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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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Reference graph

Works this paper leans on

116 extracted references · 91 canonical work pages · 1 internal anchor

  1. [1]

    Interacting quantum field theories as relativistic statistical field the- ories of local beables, 2017

    Antoine Tilloy. Interacting quantum field theories as relativistic statistical field the- ories of local beables, 2017. URLhttps://arxiv.org/abs/1702.06325. 2

    work page arXiv 2017

  2. [2]

    The symmetric phase beyond NNNNNNNNLO.Journal of High Energy Physics, 2018 (8):148, Aug 2018

    Marco Serone, Gabriele Spada, and Giovanni Villadoro.𝜆𝜙4 theory — Part I. The symmetric phase beyond NNNNNNNNLO.Journal of High Energy Physics, 2018 (8):148, Aug 2018. ISSN 1029-8479. doi: 10.1007/JHEP08(2018)148. URLhttps: //doi.org/10.1007/JHEP08(2018)148. 3, 47, 69

    work page doi:10.1007/jhep08(2018)148 2018

  3. [3]

    the broken phase beyond NNNN(NNNN)LO.Journal of High Energy Physics, 2019(5):47, May 2019

    Marco Serone, Gabriele Spada, and Giovanni Villadoro.𝜆𝜙4 theory — Part II. the broken phase beyond NNNN(NNNN)LO.Journal of High Energy Physics, 2019(5):47, May 2019. ISSN 1029-8479. doi: 10.1007/JHEP05(2019)047. URLhttps://doi.org/ 10.1007/JHEP05(2019)047

    work page doi:10.1007/jhep05(2019)047 2019

  4. [4]

    Heymans and Marcus Benghi Pinto

    Gustavo O. Heymans and Marcus Benghi Pinto. Critical behavior of the 2d scalar theory: resumming the N8LO perturbative mass gap.Journal of High Energy Physics, 2021(7):163, Jul 2021. ISSN 1029-8479. doi: 10.1007/JHEP07(2021)163. URLhttps: //doi.org/10.1007/JHEP07(2021)163. 3, 47

    work page doi:10.1007/jhep07(2021)163 2021

  5. [5]

    & Hutzler, N

    Riccardo Rossi. Determinant Diagrammatic Monte Carlo Algorithm in the Thermo- dynamic Limit.Phys. Rev. Lett., 119:045701, Jul 2017. doi: 10.1103/PhysRevLett.119. 045701. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.119.045701. 3

    work page doi:10.1103/physrevlett.119 2017

  6. [6]

    Núñez Fernández, M

    Yuriel Núñez Fernández, Matthieu Jeannin, Philipp T. Dumitrescu, Thomas Kloss, Jason Kaye, Olivier Parcollet, and Xavier Waintal. Learning Feynman Diagrams with Tensor Trains.Phys. Rev. X, 12:041018, Nov 2022. doi: 10.1103/PhysRevX.12.041018. URLhttps://link.aps.org/doi/10.1103/PhysRevX.12.041018. 3

    work page doi:10.1103/physrevx.12.041018 2022

  7. [7]

    Tropical Monte Carlo quadrature for Feynman integrals.Annales de l’Institut Henri Poincaré D, 10(4):635–685, 2023

    Michael Borinsky. Tropical Monte Carlo quadrature for Feynman integrals.Annales de l’Institut Henri Poincaré D, 10(4):635–685, 2023. doi: 10.4171/AIHPD/158. 3

    work page doi:10.4171/aihpd/158 2023

  8. [8]

    Advanced Lattice QCD, 1998

    Martin Lüscher. Advanced Lattice QCD, 1998. URLhttps://arxiv.org/abs/ hep-lat/9802029. 4

    work page internal anchor Pith review arXiv 1998

  9. [9]

    Petreczky and J

    P. Petreczky and J. H. Weber. Strong coupling constant and heavy quark masses in (2+1)-flavor QCD.Phys. Rev. D, 100:034519, Aug 2019. doi: 10.1103/PhysRevD.100. 034519. URLhttps://link.aps.org/doi/10.1103/PhysRevD.100.034519. 4 113 114BIBLIOGRAPHY

    work page doi:10.1103/physrevd.100 2019

  10. [10]

