Emergent geometry and path integral optimization for a Lifshitz action

TL;DR

This paper extends path integral optimization to a Lifshitz field theory, revealing optimal geometries that resemble AdS and Lifshitz metrics, and suggesting tensor networks as effective descriptions.

Contribution

It generalizes the background metric optimization procedure to Lifshitz theories, identifying optimal geometries for static and dynamic correlators.

Findings

Optimal geometry for static correlators is AdS-like.

Dynamic correlators favor Lifshitz-like geometry.

Tensor networks may serve as effective background configurations.

Abstract

Extending the background metric optimization procedure for Euclidean path integrals of two-dimensional conformal field theories, introduced by Caputa et al. (Phys. Rev. Lett. 119, 071602 (2017)), to a $z=2$ anisotropically scale-invariant $(2+1)$-dimensional Lifshitz field theory of a free massless scalar field, we find optimal geometries for static and dynamic correlation functions. For the static correlation functions, the optimal background metric is equivalent to an AdS metric on a Poincare patch, while for dynamical correlation functions, we find Lifshitz like metric. This results suggest that a MERA-like tensor network, perhaps without unitarity, would still be considered an optimal background spacetime configuration for the numerical description of this system, even though the classical action we start with is not a conformal field theory.