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arXiv:2003.04328 · Physical review. B./Physical review. B

Bulk anyons as edge symmetries: Boundary phase diagrams of topologically ordered states

Tsuf Lichtman, Ryan Thorngren, Netanel H. Lindner, Ady Stern, Erez Berg

Published Aug 23, 2021• 64 citations
arXivHTMLPDF

Abstract

We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with emergent symmetries of the edge, which play a crucial role in the structure of its phase diagram. Using this symmetry principle, transitions between distinct gapped phases at the boundaries of Abelian states can be understood in terms of symmetry-breaking transitions or transitions between symmetry-protected topological phases. Yet more exotic phenomena occur when the bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the phase diagrams of the edges of a single and a double layer of the toric code, as well as those of domain walls in a single- and double-layer Kitaev spin liquid (KSL). In the case of the KSL, we find that the presence of a non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a symmetry of the effective boundary theory. These examples illustrate a number of surprising phenomena, such as spontaneous duality breaking, two-sector phase transitions, and unfreezing of marginal operators at a transition between different gapless phases.

DOI
10.1103/physrevb.104.075141

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We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with emergent symmetries of the edge, which play a crucial role in the structure of its phase diagram. Using this symmetry principle, transitions between distinct gapped phases at the boundaries of Abelian states can be understood in terms of symmetry-breaking transitions or transitions between symmetry-protected topological phases. Yet more exotic phenomena occur when the bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the phase diagrams of the edges of a single and a double layer of the toric code, as well as those of domain walls in a single- and double-layer Kitaev spin liquid (KSL). In the case of the KSL, we find that the presence of a non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a symmetry of the effective boundary theory. These examples illustrate a number of surprising phenomena, such as spontaneous duality breaking, two-sector phase transitions, and unfreezing of marginal operators at a transition between different gapless phases.
The algebraic tools used to study topological phases of matter are not clearly suited to studying processes in which the bulk energy gap closes, such as phase transitions. We describe an elementary two edge thought experiment, which reveals the effect of an anyon condensation phase transition on the robust edge properties of a sample, bypassing a limitation of the algebraic description. In particular, the two edge construction allows some edge degrees of freedom to be tracked through the transition, despite the bulk gap closing. The two edge model demonstrates that bulk anyon condensation induces symmetry breaking in the edge model. Further, the construction recovers the expected result that the number of chiral current carrying modes at the edge cannot change through anyon condensation. We illustrate the construction through detailed analysis of anyon condensation transitions in an achiral phase, the toric code, and in chiral phases, the Kitaev spin liquids.
We study (2+1)-dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry-enriched topological (SET) phases. Here, we present a KKK-matrix Chern-Simons approach to identify distinct phases with Abelian topological order, in the presence of unitary or antiunitary global symmetries. A key step is the identification of a smooth edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method by classifying Z2{Z}_{2}Z2​ topological order (Z2{Z}_{2}Z2​ spin liquids) in the presence of an internal Z2{Z}_{2}Z2​ global symmetry for which we find six distinct phases. These include two phases with an unconventional action of symmetry that permutes anyons leading to symmetry-protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry-enriched Laughlin states and double-semion theories are also discussed. Somewhat surprisingly, we observe that (i) gauging the global symmetry of distinct SET phases leads to topological orders with the same total quantum dimension, and (ii) a pair of distinct SET phases can yield the same topological order on gauging the symmetry.
The focus of our investigations in this thesis is quantum spin systems in two dimensions. We examine phase transitions between topologically distinct phases of quantum matter using the framework of tensor network states. We study the phenomena of anyon condensation and confinement in the context of tensor network states where these two notions offer a robust probe to characterize the universal features of quantum phase transitions. We do a comprehensive study of the numerical methods enabled by the tensor network formalism for the study of quantum phase transitions. We map out the phase diagram of certain exotic phases of quantum matter (namely the D(mathbbZ4)D(\\mathbb{Z}_4)D(mathbbZ4​) quantum double, the toric code, and the double semion model) and identify the order and the universality classes