    Miranda C. N. Cheng and Niki Stratikopoulou. Lecture notes on normalizing flows for lattice quantum field theories.SciPost Phys. Lect. Notes, page 110, 2026. doi: 10.21468/SciPostPhysLectNotes.110. URLhttps://scipost.org/10.21468/ SciPostPhysLectNotes.110. 4

    work page doi:10.21468/scipostphyslectnotes.110 2026

  11. [11]

    Feynman.Difficulties in Applying the Variational Principle to Quantum Field Theories, pages 28–40

    Richard P. Feynman.Difficulties in Applying the Variational Principle to Quantum Field Theories, pages 28–40. World Scientific Publishing, Singapore, 1987. doi: 10. 1142/9789814390187_0003. 8, 9

    1987

  12. [12]

    Simulating exci- tation spectra with projected entangled-pair states.Phys

    Laurens Vanderstraeten, Jutho Haegeman, and Frank Verstraete. Simulating exci- tation spectra with projected entangled-pair states.Phys. Rev. B, 99:165121, Apr

  13. [13]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevB.99.165121

    doi: 10.1103/PhysRevB.99.165121. URLhttps://link.aps.org/doi/10.1103/ PhysRevB.99.165121. 9

    work page doi:10.1103/physrevb.99.165121

  14. [14]

    Tangent-space methods for uniform matrix product states.SciPost Phys

    Laurens Vanderstraeten, Jutho Haegeman, and Frank Verstraete. Tangent-space methods for uniform matrix product states.SciPost Phys. Lect. Notes, page 7, 2019. doi: 10.21468/SciPostPhysLectNotes.7. 9, 39, 40, 74, 102

    work page doi:10.21468/scipostphyslectnotes.7 2019

  15. [15]

    Verstraete, J

    F. Verstraete, J. J. García-Ripoll, and J. I. Cirac. Matrix Product Density Opera- tors: Simulation of Finite-Temperature and Dissipative Systems.Phys. Rev. Lett., 93:207204, Nov 2004. doi: 10.1103/PhysRevLett.93.207204. URLhttps://link.aps. org/doi/10.1103/PhysRevLett.93.207204. 9

    work page doi:10.1103/physrevlett.93.207204 2004

  16. [16]

    Kshetrimayum, M

    A. Kshetrimayum, M. Rizzi, J. Eisert, and R. Orús. Tensor Network Annealing Al- gorithm for Two-Dimensional Thermal States.Phys. Rev. Lett., 122:070502, Feb

  17. [17]

    URLhttps://link.aps.org/doi/10

    doi: 10.1103/PhysRevLett.122.070502. URLhttps://link.aps.org/doi/10. 1103/PhysRevLett.122.070502. 9

    work page doi:10.1103/physrevlett.122.070502

  18. [18]

    Geometry of Variational Methods: Dynamics of Closed Quantum Systems , shorttitle =

    Lucas Hackl, Tommaso Guaita, Tao Shi, Jutho Haegeman, Eugene Demler, and J. Ignacio Cirac. Geometry of variational methods: dynamics of closed quan- tum systems.SciPost Phys., 9:48, 2020. doi: 10.21468/SciPostPhys.9.4.048. URL https://scipost.org/10.21468/SciPostPhys.9.4.048. 9

    work page doi:10.21468/scipostphys.9.4.048 2020

  19. [19]

    Non-perturbative methodologies for low-dimensional strongly- correlated systems: From non-Abelian bosonization to truncated spectrum methods

    Andrew J A James, Robert M Konik, Philippe Lecheminant, Neil J Robinson, and Alexei M Tsvelik. Non-perturbative methodologies for low-dimensional strongly- correlated systems: From non-Abelian bosonization to truncated spectrum methods. Reports on Progress in Physics, 81(4):046002, feb 2018. doi: 10.1088/1361-6633/aa91ea. 10

    work page doi:10.1088/1361-6633/aa91ea 2018

  20. [20]

    Elias-Miró, M

    J. Elias-Miró, M. Montull, and M. Riembau. The renormalized Hamiltonian trun- cation method in the large E T expansion.Journal of High Energy Physics, 2016 (4):1–34, Apr 2016. ISSN 1029-8479. doi: 10.1007/jhep04(2016)144. URLhttp: //dx.doi.org/10.1007/JHEP04(2016)144. 47

    work page doi:10.1007/jhep04(2016)144 2016

  21. [21]

