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arxiv: 2606.20029 · v1 · pith:3RSQUY7Znew · submitted 2026-06-18 · ✦ hep-lat · quant-ph

A Finite-Volume Scheme for the Continuum Extrapolation of Lattice Step-Scaling in (2+1)D Hamiltonian U(1) Gauge Theory

Alessio Negro , Emil Otis Rosanowski , Lena Funcke , Timo Jakobs , Karl Jansen , Paul Ludwig , Carsten Urbach This is my paper

Pith reviewed 2026-06-26 15:12 UTC · model grok-4.3

classification ✦ hep-lat quant-ph
keywords finite-volume schemestep-scaling functionHamiltonian lattice gauge theoryU(1) gauge theorycontinuum extrapolationmatrix product statesstatic potentialrenormalized coupling
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The pith

A finite-volume scheme enables controlled continuum extrapolations of the step-scaling function in Hamiltonian U(1) gauge theory.

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

The paper proposes a finite-volume scheme to achieve controlled continuum extrapolations of the lattice step-scaling function, a quantity needed to define the running coupling in small-volume Hamiltonian lattice gauge theories. In a dual Hamiltonian formulation of pure U(1) gauge theory in (2+1) dimensions, the authors use matrix product states to place static external charges, study the resulting confining string, and extract a force-based renormalized coupling. They demonstrate that the scheme produces a stable continuum limit for the step-scaling function on the lattice sizes reachable by current Hamiltonian simulations. The method is presented as directly extendable to other gauge groups and dimensions.

Core claim

The authors claim that their finite-volume scheme, applied to the dual Hamiltonian formulation of (2+1)D pure U(1) gauge theory with matrix-product-state computations of the confining string, produces a stable continuum limit of the step-scaling function on presently accessible lattice sizes, thereby providing a controlled route to the running coupling via a force-based definition.

What carries the argument

The finite-volume scheme for continuum extrapolation of the step-scaling function, which organizes volume and coupling data to isolate the continuum limit while the confining string supplies the force used to define the renormalized coupling.

If this is right

  • The running coupling can be determined in Hamiltonian lattice gauge theories using step-scaling on accessible volumes.
  • The scheme remains stable for the lattice sizes currently reachable by Hamiltonian simulations with matrix product states.
  • Static external charges can be implemented and the confining string studied directly in the dual formulation.
  • The approach extends to other gauge groups and spacetime dimensions without fundamental changes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar finite-volume control could be tested in non-Abelian theories once efficient Hamiltonian formulations become available.
  • The method may reduce the volume range needed to reach the continuum compared with traditional Euclidean step-scaling analyses.
  • Force-based definitions extracted from the string tension offer an alternative route to renormalization that avoids direct use of Wilson loops at large separations.

Load-bearing premise

The dual Hamiltonian formulation together with the chosen operator basis remains efficient and accurate toward weak coupling so that the static potential can be extracted reliably from the confining string.

What would settle it

A computation on significantly larger lattices or with an independent formulation that yields a step-scaling function whose extrapolated continuum value differs from the one obtained with the proposed scheme would falsify the claim of stability.

Figures

Figures reproduced from arXiv: 2606.20029 by Alessio Negro, Carsten Urbach, Emil Otis Rosanowski, Karl Jansen, Lena Funcke, Paul Ludwig, Timo Jakobs.

Figure 1
Figure 1. Figure 1: FIG. 1: Dualization of the lattice in the absence of static [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Dualization in the presence of static charges. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Snake ordering of the lattice used in the MPS [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Formation of a charged flux tube between two [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Schematic of the finite-volume continuum [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Potential (top) and force (bottom) between two [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Continuum limit extrapolation of the lattice [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Universality check: dimensionless potential [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Absolute value of the subtracted renormalized [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Discrete difference [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Finite- [PITH_FULL_IMAGE:figures/full_fig_p009_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Schematic of the lattice setup for two static [PITH_FULL_IMAGE:figures/full_fig_p012_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Schematic of the lattice setup for two static [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Schematic of the Hodge dual of a single [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Uncontrolled continuum extrapolation of the [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗
read the original abstract

We propose a finite-volume scheme to perform controlled continuum extrapolations of the lattice step-scaling function, a key ingredient for determining the running coupling in a Hamiltonian lattice gauge theory in small volumes. As a testbed, we employ a dual Hamiltonian formulation of pure U(1) gauge theory in (2+1) dimensions and an operator basis that remains efficient toward weak coupling. We describe the implementation of static external charges on the spatial lattice and study, using matrix product states, the resulting confining string, from which we extract the static potential and a force-based renormalized coupling. Using the proposed finite-volume scheme, we demonstrate a stable continuum limit of the step-scaling function on the lattice sizes accessible to present Hamiltonian simulations. The method is readily extendable to other gauge groups and dimensions, providing a pathway toward Hamiltonian step-scaling studies in other theories.

