Symmetry protected topological orders and the group cohomology of their symmetry group

TL;DR

The paper provides a systematic framework for classifying interacting bosonic SPT phases with on-site symmetry G in d dimensions via the (1+d)-th Borel cohomology group H^{1+d}[G, U_T(1)]. It develops both Hamiltonian and Lagrangian (path-integral) formalisms, showing that fixed-point SPT orders arise from cocycle data ν_{1+d} in H^{1+d}[G, U_T(1)], with boundary theories described by non-local Lagrangians that encode gapless or degenerate edge modes. The authors construct explicit lattice models and wavefunctions from group cocycles, demonstrate that equivalent cocycles define the same phase, and extend the scheme to translation-symmetric cases, thereby mapping a broad landscape of bosonic SPTs (including SO(3), SU(2), and U(1) cases) to cohomology data. This framework clarifies how symmetry, topology, and entanglement interplay in gapped quantum states and provides a concrete toolkit for identifying and engineering SPT phases in interacting systems. The work paves the way for exploring symmetry-protected phenomena in higher dimensions and in fermionic settings via generalized cohomology theories.

Abstract

Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H, G_Ψ, H^{1+d}[G_Ψ, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_Ψ the symmetry group of the ground states.

Figures (42)

• Figure 1: (Color online) (a) The possible gapped phases for a class of Hamiltonians $H(g_1,g_2)$ without any symmetry restriction. (b) The possible gapped phases for the class of Hamiltonians $H_\text{symm}(g_1,g_2)$ with symmetry. Each phase is labeled by its entanglement properties and symmetry breaking properties. SRE stands for short range entanglement, LRE for long range entanglement, SB for symmetry breaking, SY for no symmetry breaking. SB-SRE phases are the Landau symmetry breaking phases, which are understood by introducing group theory. The SY-SRE phases are the SPT phases, and we will show that they can be understood by introducing group cohomology theory. The SY-LRE phases are the SET phases.
• Figure 2: (a) A triangular lattice. The Hamiltonian term Hi acts on the seven sites in the shaded area. (b) A geometric representation of the the phase factors in Hi.
• Figure 3: (Color online) (a) A graphic representation of a quantum circuit, which is form by (b) unitary operations on blocks of finite size $l$. The green shading represents a causal structure.
• Figure 4: (Color online) Transforming a SRE state to a tensor-network state which take simple canonical form. (a) A SRE state. (b) Using the unitary transformations that act within each block, we can transform the SRE state to a tensor-network state. Entanglements exist only between the degrees of freedom on the connected tensors.
• Figure 5: (Color online) Graphic representations of tensors: (a) $A^m_{\al}$, (b) $A^m_{\al\bt}$, and (c) $A^m_{\al\bt\ga\la}$. (d) A corner represents a special rank-2 tensor $A_{\al\bt}=\del_{\al\bt}$.

Theorems & Definitions (5)

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