    Joan Elias-Miró, Slava Rychkov, and Lorenzo G. Vitale. NLO renormalization in the Hamiltonian truncation.Phys. Rev. D, 96:065024, Sep 2017. doi: 10.1103/PhysRevD. 96.065024. URLhttps://link.aps.org/doi/10.1103/PhysRevD.96.065024. BIBLIOGRAPHY115

    work page doi:10.1103/physrevd 2017

  22. [22]

    Joan Elias-Miró, Slava Rychkov, and Lorenzo G. Vitale. High-precision calculations in strongly coupled quantum field theory with next-to-leading-order renormalized Hamiltonian Truncation.JHEP, 2017(10):213, 2017. ISSN 1029-8479. doi: 10.1007/ JHEP10(2017)213. URLhttps://doi.org/10.1007/JHEP10(2017)213. 82, 85

    work page doi:10.1007/jhep10(2017)213 2017

  23. [23]

    Slava Rychkov and Lorenzo G. Vitale. Hamiltonian truncation study of the𝜑4 theory in two dimensions.Phys. Rev. D, 91:085011, Apr 2015. doi: 10.1103/PhysRevD.91. 085011. 10, 14, 47, 50

    work page doi:10.1103/physrevd.91 2015

  24. [24]

    Carleo and M

    Giuseppe Carleo and Matthias Troyer. Solving the quantum many-body problem with artificial neural networks.Science, 355(6325):602–606, 2017. ISSN 0036-8075. doi: 10.1126/science.aag2302. 10

    work page doi:10.1126/science.aag2302 2017

  25. [25]

    Fermionic neural-network states for ab-initio electronic structure.Nature communications, 11(1):1–7, 2020

    Kenny Choo, Antonio Mezzacapo, and Giuseppe Carleo. Fermionic neural-network states for ab-initio electronic structure.Nature communications, 11(1):1–7, 2020. 10

    2020

  26. [27]

    M. M. Wolf, F. sredVerstraete, M. B. Hastings, and J. I. Cirac. Area Laws in Quantum Systems: Mutual Information and Correlations.Phys. Rev. Lett., 100:070502, Feb

  27. [28]

    URLhttps://link.aps.org/doi/10

    doi: 10.1103/PhysRevLett.100.070502. URLhttps://link.aps.org/doi/10. 1103/PhysRevLett.100.070502

    work page doi:10.1103/physrevlett.100.070502

  28. [29]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio. Colloquium: Area laws for the entanglement entropy.Rev. Mod. Phys., 82:277–306, Feb 2010. doi: 10.1103/RevModPhys.82.277. URLhttps://link.aps.org/doi/10.1103/RevModPhys.82.277. 11

    work page doi:10.1103/revmodphys.82.277 2010

  29. [30]

    Don N. Page. Average entropy of a subsystem.Phys. Rev. Lett., 71:1291–1294, Aug

  30. [31]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.71.1291

    doi: 10.1103/PhysRevLett.71.1291. URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.71.1291. 11

    work page doi:10.1103/physrevlett.71.1291

  31. [32]

    Entangle- ment typicality.Journal of Physics A: Mathematical and Theoretical, 47(36):363001, aug 2014

    Oscar C O Dahlsten, Cosmo Lupo, Stefano Mancini, and Alessio Serafini. Entangle- ment typicality.Journal of Physics A: Mathematical and Theoretical, 47(36):363001, aug 2014. doi: 10.1088/1751-8113/47/36/363001. 11

    work page doi:10.1088/1751-8113/47/36/363001 2014

  32. [33]

    Journal of Physics A: Mathematical and Theoretical , abstract =

    Jacob C Bridgeman and Christopher T Chubb. Hand-waving and interpretive dance: an introductory course on tensor networks.Journal of Physics A: Mathematical and Theoretical, 50(22):223001, may 2017. doi: 10.1088/1751-8121/aa6dc3. 11

    work page doi:10.1088/1751-8121/aa6dc3 2017

  33. [34]

    Fannes , author B

    M. Fannes, B. Nachtergaele, and R. F. Werner. Finitely correlated states on quantum spin chains.Commun. Math. Phys., 144(3):443–490, Mar 1992. ISSN 1432-0916. doi: 10.1007/BF02099178. URLhttps://doi.org/10.1007/BF02099178. 11 116BIBLIOGRAPHY

    work page doi:10.1007/bf02099178 1992

  34. [36]

    Computing energy density in one dimension, 2015

    Yichen Huang. Computing energy density in one dimension, 2015. 12

    2015

  35. [37]

