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

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.