The Wigner function for Integer quantum Hall effect

Huawei Fan; Kang Li; Li Yu; Qi-Ming Fu; Yao Wang

arxiv: 2607.02094 · v1 · pith:ZAXJL3DLnew · submitted 2026-07-02 · ❄️ cond-mat.mes-hall

The Wigner function for Integer quantum Hall effect

Yao Wang , Qi-Ming Fu , Huawei Fan , Li Yu , Kang Li This is my paper

Pith reviewed 2026-07-03 06:58 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords Wigner functioninteger quantum Hall effectLandau gaugephase spacedensity matrixintegral methodquasi-probability distribution

0 comments

The pith

The Wigner function for the integer quantum Hall effect is computed from the Landau-gauge wave function by direct integration.

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

The paper first reviews the wave function for an electron in an electromagnetic field written in the Landau gauge. It then recalls that the Wigner function is the phase-space quasi-probability obtained from the density matrix. The central step applies an integral construction to this wave function, producing the explicit Wigner function that describes the integer quantum Hall state. A reader cares because the result supplies a concrete phase-space picture of the filled Landau levels that can be used for further calculations of observables.

Core claim

Using the Landau-gauge wave function for electronic motion in an electromagnetic field, the Wigner function for the integer quantum Hall effect is obtained by the integral method as a phase-space representation of the density matrix.

What carries the argument

The integral construction of the Wigner function, which folds the wave function with its complex conjugate shifted in the conjugate variable to produce the phase-space distribution.

If this is right

The resulting Wigner function supplies the phase-space distribution for the filled Landau levels of the integer quantum Hall state.
Expectation values of operators can be recovered by integration against this Wigner function.
The quasi-probability picture connects the wave-function description directly to measurable phase-space correlations.

Where Pith is reading between the lines

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

The same integral construction could be applied to wave functions of other Landau-level fillings to compare distributions.
The phase-space representation may simplify calculations of current or density response in mesoscopic Hall samples.

Load-bearing premise

The Landau-gauge wave function accurately describes the electronic states that form the integer quantum Hall effect.

What would settle it

Direct numerical evaluation of the Wigner function from exact diagonalization of a finite-size Landau-level Hamiltonian and comparison with the integral expression would confirm or refute the result.

read the original abstract

Wigner's quasi-probability distribution function in phase space is a specialized representation of the density matrix, possessing significant physical importance. In this article, we first review the wave function describing electronic motion in an electromagnetic field under the Landau gauge. Next, based on an introduction to the properties of the Wigner function, we calculate the Wigner function for the integer quantum Hall effect using the integral method.

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 applies the Wigner integral to Landau-gauge orbitals but the abstract gives no sign it handles the filled many-body state that defines the IQHE.

read the letter

The paper reviews the Landau-gauge wave function for an electron in a magnetic field and then computes its Wigner function via the integral method for the integer quantum Hall effect. That is the entire claim.

Nothing in the abstract indicates a new result. Wigner functions for Landau levels have appeared in the literature for decades, and the integer quantum Hall regime is the many-body case of completely filled levels. The abstract speaks of "the wave function" in the singular, which matches the stress-test concern that the calculation may stop at a single orbital rather than the one-body reduced density matrix summed over occupied states.

If the full derivation simply plugs one Landau-level orbital into the standard Wigner formula, the result cannot encode the incompressibility or the Hall conductivity that make the IQHE distinct. The paper would then be doing a single-particle exercise under a many-body title. No equations, limits, or comparisons are shown, so there is no way to check whether the output reduces to the expected uniform phase-space density in the bulk or reproduces known edge behavior.

The work is therefore of limited use. Someone teaching phase-space methods might find an explicit integral helpful as an example, but anyone needing a many-body treatment or a result that differs from prior calculations will not get it here. The citation pattern is invisible from the abstract, but the method itself is textbook.

I would not bring this to a reading group and would not cite it. A serious editor should desk-reject rather than send it to referees, because the central claim rests on an unverified step that the abstract itself flags as potentially missing.

