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arxiv: 2508.16363 · v3 · submitted 2025-08-22 · ✦ hep-th · cond-mat.str-el· hep-lat

Infinite matrix product states for (1+1)-dimensional gauge theories

Ross Dempsey , Anna-Maria E. Gl\"uck , Silviu S. Pufu , Benjamin T. S{\o}gaard This is my paper

Pith reviewed 2026-05-18 21:46 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elhep-lat
keywords matrix product statesmatrix product operatorslattice gauge theoriesSchwinger modelinfinite latticesgauge symmetrytensor networksHamiltonian formulation
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The pith

A link-enhanced matrix product operator construction represents gauge theory lattice Hamiltonians locally and with manifest translation invariance on infinite lattices.

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

The paper develops a matrix product operator approach for one-plus-one dimensional gauge theories that keeps the Hamiltonians local while preserving translation invariance. It relies on symmetric matrix product states paired with link-enhanced matrix product operators that act on both physical and virtual degrees of freedom. This setup is needed because standard tensor-network methods have been restricted to finite lattices where boundary effects distort physical results. The construction applies to abelian and non-abelian groups alike and is shown explicitly for the massless and massive Schwinger model plus adjoint QCD in two dimensions.

Core claim

The central claim is that symmetric matrix product states together with link-enhanced matrix product operators (LEMPOs) can represent the lattice Hamiltonians of abelian or non-abelian gauge theories in a form that is both local and manifestly translation-invariant, thereby allowing Hamiltonian studies on infinite lattices without boundary artifacts.

What carries the argument

Link-enhanced matrix product operators (LEMPOs) that act simultaneously on the physical and virtual spaces of symmetric matrix product states while preserving gauge symmetry and locality.

If this is right

  • The construction directly implements the massless and massive one-flavor Schwinger model.
  • It extends to adjoint QCD in two dimensions.
  • Hamiltonian lattice gauge theories become accessible on infinite lattices.
  • The resulting operators remain local and manifestly translation-invariant for any gauge group treated.

Where Pith is reading between the lines

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

  • Ground-state and spectral calculations could be performed directly in the infinite-volume limit, removing the need for finite-size extrapolations.
  • The same operator structure might support real-time evolution studies of gauge-theory dynamics once combined with time-evolution algorithms.
  • Similar link-enhancement ideas could be tested in higher-dimensional lattice models where translation invariance is harder to maintain.

Load-bearing premise

Link-enhanced matrix product operators can be constructed to act simultaneously on physical and virtual spaces while preserving the required gauge symmetry and locality for arbitrary gauge groups.

What would settle it

An explicit counterexample showing that no local, gauge-symmetric LEMPO exists for the Hamiltonian of a chosen non-abelian gauge theory on an infinite lattice.

Figures

Figures reproduced from arXiv: 2508.16363 by Anna-Maria E. Gl\"uck, Benjamin T. S{\o}gaard, Ross Dempsey, Silviu S. Pufu.

Figure 1
Figure 1. Figure 1: The spatial arrangement of the degrees of freedom in the Schwinger model Hamil [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The lattice strong-coupling expansions ( [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Precision lattice estimates of the lightest particle mass and the chiral condensate [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The spectrum of the Schwinger model as a function of the electron mass [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The energy density difference ∆ε(m, θ) in the Schwinger model, defined in (3.6). The black dashed line is the weak-coupling expression (3.8) for θ = π. The lattice data also agrees well with the weak-coupling expansion at other values of θ, as well as the small-mass expansion (3.7), but we do not show these approximations in this figure in order to avoid clutter. for −π ≤ θ ≤ π; this formula can be extende… view at source ↗
Figure 6
Figure 6. Figure 6: Zero-temperature phase diagram of the flux tube sectors of the SU(2) and SU(3) [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The low-lying spectrum of SU(2) adjoint QCD [PITH_FULL_IMAGE:figures/full_fig_p036_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The fundamental string tension in the SU(2) and SU(3) theories. The blue dashed [PITH_FULL_IMAGE:figures/full_fig_p037_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The chiral condensate and the lowest fermion mass for the SU(2) and SU(3) theories [PITH_FULL_IMAGE:figures/full_fig_p039_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The low-lying spectrum of SU(3) adjoint QCD [PITH_FULL_IMAGE:figures/full_fig_p040_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The contraction pattern in generalized MPS (a) and PEPS (b) for lattice gauge [PITH_FULL_IMAGE:figures/full_fig_p042_11.png] view at source ↗
read the original abstract

We present a matrix product operator construction that allows us to represent the lattice Hamiltonians of (abelian or non-abelian) gauge theories in a local and manifestly translation-invariant form. In particular, we use symmetric matrix product states and introduce link-enhanced matrix product operators (LEMPOs) that can act on both the physical and virtual spaces of the matrix product states. This construction allows us to study Hamiltonian lattice gauge theories on infinite lattices. As examples, we show how to implement this method to study the massless and massive one-flavor Schwinger model and adjoint QCD$_2$.

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 link-enhanced matrix product operators (LEMPOs) that act jointly on the physical (matter and gauge) and virtual indices of symmetric matrix product states. This construction is claimed to represent the lattice Hamiltonians of (1+1)-dimensional abelian and non-abelian gauge theories in a strictly local and manifestly translation-invariant form, thereby enabling Hamiltonian studies on infinite lattices. Explicit implementations are presented for the massless and massive one-flavor Schwinger model (U(1)) and for adjoint QCD2.

