Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Heat capacity of the S= ∞ Ising ferromagnet studied by Monte Carlo simulation

Svend J. Knak Jensen

Published Apr 1, 1984• 2 citations
Publisher

Abstract

The heat capacity of the S= ∞ Ising ferromagnet arrayed on a simple cubic lattice is investigated by analysis of Monte Carlo data for the internal energy. A function is least-squares fitted to the energy data close to the critical temperature T c . The function is consistent with the expected singularity of the heat capacity C(t) = A ± | t | -α (1 + D ± | t | Δ 1 ) + B ± , where t = (T - T c )/T c and the ± superscripts refer to t > 0 and t < 0, respectively. The results of the data analysis show that the relation B + = B - is supported and that A + /A - accords with the expected universal value for a system with Ising symmetry. The ratio of the confluent singularity amplitudes D + /D - is somewhat lower than the expected universal value. The amplitudes of the leading and confluent singularities of the order parameter are also reported.

DOI
10.1088/0305-4470/17/5/035

Related papers

Critical parameters for the heat capacity of three-dimensional Ising ferromagnets

Svend J. Knak Jensen, Ole G. Mouritsen

The singularity of the heat capacity for the spin- 1 / 2 Ising ferromagnet arrayed on simple cubic and diamond lattices is investigated through the analysis of high-precision Monte Carlo data for the internal energy. The analysis incorporates confluent singularities. Two least-squares fitting procedures are considered. In the first one, the leading confluent singularity amplitude is zero and the next confluent singularity is included. For both lattices this procedure gives accordance with: (i) the universal value for the ratio of the leading singularity amplitudes predicted from the epsilon expansion and (ii) the theoretical expectation of equality between the non-singular parts of the heat capacity above and below the critical temperature. In the second least-squares fitting procedure, the leading confluent singularity is finite. This leads to a fit which disagrees with both (i) and (ii). The Monte Carlo results suggest that the failure of the series analysis to accord with (i) and (ii) may be due to neglect of a rather large confluent singularity in the ferromagnetic phase.

Journal of Physics A Mathematical and General•Aug 1, 1982•8 citations

Studies of 3D-Ising model with long-range interaction

Boris Kryzhanovsky, Olga G. Maksimova, Leonid B. Litinskii +1 more

We use the Monte-Carlo simulations to study thermodynamic properties of Ising models on cubic lattices accounting for interaction with next-nearest neighbors. The results of simulations are in good agreement with an analytic formula that defines the critical inverse temperature as a function of the interaction parameter. The behaviors of the internal energy and the heat capacity indicate that at the critical point the heat capacity has an infinite singularity.

International Conference on Information Technology and Nanotechnology (ITNT)•Sep 20, 2021•1 citation

Monte Carlo simulation of strongly disordered Ising ferromagnets

Hans-Otto Heuer

We report extensive Monte Carlo simulations of disordered Ising systems in the ferromagnetic region with concentrations of magnetic sites between p=1.0 and 0.5. The magnetization, the susceptibility, and the caloric properties have been studied in the critical region. The critical exponents \ensuremath{\beta} and \ensuremath{\gamma} as well as the universal amplitude A of the magnetization and the ratio ${\mathit{C}}_{+}$/${\mathit{C}}_{\mathrm{\ensuremath{-}}}$ of the susceptibility amplitudes have been determined with high precision. In addition to the cusplike specific heat, we have also measured the magnetization-energy correlation function \ensuremath{\Gamma}, which is a divergent thermal quantity. The corresponding critical exponent \ensuremath{\zeta}, which is related to the other exponents by \ensuremath{\zeta}=1-\ensuremath{\beta}=(\ensuremath{\alpha}+\ensuremath{\gamma})/2, has been determined. We have found that all quantities show power-law behavior within the temperature range of our simulation. All critical exponents change continuously with dilution. Even in the range of weak dilution (p\ensuremath{\ge}0.8), the effective critical exponents are concentration dependent and are clearly different from their pure system values. In the strongly diluted regime the critical exponents gradually reach new asymptotic behavior at p=0.5--0.6 with values of \ensuremath{\beta}=0.335\ifmmode\pm\else\textpm\fi{}0.01 and \ensuremath{\gamma}=1.49\ifmmode\pm\else\textpm\fi{}0.02. The exponent \ensuremath{\alpha} of the specific heat becomes -0.17\ifmmode\pm\else\textpm\fi{}0.04, which corresponds to a cusplike singularity. We conclude that disorder profoundly changes the critical behavior for weakly as well as strongly disordered spin systems.