of the phase transitions between these distinct topological phases. We analyze the phase boundaries of the toric code and the double semion model which exhibit first and second order phase transitions. We investigate the class of mathbbZ2\\mathbb{Z}_2mathbbZ2​-invariant tensor network states while taking in to account the spin rotation and lattice symmetries. The resulting tensor network states allow us to map the phase diagram of mathbbZ2\\mathbb{Z}_2mathbbZ2​ spin liquids. The behavior of the system at the phase boundaries of mathbbZ2\\mathbb{Z}_2mathbbZ2​ spin liquid is governed by the condensation of spinons and visions. We also present our preliminary findings regarding the mathbbZ4\\mathbb{Z}_4mathbbZ4​-invariant tensors with SU(2)SU(2)SU(2) symmetry. This approach enables us to study the spin liquid states of the toric code and the double semion model in a unified framework. We give a local tensor description for the approximate ground states of the Heisenberg antiferromagnet on the kagome lattice by using the variational manifolds of just three and five parameters in the gapped mathbbZ2\\mathbb{Z}_2mathbbZ2​ spin liquid phase. The approximation of the ground states we construct have an energy density that is remarkably close to the results from the state-of-the-art density matrix renormalization group and the exact diagonalization method. By analyzing the deconfinement of anyonic excitations, we also present our findings regarding the vicinity of the variational ground state in gapped mathbbZ2\\mathbb{Z}_2mathbbZ2​ spin liquid phase.
Symmetry-protected topological (SPT) phases are gapped quantum phases which host symmetry-protected gapless edge excitations. On the other hand, the edge states can be gapped by spontaneously breaking symmetry. We show that topological defects on the symmetry-broken edge cannot proliferate due to their fractional statistics. A gapped symmetric boundary, however, can be achieved between an SPT phase and certain fractionalized phases by condensing the bound state of a topological defect and an anyon. We demonstrate this by two examples in two dimensions: an exactly solvable model for the boundary between a topological Ising paramagnet and the double-semion model, and a fermionic example about the quantum spin Hall edge. Such a hybrid structure containing both SPT phase and fractionalized phase generally support ground-state degeneracy on a torus.
Boundary conformal field theory is brought to bear on the study of topological insulating phases of non-Abelian anyonic chains. These phases display protected anyonic end modes. We consider spin-1/2 su(2)k{}_{k}k​ chains at any level kkk, focusing on the most prominent examples: the case k=2k=2k=2 describes Ising anyons (equivalent to Majorana fermions) and k=3k=3k=3 corresponds to Fibonacci anyons. The method we develop is quite general and rests on a deep connection between boundary conformal field theory and topological symmetry. This method tightly constrains the nature of the topological insulating phases of these chains for general kkk. Emergent anyons which arise at domain walls are shown to have the same braiding properties as the physical quasiparticles. This suggests a ``solid-state'' topological quantum computation scheme in which emergent anyons are braided by tuning the couplings of non-Abelian quasiparticles in a fixed network.
Non-Abelian excitations are sought after because of the promise they hold for topological quantum computation. However, in spite of many proposals of systems in which they can occur intrinsically, observation has proved to be elusive. In this thesis, we study ways in which we can engineer such excitations by combining two parent Abelian theories, and as a result allow for defects which host non-Abelian zero modes. We first formulate the symmetries of the anyonic excitations of Abelian topological states. Then, we study properties of semiclassical point defects, called twist defects, which permute orbiting anyons by their symmetries. We see how to realize these defects at domain walls on the gapped edges of non-chiral fractional quantum Hall states. We have distinct gapped edge phases in one to one correspondence with the anyonic symmetries. Non-Abelian twist defects appear between different gapped phases. We study the twist defects for a variety of bosonic ADE fractional quantum spin Hall systems for which the anyonic symmetries turn out to be given by their Dynkin diagrams. We also look at superconducting heterostructures with fermionic quantum Hall states and show the presence of anyonic symmetries and twist defects which physically correspond to a fermion parity flip. This can result in Majorana bound states at such defects instead of more exotic parafermions as was believed before. This is because the presence of the superconductor necessitates the consideration of a larger bosonic topological order. We then investigate the twist defects further by promoting the anyonic symmetries from global to local. This deconfines the defects and results in non-Abelian topological states called twist liquids. Finally, we study the emergence of new topological order from two constituent systems driven by the condensation of a composite boson. This allows us to study a variety of systems with non-Abelian excitations.