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Physical Review Letters

Order-Parameter Profiles and Casimir Amplitudes in Critical Slabs

Zoran Borjan, Paul J. Upton

Published Nov 30, 1998• 56 citations
Publisher

Abstract

A critical phase confined between two parallel plates, with symmetry-breaking fields ${h}_{1}$ and ${h}_{2}$ acting on each plate, is considered for (i) ${h}_{1}{h}_{2}>0$ (denoted $ab\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}++$) and (ii) ${h}_{1}{h}_{2}<0$ ( $ab\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+\ensuremath{-}$). Using local-functional methods, we calculate order-parameter scaling functions ${\ensuremath{\Psi}}_{\mathrm{ab}}(x)$ and Casimir amplitudes ${A}_{\mathrm{ab}}$, for general dimension $d$. At $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$, our ${\ensuremath{\Psi}}_{++}(x)$ almost coincides with exact conformal predictions. For the Ising universality class, we obtain expansions for ${A}_{\mathrm{ab}}$ in $\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4\ensuremath{-}d\ensuremath{\downarrow}0$ in excellent agreement with those obtained from field theory and, in $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$, new results are presented for ${A}_{\mathrm{ab}}$ and ${\ensuremath{\Psi}}_{\mathrm{ab}}(x)$.

DOI
10.1103/physrevlett.81.4911

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A critical phase confined between two parallel plates, with symmetry-breaking fields h1{h}_{1}h1​ and h2{h}_{2}h2​ acting on each plate, is considered for (i) h1h2>0{h}_{1}{h}_{2}>0h1​h2​>0 (denoted ab=++ab\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}++ab=++) and (ii) h1h2<0{h}_{1}{h}_{2}<0h1​h2​<0 ( ab=+\ensuremath−ab\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+\ensuremath{-}ab=+\ensuremath−). Using local-functional methods, we calculate order-parameter scaling functions \ensuremathΨab(x){\ensuremath{\Psi}}_{\mathrm{ab}}(x)\ensuremathΨab​(x) and Casimir amplitudes Aab{A}_{\mathrm{ab}}Aab​, for general dimension ddd. At d=2d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2d=2, our \ensuremathΨ++(x){\ensuremath{\Psi}}_{++}(x)\ensuremathΨ++​(x) almost coincides with exact conformal predictions. For the Ising universality class, we obtain expansions for Aab{A}_{\mathrm{ab}}Aab​ in \ensuremathϵ=4\ensuremath−d\ensuremath↓0\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4\ensuremath{-}d\ensuremath{\downarrow}0\ensuremathϵ=4\ensuremath−d\ensuremath↓0 in excellent agreement with those obtained from field theory and, in d=3d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3d=3, new results are presented for Aab{A}_{\mathrm{ab}}Aab​ and \ensuremathΨab(x){\ensuremath{\Psi}}_{\mathrm{ab}}(x)\ensuremathΨab​(x).
Scaling density profiles O(N) symmetric systems confined between two parallel plates with a large separation L are analyzed by field-theoretical methods for a continuous phase transition at bulk criticality Tc,{\mathit{T}}_{\mathit{c},}Tc,​b in spatial dimension d=4-\ensuremath{\varepsilon}. At distances z\ensuremath{\ll}L from one wall a the profile contains a perturbation Ba,{\mathit{B}}_{\mathit{a},}Ba,​b(z/L)d{)}^{\mathit{d}})d induced by the distant wall b. In contrast to previous expectations the ratio of the universal amplitude Ba,{\mathit{B}}_{\mathit{a},}Ba,​b and the corresponding Casimir amplitude is not hyperuniversal, i.e., not independent of both surface universality classes characterizing the walls a and b.
Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously.
The exact critical Casimir amplitude is derived for anisotropic systems within the d=2 Ising universality class by combining conformal field theory with anisotropic φ^{4} theory. Explicit results are presented for the general anisotropic scalar φ^{4} model and for the fully anisotropic triangular-lattice Ising model in finite rectangular and infinite strip geometries with periodic boundary conditions. These results demonstrate the validity of multiparameter universality for confined anisotropic systems and the nonuniversality of the critical Casimir amplitude. We find an unexpected complex form of self-similarity of the anisotropy effects near the instability where weak anisotropy breaks down. This can be traced back to the property of modular invariance of isotropic conformal field theory for d=2. More generally, for d>2 we predict the existence of self-similar structures of the finite-size scaling functions of O(n)-symmetric systems with planar anisotropies and periodic boundary conditions both in the critical region for n≥1 as well as in the Goldstone-dominated low-temperature region for n≥2.
We consider the classical O(N)O(N)O(N)-symmetric models confined in a ddd-dimensional slablike geometry and subject to periodic boundary conditions. Applying the one-particle-irreducible variant of functional renormalization group (RG), we compute the critical Casimir forces acting between the slab boundaries. The applied truncation of the exact functional RG flow equation retains interaction vertices of arbitrary order. We evaluate the critical Casimir amplitudes ${\ensuremath{\Delta}}_{f}(d,N)$ for continuously varying dimensionality between two and three and N=1,2N=1,2N=1,2. Our findings are in very good agreement with exact results for d=2d=2d=2 and N=1N=1N=1. For d=3d=3d=3, our results are closer to Monte Carlo predictions than earlier field-theoretic RG calculations. Inclusion of the wave function renormalization and the corresponding anomalous dimension in the calculation has negligible impact on the computed Casimir forces.
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 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.