Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions

TL;DR

<3-5 sentence high-level summary>The paper develops a comprehensive framework to identify and measure symmetry-protected topological invariants for interacting bosonic and fermionic SP T phases by gauging on-site symmetries and probing topological responses. It unifies invariants across dimensions using cup products and the Künneth formula, providing explicit constructions for simple groups (e.g., Z_n, U(1), Z_2^T) and for composite groups, with both bosonic and fermionic realizations. The work derives tangible diagnostics such as defect monodromies, ground-state charges, defect statistics, and CS-type topological terms, linking cocycles in ${\cal H}^d(G,\mathbb{R}/\mathbb{Z})$ to measurable quantities. It also discusses boundary phenomena and dimensional reduction, showing how high-dimensional SP T data manifest as lower-dimensional boundary physics, and outlines a path toward experimental and numerical detection of SP T order in interacting systems.

Abstract

Recently, it was realized that quantum states of matter can be classified as long-range entangled (LRE) states (i.e. with non-trivial topological order) and short-range entangled (SRE) states (\ie with trivial topological order). We can use group cohomology class ${\cal H}^d(SG,R/Z)$ to systematically describe the SRE states with a symmetry $SG$ [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in $d$-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects, and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e. to measure the elements in ${\cal H}^d(G, R/Z)$ that label different SPT states. For example, 2+1D bosonic SPT states with $Z_n$ symmetry are classified by a $Z_n$ integer $m \in {\cal H}^3(Z_n, R/Z)=Z_n$. We find that $n$ identical monodromy defects, in a $Z_n$ SPT state labeled by $m$, carry a total $Z_n$-charge $2m$ (which is not a multiple of $n$ in general).

Figures (12)

• Figure 1: (Color online) (a) The possible gapped phases for a class of Hamiltonians $H(g_1,g_2)$ without any symmetry. (b) The possible gapped phases for the class of Hamiltonians $H_\text{symm}(g_1,g_2)$ with a symmetry. The yellow regions in (a) and (b) represent the phases with long range entanglement. 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. The SY-SRE phases are the SPT phases. The SY-LRE phases are the SET phases.
• Figure 2: (Color online) A 2D lattice on a torus. A $Z_n$ transformation is performed on the sites in the shaded region. The $Z_n$ transformation changes the Hamiltonian term on the triangle $(ijk)$ across the boundary from $H_{ijk}$ to $H'_{ijk}$.
• Figure 3: (Color online) A $Z_2$-gauge configuration with two identical$Z_2$ vertices (or two monodromy defects) on a torus. Such a $Z_2$-gauge configuration has $U^\text{bulk}_{-1}=-1$ (each yellow triangle contributes a factor $-1$). Thus $U^\text{bulk}_g$ forms a 1D representation of $Z_2$ with a $Z_2$-charge 1.
• Figure 4: (Color online) A $Z_2$-gauge configuration with two identical holes on a torus that contains a $Z_2$-monodromy defect in each hole. Such a $Z_2$-gauge configuration has $U(-1)=-1$ (each yellow triangle contributes a factor $-1$) (see Fig. \ref{['z2gauge']}).
• Figure 5: (Color online) The circle represents $2\pi n$ flux which induce an integer $U(1)$-charge. As we move the flux into the hole, the induced $U(1)$ charge disappears.