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Physical Review Letters

Exact variational analysis of layered planar Ising models

Lev V. Mikheev, Michael Ellis Fisher

Published Jan 11, 1993• 22 citations
Publisher

Abstract

A quest for microcanonical functionals describing near-critical behavior has led to an exact local variational functional for square lattice Ising models in zero field with varying bond strengths along the z axis. In the continuum limit near ${\mathit{T}}_{\mathit{C}}$ this yields a second-order differential equation for the transverse Fourier coefficients of the energy density. Applications to a ``grain boundary'' between differing regions and to a wall with a surface field ${\mathit{h}}_{1}$ give bulk and surface exponents and scaling-limit energy-density profiles which exhibit rich crossovers and nonmonotonic variation.

DOI
10.1103/physrevlett.70.186

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A quest for microcanonical functionals describing near-critical behavior has led to an exact local variational functional for square lattice Ising models in zero field with varying bond strengths along the z axis. In the continuum limit near TC{\mathit{T}}_{\mathit{C}}TC​ this yields a second-order differential equation for the transverse Fourier coefficients of the energy density. Applications to a ``grain boundary'' between differing regions and to a wall with a surface field h1{\mathit{h}}_{1}h1​ give bulk and surface exponents and scaling-limit energy-density profiles which exhibit rich crossovers and nonmonotonic variation.
Ising models on the plane square lattice with an arbitrary variation of the bond strengths, ${\mathit{J}}^{\mathrm{\ensuremath{\parallel}}}$(z) and ${\mathit{J}}^{\mathrm{\ensuremath{\perp}}}$(z), with one of the two axial coordinates, z, are considered. The total entropy is exactly represented as a functional of contributions ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z) to the local energy density arising from the Onsager fermions with wave vector q parallel to the layer axis, y. The resulting explicit local expression provides an effective variational principle for the free energy and energy-density profiles. In the scaling limit the problem reduces to a set of independent second-order differential equations for each ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z). The power of the method is demonstrated by application to an interface between two uniform but distinct regions; this includes the problem of a wall with a surface field, h1{\mathit{h}}_{1}h1​, as a special case. Previous results for the bulk and surface exponents and for the energy-energy correlation function are easily recovered. Near criticality the method yields, in addition, universal, scaled energy-density profiles, \ensuremath{\varepsilon}(z;T), which exhibit rich crossovers and nonmonotonic variation with z.
I use an exact variational formulation of Mikheev and Fisher to study the critical Ising strip with a grain boundary and confining surfaces characterized by arbitrary and different surface magnetic fields. Energy density profiles that serve as order parameters of the system within the used method show strong nonmonotonous behavior in the vicinity of confining surfaces. I consider short-distance expansion of energy density profiles. New universal amplitudes associated with distant-wall corrections exhibit nontrivial crossover behaviors from positive to negative values as the field's variables are continuously varied. Casimir amplitudes calculated in a self-contained manner are characterized by complex manifolds which may comprise two disconnected positive wings separated by an area of negative values. I define and determine the generalized de Gennes-Fisher amplitude, which strongly suggests that a stress tensor is present within one of the distant-wall corrections. This is an unanticipated result given that a similar discovery for standard extraordinary (E) and ordinary (O) surface universality classes was based on the conformal invariance symmetry, which is broken under the present boundary conditions. A grain boundary essentially influences both energy density profiles and Casimir amplitudes, besides surface fields with a number of accompanying features that are examined in detail. We present closed analytic forms of energy density profiles for standard (N) and (O) BCs: ɛNN(x,g),ɛOO(x,g),ɛNO(x,g) [(N) is the normal surface universality class, g is the grain boundary's strength, x is the relative distance x:=z/L with L as a film width], thus enabling us to deduce their important symmetry properties and to have a detailed insight into their general behaviors.
A critical Ising strip with confining boundaries characterized by arbitrary and different surface magnetic field variables is considered by the exact variational formulation of Mikheev and Fisher. As one of the most important scaling densities in films we study the universal functions of the energy density profiles which exhibit strong nonmonotonous behavior near the confining boundaries. We also examine their short-distance expansion that reveals novel universal amplitudes associated with the distant-wall corrections. They manifest nontrivial crossover behaviors from positive to negative values as the fields variables are freely varied. As an indispensable part of our new formulation of the generalized distant-wall correction de Gennes-Fisher universal amplitude we derive in a self-contained manner within the present approach the Casimir amplitudes that may change their nature twice under the strong influence of surface fields: from repulsive to attractive and vice versa along certain trajectories. Our findings strongly suggest a surprising role of the stress-tensor within one of the distant-wall corrections, given that similar discovery for standard extraordinary and ordinary surface universality classes was based on the conformal invariance symmetry which is broken under the present boundary conditions. We also show that a constituent part of the de Gennes-Fisher amplitude, defined in a generalized sense, is not hyperuniversal in two dimensions, contrary to earlier results for standard boundary conditions in this dimension.