Motivated by symmetry-protected topological phases (SPTs) with both spatial symmetry (e.g., lattice rotation) and internal symmetry (e.g., spin rotation), we propose a class of exotic topological terms, which generalize the well-known Wen-Zee topological terms of quantum Hall systems [X.-G. Wen and A. Zee, Phys. Rev. Lett. 69, 953 (1992)]. These generalized Wen-Zee terms are expressed as wedge product of spin connection and usual gauge fields (1-form or higher) in various dimensions. In order to probe SPT orders, we externally insert symmetry twists such as domain walls of discrete internal symmetry and disclinations that are geometric defects with nontrivial Riemann curvature. Then, generalized Wen-Zee terms simply tell us how SPTs respond to those symmetry twists. Classifying these exotic topological terms thus leads to a complete classification and characterization of SPTs within the present framework. We also propose SPT low-energy field theories, from which generalized Wen-Zee terms are deduced as topological response actions. Following the Abstract of the Wen-Zee paper, our work enriches alternative possibilities of condensed-matter realization of unification of electromagnetism and gravity.
A large class of symmetry-protected topological phases (SPT) in boson/spin systems have been recently predicted by the group cohomology theory. In this work, we consider bosonic SPT states at least with charge symmetry [U(1) or Z
N] or spin-
Sz rotation symmetry [U(1) or Z
N] in two (2D) and three dimensions (3D) and the surface of 3D. If both are U(1), we apply external electromagnetic field/``spin gauge field'' to study the charge/spin response. For the SPT examples we consider {i.e., U
c(1)$\ensuremath{\rtimes}$Z
2T, U
s(1)$\ifmmode\times\else\texttimes\fi{}$Z
2T, U
c(1)$\ifmmode\times\else\texttimes\fi{}$[U
s(1)$\ensuremath{\rtimes}$Z
2]; subscripts
c and
s are short for charge and spin; Z
2T and Z
2 are time-reversal symmetry and $\ensuremath{\pi}$ rotation about
Sy, respectively}, many variants of Witten effect in the 3D SPT bulk and various versions of anomalous surface quantum Hall effect are defined and systematically investigated. If charge or spin symmetry reduces to Z
N by considering charge-
N or spin-
N condensate, instead of the linear response approach, we gauge the charge/spin symmetry, leading to a dynamical gauge theory with some remaining global symmetry. The 3D dynamical gauge theory describes a symmetry-enriched topological phase (SET), i.e., a topologically ordered state with global symmetry which admits nontrivial ground-state degeneracy depending on spatial manifold topology. For the SPT examples we consider, the corresponding SET states are described by dynamical topological gauge theory with topological
BF term and axionic $\ensuremath{\Theta}$ term in 3D bulk. In addition, the surface of SET is described by the chiral boson theory with quantum anomaly.
A bstract We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.
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.
We study the classification of interacting fermionic symmetry-protected topological (SPT) phases with both rotation symmetry and Abelian internal symmetries in one, two, and three dimensions. By working out this classification, on the one hand, we demonstrate the recently proposed correspondence principle between crystalline topological phases and those with internal symmetries through explicit block-state constructions. We find that for the precise correspondence to hold it is necessary to change the central extension structure of the symmetry group by the
Z2 fermion parity. On the other hand, we uncover new classes of intrinsically fermionic SPT phases that are only enabled by interactions, both in 2D and 3D. Several 3D Majorana-type fermionic SPT phases are identified. Moreover, several new instances of Lieb-Schultz-Mattis-type theorems for Majorana-type fermionic SPT phases are obtained and we discuss their interpretations from the perspective of bulk-boundary correspondence.
It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with Zm symmetry and find the classification of SPT phases depends on the parity of m: for even m there are 2m classes, m out of which is intrinsically fermionic SPT phases and cannot be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.