We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e., with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer quantum Hall states, topological insulators, and superconductors. We adapt the well-known Chern-Simons
K-matrix description of quantum Hall states to classify such ``integer'' topological phases. Our main result is a general formalism that incorporates symmetries into the
K-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder, include (i) bosonic insulators with a Hall conductance quantized to even integers, (ii) a bosonic analog of quantum spin Hall insulators, and (iii) a bosonic analog of a chiral topological superconductor, whose
K matrix is the Cartan matrix of Lie group
E8. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly, we construct microscopic models of these phases from coupled one-dimensional systems.
We examine the interplay of symmetry and topological order in
2+1 dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the
Gf fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry
G=Gf/Z2f on the physical fermions is essentially the central extension of
G by the
Z2f fermion parity conservation that yields the fermionic symmetry
Gf. We develop an algebraic theory of
Gf symmetry defects for fermionic topological phases using
G-crossed braided tensor category theory. This formalism allows us to fully characterize and classify
2+1 dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group
Gf. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed
Gf form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.
While free fermion topological crystalline insulators have been largely classified, the analogous problem in the strongly interacting case has been only partially solved, and the relationship between the free and interacting classifications is not well understood. In this paper, we develop a characterization and classification of interacting, invertible fermionic topological phases in (
2+1) dimensions with charge conservation, discrete magnetic translation, and
M-fold point group rotation symmetries, which form the group ${G}_{f}=\text{U}{(1)}^{f}{\ifmmode\times\else\texttimes\fi{}}_{\ensuremath{\phi}}[{\mathbb{Z}}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{M}]$ for
M=1,2,3,4,and6. $\ensuremath{\phi}$ is the magnetic flux per unit cell. We derive a topological response theory in terms of background crystalline gauge fields, which gives a complete classification of different phases and a physical characterization in terms of quantized response to symmetry defects. We then derive the same classification in terms of a set of real-space invariants ${{\mathrm{\ensuremath{\Theta}}}_{\text{o}}^{\ifmmode\pm\else\textpm\fi{}}}$ that can be obtained from ground state expectation values of suitable partial rotation operators. We explicitly relate these real-space invariants to the quantized coefficients in the topological response theory, and find the dependence of the invariants on the chiral central charge ${c}_{\ensuremath{-}}$ of the invertible phase. Finally, when $\ensuremath{\phi}=0,$ we derive an explicit map between the free and interacting classifications.
Symmetry-protected topological (SPT) phases (e.g., the well known topological insulators) are a class of energetically gapped condensed matter systems that exhibit interesting topological properties only in the presence of certain global symmetries. While a great understanding of noninteracting fermionic SPT phases has been achieved recently, interacting SPT phases, in particular those that exist only in strongly interacting systems, are much less understood in general. Here, the authors study strongly interacting fermionic SPT phases in two spatial dimensions with finite Abelian unitary symmetries and provide a potentially complete classification of them.
After briefly recalling the quantum entanglement-based view of topological phases of matter in order to outline the general context, we give an overview of different approaches to the classification problem of topological insulators and superconductors of non-interacting Fermions. In particular, we review in some detail general symmetry aspects of the "ten-fold way" which forms the foundation of the classification, and put different approaches to the classification in relationship with each other. We end by briefly mentioning some of the results obtained on the effect of interactions, mainly in three spatial dimensions.
In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of one-dimensional systems. We focus on the time-reversal-invariant Majorana chain (BDI symmetry class). While the band classification yields an integer topological index
k, it is known that phases characterized by values of
k in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half chains. We generalize these results to the classification of all one-dimensional gapped phases of fermionic systems with possible antiunitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.