Significance. If the LEMPO construction is shown to preserve exact gauge invariance and locality for arbitrary groups, the method would provide a valuable tool for tensor-network studies of (1+1)D gauge theories in the infinite-volume limit, removing finite-size artifacts that complicate continuum extrapolations. The explicit treatment of both abelian and non-abelian cases, together with the use of symmetric MPS, strengthens the potential applicability to models with confinement and chiral symmetry breaking.

major comments (2)
  1. [LEMPO construction for non-abelian groups (around the definition of the enhanced link operators)] The central claim that LEMPOs can be defined to act simultaneously on physical and virtual spaces while exactly preserving gauge symmetry and strict locality for arbitrary (including non-abelian) gauge groups is load-bearing. The provided Schwinger-model example is abelian and therefore does not test the required closure of virtual indices under irreducible representations when the enhanced link operators act jointly with the non-abelian gauge action. An explicit algebraic verification or projector construction that guarantees this closure without introducing non-local virtual contractions is needed.
  2. [Numerical results and symmetry checks for adjoint QCD2] In the adjoint QCD2 implementation, the numerical convergence data and explicit checks that the infinite-lattice ground state remains exactly gauge-invariant under the full non-abelian group action are not sufficient to confirm that the virtual-space representation closes properly. Any residual gauge violation or effective non-locality would undermine the translation-invariant infinite-lattice claim.
minor comments (2)
  1. [Notation and definitions] Notation for the virtual indices and the precise embedding of the group representations into the MPO tensors should be clarified with an explicit example for a small non-abelian group (e.g., SU(2)).
  2. [Discussion] A short comparison table of computational cost versus conventional finite-lattice MPS or DMRG would help readers assess the practical advantage of the infinite-lattice formulation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of the LEMPO construction. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [LEMPO construction for non-abelian groups (around the definition of the enhanced link operators)] The central claim that LEMPOs can be defined to act simultaneously on physical and virtual spaces while exactly preserving gauge symmetry and strict locality for arbitrary (including non-abelian) gauge groups is load-bearing. The provided Schwinger-model example is abelian and therefore does not test the required closure of virtual indices under irreducible representations when the enhanced link operators act jointly with the non-abelian gauge action. An explicit algebraic verification or projector construction that guarantees this closure without introducing non-local virtual contractions is needed.

    Authors: We agree that an explicit algebraic verification strengthens the claim for non-abelian groups. The general LEMPO construction in the manuscript is formulated using the representation theory of the gauge group, with enhanced link operators defined via intertwiners that map between physical and virtual spaces while preserving invariance. To address the referee's concern directly, the revised manuscript will include an explicit algebraic verification for the SU(2) adjoint case, showing closure of the virtual indices under the group action via projectors without non-local contractions. revision: yes

  2. Referee: [Numerical results and symmetry checks for adjoint QCD2] In the adjoint QCD2 implementation, the numerical convergence data and explicit checks that the infinite-lattice ground state remains exactly gauge-invariant under the full non-abelian group action are not sufficient to confirm that the virtual-space representation closes properly. Any residual gauge violation or effective non-locality would undermine the translation-invariant infinite-lattice claim.

    Authors: We acknowledge that additional explicit checks would be useful to confirm the closure property. The current numerical results already demonstrate gauge invariance to machine precision due to the use of symmetric MPS combined with LEMPOs. In the revised manuscript we will augment the adjoint QCD2 section with further data on gauge-invariance residuals across system sizes and an explanation of how the virtual representation is guaranteed to close by the projector structure of the LEMPOs. revision: yes

Circularity Check

0 steps flagged

New LEMPO construction for gauge-theory Hamiltonians is a self-contained operator definition

full rationale

The paper introduces link-enhanced matrix product operators (LEMPOs) as a novel construction to represent lattice Hamiltonians in a local, translation-invariant manner for both abelian and non-abelian gauge theories. This is presented directly as a definitional method acting on physical and virtual spaces of symmetric MPS, without any equations that reduce the claimed representation to a fitted parameter, prior result by the same authors, or self-referential input. The Schwinger-model and adjoint QCD2 examples are applications of the construction rather than validations that force the result by construction. No load-bearing derivation step collapses to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method relies on standard properties of matrix product states and gauge symmetry but introduces no new fitted parameters or invented particles in the abstract description.

axioms (1)
  • domain assumption Gauge symmetry must be preserved by the link-enhanced operators acting on both physical and virtual indices.
    Invoked when the authors state that the construction works for abelian or non-abelian gauge theories while maintaining manifest translation invariance.

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Forward citations

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  2. Quantum dynamics of cosmological particle production: interacting quantum field theories with matrix product states

    hep-th 2026-01 unverdicted novelty 7.0

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  3. Real-time Scattering in \phi^4 Theory using Matrix Product States

    hep-th 2025-11 unverdicted novelty 7.0

    uMPS simulations of φ⁴ theory in 1+1 dimensions extract elastic scattering probabilities and time delays that diverge near the critical point, serving as a dynamical signature of the quantum phase transition.

  4. Dense $\mathrm{QC_2D_2}$ with uniform matrix product states

    hep-lat 2026-05 unverdicted novelty 6.0

    Uniform MPS simulations of dense 1+1D SU(2) gauge theory find Tomonaga-Luttinger liquid infrared behavior with central charge 1, density modulations at the predicted wavenumber, and a smooth crossover in the Luttinger...

Reference graph

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