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Journal of Physics A Mathematical and General

Domain boundary energies in finite regions at 2D criticality via conformal field theory

Peter H. Kleban, I Vassileva

Published Nov 21, 1992• 10 citations
Publisher

Abstract

By use of conformal field theory, the authors calculate universal results for the energies of curved domain boundaries at two-dimensional critical points in circular and rectangular regions. This work extends the theory of finite-size effects at criticality, complementing previous exact results for the energies of domain boundaries in infinite strips.

DOI
10.1088/0305-4470/25/22/007

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By use of conformal field theory, the authors calculate universal results for the energies of curved domain boundaries at two-dimensional critical points in circular and rectangular regions. This work extends the theory of finite-size effects at criticality, complementing previous exact results for the energies of domain boundaries in infinite strips.
By use of conformal field theory, we calculate the energies of one or more curved domain boundaries at two-dimensional critical points. We find the interaction of two separated boundaries of the same type to be weakly attractive, and proportional to the square of an operator-product expansion coefficient. In a strip of width L, the interaction vanishes exponentially with separation, over a distance set by L. It is also small for nearby domains when one of them is much smaller than L. Many-body interaction energies are weak as well.
Using recent results by Cardy (1984) based on conformal invariance of critical correlation functions the authors calculate universal results for scattering functions S(k), susceptibilities, correlation lengths and specific heat correction terms for finite Ising systems in two dimensions with circular, elliptical and rectangular shapes and free boundary conditions. Results show the effects of critical order on these quantities. For a circle, S(k) decays as 1/k 2- eta for a large range of intermediate k values with an 'apparent' exponent eta =0.09. The probable influence of end, edge and domain wall effects in the rectangular geometry is discussed. Application of the results to experimental systems and other theoretical models is discussed.
We show that for conformally invariant two-dimensional systems, the amplitude of the finite-size corrections to the free energy of an infinitely long strip of width L at criticality is linearly related to the conformal anomaly number c, for various boundary conditions. The result is confirmed by renormalization-group arguments and numerical calculations. It is also related to the magnitude of the Casimir effect in an interacting one-dimensional field theory, and to the low-temperature specific heat in quantum chains.
We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.
We show how the apparent power-law scaling of classical two-dimenional structure factors in bounded domains at criticality is related to fixed and free boundary conditions on the order parameter. This enables us to predict the behavior of the corresponding apparent scaling dimensions based on a simple symmetry analysis. With the use of correlation functions from conformal field theory, our predictions are confirmed with exact calculations for multicritical Ising models. Effects of random boundary fields are also discussed.