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

arXiv:1212.2023 · Physical Review E

Shape-dependent finite-size effect of the critical two-dimensional Ising model on a triangular lattice

Xintian Wu, N. Sh. Izmailian, Wenan Guo

Published Feb 19, 2013• 27 citations
arXivHTMLPDF

Abstract

Using a bond-propagation algorithm, we study the finite-size behavior of the critical two-dimensional Ising model on a finite triangular lattice with free boundaries in five shapes: triangular, rhomboid, trapezoid, hexagonal, and rectangular. The critical free energy, internal energy, and specific heat are calculated. The accuracy of the free energy reaches 10(-26). Based on accurate data on several finite systems with linear size up to N=2000, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat accurately. We confirm the conformal field theory prediction that the corner free energy is universal and find logarithmic corrections in higher-order terms in the critical free energy for the rhomboid, trapezoid, and hexagonal systems, which are absent for the triangular and rectangular systems. The logarithmic edge corrections due to edges parallel or perpendicular to the bond directions in the internal energy are found to be identical, while the logarithmic edge corrections due to corresponding edges in the specific heat are different. The corner internal energy and corner specific heat for angles π/3, π/2, and 2π/3 are obtained, as well as higher-order corrections. Comparing with the corner internal energy and corner specific heat we previously found on a rectangle of the square lattice [Phys. Rev. E 86, 041149 (2012)], we conclude that the corner internal energy and corner specific heat for the rectangular shape are not universal.

DOI
10.1103/physreve.87.022124

Related papers

Critical two-dimensional Ising model with free, fixed ferromagnetic, fixed antiferromagnetic, and double antiferromagnetic boundaries

Xintian Wu, Nickolay Izmailyan

The critical two-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, and fixed double antiferromagnetic. Using bond propagation algorithms with surface fields, we obtain the free energy, internal energy, and specific heat numerically on square lattices with a square shape and various combinations of the four types of boundary conditions. The calculations are carried out on the square lattices with size N×N and 30<N<1000. The numerical data are analyzed with finite-size scaling. The bulk, edge, and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge logarithmic term in the internal energy, and the corner logarithmic term in the specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.

Physical Review E•Jan 5, 2015•14 citations

Finite-size behavior of the critical Ising model on a rectangle with free boundaries

Xintian Wu, N. Sh. Izmailian, Wenan Guo

Using the bond-propagation algorithm, we study the Ising model on a rectangle of size $M\ifmmode\times\else\texttimes\fi{}N$ with free boundaries. For five aspect ratios, $\ensuremath{\rho}=M/N=1$, 2, 4, 8, and 16, the critical free energy, internal energy and specific heat are calculated. The largest size reached is $M\ifmmode\times\else\texttimes\fi{}N=64\ifmmode\times\else\texttimes\fi{}{10}^{6}$. The accuracy of the free energy reaches ${10}^{\ensuremath{-}26}$. Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The fitted bulk free energy density is given by ${f}_{\ensuremath{\infty}}=0.92969539834161021499(1)$, compared with Onsager's exact result ${f}_{\ensuremath{\infty}}=0.929695398341610214985\ensuremath{\cdots}.$ We confirm the conformal field theory (CFT) prediction of the corner free energy, in which the central charge of the Ising model is found to be $c=0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$, compared with the CFT result $c=0.5$. We find that not only the corner free energy but also the corner internal energy and specific heat are geometry independent, i.e., independent of aspect ratio. The implication of this finding for finite-size scaling is discussed. In the second-order correction of the free energy, we confirm the geometry dependence predicted by CFT and determine a geometry-independent constant beyond CFT. High-order corrections are also obtained.

arXiv:1207.4540•Physical Review E•Oct 26, 2012•39 citations

The critical 2-dimensional Ising model with fixed boundaries

Xintian Wu, Nickolay Izmailyan

The critical 2-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, fixed double antiferromagnetic. Using Bond Propagation algorithms with surface fields, we obtained the free energy, internal energy and specific heat numerically on square lattices with square shape and various combinations of the four types boundary conditions. The numerical data are analyzed with finite size scaling. The bulk, edge and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge and corner logarithmic terms in the internal energy and specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.

arXiv:1409.6153•Sep 22, 2014•0 citations

Exact finite-size-scaling corrections to the critical two-dimensional Ising model on a torus: II. Triangular and hexagonal lattices

Jesús Salas

We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulae for the first coefficients of each expansion. We analyse the implications of these findings for the renormalization-group description of the model.

arXiv:cond-mat/0110287•Journal of Physics A Mathematical and General•Feb 18, 2002•35 citations

Exact Finite-Size-Scaling Corrections to the Critical Two-Dimensional Ising Model on a Torus. II. Triangular and hexagonal lattices

Jesús Salas

We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a critical

•Jan 1, 2001•20 citations

Critical behavior of the two-dimensional Ising model with a slit

Xintian Wu

The two-dimensional Ising model with a slit is studied. The slit free energy is defined, in which the bulk term, edge terms, and corner terms other than that of the slit are canceled. The bond propagation algorithm is used to calculate the free energy, internal energy, and heat capacity numerically. At the critical point, the slit free energy has a logarithmic correction L^{-1}lnL, in addition to the correction lnL, where L is the typical length of the system. The fitted coefficients of these terms agree with the conformal field theory very well. The integral constant and extrapolation length, which are pending in the conformal field theory, are obtained. In the slit internal energy, the leading term is proportional to L. There is a correction term, ln^{2}L, that has a geometry-independent coefficient, the exact value of which is conjectured to be -sqrt[2]/(2π^{2}). The leading term in the slit heat capacity is proportional to L^{2}. There is also a ln^{2}L term; however, its coefficient depends on the geometry. In the infinitely long strip limit, its exact value is conjectured to be 5/π^{2}. Near the critical point, for different size lattices, the rescaled slit free energies, internal energies, and heat capacities collapse onto the corresponding universal curves. According to the scaling and conformal field theory, the leading term of the slit free energy should be -c/8ln|t| for L|t|≫1 and |t|≪1, where t is the reduced temperature. The amplitude c/8 is universal, and c is the central charge. Our results validate these predictions qualitatively. For L|t|≫1 and |t|≪1, the slit internal energy and heat capacity scale as t^{-1} and t^{-2}, respectively.

