We study fourfold rotation-invariant gapped topological systems with time-reversal symmetry in two and three dimensions (d=2, 3). We show that in both cases nontrivial topology is manifested by the presence of the (d-2)-dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through the explicit construction of microscopic models having robust (d-2)-dimensional edge states.
We study fourfold rotation-invariant gapped topological systems with time-reversal symmetry in two and three dimensions (d=2, 3). We show that in both cases nontrivial topology is manifested by the presence of the (d-2)-dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through the explicit construction of microscopic models having robust (d-2)-dimensional edge states.
We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single-particle excitations in the bulk. We consider specific microscopic models and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically nontrivial phase.
Recently, it has been found that there exist symmetry-protected topological phases of fermions, which have no realizations in non-interacting fermionic systems or bosonic models. We study the edge states of such an intrinsically interacting fermionic SPT phase in two spatial dimensions, protected by \mathbb{Z}_4\times\mathbb{Z}_2^\mathsf{T} ℤ 4 × ℤ 2 𝖳 symmetry. We model the edge Hilbert space by replacing the internal \mathbb{Z}_4 ℤ 4 symmetry with a spatial translation symmetry, and design an exactly solvable Hamiltonian for the edge model. We show that at low-energy the edge can be described by a two-component Luttinger liquid, with nontrivial symmetry transformations that can only be realized in strongly interacting systems. We further demonstrate the symmetry-protected gaplessness under various perturbations, and the bulk-edge correspondence in the theory.
Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system with nontrivial onsite Z2 symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (1D) system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the Z2 group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry-protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if Z2 symmetry is not broken. We prove the latter by developing the tool of a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with onsite Z2 symmetry-protected topological order.
We introduce a novel class of interaction-enabled topological crystalline insulators in two- and three-dimensional electronic systems, which we call ``topological crystalline magnet.'' It is protected by the product of the time-reversal symmetry T and a mirror symmetry or a rotation symmetry R. A topological crystalline magnet exhibits two intriguing features: (i) it cannot be adiabatically connected to any Slater insulator and (ii) the edge state is robust against coupling electrons to the edge. These features are protected by the anomalous symmetry transformation property ${(\mathcal{R}\mathcal{T})}^{2}=\ensuremath{-}1$ of the edge state. An anisotropic response to the external magnetic field can be an experimental signature.
Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with nontrivial strong topological invariants (protected by internal symmetries) with additional ``weak'' topological indices. In this paper, we study the edge physics of systems with a nontrivial shift invariant, which is protected by either a continuous U(1)r or discrete Cn rotation symmetry, along with internal U(1)c charge conservation. Specifically, we construct an interface between two systems that have the same Chern number but are distinguished by their Wen-Zee shift, and, through analytic arguments supported by numerics, we show that the interface hosts counterpropagating gapless edge modes that cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian subalgebra framework for gapped boundaries, while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and we compute them in the case in which the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in strongly interacting symmetry enriched topological phases.
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.