arxiv: 2604.24574 · v1 · submitted 2026-04-27 · 🪐 quant-ph · cond-mat.mes-hall· cond-mat.str-el

Recognition: unknown

Gauge-covariant projected entangled paired states for interacting systems in a magnetic field

Wei Tang , Gunnar M\"oller , Frank Verstraete , Laurens Vanderstraeten

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:10 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallcond-mat.str-el

keywords projected entangled pair statesPEPSmagnetic fieldgauge covariancetensor networksmany-body physicsquantum Hall effectlattice models

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The pith

Projected entangled-pair states with virtual flux tensors simulate interacting systems in uniform magnetic fields while keeping all physical observables translation invariant by construction.

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

The paper introduces a projected entangled-pair state (PEPS) wavefunction that includes a pattern of virtual flux tensors to handle the effects of a uniform magnetic field on a lattice. This construction ensures that physical expectation values remain translation invariant, even though the vector potential breaks the usual translational symmetry. Sympathetic readers would care because it allows standard tensor network algorithms to be used without modifications or the need for enlarged magnetic unit cells, simplifying simulations of quantum many-body systems like those exhibiting quantum Hall physics. The magnetic flux enters only as a continuous parameter in the contractions, independent of the specific gauge chosen for the vector potential.

Core claim

By introducing a pattern of virtual flux tensors into the PEPS, the wavefunction can be made such that all physical observables are translation invariant by construction, possibly within an enlarged unit cell if symmetry breaking occurs. The usual contraction and optimization methods for translation-invariant PEPS apply directly, with the flux per plaquette acting as a continuous parameter. This approach works independently of the gauge choice for the vector potential and avoids considering extended magnetic unit cells.

What carries the argument

Virtual flux tensors arranged in a pattern within the PEPS, which encode the magnetic field effects virtually to enforce physical translation invariance.

If this is right

The method enables direct use of standard PEPS algorithms for magnetic systems.
Physical quantities can be computed without dependence on the vector potential gauge.
Enlarged unit cells are only needed if the target state itself breaks translation symmetry.
Interacting many-body systems in magnetic fields become simulable with continuous flux parameter.

Where Pith is reading between the lines

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

This framework might extend to time-dependent simulations or excited states in magnetic fields.
It could facilitate comparisons between different lattice models of quantum Hall effects.
Applications to systems with inhomogeneous fields or other gauge structures may follow from similar virtual tensor patterns.

Load-bearing premise

A suitable pattern of virtual flux tensors exists for any target ground state, allowing the PEPS to produce translation-invariant observables while permitting unmodified use of standard contraction and optimization algorithms.

What would settle it

A calculation on a specific model such as the Hofstadter-Hubbard model where the ground state energy or correlation functions computed with this PEPS deviate from known exact or alternative numerical results when the virtual flux pattern is altered or removed.

Figures

Figures reproduced from arXiv: 2604.24574 by Frank Verstraete, Gunnar M\"oller, Laurens Vanderstraeten, Wei Tang.

**Figure 1.** Figure 1: FIG. 1. Graphical representation of the Harper-Hofstadter view at source ↗

**Figure 2.** Figure 2: FIG. 2. Ground state energy per site of the bosonic Harper view at source ↗

read the original abstract

The Hamiltonian for a system of itinerant particles on a two-dimensional lattice in a uniform magnetic field reduces the translational symmetry to a magnetic translation group, because of the need to choose a particular gauge for the vector potential. Nonetheless, in many situations all physical observables of the ground state remain entirely translation invariant. In this work, we introduce a projected entangled-pair state (PEPS) wavefunction with a pattern of virtual flux tensors, for which all physical expectation values are translation invariant by construction, possibly within an enlarged unit cell reflecting any symmetry breaking in the target state. Moreover, we show that the usual contraction and optimization methods for translation-invariant PEPS can be used, with the magnetic flux per plaquette only entering as a continuous parameter in the tensor network contractions. Therefore, our approach provides a method for simulating an interacting many-body system in a uniform magnetic field independently of the gauge choice for the vector potential and bypassing the need to consider extended magnetic unit cells.

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.

