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arXiv:2307.12509 · Physical review. B./Physical review. B

Edge theories for anyon condensation phase transitions

David M. Long, Andrew C. Doherty

Published Feb 20, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.109.075140

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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 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.
Condensable algebras in 2+1d non-chiral topological orders characterize gapped boundary conditions and interfaces. Applied to the Symmetry Topological Field Theory, they allow classification of symmetric gapped phases and impose sharp constraints on possible phase transitions. A condensable algebra is specified not only by its underlying set of anyons, which end on the boundary or interface, but also by its algebra structure. We introduce the concept of twin condensable algebras, which have the same anyon decomposition, but inequivalent algebra structure. We revisit the classification of condensable algebras in Z(VecGω)\mathcal{Z}(\text{Vec}_G^ω)Z(VecGω​), i.e. in group-theoretical topological orders for finite groups GGG with anomaly ωωω. In this context we are able to identify twin algebras that arise from different mechanisms, such as subgroup data, SPT cocycles, and symmetry actions. In particular, we construct infinite families of examples of twins from so-called Gassmann triples, and exhibit cases in which the reduced topological orders are inequivalent despite having identical anyon content. Physically, twin algebras describe distinct symmetric phases that have isomorphic spaces of ground states, but inequivalent order parameters. Such twin phases never exhibit relative spontaneous symmetry breaking, and can be used to construct phase transitions without hidden symmetry breaking, which are intrinsically beyond Landau transitions.
Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.
Instead of constructing anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relations between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. More general situation is also discussed in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin-Wen types of lattice models and some chiral topological phases.
In this thesis we investigate topological phases of matter that have a global, unbroken symmetry group -- also known as symmetry enriched topological (SET) phases. We address three questions about these phases: (1) how can we build exactly solvable models that realize them? (2) how can we determine if their edge theories can be gapped without breaking the symmetry? and (3) how do we understand the phenomenon of decoupled charge and neutral modes which occurs in certain fractional quantum Hall states? ,More specifically, we address the first question by constructing exactly solvable models for a wide class of symmetry enriched topological (SET) phases, which we call symmetry-enriched string nets. The construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group G, and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2 symmetry which exchanges the 'e' and 'm' type anyons. We further illustrate our construction with a number of additional examples. ,For the second question, we focus on the edge theories of 2D SET phases with Z2 symmetry. The central problem we seek to solve is to determine which edge theories can be gapped without breaking the symmetry. Previous attempts to answer this question in special cases relied on constructing perturbations of a particular type to gap the edge. This method proves the edge can be gapped when the appropriate perturbations can be found, but is inconclusive if they cannot be found. We build on this previous work by deriving a necessary and sufficient algebraic condition for when the edge can be gapped. Our results apply to Z2 symmetry protected topological phases as well as Abelian Z2 SET phases,Finally, in the fourth chapter, we describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode --- i.e. an edge mode that does not carry electric charge. These models consist of two counter-propagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the nu = 2/3 state is the most prominent example.
This work explores a deformation of Kitaev's toric code that induces a phase transition out of the topologically ordered phase. By placing the model on a cylinder, the bulk global 1-form symmetries separate into distinct boundary operators, allowing us to show that the transition is accompanied by the condensation of all anyons in a given boundary. Using a lower dimensional (1 + 1)-dimensional boundary Hamiltonian, we extract an effective central charge and find a pronounced suppression near βcβ_cβc​, followed by its restoration at strong coupling.