Symmetry acting on a (2+1)$D$ topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)$D$ on-site symmetry. In this paper, we develop a (3+1)$D$ topological quantum field theory to calculate the anomaly indicators of a (2+1)$D$ topological order with a general symmetry group $G$, which may be discrete or continuous, Abelian or non-Abelian, contain anti-unitary elements or not, and permute anyons or not. These anomaly indicators are partition functions of the (3+1)$D$ topological quantum field theory on a specific manifold equipped with some $G$-bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Our framework is applied to derive the anomaly indicators for various symmetry groups, including $\mathbb{Z}_2\times\mathbb{Z}_2$, $\mathbb{Z}_2^T\times\mathbb{Z}_2^T$, $SO(N)$, $O(N)^T$, $SO(N)\times \mathbb{Z}_2^T$, etc, where $\mathbb{Z}_2$ and $\mathbb{Z}_2^T$ denote a unitary and anti-unitary order-2 group, respectively, and $O(N)^T$ denotes a symmetry group $O(N)$ such that elements in $O(N)$ with determinant $-1$ are anti-unitary. In particular, we demonstrate that some anomaly of $O(N)^T$ and $SO(N)\times \mathbb{Z}_2^T$ exhibit symmetry-enforced gaplessness, i.e., they cannot be realized by any symmetry-enriched topological order. As a byproduct, for $SO(N)$ symmetric topological orders, we derive their $SO(N)$ Hall conductance.
We show that a large class of symmetry enriched (topological) phases of matter in 2+1 dimensions can be embedded in "larger" topological phases- phases describable by larger hidden Hopf symmetries. Such an embedding is analogous to anyon condensation, although no physical condensation actually occurs. This generalizes the Landau-Ginzburg paradigm of symmetry breaking from continuous groups to quantum groups- in fact algebras- and offers a potential classification of the symmetry enriched (topological) phases thus obtained, including symmetry protected trivial phases as well, in a unified frame…
We construct in the K matrix formalism concrete examples of symmetry enriched topological phases, namely intrinsically topological phases with global symmetries. We focus on the Abelian and non-chiral topological phases and demonstrate by our examples how the interplay between the global symmetry and the fusion algebra of the anyons of a topologically ordered system determines the existence of gapless edge modes protected by the symmetry and that a (quasi)-group structure can be defined among these phases. Our examples include phases that display charge fractionalization and more exotic non-lo…
We propose a general construction of commuting projector lattice models for 2D and 3D topological phases enriched by U(1) symmetry, with finite-dimensional Hilbert space per site. The construction starts from a commuting projector model of the topological phase and decorates U(1) charges to the state space in a consistent manner. We show that all 2D U(1) symmetry-enriched topological phases which allow gapped boundary without breaking symmetry, can be realized through our construction. We also construct a large class of 3D topological phases with U(1) symmetry fractionalized on particles or lo…
Although the mathematics of anyon condensation in topological phases has been studied intensively in recent years, a proof of its physical existence is tantamount to constructing an effective Hamiltonian theory. In this paper, we concretely establish the physical foundation of anyon condensation by building the effective Hamiltonian and the Hilbert space, in which we explicitly construct the vacuum of the condensed phase as the coherent states that are the eigenstates of the creation operators that create the condensate anyons. Along with this construction, which is analogous to Laughlin's con…
In topological phases in $2+1$ dimensions, anyons fall into representations of quantum group symmetries. As proposed in our work (arXiv:1308.4673), physics of a symmetry enriched phase can be extracted by the Mathematics of (hidden) quantum group symmetry breaking of a "parent phase". This offers a unified framework and classification of the symmetry enriched (topological) phases, including symmetry protected trivial phases as well. In this paper, we extend our investigation to the case where the "parent" phases are non-Abelian topological phases. We show explicitly how one can obtain the topo…
In topological phases of matter, the interplay between intrinsic topological order and global symmetry is an interesting task. In the study of topological orders with discrete global symmetry, an important systematic approach is the construction of exactly soluble lattice models. However, for continuous global symmetry, in particular the electromagnetic $U(1)$, the lattice approach has been less systematically developed. In this paper, we introduce a systematic construction of effective theories for a large class of abelian topological orders on three-dimensional spacetime lattice with electro…
We examine the interplay of symmetry and topological order in $2+1$ dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the ${\cal G}^{\rm f}$ fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with r…
Recently a new class of quantum phases of matter: symmetry protected topological states, such as topological insulators, attracted much attention. In presence of interactions, group cohomology provides a classification of these [X. Chen et al., arXiv:1106.4772v5 (2011)]. These phases have short-ranged entanglement, and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or with…
