Anomalies of global symmetries provide important information on the quantum dynamics. We show the dynamical constraints can be organized into three classes: genuine anomalies, fractional topological responses, and integer responses that can be realized in symmetry-protected topological (SPT) phases. Coset symmetry can be present in many physical systems including quantum spin liquids, and the coset symmetry can be a non-invertible symmetry. We introduce twists in coset symmetries, which modify the fusion rules and the generalized Frobenius-Schur indicators. We call such coset symmetries twisted coset symmetries, and they are labeled by the quadruple (G,K,\omega_{D+1},\alpha_D) ( G , K , ω D + 1 , α D ) in D D spacetime dimensions where G G is a group and K\subset G K ⊂ G is a discrete subgroup, \omega_{D+1} ω D + 1 is a (D+1) ( D + 1 ) -cocycle for group G G , and \alpha_{D} α D is a D D -cochain for group K K . We present several examples with twisted coset symmetries using lattice models and field theory, including both gapped and gapless systems (such as gapless symmetry-protected topological phases). We investigate the anomalies of general twisted coset symmetry, which presents obstructions to realizing the coset symmetry in (gapped) symmetry-protected topological phases. We show that finite coset symmetry G/K G / K becomes anomalous when G G cannot be expressed as the bicrossed product G=H\Join K G = H ⋈ K , and such anomalous coset symmetry leads to symmetry-enforced gaplessness in generic spacetime dimensions. We illustrate examples of anomalous coset symmetries with A_5/\Z_2 symmetry, with realizations in lattice models.
Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of
G-crossed braided tensor categories (
G×BTCs) for SETs. We consider systems with
U(1) charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all
5 orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains
9 terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the
G×BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the
G×BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.
Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d “inflow” symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the “Frobenius-Schur indicator” of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing \mathbb{Z}_{4} ℤ 4 ordinary symmetry host worldvolume theories with time-reversal symmetry \mathsf{T} 𝖳 obeying the algebra \mathsf{T}^{2}=C 𝖳 2 = C or \mathsf{T}^{2}=(-1)^{F}C, 𝖳 2 = ( − 1 ) F C , with C C a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.
A bstract We use the intrinsic one-form and two-form global symmetries of (3+1) d bosonic field theories to classify quantum phases enriched by ordinary (0-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the 0-form symmetry on the lines and surfaces of the theory. From these data we classify the couplings to the background gauge field by the 0-form symmetry defects constructed from the higher-form symmetry defects. For trivial two-form symmetry the classification coincides with the classification for symmetry fractionalizations in (2 + 1) d . We also provide a systematic method to obtain the symmetry protected topological phases that can be absorbed by the coupling, and we give the relative ’t Hooft anomaly for different couplings. We discuss several examples including the gapless pure U(1) gauge theory and the gapped Abelian finite group gauge theory. As an application, we discover a tension with a conjectured duality in (3 + 1) d for SU(2) gauge theory with two adjoint Weyl fermions.
We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic
Z4F symmetry in (2+1) dimensions. For an onsite
Z4F symmetry, symmetric invertible phases have integer chiral central charge
c−∈Z, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local
Z4F symmetry of a
p+ip superconductor with
c−=1/2. We compute its lattice anomaly, and find that the symmetry indeed forbids
c−∈Z invertible states. The allowed invertible phases are consequently shifted to
c−∈Z+1/2. By gauging the fermion parity of the
p+ip superconductor, one obtains an Ising topological order enriched by a
Z4 symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the
p+ip state admits an enlarged non-onsite
U(1)f symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite
U(1)f symmetry again forbids Chern insulators carrying
c−∈Z, while being compatible with invertible states with
c−∈Z+1/2. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.
The topological defects of Spin(
n+1) nematics in two spatial dimensions, known as disclinations, are characterized by the ${\ensuremath{\pi}}_{1}(\mathbb{R}{P}^{n})={\text{Z}}_{2}$ homotopy group for $n\ensuremath{\ge}2$. We argue that incompressible quantum liquids of disclinations can exist as stable low-temperature phases and host composite quasiparticles which combine a fractional amount of fundamental
Z2 charge with a unit of topological charge. The fourfold topological ground state degeneracy on a torus admits a fermionic or semionic quasiparticle exchange statistics. The topological nontriviality of these states is visible in the existence of protected gapless edge modes. While the fermionic nematic and gapped
Z2 spin liquids have equivalent topological orders, they are still thermodynamically distinct due to having different edge modes, in analogy to the topologically nontrivial and trivial states of quantum spin-Hall systems. The analysis proceeds by recasting the
Z2 gauge theory of spin nematics as a continuum limit theory with a larger gauge structure. Nematic fractionalization parallels that of a quantum Hall liquid, but the large gauge symmetry restores the time-reversal symmetry and restricts the quasiparticle fusion rules and statistics. The conclusions from field theory analysis are complemented with the construction of plausible host microscopic models.