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 C^{d+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(C^{2+1}) on several space-time lattices with related topologies to extract (M_{f})_{11} and Tr(M_{f}), where M_{f} 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 C^{d+1}. We show that V=lnsqrt[〈Ψ|Ψ〉] satisfies the inclusion-exclusion property V(A∪B)+V(A∩B)/V(A)+V(B)=1 and behaves like a volume of the space-time C^{d+1} in the thermodynamics limit if the system is short range correlated. This leads to a proposal that the vector |Ψ〉 itself is the quantum volume of the space-time C^{d+1}. The subleading term of thermodynamics limit gives rise to the topological invariants.