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Physical Review Letters

Ordering of chirality for many planes of anyons

Alberto G. Rojo, Geoffrey Canright

Published Feb 18, 1991• 21 citations
Publisher

Abstract

We consider the effect of interplane coupling on the energetics of the relative sign of statistics of anyons in adjacent planes. We offer qualitative arguments as well as numerical results which support the idea that opposing chirality (i.e., ``antiferromagnetic'' order of the statistics) is favored over a wide range of interplane coupling V--including both attractive and repulsive interactions between planes. We argue that our results apply to the normal (nonsuperfluid) phase in the thermodynamic limit.

DOI
10.1103/physrevlett.66.949

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We consider the effect of interplane coupling on the energetics of the relative sign of statistics of anyons in adjacent planes. We offer qualitative arguments as well as numerical results which support the idea that opposing chirality (i.e., ``antiferromagnetic'' order of the statistics) is favored over a wide range of interplane coupling V--including both attractive and repulsive interactions between planes. We argue that our results apply to the normal (nonsuperfluid) phase in the thermodynamic limit.
We present a multiplane model for the anyon problem. Anyons on different planes are coupled via a Coulomb–type interaction. We solve exactly finite clusters and show that the "antiferromagnetic" order of the chirality is favored for both attractive and repulsive interplane coupling. We also discuss a simple model that can be treated analytically and that has the same qualitative behavior as the exact results. Our results apply to the low density limit, in which finite currents exist in each plane. These currents also occur in the high temperature (nonsuperfluid) phase of the anyon system, and so our results should apply also to that regime.
We consider the effect of interplane coupling J' on a pair of layers in the Kalmeyer-Laughlin model of anyon superconductivity. We show by exact diagonalization of an effective continuum model that this coupling has little effect on the chiral-spin-liquid ground state for small J'. For J' larger than twice the gap, one sees a transition away from the chiral spin liquid. We argue that the J' term should introduce short-range attractive interactions between holons in different layers. A model of anyons with such an interaction yields a phase diagram in qualitative agreement with what one expects when working directly with the spin system.
We obtain the ground-state phase diagram of two spin chains consisting in a set of two-level systems asymmetrically coupled to an XX chain through a chiral interaction. The interaction is parametrized by its magnitude and an angle defined by the relative orientation of the spins in different chains. From the entanglement spectrum, we identify the critical lines separating distinct magnetically ordered phases, with the interaction angle able to shift or fully suppress the transition. By increasing the coupling strength, the system is driven through a quantum phase transition, leading to the formation of two types of in-plane antiferromagnetic stripes. The interaction strength sets stripe formation, while the angle controls the spins' orientations. The chiral interaction also induces a nontrivial finite vector spin chirality with opposite orientation on the chains. We show that the vector spin chirality emerges smoothly from the decoupled limit and occurs for angles different from zero and π / 2 , where collinear order is favored instead.
A large-S approach to the description of chiral phases in frustrated antiferromagnetic spin-S chains with easy-plane anisotropy is proposed. The approach is valid in the vicinity of the classical Lifshitz point j=1/4, where j is the relative strength of the frustrating next-nearest-neighbor coupling. The effective theory turns out to be the nonlinear σ-model with the additional fourth-order term in spatial derivatives. The topological term persists for j>1/4, which leads to disappearance of the chiral gapped phase for half-integer S, in accordance with the recent numerical results.