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Physical Review Letters

Density profiles in confined critical systems and conformal invariance

Theodore W. Burkhardt, Tianyou Xue

Published Feb 18, 1991• 71 citations
Publisher

Abstract

The conformal-invariance approach is extended to two-dimensional critical systems with mixed ab boundary conditions. The profiles 〈\ensuremath{\varphi}(r)${\mathrm{〉}}_{\mathit{a}\mathit{b}}$ of the magnetization and the energy density in Ising and Q-state Potts strips with +- and fixed-free boundary conditions are derived. The profile of an arbritrary operator degenerate at level 2 is expressed in terms of hypergeometric functions. The universal amplitudes ${\mathit{A}}_{\mathit{a}\mathit{b}}$${\mathit{B}}_{\mathit{a}\mathit{b}}$ of the Casimir force and the Fisher--de Gennes distant-wall correction to the profile satisfy ${\mathit{B}}_{\mathit{a}\mathit{b}}$=-4\ensuremath{\pi}${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$${\mathit{c}}^{\mathrm{\ensuremath{-}}1}$${\mathit{A}}_{\mathit{a}\mathit{b}}$. Here ${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$ is the scaling dimension of \ensuremath{\varphi}(r) and c is the central charge.

DOI
10.1103/physrevlett.66.895

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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.