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arXiv:cond-mat/0107235 · Journal of Physics A Mathematical and General

Conformal off-diagonal boundary density profiles on a semi-infinite strip

Loïc Turban

Published Sep 14, 2001• 2 citations
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Abstract

The off-diagonal profile phi(v) associated with a local operator (order parameter or energy density) close to the boundary of a semi-infinite strip with width L is obtained at criticality using conformal methods. It involves the surface exponent x_phi^s and displays a simple universal behaviour which crosses over from surface finite-size scaling when v/L is held constant to corner finite-size scaling when v/L -> 0.

DOI
10.1088/0305-4470/34/38/101

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The off-diagonal profile phi(v) associated with a local operator (order parameter or energy density) close to the boundary of a semi-infinite strip with width L is obtained at criticality using conformal methods. It involves the surface exponent x_phi^s and displays a simple universal behaviour which crosses over from surface finite-size scaling when v/L is held constant to corner finite-size scaling when v/L -> 0.
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.
The authors present a simple argument showing that the surface energy density of a semi-infinite d-dimensional spin system has, in general, a leading thermal singularity of the same form, mod T-T c mod 2- alpha , as the bulk free energy. At bulk criticality energy-energy correlations decay as r -2d parallel to the surface. These results hold for both free and fixed boundary spins, i.e. the 'ordinary' and 'extraordinary' transitions. The extraordinary critical behaviour of the magnetisation is the same as that of the energy density. They also confirm and generalise these results in two dimensions with an independent approach based on conformal invariance.
The conformal-invariance approach is extended to two-dimensional critical systems with mixed ab boundary conditions. The profiles 〈\ensuremath{\varphi}(r)〉ab{\mathrm{〉}}_{\mathit{a}\mathit{b}}〉ab​ 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 Aab{\mathit{A}}_{\mathit{a}\mathit{b}}Aab​Bab{\mathit{B}}_{\mathit{a}\mathit{b}}Bab​ of the Casimir force and the Fisher--de Gennes distant-wall correction to the profile satisfy Bab{\mathit{B}}_{\mathit{a}\mathit{b}}Bab​=-4\ensuremath{\pi}${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$${\mathit{c}}^{\mathrm{\ensuremath{-}}1}$Aab{\mathit{A}}_{\mathit{a}\mathit{b}}Aab​. Here ${\mathit{x}}_{\mathrm{\ensuremath{\varphi}}}$ is the scaling dimension of \ensuremath{\varphi}(r) and c is the central charge.
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.
The critical behaviour of semi-infinite d -dimensional systems with short-range interactions and an O( n ) invariant Hamiltonian is investigated at an m -axial Lifshitz point with an isotropic wave-vector instability in an m -dimensional subspace of d parallel to the surface. Continuum |ϕ| 4 models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant λ) must be included in addition to the familiar ones ∝ϕ 2 . Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in d = 4 + m /2 − dimensions (with > 0) are located at λ = λ* = O( ). At second order in , the surface critical exponents of both the ordinary and the special transitions start to deviate from their m = 0 analogues. Results to order 2 are presented for the surface critical exponent β ord 1 of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by d + m (θ − 1), where θ = ν l 4 /ν l 2 is the bulk anisotropy exponent.