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Journal of Physics A Mathematical and General

A test of conformal invariance: correlation functions on a disc

R Badke, V. Rittenberg, H. Ruegg

Published Oct 1, 1985• 5 citations
Publisher

Abstract

Using conformal invariance one can derive the correlation functions on a disc from those in the half-plane. The correlation function in the half-plane is determined by the 'small' conformal invariance up to an unknown function of one variable. By measuring, using the Monte Carlo method, the correlation function for two different configurations, the unknown function can be eliminated and one obtains a test of conformal invariance. It is shown that the Ising and the three-state Potts model pass the test for very small lattices.

DOI
10.1088/0305-4470/18/14/008

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Using conformal invariance one can derive the correlation functions on a disc from those in the half-plane. The correlation function in the half-plane is determined by the 'small' conformal invariance up to an unknown function of one variable. By measuring, using the Monte Carlo method, the correlation function for two different configurations, the unknown function can be eliminated and one obtains a test of conformal invariance. It is shown that the Ising and the three-state Potts model pass the test for very small lattices.
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.
The conformal covariance of correlation functions is checked in the second-order transition induced by random bonds in the two-dimensional eight-state Potts model. The decay of correlations is obtained via transfer matrix calculations in a cylinder geometry, and large-scale Monte Carlo simulations provide access to the correlations and the profiles inside a square with free or fixed boundary conditions. In both geometries, conformal transformations constrain the form of the spatial dependence, leading to accurate determinations of the order parameter scaling index, in good agreement with previous independent determinations obtained through standard techniques. The energy density exponent is also computed.
Off-diagonal profiles of local densities (e.g. order parameter or energy density) are calculated at the bulk critical point, by conformal methods, on a strip with transverse coordinate v, for different types of boundary conditions (free, fixed and mixed). Such profiles, which are defined by the non-vanishing matrix element of the appropriate operator between the ground state and the corresponding lowest excited state of the strip Hamiltonian, enter into the expression of two-point correlation functions on a strip. They are of interest in the finite-size scaling study of bulk and surface critical behaviour since they allow the elimination of regular contributions. The conformal profiles, which are obtained through a conformal transformation of the correlation functions from the half-plane to the strip, are in agreement with the results of a direct calculation, for the energy density of the two-dimensional Ising model.
We explain how the axioms of Conformal Field Theory are used to make predictions about critical exponents of continuous phase transitions in three dimensions, via a procedure called the conformal bootstrap. The method assumes conformal invariance of correlation functions, and imposes some relations between correlation functions of different orders. Numerical analysis shows that these conditions are incompatible unless the critical exponents take particular values, or more precisely that they must belong to a small island in the parameter space.
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.