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arxiv: 2604.10284 · v1 · submitted 2026-04-11 · ✦ hep-th · cond-mat.str-el· quant-ph

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

Antoine Tilloy
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:22 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords variational methodsquantum field theorycontinuous matrix product statesnon-perturbative approximationsSine-Gordon modelphi^4 theoryRiemannian optimization
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The pith

Relativistic continuous matrix product states optimized via Riemannian methods give accurate non-perturbative results for strongly coupled 1+1D quantum field theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops variational techniques for solving strongly coupled quantum field theories in one spatial dimension plus time. It introduces relativistic continuous matrix product states as a suitable ansatz for the relativistic setting. Through Riemannian optimization on this manifold, competitive approximations to ground state energies and observables are obtained for models like phi-four and Sine-Gordon, even where standard perturbation theory breaks down. The approach is extended to multiple fields, defects, and spectral properties.

Core claim

Using Riemannian optimization on the manifold of relativistic continuous matrix product states, competitive non-perturbative approximations are obtained to ground state energies and local observables in the phi^4, Sine-Gordon, and Sinh-Gordon models in strongly coupled regimes.

What carries the argument

Riemannian optimization on the manifold of relativistic continuous matrix product states (RCMPS), a variational ansatz tailored to (1+1)-dimensional QFT.

Load-bearing premise

Finite-parameter relativistic continuous matrix product states are sufficiently expressive to approximate the true ground states of these models to the accuracy achieved.

What would settle it

A high-precision numerical computation or exact result for the ground state energy in one of the models that significantly disagrees with the variational approximation obtained from RCMPS optimization.

Figures

Figures reproduced from arXiv: 2604.10284 by Antoine Tilloy.

