We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman's spinor method to calculate low-temperature series expansions for the partition function to high order. From these we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk free energy. For the isotropic case, they also agree with Vernier and Jacobsen's conjecture for the $60^{\circ}$ corners, and with Cardy and Peschel's conformal invariance predictions for the dominant behaviour at criticality.
We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman's spinor method to calculate low-temperature series expansions for the partition function to high order. From these we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk free energy. For the isotropic case, they also agree with Vernier and Jacobsen's conjecture for the 60∘ corners, and with Cardy and Peschel's conformal invariance predictions for the dominant behaviour at criticality.
We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.
Abstract We use Kaufman’s spinor method to calculate the bulk, surface and corner free energies f b , f s , f s ′ , f c of the anisotropic square lattice zero-field Ising model for the ordered ferromagnetic case. For f b , f s , f s ′ our results of course agree with the early work of Onsager, McCoy and Wu. We also find agreement with the conjectures made by Vernier and Jacobsen (VJ) for the isotropic case. We note that the corner free energy f c depends only on the elliptic modulus k that enters the working, and not on the argument v , which means that VJ’s conjecture applies for the full anisotropic model. The only aspect of this paper that is new is the actual derivation of f c , but by reporting all four free energies together we can see interesting structures linking them.
We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass all integrable curves of the Q -state Potts model on the square and triangular lattices, including the antiferromagnetic transition curves and the Ising model ( Q = 2) at temperature T , as well as a fully packed O( n ) type loop model on the square lattice. The expansions are around the trivial fixed points at infinite Q , n or 1/ T . By using a carefully chosen expansion parameter, q ≪ 1, all expansions turn out to be of the form , where the coefficients α k and β k are periodic functions of k . Thanks to this periodicity property, we can conjecture the form of the expansions to all orders (except in a few cases where the periodicity is too large). These expressions are then valid for all 0 ⩽ q < 1. We analyse in detail the q → 1 − limit in which the models become critical. In this limit the divergence of the corner free energy defines a universal term which can be compared with the conformal field theory (CFT) predictions of Cardy and Peschel. This allows us to deduce the asymptotic expressions for the correlation length in several cases. Finally we work out the FLM formulae for the case where some of the system's boundaries are endowed with particular (non-free) boundary conditions. We apply this in particular to the square-lattice Potts model with Jacobsen–Saleur boundary conditions, conjecturing the expansions of the surface and corner free energies to arbitrary order for any integer value of the boundary interaction parameter r . These results are in turn compared with CFT predictions.
We derive an exact formula for the corner free-energy contribution of weakly anisotropic two-dimensional critical systems in the Ising universality class on rectangular domains, expressed in terms of quantities that specify the anisotropic fluctuations. The resulting expression agrees with numerical exact calculations that we perform for the anisotropic triangular Ising model and quantifies the nonuniversality of the corner term for anisotropic critical two-dimensional systems. Our generic formula is expected to apply also to other weakly-anisotropic critical two-dimensional systems that allow for a conformal field theory description in the isotropic limit. We consider the three-states and four-states Potts models as further specific examples.
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.
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.