    S. R. White. Density-matrix algorithms for quantum renormalization groups.Phys. Rev. B, 48:10345–10356, Oct 1993. doi: 10.1103/PhysRevB.48.10345. URLhttps: //link.aps.org/doi/10.1103/PhysRevB.48.10345. 12

    work page doi:10.1103/physrevb.48.10345 1993

  36. [38]

    Verstraete and J

    F. Verstraete and J. I. Cirac. Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions, 2004. 12

    2004

  37. [39]

    G. Vidal. Entanglement Renormalization.Phys. Rev. Lett., 99:220405, Nov 2007. doi: 10.1103/PhysRevLett.99.220405. URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.99.220405. 13

    work page doi:10.1103/physrevlett.99.220405 2007

  38. [40]

    Montangero, M

    S. Montangero, M. Rizzi, V. Giovannetti, and Rosario Fazio. Critical exponents with a multiscale entanglement renormalization Ansatz channel.Phys. Rev. B, 80:113103, Sep 2009. doi: 10.1103/PhysRevB.80.113103. URLhttps://link.aps.org/doi/10. 1103/PhysRevB.80.113103

    work page doi:10.1103/physrevb.80.113103 2009

  39. [41]

    Evenbly and G

    G. Evenbly and G. Vidal. Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz. In A. Avella and F. Mancini, editors,Strongly Correlated Systems. Numerical Methods, chapter 4, pages 99–130. Springer-Verlag, Berlin DE,

  40. [42]

    doi: 10.1007/978-3-642-35106-8

    ISBN 978-3-642-35105-1. doi: 10.1007/978-3-642-35106-8. 13

    work page doi:10.1007/978-3-642-35106-8

  41. [43]

    Contracting projected entangled pair states is average-case hard.Phys

    Jonas Haferkamp, Dominik Hangleiter, Jens Eisert, and Marek Gluza. Contracting projected entangled pair states is average-case hard.Phys. Rev. Research, 2:013010, Jan 2020. doi: 10.1103/PhysRevResearch.2.013010. URLhttps://link.aps.org/doi/ 10.1103/PhysRevResearch.2.013010. 13

    work page doi:10.1103/physrevresearch.2.013010 2020

  42. [44]

    Springer International Publishing, Cham, 2020

    Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo, Gang Su, and Maciej Lewenstein.Two-Dimensional Tensor Networks and Contraction Algo- rithms, pages 63–86. Springer International Publishing, Cham, 2020. ISBN 978-3- 030-34489-4. doi: 10.1007/978-3-030-34489-4_3. URLhttps://doi.org/10.1007/ 978-3-030-34489-4_3. 13

    work page doi:10.1007/978-3-030-34489-4_3 2020

  43. [45]

    E. D. Brooks and S. C. Frautschi. Scalars coupled to fermions in 1+1 dimensions. Zeitschrift für Physik C Particles and Fields, 23(3):263–273, Sep 1984. ISSN 1431-5858. doi: 10.1007/BF01546194. URLhttps://doi.org/10.1007/BF01546194. 13

    work page doi:10.1007/bf01546194 1984

  44. [46]

    V. P. YUROV and AL. B. ZAMOLODCHIKOV. TRUNCATED COMFORMAL SPACE APPROACH TO SCALING LEE-YANG MODEL.International Journal of Modern Physics A, 05(16):3221–3245, 1990. doi: 10.1142/S0217751X9000218X. URLhttps: //doi.org/10.1142/S0217751X9000218X. 13 BIBLIOGRAPHY117

    work page doi:10.1142/s0217751x9000218x 1990

  45. [47]

    Verstraete and J

    F. Verstraete and J. I. Cirac. Continuous Matrix Product States for Quantum Fields. Phys. Rev. Lett., 104:190405, May 2010. doi: 10.1103/PhysRevLett.104.190405. URL https://link.aps.org/doi/10.1103/PhysRevLett.104.190405. 17, 87

    work page doi:10.1103/physrevlett.104.190405 2010

  46. [48]

    Yarykin, J

    Jutho Haegeman, J. Ignacio Cirac, Tobias J. Osborne, and Frank Verstraete. Calculus of continuous matrix product states.Phys. Rev. B, 88:085118, Aug 2013. doi: 10. 1103/PhysRevB.88.085118. URLhttps://link.aps.org/doi/10.1103/PhysRevB.88. 085118. 17, 30, 65, 94

    work page doi:10.1103/physrevb.88 2013

  47. [50]

