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.

We describe in detail a counterexample to the topological classification of free fermion systems. We deal with a one-dimensional chain of Majorana fermions with an unusual T symmetry. The topological invariant for the free fermion classification lies in Z, but with the introduction of interactions the Z is broken to Z 8 . We illustrate this in the microscopic model of the Majorana chain by constructing an explicit path between two distinct phases whose topological invariants are equal modulo 8, along which the system remains gapped. The path goes through a strongly interacting region. We also find the field-theory interpretation of this phenomenon. There is a second-order phase transition between the two phases in the free theory, which can be avoided by going through the strongly interacting region. We show that this transition is in the two-dimensional Ising universality class, where a first-order phase transition line, terminating at a second-order transition, can be avoided by going through the analog of a high-temperature paramagnetic phase. In fact, we construct the full phase diagram of the system as a function of the thermal operator ͑i.e., the mass term that tunes between the two phases in the free theory͒ and two quartic operators, obtaining a first-order Peierls transition region, a second-order transition region, and a region with no transition.

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We study two a priori unrelated constructions: the spectrum of edge modes in a band topological insulator or superconductor with a physical edge, and the ground state entanglement spectrum in an extended system where an edge is simulated by an entanglement bipartition. We prove an exact relation between the ground state entanglement spectrum of such a system and the spectrum edge modes of the corresponding spectrally flattened Hamiltonian. In particular, we show that degeneracies of the entanglement spectrum correspond to gapless edge modes.

Three dimensional topological superconductors (TScs) protected by time reversal (T ) symmetry are characterized by gapless Majorana cones on their surface. Free fermion phases with this symmetry (class DIII) are indexed by an integer ν, of which ν = 1 is realized by the B-phase of superfluid 3 He. Previously it was believed that the surface must be gapless unless time reversal symmetry is broken. Here we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of topological order appears on the surface. The topological order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd ν TScs, the surface topological order must be non-Abelian. We propose the simplest non-Abelian topological order that contains electron like excitations, SO(3)6, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance which however is broken in any 2D realization. By explicitly constructing an exactly soluble Walker-Wang model we show that it can be realized at the surface of a short ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons trasforming as Kramers pairs, i.e. T 2 = −1 undert time reversal. We also propose an Abelian theory, the semion-fermion topological order, to realize an even ν TSc surface, for which an explicit model is derived using a coupled layer construction. We argue that this is related to the ν = 2 TSc, and use this to build candidate surface topological orders for ν = 4 and ν = 8 TScs. The latter is equivalent to the three fermion state which is the surface topological order of a Z2 bosonic topological phase protected by T invariance. One particular consequence of this is that a ν = 16 TSc admits a trivially gapped T -symmetric surface.arXiv:1305.5851v4 [cond-mat.str-el]