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arXiv:cond-mat/9910467 · Physical review. B, Condensed matter

Order parameter for two-dimensional critical systems with boundaries

Ivica Reš, Joseph P. Straley

Published Jun 1, 2000• 13 citations
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Abstract

Conformal transformations can be used to obtain the order parameter for two-dimensional systems at criticality in finite geometries with fixed boundary conditions on a connected boundary. To the known examples of this class (such as the disk and the infinite strip) we contribute the case of a rectangle. We show that the order parameter profile for simply connected boundaries can be represented as a universal function (independent of the criticality model) raised to the power $\frac{1}{2}\ensuremath{\eta}.$ The universal function can be determined from the Gaussian model or equivalently a problem in two-dimensional electrostatics. We show that fitting the order parameter profile to the theoretical form gives an accurate route to the determination of $\ensuremath{\eta}.$ We perform numerical simulations for the Ising model and percolation for comparison with these analytic predictions, and apply this approach to the study of the planar rotor model.

DOI
10.1103/physrevb.61.14425

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Conformal transformations can be used to obtain the order parameter for two-dimensional systems at criticality in finite geometries with fixed boundary conditions on a connected boundary. To the known examples of this class (such as the disk and the infinite strip) we contribute the case of a rectangle. We show that the order parameter profile for simply connected boundaries can be represented as a universal function (independent of the criticality model) raised to the power 12\ensuremathη.\frac{1}{2}\ensuremath{\eta}.21​\ensuremathη. The universal function can be determined from the Gaussian model or equivalently a problem in two-dimensional electrostatics. We show that fitting the order parameter profile to the theoretical form gives an accurate route to the determination of \ensuremathη.\ensuremath{\eta}.\ensuremathη. We perform numerical simulations for the Ising model and percolation for comparison with these analytic predictions, and apply this approach to the study of the planar rotor model.
We consider Ising strips with width N and periodic boundary conditions and Ising squares with edge length N and special partially periodic boundaries. Assuming invariance of the spin correlations under conformal mappings of the infinite plane onto the strip and square, we determine the second and fourth moments of the total magnetization M from the known bulk two- and four-spin correlation functions at criticality. For both geometries the predictions of conformal invariance for universal asymptotic forms involving the ratio 〈M4{M}^{4}M4〉〈M2{M}^{2}M2${〉}^{\mathrm{\ensuremath{-}}2}$ as N\ensuremath{\rightarrow}\ensuremath{\infty} are in excellent agreement with transfer-matrix results.
The critical behavior of site percolation confined to the interior of a parabola is investigated. At the percolation threshold, conformal mapping is used to derive analytical expressions for the pair correlation function and the local order parameter in the case of free and fixed boundary conditions, respectively. Numerical results obtained by intensive Monte Carlo simulations are in excellent agreement with the analytical expressions. The numerical data for the off-critical local order parameter with free boundary conditions show that correction to scaling terms are important and the leading correction exponent is estimated to be \ensuremath{\epsilon}\ensuremath{\simeq}0.18.
We study the scaling behavior of the entanglement entropy of two-dimensional conformal quantum critical systems, i.e., systems with scale-invariant wave functions. They include two-dimensional generalized quantum dimer models on bipartite lattices and quantum loop models, as well as the quantum Lifshitz model and related gauge theories. We show that under quite general conditions, the entanglement entropy of a large and simply connected subsystem of an infinite system with a smooth boundary has a universal finite contribution, as well as scale-invariant terms for special geometries. The universal finite contribution to the entanglement entropy is computable in terms of the properties of the conformal structure of the wave function of these quantum critical systems. The calculation of the universal term reduces to a problem in boundary conformal field theory
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.
Density profiles are investigated arising in a critical Ising model in two dimensions which is confined to a rectangular domain with uniform or mixed boundary conditions and arbitrary aspect ratio. For the cases in which the two vertical sides of the rectangle have up-spin boundary conditions + and the two horizontal sides with either down-spin boundary conditions - or with free-spin boundary conditions f, exact results are presented for the density profiles of the energy and the order parameter which display a surprisingly rich behavior. The new results follow by means of conformal transformations from known results in the half plane with +-+-+ and +f+f+ boundary conditions. The corners with mixed boundary conditions lead to interesting behavior, even in the limit of a half-infinite strip. The behavior near these corners can be described by a "corner-operator-expansion," which is discussed in the second part of the paper. The analytic predictions agree very well with simulations, with no adjustable parameters.