Volume and Topological Invariants of Quantum Many-body Systems

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 $\mathrm{\partial }{C}^{d+1}=\mathrm{\varnothing }$, and gives rise to a vector $\mathrm{\mid }\mathrm{\Psi }⟩$ on the boundary of space-time if $\mathrm{\partial }{C}^{d+1}\mathrm{\ne }\mathrm{\varnothing }$. We show that $V = $\text{log}$ \sqrt{\langle\Psi|\Psi\rangle}$ 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 $\mathrm{\mid }\mathrm{\Psi }⟩$ 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 $V=\text{log}\sqrt{⟨\mathrm{\Psi }\mathrm{\mid }\mathrm{\Psi }⟩}$ 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.