Symmetry-Protected Topological Orders in Interacting Bosonic Systems

Abstract

Symmetry protected topological (SPT) states are bulk gapped states with gapless edge excitations protected by certain symmetries. The SPT phases in free fermion systems, like topological insulators, can be classified by the K-theory. However, it is not known what SPT phases exist in general interacting systems. In this paper, we present a systematic way to construct SPT phases in interacting bosonic systems, which allows us to identify many new SPT phases, including three bosonic versions of topological insulators in three dimension and one in two dimension protected by particle number conservation and time reversal symmetry. Just as group theory allows us to construct 230 crystal structures in 3D, we find that group cohomology theory allows us to construct different interacting bosonic SPT phases in any dimensions and for any symmetry groups. In particular, we are going to show how topological terms in the path integral description of the system can be constructed from nontrivial group cohomology classes, giving rise to exactly soluble Hamiltonians, explicit ground state wave functions and symmetry protected gapless edge excitations. We used to believe that different phases of matter are different because they have different symmetries.1-3 Recently, we see a deep connection between quantum phases and quantum entanglement 4-6 which allows us to go beyond this framework. First it was realized that even in systems without any symmetry there can be distinct quantum phases -topological phases 7,8 due to different patterns of long-range entanglement in the states.6 For systems with symmetries, difference in long-range entanglement and in symmetry still lead to distinct phases. Moreover, even short-range entangled states with the same symmetry can belong to different phases. These symmetric short-range entangled states are said to contain a new kind of order -symmetry protected topological (SPT) order.9 The SPT phases have symmetry protected gapless edge modes despite the bulk gap, which clearly indicates the topological nature of this order. On the other hand, the gapless edge modes disappear when the symmetry of the system is broken, indicating that this is a different type of topological order than that found in fractional quantum Hall systems 10,11 whose edge modes cannot be removed with any local perturbation. 12Also, SPT orders have no factional statistics or fractional charges, while intrinsic topological orders from long range entanglement can have them. The discovery of SPT order hence greatly expands our original understanding of possible phases in many-body systems.One central issue is to understand what SPT phases exist and much progress has been made in this regard. The first system with SPT order was discovered decades ago in spin-1 Haldane chains. The Haldane chain with antiferromagnetic interactions was shown to have a gapped bulk 13,14 and degenerate modes at the ends of the chain 15-17 which are protected by spin rotation or time reversal symmetry of the system.9,18 This model has been generalized, l...

Framed Wilson operators, fermionic strings, and gravitational anomaly in 4d

“…These are precisely the phases labeled by the effective theory. It was originally proposed that the only effective theories that appeared were Dijkgraaf-Witten theories [3]. Later, however, it became clear that there were richer SPT phases even in the bosonic case [4][5][6].…”

“…A generalization of Z 2 Dijkgraaf-Witten theory attempting to describe these phases can be constructed using group cohomology with twisted coefficients [3]. However, it does completely describe these phases.…”

Construction and field theory of bosonic-symmetry-protected topological states beyond group cohomology

“…Now we fix λ at a negative value, and tune β from negative to positive. With negative β, effectively n (1) and n (2) will align with each other, thus n 2,(1) = n 2,(2) , C (1) = C (2) , then the two theories will "constructively interfere" with each other, and the final theory effectively has Θ = 4π; with positive β, effectively n 2,(1) = −n 2,(2) , C (1) = C (2) , thus the two theories will "destructively interfere" with each other, and the final theory effectively has Θ = 0. Because both theories are fully gapped and nondegenerate in the bulk, tuning the coupling between them does not close the bulk gap (as long as the coupling is not too strong to overcome the bulk gap), thus the two effective coupled theories with Θ = 0 and Θ = 4π are smoothly connected without going through a bulk phase transition.…”

“…Imagine we have two copies of the theory, the only U (1) symmetry allowed coupling between these two theories would be β n (1) · n (2) . Then for either sign of β, i.e.…”

“…There are roughly two types of BSPT states, their mathematical difference is whether they can be classified and described by group cohomology [1,2] and semiclassical nonlinear sigma model field theory [3]. For example, the well-known E 8 bosonic short range entangled (BSRE) state [18] [4,5] in 2d space [19], and its higher dimensional generalizations [6] cannot be classified by group cohomology.…”