Volume and topological invariants of quantum many-body systems

Abstract

Quantum many-body systems described by Lagrangian path integrals can realize many different topological phases of matter. One of the most important problems in condensed-matter physics is to extract topological invariants from the Lagrangian and to determine the topological order in the systems. In this paper, we suggest a general solution to this problem. Given a path integral on space-time lattice Cd+1 that describes a short-range correlated (i.e., gapped) system, we design systematic ways to extract topological invariants. For example, we show how to use nonuniversal partition functions Z(C2+1) on several space-time lattices with related topologies to extract (Mf)11 and Tr(Mf), where Mf is a representation of the modular group SL(2,Z), a topological invariant that characterizes (2+1)-dimensional topological orders. Our approach is guided by a notion of quantum volume. A path integral gives rise to a wave function |ψ) on the boundary of (d+1)-dimensional space-time Cd+1. We show that V=ln(ψ|ψ) satisfies the inclusion-exclusion property V(AaB)+V(Aa).