Relativistic continuous matrix product states for quantum fields without cutoff

TL;DR

This paper introduces relativistic continuous matrix product states (CMPS) tailored for 1+1 dimensional quantum field theories, enabling cutoff-free, precise, variational solutions directly in the thermodynamic limit.

Contribution

The work develops a modified CMPS framework adapted to relativistic QFTs, allowing for accurate, cutoff-free calculations in the continuum without additional UV regularization.

Findings

Successfully applied to the $_2$ scalar field model

Provides rigorous energy upper bounds with high precision

Maintains computational feasibility with standard CMPS cost complexity

Abstract

I introduce a modification of continuous matrix product states (CMPS) that makes them adapted to relativistic quantum field theories (QFT). These relativistic CMPS can be used to solve genuine 1+1 dimensional QFT without UV cutoff and directly in the thermodynamic limit. The main idea is to work directly in the basis that diagonalizes the free part of the model considered, which allows to fit its short distance behavior exactly. This makes computations slightly less trivial than with standard CMPS. However, they remain feasible and I present all the steps needed for the optimization. The asymptotic cost as a function of the bond dimension remains the same as for standard CMPS. I illustrate the method on the self-interacting scalar field, a.k.a. the $\phi^4_2$ model. Aside from providing unequaled precision in the continuum, the numerical results obtained are truly variational, and thus provide rigorous energy upper bounds.