Classification Of Symmetry-protected Topological Phases Research Articles

Construction and Classification of Symmetry-Protected Topological Phases in Interacting Fermion Systems

The classification and lattice model construction of symmetry protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group $G_f=\mathbb{Z}_2^f \times_{\omega_2} G_b$ which is a central extension of bosonic symmetry group $G_b$ (may contain time reversal symmetry) by the fermion parity symmetry group $\mathbb{Z}_2^f = \{1,P_f\}$. Our construction is based on the concept of equivalence class of finite depth fermionic symmetric local unitary (FSLU) transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We will also discuss the systematical construction and classification of boundary anomalous SPT (ASPT) states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction-free and trivialization-free constructions naturally lead to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point/space group symmetry as well as continuum Lie group symmetry.

Classification of symmetry-protected topological phases in two-dimensional many-body localized systems

We use low-depth quantum circuits, a specific type of tensor networks, to classify two-dimensional symmetry-protected topological many-body localized phases. For (anti-)unitary on-site symmetries we show that the (generalized) third cohomology class of the symmetry group is a topological invariant; however our approach leaves room for the existence of additional topological indices. We argue that our classification applies to quasi-periodic systems in two dimensions and systems with true random disorder within times which scale superexponentially with the inverse interaction strength. Our technique might be adapted to supply arguments suggesting the same classification for two-dimensional symmetry-protected topological ground states with a rigorous proof.

Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory

Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this work, we revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. We construct very general fixed point SPT wavefunctions for interacting fermion systems. We naturally reproduce the partial classifications given by special group super-cohomology theory, and we show that with an additional $\tilde{B}H^2(G_b, \mathbb Z_2)$ (the so-called obstruction free subgroup of $H^2(G_b, \mathbb Z_2)$) structure, a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group $G_f=G_b\times \mathbb Z_2^f$ can be obtained for unitary symmetry group $G_b$. We also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions.

Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries

We propose the definitions of many-body topological invariants to detect symmetry-protected topological phases protected by point group symmetry, using partial point group transformations on a given short-range entangled quantum ground state. Partial point group transformations $g_D$ are defined by point group transformations restricted to a spatial subregion $D$, which is closed under the point group transformations and sufficiently larger than the bulk correlation length $\xi$. By analytical and numerical calculations,we find that the ground state expectation value of the partial point group transformations behaves generically as $\langle GS | g_D | GS \rangle \sim \exp \Big[ i \theta+ \gamma - \alpha \frac{{\rm Area}(\partial D)}{\xi^{d-1}} \Big]$. Here, ${\rm Area}(\partial D)$ is the area of the boundary of the subregion $D$, and $\alpha$ is a dimensionless constant. The complex phase of the expectation value $\theta$ is quantized and serves as the topological invariant, and $\gamma$ is a scale-independent topological contribution to the amplitude. The examples we consider include the $\mathbb{Z}_8$ and $\mathbb{Z}_{16}$ invariants of topological superconductors protected by inversion symmetry in $(1+1)$ and $(3+1)$ dimensions, respectively, and the lens space topological invariants in $(2+1)$-dimensional fermionic topological phases. Connections to topological quantum field theories and cobordism classification of symmetry-protected topological phases are discussed.

Matrix product operators for symmetry-protected topological phases: Gauging and edge theories

Projected entangled pair states (PEPS) provide a natural ansatz for the ground states of gapped, local Hamiltonians in which global characteristics of a quantum state are encoded in properties of local tensors. We develop a framework to describe on-site symmetries, as occurring in systems exhibiting symmetry-protected topological (SPT) quantum order, in terms of virtual symmetries of the local tensors expressed as a set of matrix product operators (MPOs) labeled by distinct group elements. These MPOs describe the possibly anomalous symmetry of the edge theory, whose local degrees of freedom are concretely identified in a PEPS. A classification of SPT phases is obtained by studying the obstructions to continuously deforming one set of MPOs into another, recovering the results derived for fixed-point models [X. Chen et al., Phys. Rev. B 87, 155114 (2013)]. Our formalism accommodates perturbations away from fixed point models, opening the possibility of studying phase transitions between different SPT phases. We also demonstrate that applying the recently developed quantum state gauging procedure to a SPT PEPS yields a PEPS with topological order determined by the initial symmetry MPOs. The MPO framework thus unifies the different approaches to classifying SPT phases, via fixed-points models, boundary anomalies, or gauging the symmetry, into the single problem of classifying inequivalent sets of matrix product operator symmetries that are defined purely in terms of a PEPS.

Braiding statistics and classification of two-dimensional charge-2msuperconductors

We study braiding statistics between quasiparticles and vortices in two-dimensional charge-$2m$ (in units of $e$) superconductors that are coupled to a $\mathbb Z_{2m}$ dynamical gauge field, where $m$ is any positive integer. We show that there exist $16m$ types of braiding statistics when $m$ is odd, but only $4m$ types when $m$ is even. Based on the braiding statistics, we obtain a classification of topological phases of charge-$2m$ superconductors---or formally speaking, a classification of symmetry-protected topological phases, as well as invertible topological phases, of two-dimensional gapped fermions with $\mathbb Z_{2m}^f$ symmetry. Interestingly, we find that there is no nontrivial fermionic symmetry-protected topological phase with $\mathbb Z_4^f$ symmetry.

CPT theorem and classification of topological insulators and superconductors

We present a systematic topological classification of fermionic and bosonic topological phases protected by time-reversal, particle-hole, parity, and combination of these symmetries. We use two complementary approaches: one in terms of K-theory classification of gapped quadratic fermion theories with symmetries, and the other in terms of the K-matrix theory description of the edge theory of (2+1)-dimensional bulk theories. The first approach is specific to free fermion theories in general spatial dimensions while the second approach is limited to two spatial dimensions but incorporates effects of interactions. We also clarify the role of CPT theorem in classification of symmetry-protected topological phases, and show, in particular, topological superconductors dis- cussed before are related by CPT theorem.