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arxiv: 2605.23613 · v1 · pith:NKXFRHY7new · submitted 2026-05-22 · 🪐 quant-ph · cond-mat.mes-hall

Non-Hermitian Landau Levels

Anton Montag , Tomoki Ozawa This is my paper

Pith reviewed 2026-05-25 04:20 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords non-HermitianLandau levelscomplex magnetic fieldbiorthogonal eigenstatesHarper-Hofstadter modelcomplex Lorentz forcesymmetric gauge
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The pith

Non-Hermitian Landau levels under a complex perpendicular magnetic field produce discretely spaced, highly degenerate complex spectra with biorthogonal eigenstates.

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

The paper formulates non-Hermitian Landau levels for two-dimensional systems placed in a complex perpendicular magnetic field. In the symmetric gauge it derives the discretely spaced and highly degenerate complex energy spectra together with the corresponding biorthogonal eigenstates. It also clarifies how non-unitary gauge transformations act in this setting. A non-Hermitian version of the Harper-Hofstadter lattice model is shown to reproduce the continuum results and to generate Gaussian wave-packet motion governed by semiclassical equations that include a complex Lorentz force.

Core claim

In the symmetric gauge, non-Hermitian Landau levels exhibit discretely spaced, highly degenerate complex spectra and biorthogonal eigenstates. Non-unitary gauge transformations play a clarified role. The continuum theory is confirmed by a non-Hermitian Harper-Hofstadter lattice model, which also reveals Gaussian wave packet dynamics governed by semiclassical equations with a complex Lorentz force.

What carries the argument

Non-Hermitian Landau levels formulated in the symmetric gauge under a complex perpendicular magnetic field, together with their biorthogonal eigenstates and the associated non-unitary gauge transformations.

If this is right

  • The spectra remain discretely spaced and highly degenerate even though the energies are complex.
  • Biorthogonal eigenstates replace ordinary orthonormal ones as the natural basis.
  • Gaussian wave packets obey semiclassical dynamics that include a complex Lorentz force.
  • Lattice realizations can serve as platforms for studying these levels.

Where Pith is reading between the lines

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

  • The complex Lorentz force may produce wave-packet trajectories qualitatively different from those in the Hermitian case.
  • Engineered complex magnetic fields in optical or cold-atom lattices could provide an experimental route to these levels.
  • The formulation may extend to other gauge choices or to systems with additional non-Hermitian terms.

Load-bearing premise

A complex perpendicular magnetic field can be introduced consistently into the non-Hermitian Schrödinger equation while preserving the discrete, degenerate structure of the spectrum.

What would settle it

Numerical diagonalization of the non-Hermitian Harper-Hofstadter lattice model that fails to produce a discretely spaced complex spectrum matching the continuum prediction would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.23613 by Anton Montag, Tomoki Ozawa.

Figure 1
Figure 1. Figure 1: Eigenvalues and right eigenstates of the non-Hermitian Harper-Hofstadter model on a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Trajectories of the center of mass of Gaussian wave [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Effective mass meff depending on wave packet width for a 50 × 50 square lattice for vanishing magnetic field. SM2: DISCUSSION OF WIDTH DEPENDENT EFFECTIVE MASS OF WAVE PACKET As mentioned in the letter, the effective mass of the wave packet meff depends in general on the width of the wave packet σ and for finite Im(B) it is also (weakly) position dependent. In the parameter regime used for the simulation t… view at source ↗
read the original abstract

We formulate non-Hermitian Landau levels in two-dimensional systems under a complex perpendicular magnetic field. In the symmetric gauge, we derive their discretely spaced, highly degenerate complex spectra and biorthogonal eigenstates, and clarify the role of non-unitary gauge transformations. A non-Hermitian Harper-Hofstadter lattice model confirms the continuum theory and reveals Gaussian wave packet dynamics governed by semiclassical equations with a complex Lorentz force, pointing to possible experimental realizations of complex magnetic fields.

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 formulates non-Hermitian Landau levels for two-dimensional systems subject to a complex perpendicular magnetic field. In the symmetric gauge it derives discretely spaced, highly degenerate complex spectra together with their biorthogonal eigenstates, clarifies the role of non-unitary gauge transformations, and confirms the continuum results via a non-Hermitian Harper-Hofstadter lattice model that additionally exhibits Gaussian wave-packet dynamics governed by semiclassical equations containing a complex Lorentz force.

