Boundary theories of static bulk topological phases of matter are obstructed in the sense that they cannot be realized on their own as isolated systems. The obstruction can be quantified/characterized by quantum anomalies, in particular when there is a global symmetry. Similarly, topological Floquet evolutions can realize obstructed unitary operators at their boundaries. In this paper, we discuss the characterization of such obstructions by using quantum anomalies. As a particular example, we discuss time-reversal symmetric boundary unitary operators in one and two spatial dimensions, where the anomaly emerges as we gauge the so-called Kubo-Martin-Schwinger (KMS) symmetry. We also discuss mixed anomalies between particle number conserving U(1) symmetry and discrete symmetries, such as $C$ and $\mathit{CP}$, for unitary operators in odd spatial dimensions that can be realized at the boundaries of topological Floquet systems in even spatial dimensions.
Boundary theories of static bulk topological phases of matter are obstructed in the sense that they cannot be realized on their own as isolated systems. The obstruction can be quantified/characterized by quantum anomalies, in particular when there is a global symmetry. Similarly, topological Floquet evolutions can realize obstructed unitary operators at their boundaries. In this paper, we discuss the characterization of such obstructions by using quantum anomalies. As a particular example, we discuss time-reversal symmetric boundary unitary operators in one and two spatial dimensions, where the anomaly emerges as we gauge the so-called Kubo-Martin-Schwinger (KMS) symmetry. We also discuss mixed anomalies between particle number conserving U(1) symmetry and discrete symmetries, such as C and CP, for unitary operators in odd spatial dimensions that can be realized at the boundaries of topological Floquet systems in even spatial dimensions.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
Anomalies are renormalization group invariants that constrain the dynamics of quantum field theories. We show that certain anomalies for discrete global symmetries imply that the underlying theory either spontaneously breaks its generalized global symmetry or is gapless. We identify an obstruction, formulated in terms of the anomaly inflow action, that must vanish if a symmetry preserving gapped phase, i.e. a unitary topological quantum field theory, exits with the given anomaly. Our result is similar to the 2d Lieb-Schultz-Mattis theorem but applies more broadly to continuum theories in general spacetime dimension with various types of discrete symmetries including higher-form global symmetries. As a particular application, we use our result to prove that certain 4d non-abelian gauge theories at θ=π cannot flow at long distances to a phase which simultaneously, preserves time-reversal symmetry, is confining, and is gapped. We also apply our obstruction to 4d adjoint QCD and constrain its dynamics.
We study anomalies in time-reversal-(Z2T) and U(1)-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly (2+1)-dimensional system but can be realized on the surface of a (3+1)-dimensional [(3+1)D] symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators, functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e., $U(1)\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{T}$ and $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of (3+1)D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk (3+1)D SPT phase on a particular manifold and in the presence of a particular background gauge field for the U(1) symmetry.
Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the noninteracting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)d fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\ifmmode\times\else\texttimes\fi{}$ reflection) and electromagnetic U(1) symmetries, and (ii) topological superconductors protected by reflection symmetry. For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under large U(1) gauge transformations, but picks up an anomalous sign, signaling a Z2 topological classification. Similarly, for the second example, which is related to, by CPT-theorem, class DIII topological superconductors, we discuss the invariance/noninvariance of the partition function of the surface theory, defined on the three-torus and its descendants generated by the orientifold projection, under large diffeomorphisms (coordinate transformations). The connection to the collapse of the noninteracting classification by an integer (Z) to Z16, in the presence of interactions, is discussed.
Periodically driven Floquet systems can host dynamical phases, including a range of exotic topological phases that have no static analogs. This work presents a homotopy approach to the study of driven systems, which treats unitary evolutions as paths in the space of unitary operators. By considering loop evolutions in this space, the authors obtain a topological classification, free of uncertainties and ambiguities about the long-time robustness of of behavior in specific physical models. Two classes of Floquet symmetry-protected topological phase are identified, characterized by unitary evolutions that act trivially in the bulk but nontrivially at the boundary of an open system. The first class is captured by an explicit group cohomology construction; it has the remarkable property of converting a trivial boundary state into a nontrivial symmetry-protected topological state. The second class exhibits anomalous counter propagating Hilbert-space transport at the boundary; it lies beyond the cohomology construction.
By definition, the physics of the d-dimensional (dim) boundary of a (d+1)-dim symmetry protected topological (SPT) state cannot be realized as itself on a d-dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a d-dim theory must be a boundary or not, is to couple this theory to a gauge field (or to ``gauge'' its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: Can the boundary of a SPT state be driven into a fully gapped topological order which preserves all the symmetries? We argue (conjecture) that if the gauge anomaly of the boundary is ``perturbative,'' then the boundary must remain gapless; while if the boundary only has global gauge anomaly but no perturbative anomaly, then it is possible to gap out the boundary by driving it into a topological state, when $d\ensuremath{\ge}2$. We will demonstrate this conjecture with two examples: (1) the 3d spin-1/2 chiral fermion with the well-known Witten's global anomaly [Phys. Lett. 117, 324 (1982)], which can be realized on the boundary of a 4d topological superconductor with SU(2) or $U(1)\ensuremath{\rtimes}{Z}_{2}$ symmetry; and (2) the 4d boundary of a 5d topological superconductor with the same symmetry. We show that these boundary systems can be driven into a fully gapped Z2N topological order with topological degeneracy, but this Z2N topological order cannot be future driven into a trivial confined phase that preserves all the symmetries due to some special properties of its topological defects. Our study also leads to exotic states of matter in pure 3d space.