Physical review. B, Condensed matter•Oct 1, 1990•55 citations

Energy fluctuations and the singularity of specific heat in a 3D Ising model

J. Kaupužs

We study the energy fluctuations in 3D Ising model near the phase transition point. Specific heat is a relevant quantity which is directly related to the mean squared amplitude of the energy fluctuations in the system. We have made extensive Monte Carlo simulations in 3D Ising model to clarify the character of the singularity of the specific heat Cv based on the finite-size scaling of its maximal values Cvmax depending on the linear size of the lattice L. An original iterative method has been used which automatically finds the pseudocritical temperature corresponding to the maximum of Cv. The simulations made up to L ≤ 128 with application of the Wolff's cluster algorithm allowed us to verify the possible power-like as well as logarithmic singularity of the specific heat predicted by different theoretical treatments. The most challenging and interesting result we have obtained is that the finite-size scaling of Cvmax in 3D Ising model is well described by a logarithmic rather than power-like ansatz, just like in 2D case. Another modification of our iterative method has been considered to estimate the critical coupling of 3D Ising model from the Binder cumulant data within L ε [96; 384]. Furthermore, the critical exponent β has been evaluated from the simulated magnetization data within the range of reduced temperatures t ≥ 0.000086 and system sizes L ≤ 410.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 25, 2004•2 citations

Phase Transition in 2D Ising Model with Next-Neighbor Interaction

Boris Kryzhanovsky, Leonid B. Litinskii

Abstract We used a Monte Carlo method to analyze a planar Ising model taking into account a ferromagnetic interaction with next neighbors. We found that independent of the value of the additional interaction, the character of singularity of the heat capacity did not change. Namely, at the critical point the heat capacity has a logarithmic singularity. We obtained a semi-theoretical formula relating the value of the critical temperature with the value of the additional ferromagnetic interaction.

Journal of Physics Conference Series•Nov 1, 2019•2 citations

Critical behavior of the Ising model on the four-dimensional cubic lattice

P. H. Lundow, Klas Markström

In this paper we investigate the nature of the singularity of the Ising model of the four-dimensional cubic lattice. It is rigorously known that the specific heat has critical exponent alpha=0 but a nonrigorous field-theory argument predicts an unbounded specific heat with a logarithmic singularity at Tc. We find that within the given accuracy the canonical ensemble data are consistent both with a logarithmic singularity and a bounded specific heat but that the microcanonical ensemble lends stronger support to a bounded specific heat. Our conclusion is that either much larger system sizes are needed for Monte Carlo studies of this model in four dimensions or the field-theory prediction of a logarithmic singularity is wrong.

