Symmetry protected topological orders in interacting bosonic systems

TL;DR

Just as group theory allows us to construct 230 crystal structures in three-dimensional space, group cohomology theory is used to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.

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.

Figures (11)

• Figure 1: (Color online)Dimer form of the ground state wave-function in Haldane chain. Each site (big oval) contains two spin $1/2$'s (small dot), which are connected into singlet pairs (connected dots) between neighboring sites.
• Figure 2: (Color online) (a) A branched triangularization of space-time. (For details of braching see appendix \ref{['fixed_point']}) (b) A tetrahedron -- the simplest discrete closed surface. $\prod \nu^{s_{ijk}}(g_i,g_j,g_k)=1$ on a tetrahedron is guaranteed by Eq.(\ref{['Gcoh2']}). Note that $s_{123}=s_{013}=1$ and $s_{023}=s_{012}=-1$. (c) Discretized space-time manifold $M_\text{ext}$ on an open disk with boundary manifold $M$. $g_i \in M$, $g^*$ is in the interior of $M_\text{ext}$.
• Figure 3: (Color online) Duality transformation between wave-functions in Eq.(\ref{['phi_dimer']}) and Eq.(\ref{['PsiM']}).
• Figure 4: If we extend $\v n(t)$ that traces out a loop to $\v n(t,\xi)$ that covers the shaded disk, then the WZW term $\int_{D^2} \dd t\dd \xi\; \v n(t,\xi)\cdot [\prt_t \v n(t,\xi) \times \prt_\xi \v n(t,\xi)]$ corresponds to the area of the disk.
• Figure 5: (a) The topological term $W$ describes the number of times that $\v n(x,t)$ wraps around the sphere (as we change $t$). (b) On an open chain $x \in [0,L]$, the topological term $W$ in the (1+1)D bulk becomes the WZW term for the end spin $\v n_L(t)=\v n(L,t)$ (where the end spin at $x=0$ is hold fixed).