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arXiv:2301.05721 · Physical review. B./Physical review. B

Anyon condensation and confinement transition in a Kitaev spin liquid bilayer

Kyusung Hwang

Published Apr 9, 2024• 7 citations
arXivHTMLPDF

Abstract

Transitions between quantum spin liquids (QSLs) are fundamental problems lying beyond the Landau paradigm and requiring a deep understanding of the entanglement structures of QSLs called topological orders. The novel concept of anyon condensation has been proposed as a theoretical mechanism, predicting various possible transitions between topological orders, but it has long been elusive to confirm the mechanism in quantum spin systems. Here, we introduce a concrete spin model that incarnates the mechanism of the anyon condensation transition. Our model harbors two topological QSLs in different parameter regions, a non-Abelian Kitaev spin liquid (KSL) bilayer state and a resonating valence bond (RVB) state. The bilayer-KSL-to-RVB transition indeed occurs by the mechanism of anyon condensation, which we identify by using parton theories and exact diagonalization studies. Moreover, we observe “anyon confinement” phenomena in our numerical results, akin to the quark confinement in high-energy physics. Namely, non-Abelian Ising anyons of the bilayer KSL are confined in the transition to the RVB state. Implications and extensions of this study are discussed in various aspects such as (i) anyon-condensed multilayer construction of Kitaev's 16-fold way of anyon theories, (ii) an additional vison condensation transition from the RVB to a valence bond solid in the Kitaev bilayer system, (iii) dynamical anyon condensation in a non-Hermitian Kitaev bilayer, (iv) generalizations of our model to other lattice geometries, and (v) experimental realizations. This work puts together the two fascinating QSLs that are extensively studied in modern condensed matter and quantum physics into a concrete spin model, offering a comprehensive picture that unifies the anyon physics of the Kitaev spin liquids and the resonating valence bonds. Published by the American Physical Society 2024

DOI
10.1103/physrevb.109.134412

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Transitions between quantum spin liquids (QSLs) are fundamental problems lying beyond the Landau paradigm and requiring a deep understanding of the entanglement structures of QSLs called topological orders. The novel concept of anyon condensation has been proposed as a theoretical mechanism, predicting various possible transitions between topological orders, but it has long been elusive to confirm the mechanism in quantum spin systems. Here, we introduce a concrete spin model that incarnates the mechanism of the anyon condensation transition. Our model harbors two topological QSLs in different parameter regions, a non-Abelian Kitaev spin liquid (KSL) bilayer state and a resonating valence bond (RVB) state. The bilayer-KSL-to-RVB transition indeed occurs by the mechanism of anyon condensation, which we identify by using parton theories and exact diagonalization studies. Moreover, we observe “anyon confinement” phenomena in our numerical results, akin to the quark confinement in high-energy physics. Namely, non-Abelian Ising anyons of the bilayer KSL are confined in the transition to the RVB state. Implications and extensions of this study are discussed in various aspects such as (i) anyon-condensed multilayer construction of Kitaev's 16-fold way of anyon theories, (ii) an additional vison condensation transition from the RVB to a valence bond solid in the Kitaev bilayer system, (iii) dynamical anyon condensation in a non-Hermitian Kitaev bilayer, (iv) generalizations of our model to other lattice geometries, and (v) experimental realizations. This work puts together the two fascinating QSLs that are extensively studied in modern condensed matter and quantum physics into a concrete spin model, offering a comprehensive picture that unifies the anyon physics of the Kitaev spin liquids and the resonating valence bonds. Published by the American Physical Society 2024
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Quantum spin liquids have fascinated condensed matter physicists for decades because of their unusual properties such as spin fractionalization and long-range entanglement. Unlike conventional symmetry breaking, the topological order underlying quantum spin liquids is hard to detect experimentally. Even theoretical models are scarce for which the ground state is established to be a quantum spin liquid. The Kitaev honeycomb model and its generalizations to other tricoordinated lattices are chief counterexamples—they are exactly solvable, harbor a variety of quantum spin liquid phases, and are also relevant for certain transition metal compounds including the polymorphs of (Na,Li) 2 IrO 3 iridates and RuCl 3 . In this review, we give an overview of the rich physics of the Kitaev model, including two-dimensional and three-dimensional fractionalization as well as dynamic correlations and behavior at finite temperatures. We discuss the different materials and argue how the Kitaev model physics can be relevant even though most materials show magnetic ordering at low temperatures.
We investigate the phase diagram of a bilayer Kitaev honeycomb model with Ising interlayer interactions, deriving effective models via perturbation theory and performing Majorana mean-field theory calculations. We show that a diverse array of magnetic and topological phase transitions occur, depending on the direction of the interlayer Ising interaction and the relative sign of Kitaev interactions. When two layers have the same sign of the Kitaev interaction, a first-order transition from a Kitaev spin liquid to a magnetically ordered state takes place. The magnetic order points along the Ising axis and it is (anti)ferromagnetic for (anti)ferromagnetic Kitaev interactions. However, when two layers have opposite signs of the Kitaev interaction, we observe a notable weakening of magnetic ordering tendencies and the Kitaev spin liquid survives up to a remarkably larger interlayer exchange. Our mean-field analysis suggests the emergence of an intermediate gapped Z2{\mathbb{Z}}_{2}Z2​ spin-liquid state, which eventually becomes unstable upon vison condensation. The confined phase is described by a highly frustrated ${120}^{\ensuremath{\circ}}$ compass model. We furthermore use perturbation theory to study the model with the Ising axis pointing along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ axis or lying in the xyxyxy plane. In both cases, our analysis reveals the formation of one-dimensional Ising chains, which remain decoupled in perturbation theory, resulting in a subextensive ground-state degeneracy. Our results highlight the interplay between topological order and magnetic ordering tendencies in bilayer quantum spin liquids.
Abstract Quantum spin liquids (QSLs) give rise to exotic emergent particles by weaving intricate entanglement patterns in the underlying electrons. Bipartite measures between subregions can detect the presence of anyons, but little is known about the full entanglement structure of QSLs. Here, we study the multiparty entanglement of QSLs via entanglement microscopy. We find that in contrast to conventional matter, the genuine multiparty entanglement (GME) between spins is absent in the smallest subregions, a phenomenon we call ``entanglement frustration.'' Instead, GME is more collective, and arises solely in loops. By exploiting exact results and large-scale numerics, we confirm these properties in various gapped and gapless QSLs realized in physically motivated Hamiltonians, as well as with string-net wavefunctions hosting abelian or non-abelian anyons. Our results shed new light on the phase diagram of Kitaev’s honeycomb model in a Zeeman field, and the Kagome Heisenberg model under various perturbations. Going beyond QSLs, we provide evidence that entanglement loops are a universal property of quantum gauge theories. This leads to a new understanding of fractionalization, and the means by which gauge bosons encode quantum information.