A tensor network quotient takes the vacuum to the thermal state

TL;DR

This paper demonstrates how to implement a conformal quotient in the MERA tensor network to represent thermal states from the vacuum, confirming the approach in the critical Ising model and linking tensor networks to conformal invariance and holography.

Contribution

It introduces a method to realize the conformal quotient in MERA, showing that MERA tensors perform local scale transformations and recover conformal invariance.

Findings

The quotient construction is validated in the critical Ising model.

MERA tensors can emulate conformal maps through local scale transformations.

The approach enhances understanding of tensor networks' relation to holographic duality.

Abstract

In 1+1-dimensional conformal field theory, the thermal state on a circle is related to a certain quotient of the vacuum on a line. We explain how to take this quotient in the MERA tensor network representation of the vacuum and confirm the validity of the construction in the critical Ising model. This result suggests that the tensors comprising MERA can be interpreted as performing local scale transformations, so that adding or removing them emulates conformal maps. In this sense, the optimized MERA recovers local conformal invariance, which is explicitly broken by the choice of lattice. Our discussion also informs the dialogue between tensor networks and holographic duality.