    Ashley Milsted, Jutho Haegeman, and Tobias J. Osborne. Matrix product states and variational methods applied to critical quantum field theory.Phys. Rev. D, 88:085030, Oct 2013. doi: 10.1103/PhysRevD.88.085030. URLhttps://link.aps.org/doi/10. 1103/PhysRevD.88.085030. 25, 47

    work page doi:10.1103/physrevd.88.085030 2013

  48. [51]

    Computing the renormalization group flow of two-dimensional𝜙4 theory with tensor networks.Phys

    Clement Delcamp and Antoine Tilloy. Computing the renormalization group flow of two-dimensional𝜙4 theory with tensor networks.Phys. Rev. Res., 2:033278, Aug

  49. [52]

    URLhttps://link.aps.org/doi/10

    doi: 10.1103/PhysRevResearch.2.033278. URLhttps://link.aps.org/doi/10. 1103/PhysRevResearch.2.033278. 25, 26, 27, 47, 78

    work page doi:10.1103/physrevresearch.2.033278

  50. [53]

    Will Loinaz and R. S. Willey. Monte Carlo simulation calculation of the critical cou- pling constant for two-dimensional continuum𝜑4 theory.Phys. Rev. D, 58:076003, Sep 1998. doi: 10.1103/PhysRevD.58.076003. 26, 47

    work page doi:10.1103/physrevd.58.076003 1998

  51. [54]

    Renormalization of tensor networks using graph-independent local truncations.Phys

    Markus Hauru, Clement Delcamp, and Sebastian Mizera. Renormalization of tensor networks using graph-independent local truncations.Phys. Rev. B, 97:045111, Jan

  52. [55]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevB.97.045111

    doi: 10.1103/PhysRevB.97.045111. URLhttps://link.aps.org/doi/10.1103/ PhysRevB.97.045111. 26

    work page doi:10.1103/physrevb.97.045111

  53. [56]

    Aspects of Two-Dimensional Conformal Field Theories.PoS, TASI2017:003,

    Xi Yin. Aspects of Two-Dimensional Conformal Field Theories.PoS, TASI2017:003,

  54. [57]

    doi: 10.22323/1.305.0003. 27

    work page doi:10.22323/1.305.0003

  55. [58]

    Vid Stojevic, Jutho Haegeman, I. P. McCulloch, Luca Tagliacozzo, and Frank Ver- straete. Conformal data from finite entanglement scaling.Phys. Rev. B, 91:035120, Jan 2015. doi: 10.1103/PhysRevB.91.035120. URLhttps://link.aps.org/doi/10. 1103/PhysRevB.91.035120. 29

    work page doi:10.1103/physrevb.91.035120 2015

  56. [59]

    Schmoll, J

    Philipp Schmoll, Jan Naumann, Alexander Nietner, Jens Eisert, and Spyros Sotiri- adis. Hamiltonian truncation tensor networks for quantum field theories, 2023. URL https://arxiv.org/abs/2312.12506. 30 118BIBLIOGRAPHY

    work page arXiv 2023

  57. [61]

    Fishman, J

    Ang-Kun Wu, Benedikt Kloss, Wladislaw Krinitsin, Matthew T. Fishman, J. H. Pixley, and E. M. Stoudenmire. Disentangling interacting systems with fermionic Gaussian circuits: Application to quantum impurity models.Phys. Rev. B, 111:035119, Jan

  58. [62]

    URLhttps://link.aps.org/doi/10.1103/ PhysRevB.111.035119

    doi: 10.1103/PhysRevB.111.035119. URLhttps://link.aps.org/doi/10.1103/ PhysRevB.111.035119. 31

    work page doi:10.1103/physrevb.111.035119

  59. [63]

    Variational method in relativistic quantum field theory without cut- off.Phys

    Antoine Tilloy. Variational method in relativistic quantum field theory without cut- off.Phys. Rev. D, 104:L091904, Nov 2021. doi: 10.1103/PhysRevD.104.L091904. URL https://link.aps.org/doi/10.1103/PhysRevD.104.L091904. 32, 63

    work page doi:10.1103/physrevd.104.l091904 2021

  60. [64]

    Relativistic continuous matrix product states for quantum fields without cutoff.Phys

