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
G 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
BF 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,... are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as
G 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.
The challenge of identifying symmetry-protected topological states (SPTs) is due to their lack of symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is impossible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and topology-dependent ground state degeneracy. However, the partition functions from path integrals with various symmetry twists are universal SPT invariants, fully characterizing SPTs. In this work, we use gauge fields to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology. This allows us to express the SPT invariants in terms of continuum field theory. We show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gauge-gravity actions describe the beyond-group-cohomology SPTs. We find new examples of mixed gauge-gravity actions for U(1) SPTs in (4+1)D via the gravitational Chern-Simons term. Field theory representations of SPT invariants not only serve as tools for classifying SPTs, but also guide us in designing physical probes for them. In addition, our field theory representations are independently powerful for studying group cohomology within the mathematical context.
A domain-wall pump by an extended cluster model of S=1/2 spins is proposed with local U(1) gauge invariance. In contrast to the conventional pump, the current and the gauge field are defined on sites instead of links. Its snapshot ground state is gapped and doubly degenerated due to Z_{2} invariance, which is broken by an infinitesimal boundary magnetic field. The ground state associated with the spontaneous symmetry breaking is, at the same time, symmetry-protected with additional spatial inversion that is characterized by the Z_{2} Berry phase defined by the twist on the boundary site. We investigate the topological domain-wall pump with and without boundaries. The topological pump associated with the inversion symmetry-breaking path induces a nontrivial Chern number of bulk and a singular behavior of edge states of the domain wall. Generalization to the multispin interaction is also explicitly given. These extended quantum degrees of freedom are prominent qubits that are topologically protected.
We present an exactly solvable model for one-dimensional symmetry-protected topological phases with
ZN×ZN symmetry. The model works by binding point topological defects (domain walls) of one symmetry to charges of the other and condensing these bound states. Binding single topological defects to charges leads to symmetry-protected topological phases, while binding multiple topological defects to charges leads to phases with a combination of symmetry-breaking and topological properties.
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.