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Physical Review E

Partition functions and finite-size scalings of Ising model on helical tori

Tsong‐Ming Liaw, Ming-Chang Huang, Yen-Liang Chou, S. C. Lin, Feng‐Yin Li

Published May 1, 2006• 12 citations
Publisher

Abstract

The exact closed forms of the partition functions of a two-dimensional Ising model on square lattices with twisted boundary conditions are given. The constructions of helical tori are unambiguously related to the twisted boundary conditions by virtue of the SL(2, Z) transforms. The numerical analyses on the deviations of the specific-heat peaks away from the bulk critical temperature reveal that the finite-size effect of herical tori is independent of the chirality.

DOI
10.1103/physreve.73.055101

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The exact closed forms of the partition functions of a two-dimensional Ising model on square lattices with twisted boundary conditions are given. The constructions of helical tori are unambiguously related to the twisted boundary conditions by virtue of the SL(2, Z) transforms. The numerical analyses on the deviations of the specific-heat peaks away from the bulk critical temperature reveal that the finite-size effect of herical tori is independent of the chirality.
We analyze the exact partition function of the Ising model on a square lattice under helical boundary conditions obtained by Liaw [Phys. Rev. E 73, 055101(R) (2006)]. Based on such an expression, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive an exact asymptotic expansion of the logarithm of the partition function and its first to fourth derivatives at the critical point. From such results, we find that the shift exponent for the specific heat is lambda=1 for all values of the helicity factor d . We also find that finite-size corrections for the free energy, the internal energy, and the specific heat of the model in a crucial way depend on the helicity factor of the lattice.
The Ising model in two dimensions with special toroidal boundary conditions is analyzed. These boundary conditions, which we call duality-twisted boundary conditions, may be interpreted as inserting a specific defect line ("seam") in the system, along noncontractible circles of the cylinder, before closing it into a torus. We derive exact expressions for the eigenvalues of a transfer matrix for the critical ferromagnetic Ising model on the M×N square lattice wrapped on the torus with a specific defect line. As a result we have obtained analytically the partition function for the Ising model with such boundary conditions. In the case of infinitely long cylinders of circumference L with duality-twisted boundary conditions we obtain the asymptotic expansion of the free energy and the inverse correlation lengths. We find that the ratio of subdominant finite-size correction terms in the asymptotic expansion of the free energy and the inverse correlation lengths should be universal. We verify such universal behavior in the framework of a perturbating conformal approach by calculating the universal structure constant C_{n1n} for descendent states generated by the operator product expansion of the primary fields. For such states the calculations of an universal structure constants is a difficult task, since it involves knowledge of the four-point correlation function, which in general is not fixed by conformal invariance except for some particular cases, including the Ising model.
We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and – if present – the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.
Exact partition functions of the Ising model on M timestimestimes N planar lattices with periodicndashndashndashaperiodic boundary conditions
The Grassmann path integral approach is used to calculate exact partition functions of the Ising model on M × N square (sq), plane triangular (pt) and honeycomb (hc) lattices with periodic–periodic (pp), periodic–antiperiodic (pa), antiperiodic–periodic (ap) and antiperiodic–antiperiodic (aa) boundary conditions. The partition functions are used to calculate and plot the specific heat, C / k B , as a function of temperature, θ = k B T / J . We find that for the N × N sq lattice, C / k B for pa and ap boundary conditions are different from those for aa boundary conditions, but for the N × N pt and hc lattices, C / k B for ap, pa and aa boundary conditions have the same values. Our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite-size corrections of the Ising model.
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.