Tensor networks as path integral geometry

TL;DR

This paper proposes interpreting tensor networks as discrete path integrals on curved spacetime, linking their structure to the geometry of the underlying conformal field theory in quantum critical spin chains.

Contribution

It introduces a novel perspective of viewing tensor networks as approximate path integrals on curved spacetime, connecting tensor network geometry with quantum field theory.

Findings

Tensor networks can be interpreted as path integrals on curved spacetime.

This interpretation assigns a geometric structure to tensor networks.

The approach bridges tensor network methods with conformal field theory geometry.

Abstract

In the context of a quantum critical spin chain whose low energy physics corresponds to a conformal field theory (CFT), it was recently demonstrated [A. Milsted G. Vidal, arXiv:1805.12524] that certain classes of tensor networks used for numerically describing the ground state of the spin chain can also be used to implement (discrete, approximate versions of) conformal transformations on the lattice. In the continuum, the same conformal transformations can be implemented through a CFT path integral on some curved spacetime. Based on this observation, in this paper we propose to interpret the tensor networks themselves as a path integrals on curved spacetime. This perspective assigns (a discrete, approximate version of) a geometry to the tensor network, namely that of the underlying curved spacetime.