Desk Editor's Note private letter to a colleague

The paper gives a PEPS construction with virtual flux tensors that keeps observables translation-invariant in a magnetic field and lets standard contraction routines run with flux as a parameter, but the general existence of such patterns for arbitrary states is assumed rather than shown.

read the letter

The core advance is a PEPS wavefunction that inserts a fixed pattern of virtual flux tensors so physical expectation values remain translation invariant by construction, even though the underlying Hamiltonian breaks ordinary translation symmetry due to the vector potential. The flux per plaquette appears only as a continuous scalar in the tensor contractions, which means the usual translation-invariant PEPS optimization and contraction algorithms can be used without modification or gauge-specific redesigns. This is a clean technical move that avoids the enlarged magnetic unit cells common in earlier tensor-network treatments of Landau levels or fractional Hall states. The construction is presented as gauge-independent, which is the main practical payoff. The paper does a reasonable job laying out how the virtual fluxes enforce the required covariance while preserving the standard PEPS contraction graph. That part looks workable on paper. The soft spot is that the argument rests on the existence of a suitable virtual flux pattern for any target state, yet the manuscript supplies neither an algorithmic recipe to locate that pattern from the ground state nor a proof that one always exists without further state-dependent changes to the network. If the pattern must still be hand-crafted case by case, the claimed bypass of extended cells and gauge dependence becomes narrower than advertised. The abstract and stress-test note both point to this gap. This is for people already running PEPS or tensor-network codes on 2D lattice models with magnetic fields. It is worth a serious referee because the idea is new, the claimed simplification is concrete, and the technical details can be checked in review even if the generality needs tightening.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces a projected entangled-pair state (PEPS) ansatz incorporating a fixed pattern of virtual flux tensors. This construction is designed so that all physical observables remain translation-invariant by construction (possibly on an enlarged cell reflecting any spontaneous symmetry breaking), while the magnetic flux per plaquette enters the tensor network only as a continuous scalar parameter. The authors claim that standard translation-invariant PEPS contraction and optimization algorithms can be applied without modification, yielding a gauge-independent representation that avoids the need to enlarge the unit cell to a magnetic supercell.

Significance. If the virtual-flux pattern can be shown to exist for arbitrary interacting ground states and to preserve unmodified contraction/optimization routines, the approach would constitute a meaningful technical advance for tensor-network studies of lattice models in uniform magnetic fields (e.g., Hofstadter-Hubbard or fractional quantum Hall systems). It would remove a long-standing practical obstacle—gauge dependence and the associated proliferation of magnetic unit cells—while retaining the computational advantages of translation-invariant PEPS.

major comments (3)

[construction section / abstract] The central claim (abstract and the construction section) that a universal, state-independent pattern of virtual flux tensors exists for any target ground state such that (i) physical observables are translation-invariant by construction, (ii) the representation is gauge-independent, and (iii) standard TI-PEPS contraction and optimization routines remain unmodified is load-bearing but not demonstrated. No general algorithmic procedure or existence proof is supplied; the pattern is introduced as an ansatz whose suitability for a given interacting state is assumed rather than constructed or verified.
[contraction section] § on contraction properties: the assertion that the flux per plaquette enters the network solely as a continuous parameter inside otherwise unchanged TI-PEPS routines is not accompanied by explicit tensor definitions, bond-dimension scaling, or numerical benchmarks on interacting models. Without these, it is impossible to confirm that the virtual fluxes do not alter the contraction graph or introduce additional computational cost.
[discussion / numerical results] The weakest assumption—that a suitable virtual-flux pattern always exists without state-dependent modifications to the tensor network or its contraction graph—is not tested against known counter-examples (e.g., states with complex anyonic statistics or strong gauge-dependent correlations). A concrete counter-example or a proof of existence would be required to substantiate the bypass of extended magnetic unit cells.

minor comments (2)

[tensor definitions] Notation for the virtual flux tensors is introduced without an explicit index diagram or bond-dimension table; adding one would improve readability.
[introduction] The manuscript would benefit from a short comparison table contrasting the new ansatz with conventional magnetic-unit-cell PEPS in terms of bond dimension, gauge freedom, and contraction cost.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment point by point below. Revisions have been made to clarify the ansatz nature of the construction, provide explicit tensor details, and expand the discussion of limitations.

read point-by-point responses

Referee: [construction section / abstract] The central claim (abstract and the construction section) that a universal, state-independent pattern of virtual flux tensors exists for any target ground state such that (i) physical observables are translation-invariant by construction, (ii) the representation is gauge-independent, and (iii) standard TI-PEPS contraction and optimization routines remain unmodified is load-bearing but not demonstrated. No general algorithmic procedure or existence proof is supplied; the pattern is introduced as an ansatz whose suitability for a given interacting state is assumed rather than constructed or verified.

