Variational Method in Quantum Field Theory

TL;DR

This work develops a variational approach that leverages exact data from the integrable sinh–Gordon model to study the non-integrable 1+1D φ^4 theory. By minimizing the φ^4 energy on the sinh–Gordon vacuum and using finite-volume tools (TBA, LM series, and TSM with an integrable basis), the authors obtain controlled estimates of the ground-state energy, physical mass, and finite-volume spectra, including S-matrix phases below threshold, in the weak-coupling regime. The results show strong agreement with Borel-resummed perturbation theory and reveal the benefits and limitations of the variational strategy, notably that energy-optimized parameters do not universally optimize scattering data. The framework demonstrates a principled bridge between integrable and non-integrable QFTs and opens avenues for exploring critical behavior and other Z2-symmetric phases with non-perturbative control.

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.

Figures (9)

• Figure 1: Plot of vacuum expectation values of the field powers, Eq. \ref{['eq_vev_powers_shG']} and of the vacuum energy density Eq. \ref{['eq_vacuum_energy_density_shG']}. In the free limit $b=0$, all plotted quantities vanish for the chosen normal order mass. Here $\braket{:\varphi^2:}$ is monotonically increasing, while $\braket{:\varphi^4:}$ is zero at $b\approx 4.83764$. Because of the kinetic contribution, the vacuum energy density is overall monotonically decreasing.
• Figure 2: Plot of the function $b(\lambda)$ for various Hamiltonians of type Eq. \ref{['eq_truncation_shG']}, i.e $\mathcal{H}_{4}$ (blue), $\mathcal{H}_{6}$ (orange) and $\mathcal{H}_{8}$ (green). As more powers are considered, the $b(\lambda)$ curves approach Eq. \ref{['eq_g_star']} (red dashed). For $\mathcal{H}_{4}$, whose solution is given formally by Eq. \ref{['eq_formal_g_lambda']}, the agreement with the classical solution is satisfactory for $\lambda\lesssim 10$.
• Figure 3: a) Plot of the vacuum energy density obtained from Eq. \ref{['eq_vacuum_energy_phi4']} through minimization (blue) and in the leading order approximation $b=\sqrt{\lambda}$ (orange). It is also plotted the vacuum energy from perturbation theory up to $O(\lambda^8)$ Eq. \ref{['eq_vacuum_mass_phi4_perturbative']} (green) and its Borel resummation (red), obtained through the technique of Serone1. The energy obtained through the variational principle agrees with the state-of-the-art Borel resummation within $2\cdot 10^{-3}$ for $\lambda \lesssim 8$. b) Plot of the sinh-Gordon mass Eq. \ref{['eq_shG_mass_squared']} through the coupling relation Eq. \ref{['eq_formal_g_lambda']}(blue) versus the leading order approximation (orange), in the range $\lambda \leq 8$. Here too is shown the estimate from perturbation theory up to $O(\lambda^8)$ (green) and its Borel resummation (red). In the range, the mass estimated obtained through the variational principle agrees with the Borel resummation within $1\cdot 10^{-2}$.
• Figure 4: a) Ground state energy of $\mathscr H_{\varphi^4}$ with $\lambda=4.8$ on the cylinder (blue dots) computed using the combination of the TBA and the LeClair-Mussardo series. The results is compared with the results coming from TSM (see section \ref{['s_TSM']}), shown as the orange curve. Bulk energy \ref{['eq_vacuum_energy_density_shG_divergent']} is subtracted. b) The same for $\lambda=9.6$.
• Figure 5: a) Plot of the overlap $\mathscr{P}_b(g;R)$ for $R=8$, $N_c=8$, as function of the sinh-Gordon coupling $b$, varying the $\varphi^4$ coupling $g$. Each individual curve presents nontrivial minima $b_o (g;R)$. b) Plot of the minima $b_o(g;R)$ (blue dots), with numerical uncertainties, varying $g$. They line agrees with the optimal curve Eq. \ref{['eq_formal_g_lambda']} (here the gray solid curve). The deviations are due to the finite volume and we checked that increasing the volume, the curve collapses to the variational one.