Volume and Topological Invariants of Quantum Many-body Systems

Abstract

A gapped many-body system is described by path integral on a space-time lattice $C^{d+1}$, which gives rise to a partition function $Z(C^{d+1})$ if $\partial C^{d+1}=\varnothing$, and gives rise to a vector $|\Psi ⟩$ on the boundary of space-time if $\partial C^{d+1}\ne \varnothing$. We show that $V=\text{log}\sqrt{⟨\Psi |\Psi ⟩}$ satisfies the inclusion-exclusion property $\frac{V(A\cup B)+V(A\cap B)}{V(A)+V(B)}=1$ and behaves like a volume of the space-time lattice $C^{d+1}$ in large lattice limit (i.e. thermodynamics limit). This leads to a proposal that the vector $|\Psi ⟩$ is the quantum-volume of the space-time lattice $C^{d+1}$. The inclusion-exclusion property does not apply to quantum-volume since it is a vector. But quantum-volume satisfies a quantum additive property. The violation of the inclusion-exclusion property by $V=\text{log}\sqrt{⟨\Psi |\Psi ⟩}$ in the subleading term of thermodynamics limit gives rise to topological invariants that characterize the topological order in the system. This is a systematic way to construct and compute topological invariants from a generic path integral. For example, we show how to use non-universal partition functions $Z(C^{2+1})$ on several related space-time lattices $C^{2+1}$ to extract $(M_f{)}_{11}$ and $\text{Tr}(M_f)$, where $M_f$ is a representation of the modular group $SL(2,\mathbb{Z})$ -- a topological invariant that almost fully characterizes the 2+1D topological orders.

10 pages, 7 figures. arXiv admin note: text overlap with arXiv:1405.5858

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