Gaussian continuous tensor network states for simple bosonic field theories

Teresa D Karanikolaou,Antoine Tilloy,Patrick Emonts

doi:10.1103/physrevresearch.3.023059

Abstract

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.Received 18 December 2020Accepted 9 March 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023059Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum field theory (low energy)TechniquesTensor network methodsVariational approachQuantum Information

Highlights

Quantum field theories (QFTs) are difficult to solve out of the perturbative regime with deterministic techniques
There is no hope to approximate it with arbitrary precision with Gaussian continuous tensor network state (CTNS) (GCTNSs), but we may still capture qualitative features
The physics of the model depends on the adimensional coupling γ = c/ρ. This model is integrable, and with the Bethe ansatz it is possible to write an exact equation for the energy density in the ground state, which can be solved numerically to essentially arbitrary precision or expanded in a power series at weak and strong coupling [33,34]

Summary

Introduction

Quantum field theories (QFTs) are difficult to solve out of the perturbative regime with deterministic techniques. Apart from lattice Monte Carlo algorithms [1,2,3,4], an option would be to solve strongly coupled QFTs variationally. In a nutshell, this would mean guessing a “good” manifold M of states |ψν described by a manageable number of parameters ν, minimize the energy ψν|H |ψν over this class M, and hope that the answer is close enough to the real ground state |0. Tensor network states (TNSs) have essentially provided what one was looking for: a sparse and extensive parametrization of many physically relevant manybody quantum states [6,7,8]. In the translation-invariant case, all tensors are identical, making parameter economy and extensivity

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Results

Conclusion