$Z_Q$ Topological Invariants for Polyacetylene, Kagome and Pyrochlore lattices

TL;DR

This paper introduces adiabatic $Z_Q$ invariants based on quantized Berry phases to characterize topological phases in gapped electronic systems across different lattice types and dimensions, revealing fractional quantization protected by global symmetries.

Contribution

It defines a series of $Z_Q$ invariants for various lattice systems and provides explicit topological forms, including a $Z_2$ invariant for chiral symmetric cases.

Findings

Invariants characterize topological phase transitions via multimerization.

Fractional quantization protected by global $Z_Q$ symmetry.

Explicit topological form of $Z_2$ invariant for chiral symmetric systems.

Abstract

Adiabatic $Z_Q$ invariants by quantized Berry phases are defined for gapped electronic systems in $d$-dimensions ($Q=d+1$). This series includes Polyacetylene, Kagome and Pyrochlore lattice respectively for $d=1,2$ and 3. The invariants are quantum $Q$-multimer order parameters to characterize the topological phase transitions by the multimerization. This fractional quantization is protected by the global $Z_Q$ equivalence. As for the chiral symmetric case, a topological form of the $Z_2$-invariant is explicitly given as well.