Physical review. E•May 3, 2017•4 citations
Using a bond-propagation algorithm, we study the finite-size behavior of the critical two-dimensional Ising model on a finite triangular lattice with free boundaries in five shapes: triangular, rhomboid, trapezoid, hexagonal, and rectangular. The critical free energy, internal energy, and specific heat are calculated. The accuracy of the free energy reaches 10(-26). Based on accurate data on several finite systems with linear size up to N=2000, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat accurately. We confirm the conformal field theory prediction that the corner free energy is universal and find logarithmic corrections in higher-order terms in the critical free energy for the rhomboid, trapezoid, and hexagonal systems, which are absent for the triangular and rectangular systems. The logarithmic edge corrections due to edges parallel or perpendicular to the bond directions in the internal energy are found to be identical, while the logarithmic edge corrections due to corresponding edges in the specific heat are different. The corner internal energy and corner specific heat for angles π/3, π/2, and 2π/3 are obtained, as well as higher-order corrections. Comparing with the corner internal energy and corner specific heat we previously found on a rectangle of the square lattice [Phys. Rev. E 86, 041149 (2012)], we conclude that the corner internal energy and corner specific heat for the rectangular shape are not universal.
The critical two-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, and fixed double antiferromagnetic. Using bond propagation algorithms with surface fields, we obtain the free energy, internal energy, and specific heat numerically on square lattices with a square shape and various combinations of the four types of boundary conditions. The calculations are carried out on the square lattices with size N×N and 30<N<1000. The numerical data are analyzed with finite-size scaling. The bulk, edge, and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge logarithmic term in the internal energy, and the corner logarithmic term in the specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.
Using the bond-propagation algorithm, we study the Ising model on a rectangle of size $M\ifmmode\times\else\texttimes\fi{}N$ with free boundaries. For five aspect ratios, $\ensuremath{\rho}=M/N=1$, 2, 4, 8, and 16, the critical free energy, internal energy and specific heat are calculated. The largest size reached is $M\ifmmode\times\else\texttimes\fi{}N=64\ifmmode\times\else\texttimes\fi{}{10}^{6}$. The accuracy of the free energy reaches ${10}^{\ensuremath{-}26}$. Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The fitted bulk free energy density is given by ${f}_{\ensuremath{\infty}}=0.92969539834161021499(1)$, compared with Onsager's exact result ${f}_{\ensuremath{\infty}}=0.929695398341610214985\ensuremath{\cdots}.$ We confirm the conformal field theory (CFT) prediction of the corner free energy, in which the central charge of the Ising model is found to be $c=0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$, compared with the CFT result c=0.5c=0.5c=0.5. We find that not only the corner free energy but also the corner internal energy and specific heat are geometry independent, i.e., independent of aspect ratio. The implication of this finding for finite-size scaling is discussed. In the second-order correction of the free energy, we confirm the geometry dependence predicted by CFT and determine a geometry-independent constant beyond CFT. High-order corrections are also obtained.
The critical 2-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, fixed double antiferromagnetic. Using Bond Propagation algorithms with surface fields, we obtained the free energy, internal energy and specific heat numerically on square lattices with square shape and various combinations of the four types boundary conditions. The numerical data are analyzed with finite size scaling. The bulk, edge and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge and corner logarithmic terms in the internal energy and specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.
We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a critical
The two-dimensional Ising model with a slit is studied. The slit free energy is defined, in which the bulk term, edge terms, and corner terms other than that of the slit are canceled. The bond propagation algorithm is used to calculate the free energy, internal energy, and heat capacity numerically. At the critical point, the slit free energy has a logarithmic correction L^{-1}lnL, in addition to the correction lnL, where L is the typical length of the system. The fitted coefficients of these terms agree with the conformal field theory very well. The integral constant and extrapolation length, which are pending in the conformal field theory, are obtained. In the slit internal energy, the leading term is proportional to L. There is a correction term, ln^{2}L, that has a geometry-independent coefficient, the exact value of which is conjectured to be -sqrt[2]/(2π^{2}). The leading term in the slit heat capacity is proportional to L^{2}. There is also a ln^{2}L term; however, its coefficient depends on the geometry. In the infinitely long strip limit, its exact value is conjectured to be 5/π^{2}. Near the critical point, for different size lattices, the rescaled slit free energies, internal energies, and heat capacities collapse onto the corresponding universal curves. According to the scaling and conformal field theory, the leading term of the slit free energy should be -c/8ln|t| for L|t|≫1 and |t|≪1, where t is the reduced temperature. The amplitude c/8 is universal, and c is the central charge. Our results validate these predictions qualitatively. For L|t|≫1 and |t|≪1, the slit internal energy and heat capacity scale as t^{-1} and t^{-2}, respectively.