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Physical Review Letters

Dilation and Conformal Covariance of Multipoint Correlation Functions and Dimensions of Fluctuating Quantities at the Critical Point of Fluids

Alan Martin Wolsky, M. S. Green, J. D. Gunton

Published Nov 5, 1973• 4 citations
Publisher

Abstract

The operator algebra and an identity in the statistical mechanics of fluids are used to demonstrate that the cumulant average of arbitrary products of appropriate combinations of local fluctuating energy and density are scale and conformal covariant at the critical point. These combinations are uncorrelated and are found by diagonalizing a matrix whose elements are energy and density derivatives of pressure-density and pressure-energy correlations. The eigenvalues are the dimensions of the appropriate variables.

DOI
10.1103/physrevlett.31.1193

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The operator algebra and an identity in the statistical mechanics of fluids are used to demonstrate that the cumulant average of arbitrary products of appropriate combinations of local fluctuating energy and density are scale and conformal covariant at the critical point. These combinations are uncorrelated and are found by diagonalizing a matrix whose elements are energy and density derivatives of pressure-density and pressure-energy correlations. The eigenvalues are the dimensions of the appropriate variables.
We derive an identity which determines the infinitesimal response of the cumulant average of the product of mmm arbitrary local fluctuating quantities (connected mmm-point functions) to changes in its arguments engendered by a member of an arbitrary continuous group of point transformations. We illustrate the use of the identity in a variety of cases, and in particular we show that there is an intimate connection between the covariance of the connected mmm-point functions under the special conformal group and under the group of dilations. By combining our identity with the assumptions of the operator algebra, we find that at the critical point the connected density mmm-point functions are covariant under both groups. We also give another derivation for an expression for the exponent $x=(\frac{1}{2})(5\ensuremath{-}\ensuremath{\eta})$, which was previously found by Green and Gunton.
We consider variational (density functional) models of fluids confined in parallel-plate geometries (with walls situated in the planes z = 0 and z = L respectively) and focus on the structure of the pair correlation function . We show that for local variational models there exist two non-trivial identities relating both the transverse Fourier transform and the zeroth moment at different positions and . These relations form an algebra which severely restricts the possible form of the function . For the common situations in which the equilibrium one-body (magnetization/number density) profile exhibits an odd or even reflection symmetry in the plane, the algebra simplifies considerably, and is used to relate the correlation function to the finite-size excess free energy . We rederive non-trivial scaling expressions for the finite-size contribution to the free energy at bulk criticality and for systems where large-scale interfacial fluctuations are present. Extensions to non-planar geometries are also considered.
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The long distance behavior of two-point functions of critical lattice statistical mechanics models depend on detailed information of the underlying quantum field theory. Yet these universal properties can be difficult to extract because generic lattice operators couple to the infinite hierarchy of continuum scaling operators with unknown coefficients. Recently in Physical Review E 114, 024146 (2026), one approach to disentangle some of this information was proposed using the eigen-decomposition of the operator covariance matrix constructed from cross correlations of a finite set of NℓN_\ellNℓ​ lattice operators. In this work we analyze this method in detail. We provide an explicit proof that under the assumption of scale invariance the leading decay of the eigenvalues are controlled by the smallest NℓN_\ellNℓ​ scaling dimensions, even after taking irrelevant perturbations into account. Based on this understanding, we propose a new method by which the eigenvectors of the covariance matrix can be used to construct the amplitudes of the continuum two-point correlation functions. This allows for a complete determination of the structure of the continuum two-point functions from the lattice covariance matrix. We demonstrate our ideas both on synthetic data sets (drawn from the scaling hypothesis) and numerically exact data for the two dimensional Ising model.