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

1 major / 2 minor

Summary. The paper proposes a finite-volume scheme for controlled continuum extrapolations of the lattice step-scaling function in Hamiltonian lattice gauge theories. Using a dual Hamiltonian formulation of pure U(1) gauge theory in (2+1) dimensions together with an operator basis efficient toward weak coupling, the authors implement static external charges, employ matrix product states to study the confining string, extract the static potential, and define a force-based renormalized coupling. They report that the proposed scheme yields a stable continuum limit for the step-scaling function on lattice sizes accessible to present Hamiltonian simulations and note that the method extends to other gauge groups and dimensions.

Significance. If the central demonstration holds, the work supplies a practical route to step-scaling analyses in Hamiltonian formulations, where Euclidean Monte Carlo methods are unavailable or inefficient. The dual formulation and MPS approach are presented as enabling access to weak coupling on modest volumes; successful validation would therefore constitute a concrete technical advance for non-perturbative running-coupling studies in Hamiltonian lattice gauge theory.

major comments (1)
  1. [Results section (demonstration of continuum limit)] The central claim that a stable continuum limit is demonstrated rests on the accuracy of the static potential extracted from the confining string via MPS in the dual formulation. No quantitative bounds on bond-dimension convergence, truncation error, or cross-validation against known analytic or Euclidean results for the U(1) static potential at weak coupling are supplied; without such controls the observed stability could arise from uncontrolled systematics rather than true continuum behavior.
minor comments (2)
  1. [Methods] Notation for the force-based renormalized coupling and the precise definition of the finite-volume step-scaling function should be stated explicitly with an equation number in the methods section.
  2. [Figures] Figure captions should include the specific lattice sizes, bond dimensions, and coupling values used for each data set shown in the continuum-extrapolation plots.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Results section (demonstration of continuum limit)] The central claim that a stable continuum limit is demonstrated rests on the accuracy of the static potential extracted from the confining string via MPS in the dual formulation. No quantitative bounds on bond-dimension convergence, truncation error, or cross-validation against known analytic or Euclidean results for the U(1) static potential at weak coupling are supplied; without such controls the observed stability could arise from uncontrolled systematics rather than true continuum behavior.

    Authors: We agree that the current manuscript does not supply explicit quantitative bounds on bond-dimension convergence or truncation errors, nor direct cross-validation against Euclidean or analytic results. In the revised manuscript we will add a dedicated subsection (and associated figures) that reports the dependence of the extracted static potential and force on bond dimension for representative values of the coupling and lattice size, together with estimates of the truncation error obtained from the discarded singular-value weight and from direct comparisons at different bond dimensions. We will also include a comparison of the force-based renormalized coupling to one-loop perturbation theory in the weakest-coupling regime accessible to the simulations. These additions will make the control over systematics explicit and will confirm that the reported stability of the continuum extrapolation is not an artifact of insufficient MPS accuracy. revision: yes

Circularity Check

0 steps flagged

No circularity in proposed scheme or continuum extrapolation demonstration

full rationale

The paper proposes a finite-volume scheme for continuum extrapolation of the step-scaling function, implements it via a dual Hamiltonian U(1) formulation and MPS extraction of the static potential from the confining string, then reports a stable limit on accessible volumes. No quoted step reduces a claimed prediction to a fitted input by construction, invokes self-citations for load-bearing uniqueness, or renames a known result as new organization. The confining-string observable is treated as independent input for testing the scheme rather than derived from it. The central demonstration therefore remains non-circular and self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated. The work relies on standard domain assumptions of lattice gauge theory and tensor-network methods.

axioms (2)
  • domain assumption The dual Hamiltonian formulation accurately represents pure U(1) gauge theory in (2+1) dimensions.
    Invoked when describing the testbed and operator basis.
  • domain assumption Matrix product states provide a faithful representation of the confining string states for extraction of the static potential.
    Used to obtain the force-based renormalized coupling.

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discussion (0)

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Reference graph

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