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arXiv:cond-mat/0310365

The Ising model on curved lattices

Ruben Costa-Santos

Published Oct 15, 2003• 0 citations
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Abstract

We review recent results concerning finite size corrections to the Ising model free energy on lattices with non-trivial topology and curvature. From conformal field theory considerations two distinct universal terms are expected, a logarithmic term determined by the system curvature and a scale invariant term determined by the system shape and topology. Both terms have been observed numerically, using the Kasteleyn Pfaffian method, for lattices with topologies ranging from the sphere to that of a genus two surface. The constant term is shown to be expressible in terms of Riemann theta functions while the logarithmic correction reproduces the theoretical prediction by Cardy and Peschel for singular metrics.

DOI
10.48550/arxiv.cond-mat/0310365

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We review recent results concerning finite size corrections to the Ising model free energy on lattices with non-trivial topology and curvature. From conformal field theory considerations two distinct universal terms are expected, a logarithmic term determined by the system curvature and a scale invariant term determined by the system shape and topology. Both terms have been observed numerically, using the Kasteleyn Pfaffian method, for lattices with topologies ranging from the sphere to that of a genus two surface. The constant term is shown to be expressible in terms of Riemann theta functions while the logarithmic correction reproduces the theoretical prediction by Cardy and Peschel for singular metrics.
The free energy of a two-dimensional system at criticality has in general a universal part proportional to the logarithm of the system size. This term was shown by Cardy and Peschel to be related to the curvature of the system, with smooth metrics and singular points contributing in distinct ways. In this paper we present a numerical study of the effect for the Ising model on lattices with various topologies ranging sphere to genus two surfaces. The nature of this term for specific lattice models is an open problem because the distinction between the two kinds of contributions involves an interchange of the limit of singular curvature with the thermodynamic limit. For all the lattices studied we found precise agreement with the conformal field theory prediction for conical singularities.
We study the finite-size scaling of the free energy of the Ising model on lattices with the topology of the tetrahedron and the octahedron. Our construction allows us to perform changes in the length scale of the model without altering the distribution of the curvature in the space. We show that the subleading contribution to the free energy follows a logarithmic dependence, in agreement with the conformal field theory prediction. The conformal anomaly is given by the sum of the contributions computed at each of the conical singularities of the space, except when perfect order of the spins is precluded by frustration in the model.
Abstract Based on the results published recently (Hucht 2017 J. Phys. A: Math. Theor . 50 065201), the universal finite-size contributions to the free energy of the square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit L , M → ∞ , T → T c , with fixed temperature scaling variable x ∝ ( T / T c − 1 ) M and fixed aspect ratio ρ ∝ L / M . We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point T = T c we confirm predictions from conformal field theory (Cardy and Peschel 1988 Nucl. Phys . B 300 377, Kleban and Vassileva 1991 J. Phys. A: Math. Gen . 24 3407). The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.
A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.
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.
A scaling framework is developed for the emergence of the universal finite-size correction (ulnL)L2\frac{(u\mathrm{ln}L)}{{L}^{2}}L2(ulnL)​, recently found by Cardy and Peschel in the free-energy density of the two-dimensional (2D) conformally invariant models at Tc{T}_{c}Tc​. Our approach applies in the critical region $T\ensuremath{\simeq}{T}_{\mathrm{cZ}}$, and accounts for the universality of uuu, but not its value. We predict a similar universal term, (ulnL)L3\frac{(u\mathrm{ln}L)}{{L}^{3}}L3(ulnL)​, in 3D. We also find that the "corner" free energy in the bulk ($L=\ensuremath{\infty}$) has the leading singular behavior ($\ensuremath{-}\ensuremath{\nu}u\mathrm{ln}|t|$).