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arXiv:1801.05416 · Progress of Theoretical and Experimental Physics

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Juven Wang, Kantaro Ohmori, Pavel Putrov, Yunqin Zheng, Zheyan Wan, Meng Guo, Hai Lin, Peng Gao, Shing–Tung Yau

Published Apr 13, 2018• 62 citations
arXivHTMLPDF

Abstract

Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators.The possible applications include condensed matter and quantum cosmology.We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain topological quantum field theories (TQFTs), in any dimension.This information can be related to the counting of extended operators of bosonic/fermionic TQFTs.On the lattice scale, anyonic particles/strings live at the ends of line/surface operators.Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry-protected topological states (with fermion parity Z f 2 ) of symmetry groups Z 4 ×Z 2 and (Z 4 ) 2 in 3+1D, also Z 2 and (Z 2 ) 2 in 2+1D.Gauging the last three cases begets non-Abelian spin TQFTs (fermionic topological order).We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with a boundary, such that the bulk and boundary are fully gapped and short-or long-range entangled (SRE/LRE).Anyonic excitations can be deconfined on the boundary.We introduce new exotic topological interfaces on which neither particle nor string excitations alone condense, but only fuzzy-composite objects of extended operators can end (e.g., a string-like composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk).We explore the relations between group extension constructions and partially breaking constructions (e.g., 0form/higher-form/"composite" breaking) of topological boundaries, after gauging.We comment on the implications of entanglement entropy for some such LRE systems...........

