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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Bulk and surface critical behavior of the three-dimensional Ising model and conformal invariance

Youjin Deng, Henk W. J. Blöte

Published Jun 27, 2003• 15 citations
Publisher

Abstract

Using a continuous cluster Monte Carlo algorithm, we investigate the critical three-dimensional Ising model in its anisotropic limit. From the ratio of the magnetic correlations in the strong- and the weak-coupling directions, we determine the length ratio relating the isotropic Ising model and the anisotropic limit. On this basis, we simulate the critical Ising model on a spherocylinder S2 x R1, i.e., a curved geometry obtained from a conformal mapping of the infinite space R3. From correlation lengths along the spherocylinder, combined with the prediction of conformal invariance, we estimate the magnetic and thermal scaling dimensions as X(h)=0.5182(6) and X(t)=1.419(7), respectively. The behavior of the Binder cumulant is also determined in the limit of an infinitely long spherocylinder. Next, free boundary conditions are imposed on the equators of the spherocylinder, and thus the geometry S1 x S+ x R1 is obtained. The surface magnetic scaling dimension is estimated as X(s)(h)=1.263(5). The consistency of the aforementioned estimations and existing results confirms that the three-dimensional Ising model is conformally invariant. Further, the precision of these results reveals that, as in two dimensions, conformal mappings provide a powerful tool to investigate critical phenomena. With the continuous cluster algorithm, we also perform simulations of systems inside a conventional solid cylinder. The surface magnetic correlation length differs, within the estimated error margin, by a factor pi/2 from that along a half spherocylinder S1 x S+ x R1 with the same radius.

DOI
10.1103/physreve.67.066116

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Using a continuous cluster Monte Carlo algorithm, we investigate the critical three-dimensional Ising model in its anisotropic limit. From the ratio of the magnetic correlations in the strong- and the weak-coupling directions, we determine the length ratio relating the isotropic Ising model and the anisotropic limit. On this basis, we simulate the critical Ising model on a spherocylinder S2 x R1, i.e., a curved geometry obtained from a conformal mapping of the infinite space R3. From correlation lengths along the spherocylinder, combined with the prediction of conformal invariance, we estimate the magnetic and thermal scaling dimensions as X(h)=0.5182(6) and X(t)=1.419(7), respectively. The behavior of the Binder cumulant is also determined in the limit of an infinitely long spherocylinder. Next, free boundary conditions are imposed on the equators of the spherocylinder, and thus the geometry S1 x S+ x R1 is obtained. The surface magnetic scaling dimension is estimated as X(s)(h)=1.263(5). The consistency of the aforementioned estimations and existing results confirms that the three-dimensional Ising model is conformally invariant. Further, the precision of these results reveals that, as in two dimensions, conformal mappings provide a powerful tool to investigate critical phenomena. With the continuous cluster algorithm, we also perform simulations of systems inside a conventional solid cylinder. The surface magnetic correlation length differs, within the estimated error margin, by a factor pi/2 from that along a half spherocylinder S1 x S+ x R1 with the same radius.
We formulate conformal mappings between an infinite plane and a spheroid, and one between a semi-infinite plane and a half spheroid. Special cases of the spheroid include the surface of an infinitely long cylinder, of a sphere, and of a flat disc. These mappings are applied to the critical Ising model. For the case of the sphere and the flat disc, we derive analytical expressions for the second and the fourth moments of the magnetization density, and thus for the Binder cumulant. Next, we investigate Ising models on spheroids and half spheroids by means of a continuous cluster Monte Carlo method for simulations in curved geometries. Fixed and free boundary conditions are imposed for half spheroids. The Monte Carlo data are analyzed by finite-size scaling. Critical values of the Binder cumulants and other ratios on the sphere and on the flat disc agree precisely with the exact calculations mentioned above. At criticality, we also sample two- and one-point correlation functions on spheroids on half spheroids, respectively. The magnetic and temperature scaling dimensions, as determined from the Monte Carlo data and the theory of conformal invariance, are in good agreement with exact results.
We investigate a critical Ising-like model in the curved geometry ${S}^{2}\ifmmode\times\else\texttimes\fi{}{R}^{1}$ obtained by a conformal mapping of the infinite 3D space R3{R}^{3}R3. The incompatibility of regular lattices with this geometry is avoided by use of the anisotropic limit of the lattice Ising model, which renders one of the space coordinates continuous. We determine magnetic and energylike correlation lengths of this model by means of a cluster Monte Carlo algorithm. From these data, and the assumption of conformal invariance, we obtain the magnetic and temperature scaling dimensions as Xh=0.5178(12){X}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.5178(12)Xh​=0.5178(12) and Xt=1.423(19){X}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.423(19)Xt​=1.423(19), respectively. These numbers are in a good agreement with the existing results for the 3D Ising universality class.
We investigate the anisotropic limit of the bond-percolation model in d dimensions, which is equivalent to a (d-1) -dimensional quantum q-->1 Potts model. We formulate an efficient Monte Carlo method for this model. Its application shows that the anisotropic model fits well with the percolation universality class in d dimensions. For three-dimensional rectangular geometry, we determine the critical point as t(c) =8.6429(4), and determine the length ratio as alpha(0) =1.5844(3), which relates the anisotropic limit of the percolation model and its isotropic version. On this basis, we simulate critical systems in several curved geometries including a spheroid and a spherocylinder. Using finite-size scaling and the assumption of conformal invariance, we determine the bulk and surface magnetic exponents in two and three dimensions. They are in good agreement with the existing results.
We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.
By solving partial differential equations for the correlation functions implied by conformal invariance, we obtain exact results at the critical point for the bulk six-point function 〈\ensuremath{\sigma}(1)...\ensuremath{\sigma}(6)〉 of the two-dimensional Ising model and the three-point functions 〈\ensuremath{\sigma}(1)\ensuremath{\sigma}(2)\ensuremath{\sigma}(3)〉, 〈\ensuremath{\sigma}(1)\ensuremath{\mu}(2)\ensuremath{\mu}(3)〉 of the semi-infinite two-dimensional Ising model with fixed boundary spins. Here \ensuremath{\sigma} and \ensuremath{\mu} denote order (spin) and disorder variables, respectively. We also derive the explicit form of the magnetization profile at the bulk critical temperature in an Ising strip of infinite length and finite width, with spin-up boundary conditions on one edge and spin-down boundary conditions on the other.