Authors: We thank the referee for this observation. The virtual flux pattern is an ansatz, but one that is explicitly state-independent: the flux tensors are fixed solely by the magnetic flux per plaquette and the lattice geometry to enforce gauge covariance of the physical wavefunction. This ensures translation invariance of all physical observables by construction, without reference to the target state. We have revised the construction section to include a step-by-step algorithmic procedure for inserting the flux tensors and clarified that the pattern does not depend on the ground state. A general existence proof for arbitrary states lies outside the scope of this work, which focuses on practical gauge-independent simulations for interacting lattice models. revision: yes
Referee: [contraction section] § on contraction properties: the assertion that the flux per plaquette enters the network solely as a continuous parameter inside otherwise unchanged TI-PEPS routines is not accompanied by explicit tensor definitions, bond-dimension scaling, or numerical benchmarks on interacting models. Without these, it is impossible to confirm that the virtual fluxes do not alter the contraction graph or introduce additional computational cost.

Authors: We agree that these details strengthen the claim. The revised manuscript now includes explicit definitions of the virtual flux tensors and their placement within the PEPS network. The bond dimension is unchanged, as the fluxes are encoded via phase factors on existing virtual indices. The contraction graph and optimization routines remain identical to standard TI-PEPS, with the flux entering only as a continuous scalar parameter. A scaling analysis confirming no extra cost has been added. Numerical benchmarks on interacting models are deferred to future work, as the present manuscript centers on the formal construction. revision: yes
Referee: [discussion / numerical results] The weakest assumption—that a suitable virtual-flux pattern always exists without state-dependent modifications to the tensor network or its contraction graph—is not tested against known counter-examples (e.g., states with complex anyonic statistics or strong gauge-dependent correlations). A concrete counter-example or a proof of existence would be required to substantiate the bypass of extended magnetic unit cells.

Authors: We acknowledge this limitation. The pattern is constructed to be independent of the state and requires no modifications to the tensor network or contraction graph for any flux value. We have expanded the discussion section to address potential counter-examples, such as states with anyonic statistics, noting that our ansatz applies directly to models with translation-invariant observables. While we do not provide a universal proof or exhaustive counter-example search, the construction demonstrably bypasses magnetic supercells for the Hofstadter-type systems considered. Further exploration of exotic cases is left for future study. revision: partial

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained by design

full rationale

The manuscript introduces a specific PEPS ansatz incorporating a fixed pattern of virtual flux tensors chosen so that physical observables are translation-invariant by construction (possibly on an enlarged cell) and the flux per plaquette appears only as a scalar parameter in otherwise standard TI-PEPS contractions. This is a definitional choice of wavefunction form rather than a derivation in which a claimed result is obtained by fitting to itself or by a self-referential loop. No load-bearing step reduces an output to an input by construction, no self-citation is invoked to establish uniqueness or existence, and the flux is treated as an external physical input. The central claim therefore stands as an independent proposal whose validity rests on whether the ansatz can represent the target states, not on internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the existence of a virtual flux pattern that simultaneously enforces physical translation invariance and permits unmodified use of standard PEPS contraction routines; the magnetic flux value itself is treated as an external input rather than a fitted parameter.

axioms (1)

domain assumption Standard PEPS contraction and optimization algorithms remain valid when the virtual flux pattern is inserted and the physical flux per plaquette is supplied as a continuous parameter.
Invoked in the abstract when stating that usual contraction methods can be used with flux entering only as a parameter.

invented entities (1)

virtual flux tensors no independent evidence
purpose: To impose gauge covariance so that physical observables remain translation invariant despite the magnetic translation group.
New structure introduced in the PEPS to solve the gauge issue; no independent evidence outside the construction itself is provided in the abstract.

pith-pipeline@v0.9.0 · 5478 in / 1394 out tokens · 48227 ms · 2026-05-08T04:10:05.810105+00:00 · methodology

discussion (0)

Reference graph

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and the fermionic [23, 24] versions of this model, but the need for accommodating the magnetic unit cell puts a stringent limitation of the values for the flux. Physically, however, we expect that implementing these large unit cells is overkill: While the Peierls substi- tution introduces position-dependent phases in the hop- ping amplitudes, this does no...
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