arXiv:1202.3031•Physical Review E•Sep 3, 2009•82 citations
The heat capacity of the S= ∞ Ising ferromagnet arrayed on a simple cubic lattice is investigated by analysis of Monte Carlo data for the internal energy. A function is least-squares fitted to the energy data close to the critical temperature T c . The function is consistent with the expected singularity of the heat capacity C(t) = A ± | t | -α (1 + D ± | t | Δ 1 ) + B ± , where t = (T - T c )/T c and the ± superscripts refer to t > 0 and t < 0, respectively. The results of the data analysis show that the relation B + = B - is supported and that A + /A - accords with the expected universal value for a system with Ising symmetry. The ratio of the confluent singularity amplitudes D + /D - is somewhat lower than the expected universal value. The amplitudes of the leading and confluent singularities of the order parameter are also reported.
The singularity of the heat capacity for the spin- 1 / 2 Ising ferromagnet arrayed on simple cubic and diamond lattices is investigated through the analysis of high-precision Monte Carlo data for the internal energy. The analysis incorporates confluent singularities. Two least-squares fitting procedures are considered. In the first one, the leading confluent singularity amplitude is zero and the next confluent singularity is included. For both lattices this procedure gives accordance with: (i) the universal value for the ratio of the leading singularity amplitudes predicted from the epsilon expansion and (ii) the theoretical expectation of equality between the non-singular parts of the heat capacity above and below the critical temperature. In the second least-squares fitting procedure, the leading confluent singularity is finite. This leads to a fit which disagrees with both (i) and (ii). The Monte Carlo results suggest that the failure of the series analysis to accord with (i) and (ii) may be due to neglect of a rather large confluent singularity in the ferromagnetic phase.
We report extensive Monte Carlo simulations of disordered Ising systems in the ferromagnetic region with concentrations of magnetic sites between p=1.0 and 0.5. The magnetization, the susceptibility, and the caloric properties have been studied in the critical region. The critical exponents \ensuremath{\beta} and \ensuremath{\gamma} as well as the universal amplitude A of the magnetization and the ratio C+{\mathit{C}}_{+}C+​/${\mathit{C}}_{\mathrm{\ensuremath{-}}}$ of the susceptibility amplitudes have been determined with high precision. In addition to the cusplike specific heat, we have also measured the magnetization-energy correlation function \ensuremath{\Gamma}, which is a divergent thermal quantity. The corresponding critical exponent \ensuremath{\zeta}, which is related to the other exponents by \ensuremath{\zeta}=1-\ensuremath{\beta}=(\ensuremath{\alpha}+\ensuremath{\gamma})/2, has been determined. We have found that all quantities show power-law behavior within the temperature range of our simulation. All critical exponents change continuously with dilution. Even in the range of weak dilution (p\ensuremath{\ge}0.8), the effective critical exponents are concentration dependent and are clearly different from their pure system values. In the strongly diluted regime the critical exponents gradually reach new asymptotic behavior at p=0.5--0.6 with values of \ensuremath{\beta}=0.335\ifmmode\pm\else\textpm\fi{}0.01 and \ensuremath{\gamma}=1.49\ifmmode\pm\else\textpm\fi{}0.02. The exponent \ensuremath{\alpha} of the specific heat becomes -0.17\ifmmode\pm\else\textpm\fi{}0.04, which corresponds to a cusplike singularity. We conclude that disorder profoundly changes the critical behavior for weakly as well as strongly disordered spin systems.
We study the energy fluctuations in 3D Ising model near the phase transition point. Specific heat is a relevant quantity which is directly related to the mean squared amplitude of the energy fluctuations in the system. We have made extensive Monte Carlo simulations in 3D Ising model to clarify the character of the singularity of the specific heat Cv based on the finite-size scaling of its maximal values Cvmax depending on the linear size of the lattice L. An original iterative method has been used which automatically finds the pseudocritical temperature corresponding to the maximum of Cv. The simulations made up to L ≤ 128 with application of the Wolff's cluster algorithm allowed us to verify the possible power-like as well as logarithmic singularity of the specific heat predicted by different theoretical treatments. The most challenging and interesting result we have obtained is that the finite-size scaling of Cvmax in 3D Ising model is well described by a logarithmic rather than power-like ansatz, just like in 2D case. Another modification of our iterative method has been considered to estimate the critical coupling of 3D Ising model from the Binder cumulant data within L ε [96; 384]. Furthermore, the critical exponent β has been evaluated from the simulated magnetization data within the range of reduced temperatures t ≥ 0.000086 and system sizes L ≤ 410.
In this paper we investigate the nature of the singularity of the Ising model of the four-dimensional cubic lattice. It is rigorously known that the specific heat has critical exponent alpha=0 but a nonrigorous field-theory argument predicts an unbounded specific heat with a logarithmic singularity at Tc. We find that within the given accuracy the canonical ensemble data are consistent both with a logarithmic singularity and a bounded specific heat but that the microcanonical ensemble lends stronger support to a bounded specific heat. Our conclusion is that either much larger system sizes are needed for Monte Carlo studies of this model in four dimensions or the field-theory prediction of a logarithmic singularity is wrong.