  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.
