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arXiv:2604.10284

Some progress on the use of the variational method in quantum field theory

Antoine Tilloy

hep-thcond-mat.str-elquant-phSubmitted Apr 11, 2026• Announced Apr 14, 2026• 0 citations
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Abstract

Strongly coupled quantum field theories in $(1+1)$ dimensions are notoriously hard to solve non-perturbatively. Variational methods, despite their success for quantum many-body physics on the lattice, have long lacked a natural ansatz adapted to the relativistic setting. This monograph explains the intuition behind relativistic continuous matrix product states (RCMPS), a variational ansatz tailored to $(1+1)$-dimensional QFT, and reports on several years of progress in developing and applying this approach. Using Riemannian optimization on the manifold of RCMPS, we obtain competitive non-perturbative approximations to ground state energies and local observables in the $φ^4$, Sine-Gordon, and Sinh-Gordon models, including in strongly coupled regimes where perturbation theory fails. We then describe extensions to models with several interacting fields. Beyond energy density and local observables, we show how the framework can be used to evaluate non-local observables (defects) and, through an original linear programming approach, to extract spectral data such as particle masses. We close by discussing the current limitations of the method and the most promising directions for future work.

Comments
This monograph reviews the recent developments on relativistic continuous MPS previously presented in 2512.19594, 2511.20762, 2501.09797, 2209.05341, 2102.07733, 2102.07741. It also adds some hopefully pedagogical and personal comments

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A study of the quantum Sinh-Gordon model with relativistic continuous matrix product states

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I study the Sine-Gordon (SG) and Sinh-Gordon (ShG) quantum field theories with a recently introduced variational method, the relativistic continuous matrix product states (RCMPS). The main advantage is to work directly in the thermodynamic limit, and without any UV regulator. The SG model is well understood and integrable, which provides a convenient benchmark for the variational method and serves as a warm-up. RCMPS approximate the ground state of the SG model arbitrary well up to the free Fermion point [coupling $β=\sqrt{4π}$ in equal-time quantization convention, or $b=1/\sqrt{2}$ in CFT convention], where the ground energy collapses to $-\infty$, and some renormalized ansatz would be needed. The ShG model, while integrable, is less understood and its strong coupling regime $β\approx 1$ is subject to some controversy. RCMPS also fit the ground state of the ShG model up to approximately $b=1/\sqrt{2}$, after which their predictions start to deviate substantially from the "exact" results. This is more puzzling as nothing is expected to happen physically for the ShG model at that point (eg, the ground energy density does not diverge). Either the "exact" ShG results are not exact (the analytic continuation of the SG Bethe Ansatz solution is unwarranted), or, more likely, the physical structure of the ShG ground state changes in such a way that it becomes out of reach of the RCMPS manifold for reasonable bond dimensions.

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Relativistic continuous matrix product states for quantum fields without cutoff

Antoine Tilloy

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 one 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, also known as the ${\ensuremath{\phi}}_{2}^{4}$ model. Aside from providing unequaled precision in the continuum, the numerical results obtained are truly variational, and thus provide rigorous energy upper bounds.