DOI
10.1093/ptep/pty051

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Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators.The possible applications include condensed matter and quantum cosmology.We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain topological quantum field theories (TQFTs), in any dimension.This information can be related to the counting of extended operators of bosonic/fermionic TQFTs.On the lattice scale, anyonic particles/strings live at the ends of line/surface operators.Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry-protected topological states (with fermion parity Z f 2 ) of symmetry groups Z 4 ×Z 2 and (Z 4 ) 2 in 3+1D, also Z 2 and (Z 2 ) 2 in 2+1D.Gauging the last three cases begets non-Abelian spin TQFTs (fermionic topological order).We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with a boundary, such that the bulk and boundary are fully gapped and short-or long-range entangled (SRE/LRE).Anyonic excitations can be deconfined on the boundary.We introduce new exotic topological interfaces on which neither particle nor string excitations alone condense, but only fuzzy-composite objects of extended operators can end (e.g., a string-like composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk).We explore the relations between group extension constructions and partially breaking constructions (e.g., 0form/higher-form/"composite" breaking) of topological boundaries, after gauging.We comment on the implications of entanglement entropy for some such LRE systems...........
Topological quantum field theory (TQFT) provides a unifying framework for describing topological phases of matter and for constructing quantum error-correcting codes, playing a central role across high-energy physics, condensed matter, and quantum information. A central challenge is to formulate topological order on lattices and to extract the properties of topological excitations from microscopic Hamiltonians. In this work, we construct new classes of lattice gauge theories as Pauli stabilizer models, realizing a wide range of TQFTs in general dimensions. We develop a lattice description of extended excitations and systematically determine their generalized statistics. Our main example is the (4+1)D fermionic-loop toric code, obtained by condensing the e2m2e^2m^2e2m2-loop in the (4+1)D Z4\mathbb Z_4Z4​ toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of −1-1−1. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by H5(B2G,U(1))H^5(B^2G,U(1))H5(B2G,U(1)). Beyond (4+1)D, the fermionic-loop toric codes form a family of Z2\mathbb Z_2Z2​ topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using Z4\mathbb Z_4Z4​ qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of Z2\mathbb Z_2Z2​ topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that ppp-dimensional excitations in 2p+22p+22p+2 spatial dimensions can support anyonic ppp-brane statistics for only even ppp.
Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state (or atomic insulator) as long as certain global symmetry GGG is unbroken. At low energies, most of the two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of the multicomponent Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ${\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...$). In addition to the usual multicomponent BFBFBF topological term at level-1, we find that there are new topological terms with quantized coefficients (e.g., ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}$ and ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}$) in TQFT actions, where a1,a2,...{a}^{1},{a}^{2},...a1,a2,... are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as GGG is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.
This dissertation discusses some applications of conformal field theories (CFTs) in topological phases of matter. The first part is devoted to a discussion of coupled wire constructions of some novel quantum Hall systems. Through a theoretical coupled wire model, we construct strongly correlated electronic integer quantum Hall states with filling factor 16. The edge state is a bosonic chiral (E8)1 CFT, which is closely related to topological paramagnets in (3+1)d. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law. We further construct two descendant non-Abelian quantum Hall states at filling 8, each carrying bosonic chiral (G2)1 or (F4)1 edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the E8 state that relates the G2 and F4 Fibonacci states, which is reminiscent of similar physics in half-filled Landau level. The second part is focused on the surface topological orders of 3D bulk topological systems. Symmetry protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this part, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the ADE classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models. The third part is to discuss the relation between the conformal boundary state and (2+1)d SPT phases. We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group G in 2 + 1 dimensions. Our method is based on directly studying the 1 + 1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the non-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability and gappability in the presence of G. We study several examples including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 × Z2 symmetric topological superconductors.
We discuss several bosonic topological phases in (3+1) dimensions enriched by a global Z2{\mathbb{Z}}_{2}Z2​ symmetry, and gauging the Z2{\mathbb{Z}}_{2}Z2​ symmetry. More specifically, following the spirit of the bulk-boundary correspondence, expected to hold in topological phases of matter in general, we consider boundary (surface) field theories and their orbifold. From the surface partition functions, we extract the modular S\mathcal{S}S and T\mathcal{T}T matrices and compare them with (2+1)(2+1)(2+1)-dimensional toplogical phase after dimensional reduction. As a specific example, we discuss topologically ordered phases in (3+1)(3+1)(3+1) dimensions described by the BF topological quantum field theories, with abelian exchange statistics between pointlike and looplike quasiparticles. Once the Z2{\mathbb{Z}}_{2}Z2​ charge conjugation symmetry is gauged, the Z2{\mathbb{Z}}_{2}Z2​ flux becomes non-Abelian excitation. The gauged topological phases we are considering here belong to the quantum double model with non-Abelian group in (3+1)(3+1)(3+1) dimensions.
We formulate the ZN{Z}_{N}ZN​ group gauging operations (or group extensions) in (smeared) boundary conformal field theories (BCFTs) and bulk conformal field theories, and study their applications to various phenomena in topologically ordered systems. We apply the resultant theories to analyze the correspondence between the renormalization group (RG) flow of CFTs and the classification of topological quantum field theories using the testable information of general classes of partition functions. One can obtain the bulk topological properties of the 2+12+12+1 dimensional topological ordered phase corresponding to the massive RG flow of the 1+11+11+1 dimensional systems equivalent to the smeared BCFT. We present an obstruction to mass condensation for the smeared BCFT analogous to the Lieb-Shultz-Mattis theorem for noninvertible symmetry. Corresponding to the bulk topological degeneracies in 2+12+12+1 dimensions and gapped quantum phases in 1+11+11+1 dimensions, we construct a new series of BCFT. We also study the implications of the massless RG flow of the 1+11+11+1 dimensional CFTs corresponding to the domain wall or coupling problem in 2+12+12+1 dimensional topological orders, which corresponds to the earlier proposal by L. Kong and H. Zheng [Nucl. Phys. B 966, 115384 (2021)] closely related to the integer-spin simple current by Schellekens and Gato-Rivera. We reveal that two distinct kinds of massless RG flow exist, coming from anomaly cancellation and anomaly matching related to Witt equivalence of bulk CFTs. The anomaly cancellation flows produce a new series of modular invariants as coupled models, and the anomaly matching flows produce nontrivial domain walls changing the chirality of anyons. Our paper gives a unified direction for future theoretical and numerical studies of the topological phase based on the established data of classifications of conformal field theories or modular invariants (and the corresponding category theories).
Recently we conjectured that a certain set of universal topological quantities characterize topological order in any dimension. Those quantities can be extracted from the universal overlap of the ground-state wave functions. For systems with gapped boundaries, these quantities are representations of the mapping class group MCG(M)\mathtt{MCG}(\mathcal{M})MCG(M) of the space manifold M\mathcal{M}M on which the systems live. We will here consider simple examples in three dimensions and give physical interpretation of these quantities, related to the fusion algebra and statistics of particles and string excitations. In particular, we will consider dimensional reduction from 3+1D to 2+1D, and show how the induced 2+1D topological data contain information on the fusion and the braiding of non-Abelian string excitations in 3D. These universal quantities generalize the well-known modular SSS and TTT matrices to any dimension.