Significance. If the central construction is valid, the work supplies a concrete analytic framework for Landau-level physics in non-Hermitian settings and identifies a lattice realization that could be engineered experimentally. The explicit treatment of biorthogonal states and the complex Lorentz force constitute concrete, falsifiable predictions that go beyond abstract non-Hermitian extensions.

major comments (2)
  1. [Derivation of the continuum spectrum (symmetric-gauge section)] The central claim that the spectrum remains discretely spaced and exactly degenerate for arbitrary complex B rests on the continued validity of the ladder-operator algebra. The manuscript must supply an explicit verification (e.g., the commutator [a, a†] and the action of a† on the nth level) that this algebra closes without correction terms once Im(B) is nonzero; any deviation immediately lifts the degeneracy of higher levels.
  2. [Biorthogonal eigenstates and gauge transformations] Normalizability of the biorthogonal ground state (and hence of all higher states) for arbitrary complex B is asserted but not demonstrated. The Gaussian factor must be shown to remain square-integrable in the biorthogonal inner product when the imaginary part of the cyclotron frequency is present; otherwise the discrete spectrum is lost.
minor comments (2)
  1. [Abstract] The abstract states that derivations exist but supplies no equations; the main text should include at least the key commutator and the explicit form of the complex-frequency operators in the first derivation section.
  2. [Lattice-model section] The lattice-model confirmation is described only qualitatively; a quantitative comparison (e.g., energy-level spacing versus continuum prediction for several values of complex flux) would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We address each point below and will revise the manuscript accordingly to include the requested explicit verifications.

read point-by-point responses

  1. Referee: [Derivation of the continuum spectrum (symmetric-gauge section)] The central claim that the spectrum remains discretely spaced and exactly degenerate for arbitrary complex B rests on the continued validity of the ladder-operator algebra. The manuscript must supply an explicit verification (e.g., the commutator [a, a†] and the action of a† on the nth level) that this algebra closes without correction terms once Im(B) is nonzero; any deviation immediately lifts the degeneracy of higher levels.

    Authors: We agree that an explicit verification of the ladder-operator algebra for complex B strengthens the presentation. The kinetic momenta satisfy [π_x, π_y] = −i ħ e B with B complex; the ladder operators a and a† are then defined in the usual manner (normalized by the appropriate factor involving |B| or the real part as needed), yielding [a, a†] = 1 with no correction terms generated by Im(B). The action a† |n⟩ = √(n+1) |n+1⟩ likewise holds identically. To address the request directly we will insert a short dedicated paragraph in the symmetric-gauge section that computes the commutator and the raising/lowering actions explicitly for general complex B. revision: yes

  2. Referee: [Biorthogonal eigenstates and gauge transformations] Normalizability of the biorthogonal ground state (and hence of all higher states) for arbitrary complex B is asserted but not demonstrated. The Gaussian factor must be shown to remain square-integrable in the biorthogonal inner product when the imaginary part of the cyclotron frequency is present; otherwise the discrete spectrum is lost.

    Authors: We acknowledge that an explicit demonstration of normalizability in the biorthogonal inner product was omitted. The right eigenfunctions contain a Gaussian factor whose exponent involves the complex cyclotron frequency ω_c = eB/m; the corresponding left eigenfunctions involve the adjoint form. Their biorthogonal product produces an integrand whose real part is controlled by Re(B) (or Re(ω_c)), ensuring exponential decay and finite norm provided Re(B) > 0. We will add this calculation, including the explicit evaluation of the ground-state norm for general complex B, to the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation extends standard Landau-level algebra to complex B without reduction to inputs or self-citations

full rationale

The paper begins from the non-Hermitian Schrödinger equation with a complex perpendicular magnetic field inserted into the symmetric-gauge vector potential. It constructs kinetic momenta whose commutator is proportional to the complex B, defines corresponding creation/annihilation operators, and obtains the spectrum and biorthogonal states by the usual algebraic procedure. No equations are shown that define a quantity in terms of itself, rename a fit as a prediction, or rely on a load-bearing self-citation whose content is unverified. The derivation is therefore self-contained against the stated Hamiltonian and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty.

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