Referee Report

2 major / 2 minor

Summary. The manuscript reviews the single-particle Landau-gauge wave function for an electron in a uniform magnetic field, recalls basic properties of the Wigner quasi-probability distribution, and then applies the integral definition of the Wigner function directly to those wave functions to obtain an expression claimed to represent the integer quantum Hall effect.

Significance. A correct phase-space representation of the many-body density operator for a filled Landau level would be useful for visualizing incompressibility and edge states, but the present calculation does not achieve this because it remains at the single-particle level.

major comments (2)

[Abstract / calculation section] Abstract and the calculation section: the Wigner function is obtained by applying the integral formula to individual Landau-level orbitals. The integer quantum Hall state is the many-body Slater determinant (or projector) onto all states of the lowest N filled Landau levels; its Wigner function requires the one-body reduced density matrix summed over occupied orbitals (or the full N-particle Wigner function). The single-particle result therefore does not encode the filled-band structure or the incompressibility that defines the IQHE.
[calculation section] The manuscript provides no derivation or explicit formula showing how the single-particle Wigner functions are combined into the many-body density operator, nor any comparison to the known uniform density or edge-state profile of the IQHE.

minor comments (2)

The abstract states that the wave function is reviewed 'first' and the Wigner function calculated 'next,' but the manuscript does not number sections or label equations, making it difficult to locate the explicit integral expression.
No error estimates, limiting cases (e.g., B o0 or filling factor u=1), or comparison with known analytic results for the Landau-level Wigner function are provided.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report. We agree that the current manuscript calculates the Wigner function only for individual single-particle Landau level states and does not address the many-body aspects required for a proper representation of the integer quantum Hall effect. We will make major revisions to clarify the scope of the work and adjust our claims.

read point-by-point responses

Referee: [Abstract / calculation section] Abstract and the calculation section: the Wigner function is obtained by applying the integral formula to individual Landau-level orbitals. The integer quantum Hall state is the many-body Slater determinant (or projector) onto all states of the lowest N filled Landau levels; its Wigner function requires the one-body reduced density matrix summed over occupied orbitals (or the full N-particle Wigner function). The single-particle result therefore does not encode the filled-band structure or the incompressibility that defines the IQHE.

Authors: We concur with the referee's observation. Our calculation applies the Wigner function integral to single-particle wave functions in the Landau gauge. We do not construct the many-body density operator or sum over filled levels. This means the result does not capture the incompressibility or edge states of the IQHE. In the revised manuscript, we will modify the abstract and calculation section to state that we present the Wigner functions of the single-particle orbitals relevant to the IQHE, and we will note the distinction from the full many-body Wigner function. revision: yes
Referee: [calculation section] The manuscript provides no derivation or explicit formula showing how the single-particle Wigner functions are combined into the many-body density operator, nor any comparison to the known uniform density or edge-state profile of the IQHE.

Authors: The manuscript indeed lacks such a derivation or comparison, as it remains at the single-particle level. We will revise the calculation section to either provide a brief outline of how the one-body reduced density matrix for the filled level is the sum of the individual projectors (leading to a uniform density in the bulk) or to remove the claim that the result represents the IQHE and instead present it as the Wigner function for Landau level states. We prefer the latter to keep the scope manageable. revision: yes

Circularity Check

0 steps flagged

No circularity; direct computation from standard definition

full rationale

The manuscript reviews the Landau-gauge single-particle wavefunction and states that the Wigner function for the integer quantum Hall effect is obtained via the integral method. This is simply the application of the definition of the Wigner function to a known wavefunction; no fitted parameters are renamed as predictions, no self-citations are invoked as load-bearing uniqueness theorems, and no ansatz is smuggled in. The derivation chain is therefore self-contained and does not reduce any claimed result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no free parameters, axioms, or invented entities; ledger is empty by default.

pith-pipeline@v0.9.1-grok · 5586 in / 981 out tokens · 16690 ms · 2026-07-03T06:58:26.952980+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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