Figure 1.1
Figure 1.1. Figure 1.1: One of the ornaments of the Chrysler building – Norbert Nagel / Wikimedia [PITH_FULL_IMAGE:figures/full_fig_p014_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: To get closer to the Everest, we can start by climbing small hills, or by going [PITH_FULL_IMAGE:figures/full_fig_p015_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Images with random pix￾els cannot be compressed because they are generic. Images of cats can because they are not. There is a compression possible because the states we target are highly unusual, they are the low energy states of local Hamiltonians. Such states are much more locally entangled than typical states in the Hilbert space, and verify the area law. More precisely, for a low energy state of a lo… view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: Heuristic representation of entanglement between a closed subre￾gion of the lattice and the rest, for a low energy state of a local gapped Hamil￾tonian (left) and a typical random state (right). Tensor network states leverage this crucial insight from quantum information theory to target directly this corner of the Hilbert space verifying the area law. Targeting this corner with a polynomial reduction in… view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: PEPS on the left, adapted to gapped systems in 2 space dimen￾sions (or more), and MERA on the right, adapted to gapless systems in 1 space dimensions. As before, the bond in￾dices contracted over are shown in or￾ange, while physical indices are shown in blue. more, and the multiscale entanglement renormalization ansatz (MERA) [35–37], adapted to critical (gapless) systems in 1 space dimension (see [PITH… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The critical coupling 𝑓𝑐 = 4𝑔𝑐 of lattice 𝜙 4 as a function of the bare lattice coupling 𝜆, which is a proxy for the lattice spacing (𝜆 → 0 is equivalent to 𝑎 → 0). Results from [46]. proxy for the lattice spacing 𝜀, is shown in [PITH_FULL_IMAGE:figures/full_fig_p035_2_1.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Left: Energy density as a function of 𝑔, in the thermodynamic limit, compared with renormalized Hamiltonian truncation results (RHT) obtained for a size 𝐿 = 10. Right: relative error in energy density with the method explained in the text. symmetry broken phase, breaking the symmetry of the ground state does not cost energy in the thermodynamic limit compared to taking the symmetric superposition of oppo… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Spontaneous magnetization |⟨𝜙⟩| as a function of 𝑔. mode level. By choosing 𝐾, 𝑅 of the following block form 𝑅 = ( 0 𝑅1 𝑅2 0 ) and 𝐾 = ( 𝐾1 0 0 𝐾2 ) , one get the required invariance. Entanglement entropy We have defined a new entanglement entropy, which we called “free particle entanglement entropy” (for lack of a better name) in remark 6. Recall that it quantifies the amount of en￾tanglement on top of … view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Free particle entanglement entropy as a function of [PITH_FULL_IMAGE:figures/full_fig_p061_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Rescaled ground state energy density 𝜀 rq 0 /𝑏 2 of the Sine-Gordon model (con￾verted to radial quantization conventions). The dashed box on the left, corresponding to the region 𝑏 ∈ [0, 1/√ 2[ of RCMPS approximability, is magnified on the right. Note that the exact values of the energy for 𝑏 ∈]1/√ 2, 1[ do not correspond to ℎSG, but only to the resulting finite part after additional renormalization. Thi… view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Rescaled ground state energy density 𝜀 rq 0 /𝑏 2 of the Sinh-Gordon model (in radial quantization conventions). Figure from [56]. iterations (by a factor 10 to 100), and the number of iteration does seem to grow as 𝐷 is increased. This is the sign our optimizer is not capturing some of the ill-conditioning in the Hessian (the geometric intuition being insufficient), or that the ansatz itself does not cap… view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Vertex operator expectation values 𝐺(𝑎) = ⟨∶e𝑎𝜑 ∶⟩ for coupling constants 𝑏 = 0.4, 0.8, and 1.3. The dashed grey line corresponds to the FLZZ formula. Figure from [56]. consistent with this hypothesis. They also provide a rigorous energy density upper bound in this controversial region. Vertex operators Once the RCMPS approximation to the ground state has been (tediously) obtained one may directly evalua… view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Half-line entanglement in the free basis for the ground state of the Sinh-Gordon [PITH_FULL_IMAGE:figures/full_fig_p071_3_7.png] view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Left: energy density as a function of 𝐷 for 𝑔 = 𝜆 = 1 (top) and 𝑔 = 𝜆 = 2 (bottom). Right: energy density for small 𝑔 = 𝜆 compared with perturbation theory at order 𝑔 2 and 𝑔 3 . A more complex behavior, involving the two bosons in a genuine way, can be obtained when 𝑔 ≪ 𝜆. In that case the cross term ∶ 𝜙 2 1 ∶∶ 𝜙 2 2 ∶ dominates. For small enough coupling we are, again, in a fully symmetric phase. Howev… view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Minimum and maximum absolute expectation values of the fields [PITH_FULL_IMAGE:figures/full_fig_p079_3_9.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: a) original defect, interpreted as an impurity, b) equivalent rotated defect, in [PITH_FULL_IMAGE:figures/full_fig_p082_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Perturbative prediction for Γ0 ≡ ⟨0,𝑔|𝐿|0,𝑔⟩ ⟨0,0|𝐿|0,0⟩ − 1 up to (𝑔 4 ) vs. RCMPS (colored markers). The error bars are obtained from the Monte Carlo errors in the evaluation of the diagrams and are smaller than the size of the data points. Figure from [80] Remark 21 (Perturbative expansion in 𝑔 and 𝜇). From the fact that the 𝑔-expansion is exact in 𝜇, which is a consequence of working with a defect… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Bulk one-point functions from RCMPS computation as a function of the distance [PITH_FULL_IMAGE:figures/full_fig_p086_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: One-point functions obtained from RCMPS computation as a function of the [PITH_FULL_IMAGE:figures/full_fig_p087_4_4.png] view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: One-point functions from RCMPS computation as a function of the dis [PITH_FULL_IMAGE:figures/full_fig_p088_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Eigenvalue of 𝕋 with smallest real part as a function of 𝐷, compared with the mass gap estimated from renormalized Hamitonian truncation. Left, for 𝑔 = 1. Right, 𝑔 = 2. Nonetheless, we may still take 𝜆1(𝐷) as an estimate of the mass gap 𝑀, which should converge when 𝐷 → +∞. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p090_4_6.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Mass gap estimated with the bootstrap approach for [PITH_FULL_IMAGE:figures/full_fig_p093_4_7.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Coarse-graining a MPS (on the far left), the number of physical degrees of freedom (blue legs) increases while the bond dimension (orange legs) stays fixed. Coarse-graining a PEPS (on the left), both dimensions increase, meaning both need to become fields in the continuum limt. An intuition for the resulting state is shown below. Images from [93] Accepting this move from indices to fields, and taking one… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Illustration of the dimensional reduction allowed by discrete and continu￾ous tensor networks in a 𝐝 = 2 cylinder. One of the space direction of the physical theory becomes an imaginary time direction of the auxiliary (bond) relativistic field theory. In correlation functions, one gets two copies of the auxiliary theory, coupled by the connec￾tion of physical indices. Image from [93] Boundary contraction… view at source ↗
read the original 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 $\phi^4$, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces relativistic continuous matrix product states (RCMPS) as a variational ansatz for (1+1)-dimensional QFTs and applies Riemannian optimization on this manifold to compute ground-state energies and local observables for the φ⁴, Sine-Gordon, and Sinh-Gordon models, including strongly coupled regimes. It further extends the framework to multi-field models, non-local observables (defects), and spectral data extraction via a linear programming approach, while discussing limitations and future directions.

