Non-Abelian quasi-particles in electronic systems

Daniel Borcherding

Published Jan 1, 2018• 0 citations

Abstract

Ever since the proposal of Kitaev for decoherence-free quantum computing based on non-Abelian anyons physical realizations of these exotic particles have been investigated extensively. Starting from one-dimensional models of interacting fermions with different symmetries the emergence and condensation of $\mathfrak{su}(2)_{N_f}$, $\mathfrak{su}(3)_{N_f}$ and $\mathfrak{so}(5)_{N_f}$ anyons is studied in the framework of integrable perturbed WZNW models. For sufficiently small temperatures and fields non-Abelian anyons residing on massive solitonic excitations are identified by their quantum dimension. By tuning the external fields the density of anyons can be increased continuously to study the effect of interactions between them. For each model the conformal field theories describing the various collective states of the interacting anyons are determined and a phase diagram for the anyonic modes is proposed.

DOI: 10.15488/4280

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Non-Abelian quasi-particles in electronic systems

Daniel Borcherding

Published Jan 1, 2018• 0 citations

Publisher

Abstract

\mathfrak{su}(2)_{N_f}

\mathfrak{su}(3)_{N_f}

and

\mathfrak{so}(5)_{N_f}

anyons is studied in the framework of integrable perturbed WZNW models. For sufficiently small temperatures and fields non-Abelian anyons residing on massive solitonic excitations are identified by their quantum dimension. By tuning the external fields the density of anyons can be increased continuously to study the effect of interactions between them. For each model the conformal field theories describing the various collective states of the interacting anyons are determined and a phase diagram for the anyonic modes is proposed.

DOI: 10.15488/4280

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cond-mat.str-elCross-listedarXiv:2609.19282Sep 18, 20260 citations

\mathfrak{su}(2)_{N_f}

\mathfrak{su}(3)_{N_f}

and

\mathfrak{so}(5)_{N_f}

{p}_{x}+i{p}_{y}

superconductors, certain fractional quantum Hall states, and rotating Bose-Einstein condensates. Here we use a combination of exact diagonalization and conformal field theory to determine the phase diagrams of ladders with up to four chains. We discuss how phenomena familiar from ordinary SU(2) spin ladders are generalized in their anyonic counterparts, such as gapless and gapped phases, odd and even effects with the ladder width, and elementary ``magnon'' excitations. Other features are entirely due to the topological nature of the anyonic degrees of freedom. In general, two-dimensional systems of interacting localized non-Abelian anyons are anyonic generalizations of two-dimensional quantum magnets.

\mathcal D(G)

by organizing single-link perturbations in terms of non-invertible electric and magnetic 1-form symmetries that proliferate distinct anyon species. For any finite group

G

, an exact

\textit{matterization}

isometry introduces vertex degrees of freedom and maps the link-only model onto a

G

gauge--Higgs theory. Furthermore, the 1-form symmetries of the fixed point allow us to define generalized Fredenhagen--Marcu order parameters that become finite when the corresponding anyons condense. For

G = S_3

, quantum Monte Carlo simulations show that proliferating a non-Abelian electric anyon drives a first-order transition in which all nontrivial electric anyons condense. In the purely magnetic limit, the model reduces to a

(2+1)

D pure

G

gauge theory; for

G=S_3

, it exhibits a first-order confinement transition, diagnosed by the onset of a Wilson-loop area law and the restoration of an emergent magnetic 1-form symmetry. These results provide a unified numerical framework for non-Abelian anyon condensation, confinement, and generalized symmetry breaking.