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Volume and Topological Invariants of Quantum Many-body Systems · Around

2018

Volume and Topological Invariants of Quantum Many-body Systems

Wen, Xiao-Gang, Wang, Zhenghan

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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 $\partial C^{d+1} =\emptyset$, and gives rise to a vector $|\Psi\rangle$ on the boundary of space-time if $\partial C^{d+1} \neq\emptyset$.

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 $|\Psi\rangle$ is the quantum-volume of the space-time lattice $C^{d+1}$.

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