Significance. If the numerical results are substantiated, the work provides a tailored variational method for relativistic QFTs that could bridge gaps left by perturbation theory and lattice approaches in 1+1 dimensions. The use of Riemannian optimization, the linear-programming spectral extraction, and the treatment of defects represent concrete technical advances that, if controlled, would be useful for non-perturbative studies.

major comments (2)
  1. [Abstract / numerical results] Abstract and numerical-results sections: the headline claim of obtaining 'competitive non-perturbative approximations' in strongly coupled regimes is load-bearing yet unsupported by any quantitative benchmarks, error bars, or direct comparisons to independent methods or exact results; without these, it is impossible to assess whether the finite-parameter RCMPS ansatz actually captures the essential ground-state features.
  2. [RCMPS ansatz and optimization sections] Sections describing the RCMPS manifold and optimization: the central assumption that a finite bond-dimension or parameter-count RCMPS is expressive enough for the reported accuracy in the φ⁴, Sine-Gordon, and Sinh-Gordon models at strong coupling is not accompanied by convergence studies with respect to matrix size or explicit error estimates against known benchmarks; this directly affects the reliability of all subsequent claims about energies, observables, and spectra.
minor comments (2)
  1. [Introduction / ansatz definition] Notation for the RCMPS parameters and the Riemannian metric should be introduced more explicitly at first use to aid readability for readers outside the immediate subfield.
  2. [Numerical results] The manuscript would benefit from a short table summarizing the models, coupling ranges, and any available reference values used for validation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding the substantiation of our numerical claims. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract / numerical results] Abstract and numerical-results sections: the headline claim of obtaining 'competitive non-perturbative approximations' in strongly coupled regimes is load-bearing yet unsupported by any quantitative benchmarks, error bars, or direct comparisons to independent methods or exact results; without these, it is impossible to assess whether the finite-parameter RCMPS ansatz actually captures the essential ground-state features.

    Authors: We agree that quantitative benchmarks, error bars, and direct comparisons are necessary to support the headline claim. The current manuscript presents variational results for the indicated models but does not include systematic side-by-side comparisons. In the revised version we will add: (i) comparisons to weak-coupling perturbation theory for all three models, (ii) comparisons to exact results for the Sine-Gordon model (soliton masses and ground-state energy density), and (iii) comparisons to existing lattice or tensor-network data for the φ⁴ model at selected couplings. We will also report error bars obtained from the convergence of the Riemannian optimizer and from runs at neighboring bond dimensions. These additions will allow readers to evaluate the accuracy of the finite-parameter RCMPS ansatz in the strongly coupled regime. revision: yes

  2. Referee: [RCMPS ansatz and optimization sections] Sections describing the RCMPS manifold and optimization: the central assumption that a finite bond-dimension or parameter-count RCMPS is expressive enough for the reported accuracy in the φ⁴, Sine-Gordon, and Sinh-Gordon models at strong coupling is not accompanied by convergence studies with respect to matrix size or explicit error estimates against known benchmarks; this directly affects the reliability of all subsequent claims about energies, observables, and spectra.

    Authors: We acknowledge that explicit convergence studies with bond dimension and error estimates against benchmarks are required to substantiate the expressiveness of the ansatz. Although the manuscript already optimizes at several finite matrix sizes, we will expand the relevant sections with dedicated convergence tables and plots for ground-state energies and local observables as a function of bond dimension for each model. We will further include explicit error estimates obtained by comparing RCMPS results to independent benchmarks (perturbative, exact, or lattice) in regimes where such data exist, and we will propagate these uncertainties to the extracted spectral quantities obtained via the linear-programming procedure. These revisions will directly address the reliability of the reported energies, observables, and spectra. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the RCMPS variational method

full rationale

The paper describes a standard variational procedure: parameterizing trial states via the RCMPS ansatz and minimizing the energy expectation value through Riemannian optimization on the associated manifold. The reported ground-state energies and observables are outputs of this numerical minimization rather than quantities defined in terms of themselves or fitted parameters that are then relabeled as predictions. No equations or steps in the provided description reduce by construction to the inputs (e.g., no self-definitional relations where an observable is both the fit target and the reported result). Self-citations, if present for prior RCMPS development, are not invoked as load-bearing uniqueness theorems that would force the central claims; the method remains independently falsifiable via convergence with bond dimension and external benchmarks. The derivation chain is therefore self-contained as a computational approximation technique.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the variational principle plus the assumption that the RCMPS manifold is sufficiently expressive; no explicit free parameters or invented particles are named in the abstract, but the ansatz itself functions as the key new object.

free parameters (1)
  • RCMPS variational parameters (including bond dimension)
    These are optimized numerically to minimize the energy expectation value; their number and initialization are not specified in the abstract.
axioms (1)
  • domain assumption The ground state of the target QFTs can be well approximated by an RCMPS with finite parameters.
    This is the load-bearing premise that allows the variational method to succeed.
invented entities (1)
  • Relativistic continuous matrix product state (RCMPS) no independent evidence
    purpose: Variational ansatz adapted to continuous relativistic (1+1)D QFT
    Newly introduced in this line of work to fill the gap between lattice MPS and relativistic continuum theories.

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