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

A bstract 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 vacua of free Lifshitz theories with even dynamical exponent, establishing a connection between certain entanglement properties of the two. Two schemes for approximating 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 GCTNS, identifying the scale at which deviations from its conformal short-distance behavior appear. In particular, we construct an optimal rational approximation for which, at fixed accuracy, the bond dimension grows only logarithmically with the UV cutoff. These results provide a simple and analytically controlled setting to assess the strengths and limitations of GCTNS as variational ansätze for relativistic quantum fields.