We study 't Hooft anomalies of symmetry-enriched rational conformal field theories (RCFT) in (1+1)d. Such anomalies determine whether a theory can be realized in a truly (1+1)d system with on-site symmetry, or on the edge of a (2+1)d symmetry-protected topological phase. RCFTs with the identical symmetry actions on their chiral algebras may have different 't Hooft anomalies due to additional symmetry charges on local primary operators. To compute the relative anomaly, we establish a precise correspondence between (1+1)d non-chiral RCFTs and (2+1)d doubled symmetry-enriched topological (SET) ph…
We develop a folding approach to study two-dimensional symmetry-enriched topological (SET) phases with the mirror reflection symmetry. Our folding approach significantly transforms the mirror SETs, such that their properties can be conveniently studied through previously known tools: (i) it maps the nonlocal mirror symmetry to an onsite $\mathbb{Z}_2$ layer-exchange symmetry after folding the SET along the mirror axis, so that we can gauge the symmetry; (ii) it maps all mirror SET information into the boundary properties of the folded system, so that they can be studied by the anyon condensati…
We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants, establishing the notion of symmetry-enriched quantum criticality. The charges of nonlocal scaling operators, or more generally of symmetry defects, are topological and imply the presence of localized edge modes. We primarily focus on the $1+1d$ case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on …
We classify symmetry fractionalization and anomalies in a (3+1)d U(1) gauge theory enriched by a global symmetry group $G$. We find that, in general, a symmetry-enrichment pattern is specified by 4 pieces of data: $ρ$, a map from $G$ to the duality symmetry group of this $\mathrm{U}(1)$ gauge theory which physically encodes how the symmetry permutes the fractional excitations, $ν\in\mathcal{H}^2_ρ[G, \mathrm{U}_\mathsf{T}(1)]$, the symmetry actions on the electric charge, $p\in\mathcal{H}^1[G, \mathbb{Z}_\mathsf{T}]$, indication of certain domain wall decoration with bosonic integer quantum Ha…
We study general 2D fermionic topological orders enriched by the mirror symmetry with $\mathcal{M}^2=1$. It is known that certain mirror symmetry enriched fermionic topological orders (mirror SETs) are anomalous, in the sense that they cannot be realized in strict two dimensions but have to live on the surface of 3D topological crystalline superconductors. Mirror anomaly, or equivalently 3D topological crystalline superconductor, has a $\mathbb{Z}_{16}$ classification. In this work, we derive an explicit expression, namely an \emph{anomaly indicator}, for the $\mathbb{Z}_{16}$ mirror anomaly f…
We introduce a construction of symmetry-enriched topological orders on bipartite lattices in which two $\mathbb{Z}_2$ spin liquids defined on each sublattice are combined, and then anyons are condensed to reduce the topological order. By choosing different anyon condensate structures, one can vary the fractionalization pattern of the resulting spin liquid, some of which cannot be readily constructed from parton based approaches. We demonstrate the construction for $i$) a spin-1/2 honeycomb lattice where we construct a featureless state as well as intermediate states with topological order, $ii…
Time-periodic driving of a quantum system can enable new dynamical topological phases of matter that could not exist in thermal equilibrium. We investigate two related classes of dynamical topological phenomena in 2D systems: Floquet symmetry protected topological phases (FSPTs), and Floquet enriched topological orders (FETs). By constructing solvable lattice models for a complete set of 2D bosonic FSPT phases, we show that bosonic FSPTs can be understood as topological pumps which deposit loops of 1D SPT chains onto the boundary during each driving cycle, which protects a non-trivial edge sta…
We discuss three dimensional $Z_2$ topological phases enriched by the $Z_2^T$ time-reversal symmetry with bosonic bulk excitations. Some of these phases can be constructed by coupling the three dimensional symmetry protected topological phases with $Z_2 \times Z_2^T$ symmetry to a deconfined dynamical $Z_2$ gauge field. Besides these simple phases, we also construct a special root phase (phase 3) which cannot be interpreted as gauged $Z_2 \times Z_2^T$ SPT phase. The $2d$ boundary of these phases can have an extra $Z_2$ topological order in addition to their bulk topological order. In one of t…
We construct fixed-point wave functions and exactly solvable commuting-projector Hamiltonians for a large class of bosonic symmetry-enriched topological (SET) phases, based on the concept of equivalent classes of symmetric local unitary transformations. We argue that for onsite unitary symmetries, our construction realizes all SETs free of anomaly, as long as the underlying topological order itself can be realized with a commuting-projector Hamiltonian. We further extend the construction to anti-unitary symmetries (e.g. time-reversal symmetry), mirror-reflection symmetries, and to anomalous SE…
Topologically ordered systems in the presence of symmetries can exhibit new structures which are referred to as symmetry enriched topological (SET) phases. We introduce simple methods to detect the SET order directly from a complete set of topologically degenerate ground state wave functions. In particular, we first show how to directly determine the characteristic symmetry fractionalization of the quasiparticles from the reduced density matrix of the minimally entangled states. Second, we show how a simple generalization of a non-local order parameter can be measured to detect SETs. The usefu…
