Gaussian continuous tensor network states: short-distance properties and imaginary-time evolution

TL;DR

This paper introduces Gaussian continuous tensor network states (GCTNS) as a continuum class of states for quantum fields, analyzes their properties, and proposes methods to approximate ground states of free bosonic theories, highlighting their strengths and limitations.

Contribution

It establishes GCTNS as continuum limits of discrete tensor networks, connects them to free Lifshitz vacua, and develops approximation schemes for bosonic field ground states.

Findings

GCTNS correspond to free Lifshitz vacua at short distances.

Two approximation schemes for bosonic ground states are proposed.

Identifies energy scales where GCTNS approximations deviate from target theories.

Abstract

We study Gaussian continuous tensor network states (GCTNS) - a finitely-parameterized subclass of Gaussian states admitting an interpretation as continuum limits of discrete tensor network states. We show that, at short distance, GCTNS correspond to free Lifshitz vacua, establishing a connection between certain entanglement properties of the two. Two schemes to approximate ground states of (free) bosonic field theories using GCTNS are presented: rational approximants to the exact dispersion relation and Trotterized imaginary-time evolution. We apply them to Klein-Gordon theory and characterize the resulting approximations, identifying the energy scales at which deviations from the target theory appear. These results provide a simple and analytically controlled setting to assess the strengths and limitations of GCTNS as variational ans\"atze for relativistic quantum fields.