    Antoine Tilloy. Relativistic continuous matrix product states for quantum fields without cutoff.Phys. Rev. D, 104:096007, Nov 2021. doi: 10.1103/PhysRevD.104. 096007. URLhttps://link.aps.org/doi/10.1103/PhysRevD.104.096007. 32, 50, 63

    work page doi:10.1103/physrevd.104 2021

  61. [65]

    A study of the quantum Sinh-Gordon model with relativistic con- tinuous matrix product states, 2022

    Antoine Tilloy. A study of the quantum Sinh-Gordon model with relativistic con- tinuous matrix product states, 2022. URLhttps://arxiv.org/abs/2209.05341. 32, 42, 54, 59, 60, 61, 63

    work page arXiv 2022

  62. [66]

    On Automatic Differentiation

    Andreas Griewank. On Automatic Differentiation. In Masao Iri and Kunio Tanabe, editors,Mathematical Programming: Recent Developments and Applications, pages 83–108. Kluwer Academic Publishers, Dordrecht, 1989. URLhttps://softlib.rice. edu/pub/CRPC-TRs/reports/CRPC-TR89003.pdf. 40

    1989

  63. [67]

    In: Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pp

    Andreas Griewank and Andrea Walther.Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, 2 edition, 2008. doi: 10.1137/1. 9780898717761. 40

    work page doi:10.1137/1 2008

  64. [68]

    The double-exponential transformation in numerical analysis.Journal of Computational and Applied Mathematics, 127(1):287– 296, 2001

    Masatake Mori and Masaaki Sugihara. The double-exponential transformation in numerical analysis.Journal of Computational and Applied Mathematics, 127(1):287– 296, 2001. ISSN 0377-0427. doi: https://doi.org/10.1016/S0377-0427(00)00501-X. URL https://www.sciencedirect.com/science/article/pii/S037704270000501X. Nu- merical Analysis 2000. Vol. V: Quadrature a...

    work page doi:10.1016/s0377-0427(00)00501-x 2001

  65. [69]

    Riemannian opti- mization of isometric tensor networks.SciPost Phys., 10:40, 2021

    Markus Hauru, Maarten Van Damme, and Jutho Haegeman. Riemannian opti- mization of isometric tensor networks.SciPost Phys., 10:40, 2021. doi: 10.21468/ SciPostPhys.10.2.040. URLhttps://scipost.org/10.21468/SciPostPhys.10.2.040. 42, 43

    work page doi:10.21468/scipostphys.10.2.040 2021

  66. [70]

    Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia.Journal of Open Research Software, 5(1), 2017

    Christopher Rackauckas and Qing Nie. Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia.Journal of Open Research Software, 5(1), 2017. 43 BIBLIOGRAPHY119

    2017

  67. [71]

    Mathematical theory of ele- mentary particles

    E Nelson. A quartic interaction in two dimensions, in" Mathematical theory of ele- mentary particles", eds. R. Goodman and I. Segal, 1966. 46

    1966

  68. [72]

    Ed Nelson’s work in quantum theory.Diffusion, Quantum Theory, and Radically Elementary Mathematics (WG Faris, ed.), 47:75–93, 2004

    Barry Simon. Ed Nelson’s work in quantum theory.Diffusion, Quantum Theory, and Radically Elementary Mathematics (WG Faris, ed.), 47:75–93, 2004. doi: 10.1515/ 9781400865253.75. 46

    2004

  69. [73]

    Partially Alternate Derivation of a Result of Nelson.Journal of Mathematical Physics, 10(1):50–52, 01 1969

    Paul Federbush. Partially Alternate Derivation of a Result of Nelson.Journal of Mathematical Physics, 10(1):50–52, 01 1969. ISSN 0022-2488. doi: 10.1063/1.1664760. URLhttps://doi.org/10.1063/1.1664760. 46

    work page doi:10.1063/1.1664760 1969

  70. [74]

    van Rees

    Matthijs Hogervorst, Slava Rychkov, and Balt C. van Rees. Truncated conformal space approach in𝑑dimensions: A cheap alternative to lattice field theory?Phys. Rev. D, 91:025005, Jan 2015. doi: 10.1103/PhysRevD.91.025005. URLhttps://link. aps.org/doi/10.1103/PhysRevD.91.025005. 47

    work page doi:10.1103/physrevd.91.025005 2015

  71. [75]