We study how symmetry can enrich strong-randomness quantum critical points and phases, and lead to robust topological edge modes coexisting with critical bulk fluctuations. These are the disordered analogues of gapless topological phases. Using real-space and density matrix renormalization group approaches, we analyze the boundary and bulk critical behavior of such symmetry-enriched random quantum spin chains. We uncover a new class of symmetry-enriched infinite randomness fixed points: while local bulk properties are indistinguishable from conventional random singlet phases, nonlocal observab…
We study the quantized topological terms in a weak-coupling gauge theory with gauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,ν_d)$, where $G$ is an extension of $G_s$ by $G_g$ and $ν_d$ an element in group cohomology $\cH^d(G,\R/\Z)$. When $d=3$ and/or when $G_g$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e. gapped long-range entangled phases with symmetry). Thus, those SET phases are classified by…
Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionaliza…
We construct a new model of two-dimensional quantum spin systems that combines intrinsic topo- logical orders and a global symmetry called flavour symmetry. It is referred as the bilayer Doubled Semion model (bDS) and is an instance of symmetry-enriched topological order. A honeycomb bi- layer lattice is introduced to combine a Double Semion Topolgical Order with a global spin-flavour symmetry to get the fractionalization of its quasiparticles. The bDS model exhibits non-trival braid- ing self-statistics of excitations and its dual model constitutes a Symmetry-Protected Topological Order with …
We study a topological phase of interacting bosons in (3+1) dimensions which is protected by charge conservation and time-reversal symmetry. We present an explicit lattice model which realizes this phase and which can be studied in sign-free Monte Carlo simulations. The idea behind our model is to bind bosons to topological defects called hedgehogs. We determine the phase diagram of the model and identify a phase where such bound states are proliferated. In this phase we observe a Witten effect in the bulk whereby an external monopole binds half of the elementary boson charge, which confirms t…
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 K-matrix Chern-Simons approach to identify all distinct phases with Abelian topological order, in the presence of unitary or anti-unitary global symmetries . A key step is the identification of an edge sewing condition that is used to check if two putative phases are ind…
Topologically ordered phases of matter, in particular so-called symmetry enriched topological (SET) phases, can exhibit quantum number fractionalization in the presence of global symmetry. In Z_2 topologically ordered states in two dimensions, fundamental translations T_x and T_y acting on anyons can either commute or anticommute. This property, crystal momentum fractionalization, can be seen in a periodicity of the excited-state spectrum in the Brillouin zone. We present a numerical method to detect the presence of this form of symmetry enrichment given a projected entangled pair state (PEPS)…
We derive a series of quantitative bulk-boundary correspondences for 3D bosonic and fermionic symmetry-protected topological (SPT) phases under the assumption that the surface is gapped, symmetric and topologically ordered, i.e., a symmetry-enriched topological (SET) state. We consider those SPT phases that are protected by the mirror symmetry and continuous symmetries that form a group of $U(1)$, $SU(2)$ or $SO(3)$. In particular, the fermionic cases correspond to a crystalline version of 3D topological insulators and topological superconductors in the famous ten-fold-way classification, with…
Classical symmetry enriched topological orders and distinct monopole charges for dipole-octupole spin ices
International Center for Quantum Materials, Peking University（北京大学量子材料中心）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）；The University of Hong Kong, Shenzhen Institute of Research and Innovation（香港大学深圳研究院）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学量子器件北京市重点实验室）
In this work, we give a precise mathematical description of a fully chiral gapless edge of a 2d topological order (without symmetry). We show that the observables on the 1+1D world sheet of such an edge consist of a family of topological edge excitations, boundary CFT's and walls between boundary CFT's. These observables can be described by a chiral algebra and an enriched monoidal category. This mathematical description automatically includes that of gapped edges as special cases. Therefore, it gives a unified framework to study both gapped and gapless edges. Moreover, the boundary-bulk duali…
Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with the time reversal (TR) symmetry satisfying $T^{2}=1$. Without extra symmetry, it has been found that $\mathbb{Z}_{p}$ parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labelled by the divisors $n$ of $p$. In the presence of TR symmetry, however, the enriched classification is characterized by three indices $n$, $κ$ and $μ$, where $κ\in \mathb…
We describe a method to find the anomaly of the time-reversal symmetry of 2+1d topological quantum field theories, by computing the fractional anomalous momentum on the cross-cap background. This allows us, for example, to identify the parameter $ν$ mod 16 of the bulk 3+1d topological superconductor with $\mathsf{T}^2=(-1)^F$ on whose boundary a given 2+1d time-reversal-invariant topological phase can appear.
