Abstract
Collective states of interacting non-Abelian anyons have recently been studied mostly in the context of certain fractional quantum Hall states, such as the Moore–Read state proposed to describe the physics of the quantum Hall plateau at filling fraction ν=5/2. In this paper, we further expand this line of research and present non-unitary generalizations of interacting anyon models. In particular, we introduce the notion of Yang–Lee anyons, discuss their relation to the so-called ‘Gaffnian’ quantum Hall wave function and describe an elementary model for their interactions. A one-dimensional (1D) version of this model—a non-unitary generalization of the original golden chain model—can be fully understood in terms of an exact algebraic solution and numerical diagonalization. We discuss the gapless theories of these chain models for general su(2)k anyonic theories and their Galois conjugates. We further introduce and solve a 1D version of the Levin–Wen model for non-unitary Yang–Lee anyons.
Highlights
In this paper, we will study the excitations of the putative quantum liquid described by such non-unitary wave functions from an ‘anyon-model’ perspective
The non-unitary models we encounter in this paper are Galois conjugates of unitary CFTs, and the study of non-unitary anyons is closely related to the study of Galois conjugation in CFT [26, 27]
We start by giving a review of the derivation of the Hamiltonian of the golden chain [23] of Fibonacci anyons in section 2.1, and a short mathematical description of the Galois-conjugated model, the so-called Yang–Lee chain in section 2.2, including an explicit derivation of their non-Hermitian, microscopic Hamiltonians, and discuss their relation to the ‘golden chain’ models of [23] via Galois conjugation
Summary
The focus of this paper is on anyonic models that are certain non-unitary generalizations of unitary non-Abelian anyon models, which have been extensively studied in the recent past [23], [28]–[35]. The basic constituents of the generalizations considered here are nonunitary, non-Abelian anyons Like their unitary counterparts they carry a quantum number that corresponds to a generalized angular momentum in so-called su(2)k anyonic theories, which are certain deformations (see for instance [36]) of SU(2). We start by quickly reviewing the basic construction of microscopic (chain) models of interacting non-Abelian anyons, following the ideas of the ‘golden chain’ model of [23] and the detailed exposition of [31]. The construction of these models proceeds in two steps. While this second step is quite similar for the unitary and non-unitary cases, the microscopic Hamiltonians for the two cases are distinct
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