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arXiv:cond-mat/9612128 · Journal of Physics A Mathematical and General

Off-diagonal density profiles and conformal invariance

Loïc Turban, Ferenc Iglói

Published Mar 7, 1997• 13 citations
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Abstract

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.

DOI
10.1088/0305-4470/30/5/006

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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.
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 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.
With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane y>0 with different boundary conditions a and b on the negative and positive x axes. For ab=-+ and f+, they determined the one- and two-point averages of the spin σ and energy ε. Here +,-, and f stand for spin-up, spin-down, and free-spin boundaries, respectively. The case +-+-+⋯, where the boundary condition switches between + and - at arbitrary points, ζ_{1},ζ_{2},⋯ on the x axis was also analyzed. In the first half of this paper a similar study is carried out for the alternating boundary condition +f+f+⋯ and the case -f+ of three different boundary conditions. Exact results for the one- and two-point averages of σ,ε, and the stress tensor T are derived with conformal-invariance methods. From the results for 〈T〉, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is derived for mixed boundary conditions. In the second half of the paper, arbitrary two-dimensional critical systems with mixed boundary conditions are analyzed with boundary-operator expansions. Two distinct types of expansions-away from switching points of the boundary condition and at switching points-are considered. Using the expansions, we express the asymptotic behavior of two-point averages near boundaries in terms of one-point averages. We also consider the strip geometry with mixed boundary conditions and derive the distant-wall corrections to one-point averages near one edge due to the other edge. Finally we confirm the consistency of the predictions obtained with conformal-invariance methods and with boundary-operator expansions, in the the first and second halves of the paper.
We use the density matrix renormalization group (DMRG) to perform a detailed study of the critical properties of the two-dimensional QQQ state Potts model, including the magnetization and energy-density profiles, bulk and surface critical exponents, and the Casimir amplitudes. We apply symmetry breaking boundary conditions to a $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ strip and diagonalize the corresponding transfer matrix for a series of moderately large systems (L< 64)(L<~64)(L< 64) by the DMRG method. The numerically very accurate finite lattice results are then extrapolated by efficient sequence extrapolation techniques. The density profiles and the Casimir amplitudes at the critical point are found to follow precisely the conformal predictions for Q=2Q=2Q=2 and 333. Similarly, the bulk and surface critical exponents of the models are in very good agreement with the conformal and exact values: their accuracy has reached or even exceeded the accuracy of other available numerical methods. For the Q=4Q=4Q=4 model both the profiles and the critical exponents show strong logarithmic corrections, which are also studied.
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.