiroz</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="313597810" href="https://independent.academia.edu/FabioAar%C3%A3oReis">Fabio Aarão Reis</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review B, 1999</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Universality, frustration, and conformal invariance in two-dimensional random Ising 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ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/34264957/Finite_temperature_phase_transition_in_the_two_dimensional_randomly_coupled_ferromagnet">Finite-temperature phase transition in the two-dimensional randomly coupled ferromagnet</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="21394392" href="https://independent.academia.edu/LemkeNey">Ney Lemke</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 1999</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Finite-temperature phase transition in the two-dimensional randomly coupled 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\nu</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26393887" href="https://independent.academia.edu/FerencSzalma">Ferenc Szalma</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1999</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Two-dimensional Dilute Ising Models: Defect Lines and the Universality of the Critical Exponent \\nu","attachmentId":36667894,"attachmentType":"pdf","work_url":"https://www.academia.edu/10866067/Two_dimensional_Dilute_Ising_Models_Defect_Lines_and_the_Universality_of_the_Critical_Exponent_nu","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link 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Sikakana</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2013</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Phase Transitions of Random Binary Magnetic Square Lattice Ising Systems","attachmentId":69560534,"attachmentType":"pdf","work_url":"https://www.academia.edu/52173150/Phase_Transitions_of_Random_Binary_Magnetic_Square_Lattice_Ising_Systems","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/52173150/Phase_Transitions_of_Random_Binary_Magnetic_Square_Lattice_Ising_Systems"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" 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