Universal symmetry-protected topological invariants for symmetry-protected topological states

TL;DR

This work develops a systematic framework to characterize symmetry-protected topological (SPT) states by universal invariants tied to group cohomology $H^{d+1}(G,\mathbb{R}/\mathbb{Z})$. It introduces symmetry twists, gauged simulations of nearly degenerate ground states, and non-Abelian geometric phases arising from modular transformations on a torus, arguing that the universal parts of the modular matrices $S$ and $T$ (together with the group action $I(g)$) encode the cocycle data that fully specify an SPT phase. The authors provide explicit constructions in 2+1D for finite groups, connect these invariants to the topological partition function on space-time bundles, and discuss phase ambiguities and how to form robust invariants from closed transformation orbits. This approach offers a practical, dimension-generalizable route to exhaustively classify SPT orders by measurements that depend only on the group cohomology data, with potential extensions to higher dimensions and more complex symmetry groups.

Abstract

Symmetry-protected topological (SPT) states are short-range entangled states with a symmetry G. They belong to a new class of quantum states of matter which are classified by the group cohomology $H^{d+1}(G,\mathbb{R}/\mathbb{Z})$ in d-dimensional space. In this paper, we propose a class of symmetry- protected topological invariants that may allow us to fully characterize SPT states with a symmetry group G (ie allow us to measure the cocycles in $H^{d+1}(G,\mathbb{R}/\mathbb{Z})$ that characterize the SPT states). We give an explicit and detailed construction of symmetry-protected topological invariants for 2+1D SPT states. Such a construction can be directly generalized to other dimensions.

Figures (11)

• Figure 1: (Color online) A 2D lattice on a torus. A $h\in G$ transformation is performed on the sites in the shaded region. The $h$ transformation changes the Hamiltonian term on the triangles $(ijk)$ across the boundary (the loop) from $H_{ijk}$ to $H'_{h,ijk}$.
• Figure 2: (Color online) The Hamiltonian $H^\text{gauged}_{h_x,h_y}$ with two symmetry twists $h_x$ and $h_y$ along the loops in $y$- and $x$- directions respectively. The shaded triangles $(ijk)$ across the the loop contain Hamiltonian terms $H'_{h_x,ijk}$ or $H'_{h_y,ijk}$.
• Figure 3: (Color online) (a) A system on a torus with two symmetry twists in $x$- and $y$-directions. Note that the torus has the same size $L$ in $x$- and $y$-directions. (b) The $S$-transformation of the torus, and the resulting new symmetry twists.
• Figure 4: (Color online) (a) A system on a torus with two symmetry twists in $x$- and $y$-directions. Note that the torus has the same size $L$ in $x$- and $y$-directions. (b) The $T$-transformation of the torus, and the resulting new symmetry twists. (c) After local symmetry transformation $h_x$ in the shaded region, the symmetry twists in (b) become symmetry twists in $x$- and $y$-directions.
• Figure 5: (Color online) The cellurarization into three tetrahedrons of $Y \times S^1$.