Topological crystalline insulators (TCIs) are nontrivial quantum phases of matter protected by crystalline (and other) symmetries. They are originally predicted by band theories, so an important question is their stability under interactions. In this paper, by directly studying the physical bulk properties of several band-theory-based nontrivial TCIs that are conceptually interesting and/or experimentally feasible, we show they are stable under interactions. These TCIs include (1) a weak topological insulator, (2) a TCI with a mirror symmetry and its time-reversal symmetric generalizations, (3…
Inspired by experiments on the magnetic field induced phases of the spin-orbit coupled $2d$ Mott insulator $α$-RuCl$_3$, we study some general aspects of gapped $Z_2$ Quantum Spin Liquids (QSL) enriched by lattice translation symmetry. We show that there are $12$ distinct such phases with different implementations of translation symmetry. In some of these phases the vison excitations of this QSL may form topological Chern bands. We explore a phenomenological description of a putative $Z_2$ QSL as a candidate ground state at intermediate magnetic fields in $α$-RuCl$_3$. This state has broad con…
Symmetry-enriched topological order and quasifractonic behavior in $\mathbb{Z}_N$ stabilizer codes
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
We examine (3+1)D topological ordered phases with $C_k$ rotation symmetry. We show that some rotation symmetric (3+1)D topological orders are anomalous, in the sense that they cannot exist in standalone (3+1)D systems, but only exist on the surface of (4+1)D SPT phases. For (3+1)D discrete gauge theories, we propose anomaly indicator that can diagnose the $\mathbb{Z}_k$ valued rotation anomaly. Since (3+1)D topological phases support both point-like and loop-like excitations, the indicator is expressed in terms of the symmetry properties of point and loop-like excitations, and topological data…
Certain patterns of symmetry fractionalization in (2+1)D topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological (SET) states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group $G$, one can define a (3+1)D topologically invariant path integral in terms of a state sum for a $G$ symmetry-protected topological (S…
The unified mathematical theory of gapped and gapless edges of 2d topological orders was developed by two of the authors. It provides a powerful tool to study pure edge topological phase transitions on the edges of 2d topological orders (without altering the bulks). In particular, it implies that the critical points are described by enriched fusion categories. In this work, we illustrate this idea in a concrete example: the 2d $\mathbb{Z}_2$ topological order. In particular, we construct an enriched fusion category, which describes a gappable non-chiral gapless edge of the 2d $\mathbb{Z}_2$ to…
We introduce a model of three-dimensional (3D) topological order enriched by planar subsystem symmetries. The model is constructed starting from the 3D toric code, whose ground state can be viewed as an equal-weight superposition of two-dimensional (2D) membrane coverings. We then decorate those membranes with 2D cluster states possessing symmetry-protected topological order under line-like subsystem symmetries. This endows the decorated model with planar subsystem symmetries under which the loop-like excitations of the toric code fractionalize, resulting in an extensive degeneracy per unit le…
Some time-reversal symmetric topological orders are anomalous in that they cannot be realized in strictly two-dimensions without breaking time reversal symmetry; instead, they can only be realized on the surface of certain three-dimensional systems. We propose two quantities, which we call {\it anomaly indicators}, that can detect if a time-reversal symmetric topological order is anomalous in this sense. Both anomaly indicators are expressed in terms of the quantum dimensions, topological spins, and time-reversal properties of the anyons in the given topological order. The first indicator, $η_…
Symmetry plays a fundamental role in our understanding of both conventional symmetry breaking phases and the more exotic quantum and topological phases of matter. We explore the experimental signatures of symmetry enriched U(1) quantum spin liquids (QSLs) on the pyrochlore lattice. We point out that the Ce local moment of the newly discovered pyrochlore QSL candidate Ce$_2$Sn$_2$O$_7$, is a dipole-octupole doublet. The generic model for these unusual doublets supports two distinct symmetry enriched U(1) QSL ground states in the corresponding quantum spin ice regimes. These two U(1) QSLs are du…
We classify and characterize three dimensional $U(1)$ quantum spin liquids (deconfined $U(1)$ gauge theories) with global symmetries. These spin liquids have an emergent gapless photon and emergent electric/magnetic excitations (which we assume are gapped). We first discuss in great detail the case with time reversal and $SO(3)$ spin rotational symmetries. We find there are 15 distinct such quantum spin liquids based on the properties of bulk excitations. We show how to interpret them as gauged symmetry-protected topological states (SPTs). Some of these states possess fractional response to an…
In this paper, we explore the interplay between symmetry and fracton order, motivated by the analogous close relationship for topologically ordered systems. Specifically, we consider models with 3D planar subsystem symmetry, and show that these can realize subsystem symmetry protected topological phases with gapless boundary modes. Gauging the planar subsystem symmetry leads to a fracton order in which particles restricted to move along lines exhibit a new type of statistical interaction that is specific to the lattice geometry. We show that both the gapless boundary modes of the ungauged theo…