    Entanglement scaling for𝜆𝜙4 2.Phys

    Bram Vanhecke, Frank Verstraete, and Karel Van Acoleyen. Entanglement scaling for𝜆𝜙4 2.Phys. Rev. D, 106:L071501, Oct 2022. doi: 10.1103/PhysRevD.106.L071501. URLhttps://link.aps.org/doi/10.1103/PhysRevD.106.L071501. 47, 49, 78

    work page doi:10.1103/physrevd.106.l071501 2022

  72. [76]

    Liam Fitzpatrick, Emanuel Katz, Zuhair U

    Nikhil Anand, A. Liam Fitzpatrick, Emanuel Katz, Zuhair U. Khandker, Matthew T. Walters, and Yuan Xin. Introduction to Lightcone Conformal Truncation: QFT Dy- namics from CFT Data, 2020. URLhttps://arxiv.org/abs/2005.13544. 47

    work page arXiv 2020

  73. [77]

    Liam Fitzpatrick, Emanuel Katz, and Matthew T

    A. Liam Fitzpatrick, Emanuel Katz, and Matthew T. Walters. Nonperturbative matching between equal-time and lightcone quantization.Journal of High Energy Physics, 2020(10):92, Oct 2020. ISSN 1029-8479. doi: 10.1007/JHEP10(2020)092. URL https://doi.org/10.1007/JHEP10(2020)092. 48

    work page doi:10.1007/jhep10(2020)092 2020

  74. [78]

    Computing energy density in one dimension, 2015

    Yichen Huang. Computing energy density in one dimension, 2015. URLhttps: //arxiv.org/abs/1505.00772. 50

    work page arXiv 2015

  75. [79]

    Approaching the self- dual point of the sinh-Gordon model.JHEP, 2021(1):1–85, 2021

    Robert Konik, Márton Lájer, and Giuseppe Mussardo. Approaching the self- dual point of the sinh-Gordon model.JHEP, 2021(1):1–85, 2021. doi: 10.1007/ JHEP01(2021)014. 54, 55, 56, 57, 58

    2021

  76. [80]

    The sinh-Gordon model beyond the self dual point and the freezing transition in disordered systems.Journal of High Energy Physics, 2022(5):22, May 2022

    Denis Bernard and André LeClair. The sinh-Gordon model beyond the self dual point and the freezing transition in disordered systems.Journal of High Energy Physics, 2022(5):22, May 2022. ISSN 1029-8479. doi: 10.1007/JHEP05(2022)022. URLhttps: //doi.org/10.1007/JHEP05(2022)022. 54

    work page doi:10.1007/jhep05(2022)022 2022

  77. [81]

    Bl´ azquez-Salcedo, C

    Jürg Fröhlich. Quantized "Sine-Gordon" Equation with a Nonvanishing Mass Term in Two Space-Time Dimensions.Phys. Rev. Lett., 34:833–836, Mar 1975. doi: 10. 1103/PhysRevLett.34.833. URLhttps://link.aps.org/doi/10.1103/PhysRevLett. 34.833. 56 120BIBLIOGRAPHY

    work page doi:10.1103/physrevlett 1975

  78. [82]

    Remarks on exponential interactions and the quantum sine-Gordon equation in two space-time dimensions.Helvetica Physica Acta, 50(3):315–329, 1977

    Jürg Fröhlich and Yong Moon Park. Remarks on exponential interactions and the quantum sine-Gordon equation in two space-time dimensions.Helvetica Physica Acta, 50(3):315–329, 1977. URLhttps://www.e-periodica.ch/cntmng?pid=hpa-001% 3A1977%3A50%3A%3A982. 56

    1977

  79. [83]

    Construction of the two-dimensional Sine-Gordon model for𝛽 <8𝜋.Communications in mathematical physics, 156(3):547–580, 1993

    J Dimock and TR Hurd. Construction of the two-dimensional Sine-Gordon model for𝛽 <8𝜋.Communications in mathematical physics, 156(3):547–580, 1993. doi: 10.1007/BF02096863. 56

    work page doi:10.1007/bf02096863 1993

  80. [84]

    Quantum sine-Gordon equation as the massive Thirring model

    Sidney Coleman. Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev. D, 11:2088–2097, Apr 1975. doi: 10.1103/PhysRevD.11.2088. URLhttps: //link.aps.org/doi/10.1103/PhysRevD.11.2088. 57

    work page doi:10.1103/physrevd.11.2088 2088

Showing first 80 references.