arXiv:2102.07741•Physical review. D/Physical review. D.•Nov 11, 2021•16 citations
Strongly coupled quantum field theories in (1+1)(1+1)(1+1) dimensions are notoriously hard to solve non-perturbatively. Variational methods, despite their success for quantum many-body physics on the lattice, have long lacked a natural ansatz adapted to the relativistic setting. This monograph explains the intuition behind relativistic continuous matrix product states (RCMPS), a variational ansatz tailored to (1+1)(1+1)(1+1)-dimensional QFT, and reports on several years of progress in developing and applying this approach. Using Riemannian optimization on the manifold of RCMPS, we obtain competitive non-perturbative approximations to ground state energies and local observables in the φ4φ^4φ4, Sine-Gordon, and Sinh-Gordon models, including in strongly coupled regimes where perturbation theory fails. We then describe extensions to models with several interacting fields. Beyond energy density and local observables, we show how the framework can be used to evaluate non-local observables (defects) and, through an original linear programming approach, to extract spectral data such as particle masses. We close by discussing the current limitations of the method and the most promising directions for future work.
I study the Sine-Gordon (SG) and Sinh-Gordon (ShG) quantum field theories with a recently introduced variational method, the relativistic continuous matrix product states (RCMPS). The main advantage is to work directly in the thermodynamic limit, and without any UV regulator. The SG model is well understood and integrable, which provides a convenient benchmark for the variational method and serves as a warm-up. RCMPS approximate the ground state of the SG model arbitrary well up to the free Fermion point [coupling β=4πβ=\sqrt{4π}β=4π​ in equal-time quantization convention, or b=1/2b=1/\sqrt{2}b=1/2​ in CFT convention], where the ground energy collapses to −∞-\infty−∞, and some renormalized ansatz would be needed. The ShG model, while integrable, is less understood and its strong coupling regime β≈1β\approx 1β≈1 is subject to some controversy. RCMPS also fit the ground state of the ShG model up to approximately b=1/2b=1/\sqrt{2}b=1/2​, after which their predictions start to deviate substantially from the "exact" results. This is more puzzling as nothing is expected to happen physically for the ShG model at that point (eg, the ground energy density does not diverge). Either the "exact" ShG results are not exact (the analytic continuation of the SG Bethe Ansatz solution is unwarranted), or, more likely, the physical structure of the ShG ground state changes in such a way that it becomes out of reach of the RCMPS manifold for reasonable bond dimensions.
Relativistic continuous matrix product states (RCMPS) are a powerful variational ansatz for quantum field theories of a single field. However, they inherit a property of their non-relativistic counterpart that makes them divergent for models with multiple fields, unless a regularity condition is satisfied. This has so far restricted the use of RCMPS to toy models with a single self-interacting field. We address this long standing problem by introducing a Riemannian optimization framework, that allows to minimize the energy density over the regular submanifold of multi-field RCMPS, and thus to retain purely variational results. We demonstrate its power on a model of two interacting scalar fields in 1+11+11+1 dimensions. The method captures distinct symmetry-breaking phases, and the signature of a Berezinskii-Kosterlitz-Thouless (BKT) transition along an O(2)O(2)O(2)-symmetric parameter line. This makes RCMPS usable for a far larger class of problems than before.
<!--HTML-->The Sinh-Gordon model is a 1+1 dimensional quantum field theory with a potential cosh(b phi) that is quite peculiar. It is at the same time exactly solvable (for many observables) and not well understood. I will present the results of a variational exploration of its strong coupling regime with a recent generalization of continuous matrix product states. The advantage of this method is that it does not require introducing a cutoff, UV or IR, is fully non-perturbative, typically converges fast for Hamiltonians with polynomial interactions, and gives rigorous energy upper bounds. Its application to the Sinh-Gordon model is only partly successful: observables can be computed accurately up to fairly large values of the coupling, but the ultra strong coupling regime remains difficult to access without extrapolations. As a result, the behavior near the self-dual point is not yet fully settled. I will show the basics of relativistic continuous matrix product states, explain how they typically work for phi^4 like theories or even the easier Sine-Gordon model and finally discuss my attempts at taming Sinh-Gordon.
The Relativistic Continuous Matrix Product State (RCMPS) ansatz introduced in arXiv:2102.07741 is a translation-invariant variational ansatz intended to approximate the ground states of gapped one-dimensional relativistic Quantum Field Theories directly in the continuum limit and in infinite volume. In this work, we develop an ansatz, referred to as the Relativistic Quasiparticle (RQP) ansatz, for the low-lying one-particle excitations of the RCMPS vacuum, as well as for solitons interpolating between two distinct degenerate vacua. The corresponding energies of these states can be obtained as the eigenvalues of an effective Hamiltonian. The RQP ansatz generalizes constructions used in the Matrix Product State and Continuous Matrix Product State literature. We apply the RQP ansatz to the sine-Gordon model and find good agreement with results from integrability for both the soliton and breather masses. We also apply the RQP ansatz to the ϕ4ϕ^4ϕ4 theory, in both its Z2\mathbb Z_2Z2​-preserving and Z2\mathbb{Z}_2Z2​-broken phases, and determine the spectrum of particles and solitons. This enables us to determine the parameter regime for which multiple stable particles exist.
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 φ4\varphi^4φ4 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 φ4\varphi^4φ4 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.
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+11+11+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 one 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, also known as the ${\ensuremath{\phi}}_{2}^{4}$ model. Aside from providing unequaled precision in the continuum, the numerical results obtained are truly variational, and thus provide rigorous energy upper bounds.