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:2212.14358 · Physica A Statistical Mechanics and its Applications

Sensitivity of the thermodynamics of two-dimensional systems towards the topological classes of their surfaces

Oleg A. Vasilyev, Anna Maciołek, S. Dietrich

Published Jun 15, 2023• 2 citations
arXivHTMLPDF

Abstract

Using Monte Carlo simulations we study the two-dimensional Ising model on closed manifolds of various topologies near the critical point of the planar bulk system. To this end we consider triangular, square, and hexagonal lattices. We find that the corresponding universal scaling functions of the magnetic susceptibility χ, as well as those of the specific heat C, differ for distinct Euler characteristics K of the manifold. In particular, for each lattice decoration the maxima of the scaling functions of χ grow as K increases. For the specific heat this relationship is inverted: e.g., for each lattice decoration the maxima of the scaling functions of C on the spherical surface (Euler characteristic K=2) is smaller than on the projective plane (K=1) which, in turn, is smaller than on the torus and on the Klein bottle (both with K=0). We find that if the aspect ratio of the lattices is conform to a square geometrical shape, the magnetic susceptibility scaling functions for different lattice types differ only by a non-universal amplitude.

DOI
10.1016/j.physa.2023.128960

Related papers

Criticality senses topology

Oleg A. Vasilyev, Anna Maciołek, S. Dietrich

We numerically investigate the Ising model near quasi-criticality on finite two-dimensional (2D) manifolds formed by triangular and square lattices which exhibit the topology of the surface of a sphere, the projective plane, the torus, and the Klein bottle. We find that the finite-size scaling function of the magnetic susceptibility is the same for the surface of the torus and of the Klein bottle (Euler's characteristic 0 for both of them) but differs from the one for the projective plane (Euler's characteristic 1) and for the surface of the sphere (Euler's characteristic 2). This indicates that the universal properties of the continuous order-disorder phase transition in the Ising model for 2D manifolds depend on their topology.

Europhysics Letters (EPL)•Dec 9, 2019•5 citations

Monte Carlo studies of finite-size scaling in the Gibbs ensemble at criticality for the two-dimensional Ising model

K. K. Mon

In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.

Physical review. B, Condensed matter•Mar 1, 1993•8 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

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

Xintian Wu, N. Sh. Izmailian, Wenan Guo

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.

arXiv:1212.2023•Physical Review E•Feb 19, 2013•27 citations

Is Finite-Size Scaling Universal for Ising Models?

B. Müller

The sensitivity on shape and boundaries at the Curie point of the universal critical ratio LdM2/χ for the Ising model is confirmed by a comparison of square and rectangular lattices. Tests with three different algorithms give the same results for the universal critical ratio on various lattice types and always the same dependence on the system's aspect ratio.

International Journal of Modern Physics C•Jun 1, 1998•5 citations

Universality of the Ising model on spherelike lattices

Christian Hoelbling, Christian B. Lang

We study the two-dimensional (2D) Ising model on three different types of lattices that are topologically equivalent to spheres. The geometrical shapes are reminiscent of the surface of a 3D cube, a slab, and a sphere, respectively. Systems of volumes ranging up to O(${10}^{5}$) sites are simulated and finite-size scaling is analyzed. The partition function zeros and the values of various cumulants at their respective peak positions are determined and they agree with the scaling behavior expected from universality with the Onsager solution on the torus (\ensuremath{\nu}=1). For the pseudocritical values of the coupling we find significant anomalies indicating a shift exponent \ensuremath{\ne}1 for spherelike lattice topology. \textcopyright{} 1996 The American Physical Society.

arXiv:hep-lat/9602025•Physical review. B, Condensed matter•Aug 1, 1996•33 citations
Using Monte Carlo simulations we study the two-dimensional Ising model on closed manifolds of various topologies near the critical point of the planar bulk system. To this end we consider triangular, square, and hexagonal lattices. We find that the corresponding universal scaling functions of the magnetic susceptibility χ, as well as those of the specific heat C, differ for distinct Euler characteristics K of the manifold. In particular, for each lattice decoration the maxima of the scaling functions of χ grow as K increases. For the specific heat this relationship is inverted: e.g., for each lattice decoration the maxima of the scaling functions of C on the spherical surface (Euler characteristic K=2) is smaller than on the projective plane (K=1) which, in turn, is smaller than on the torus and on the Klein bottle (both with K=0). We find that if the aspect ratio of the lattices is conform to a square geometrical shape, the magnetic susceptibility scaling functions for different lattice types differ only by a non-universal amplitude.
We numerically investigate the Ising model near quasi-criticality on finite two-dimensional (2D) manifolds formed by triangular and square lattices which exhibit the topology of the surface of a sphere, the projective plane, the torus, and the Klein bottle. We find that the finite-size scaling function of the magnetic susceptibility is the same for the surface of the torus and of the Klein bottle (Euler's characteristic 0 for both of them) but differs from the one for the projective plane (Euler's characteristic 1) and for the surface of the sphere (Euler's characteristic 2). This indicates that the universal properties of the continuous order-disorder phase transition in the Ising model for 2D manifolds depend on their topology.
In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.
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
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.
We study the two-dimensional (2D) Ising model on three different types of lattices that are topologically equivalent to spheres. The geometrical shapes are reminiscent of the surface of a 3D cube, a slab, and a sphere, respectively. Systems of volumes ranging up to O(105{10}^{5}105) sites are simulated and finite-size scaling is analyzed. The partition function zeros and the values of various cumulants at their respective peak positions are determined and they agree with the scaling behavior expected from universality with the Onsager solution on the torus (\ensuremath{\nu}=1). For the pseudocritical values of the coupling we find significant anomalies indicating a shift exponent \ensuremath{\ne}1 for spherelike lattice topology. \textcopyright{} 1996 The American Physical Society.