Variational Method in Quantum Field Theory

TL;DR

This paper introduces a variational approach leveraging integrable model structures to estimate physical quantities in non-integrable quantum field theories, demonstrated on the $^4$ Landau-Ginzburg model.

Contribution

It develops a novel variational framework based on integrable models to analyze non-integrable quantum field theories, combining analytical and numerical techniques.

Findings

Controlled estimates of ground-state energy and mass as functions of coupling.

Numerical analysis of finite-volume spectrum and scattering matrix.

Validation of variational methods against Hamiltonian truncation results.

Abstract

We develop a variational framework for addressing two-dimensional non-integrable quantum field theories through the exact structure of their integrable counterparts. Concentrating on the $\varphi^4$ Landau-Ginzburg model, we use the analytical Vacuum Expectation Values and Form Factors of local operators in the sinh-Gordon theory as the foundation of a variational ansatz. In this way, we obtain controlled estimates of central physical quantities of the $\varphi^4$ theory - such as the finite-volume ground-state energy and the physical mass as a function of the coupling constant. The strengths of the variational methods are leveraged in combination with the Hamiltonian truncation techniques and the LeClair-Mussardo formula, which also allow to probe the accuracy of the variational approximation varying the system size. Within the weak-coupling regime, a detailed numerical analysis reveals the behaviour of the finite-volume spectrum, the ground-state energy, and the elastic part of the scattering matrix, showing how the rigorous machinery of integrable models can serve as a guiding light into the complex landscape of non-integrable quantum field dynamics.