Continuous Tensor Network States

Glen Evenbly, Guifré Vidal

Marco Rigobello, Erez Zohar

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.

Teresa D. Karanikolaou, Patrick Emonts, Antoine Tilloy

Tensor networks states allow one to find the low-energy states of local lattice Hamiltonians through variational optimization.Recently, a construction of such states in the continuum was put forward, providing a first step towards the goal of solving quantum field theories (QFTs) variationally.However, the proposed manifold of continuous tensor network states (CTNSs) is difficult to study in full generality, because the expectation values of local observables cannot be computed analytically.In this paper we study a tractable subclass of CTNSs, the Gaussian CTNSs (GCTNSs), and benchmark them on simple quadratic and quartic bosonic QFT Hamiltonians.We show that GCTNSs provide arbitrarily accurate approximations to the ground states of quadratic Hamiltonians and decent estimates for quartic ones at weak coupling.Since they capture the short distance behavior of the theories we consider exactly, GCTNSs even allow one to renormalize away simple divergences variationally.In the end our study makes it plausible that CTNSs are indeed a good manifold to approximate the low-energy states of QFTs.

Thibaut Lacroix

Antoine Tilloy, J. I. Cirac

We introduce a new class of states for bosonic quantum fields which extend tensor network states to the continuum and generalize continuous matrix product states to spatial dimensions d ≥ 2. By construction, they are Euclidean invariant and are genuine continuum limits of discrete tensor network states.Admitting both a functional integral and an operator representation, they share the important properties of their discrete counterparts: expressiveness, invariance under gauge transformations, simple rescaling flow, and compact expressions for the N-point functions of local observables.While we discuss mostly the continuous tensor network states extending projected entangled-pair states, we propose a generalization bearing similarities with the continuum multiscale entanglement renormalization ansatz.

Samuel Desrosiers, Glen Evenbly, Thomas E. Baker