Non-Hermitian Landau Levels

• Non-Hermitian Landau levels are a family of quantization phenomena where gain/loss and complex gauge fields modify the traditional, equidistant Landau spectrum, leading to real, complex, or continuum energy structures.
• Different mechanisms—such as complex magnetic fields, gain/loss induced pseudofields, and non-Hermitian perturbations—yield varied spectral outcomes including gap collapse, chirality redistributions, and the emergence of biorthogonal states.
• Experimental platforms like photonic systems, electric circuits, and lattice simulations validate these models by revealing unique signatures such as hidden counter-propagating channels and topologically robust responses despite non-Hermitian deformations.

Non-Hermitian Landau levels are Landau-quantized spectral and eigenmode structures that arise when the conventional magnetic-field problem is generalized to Hamiltonians with gain and loss, complex self-energies, non-reciprocal couplings, complex gauge fields, or radiative boundary dissipation. In this broader setting, the familiar Landau ladder can remain real, acquire complex energies, undergo gap collapse, redistribute chirality between bulk and boundary sectors, or even give way to Gaussian-localized states with a continuous complex spectrum rather than discrete quantization (Montag et al., 22 May 2026, Xue et al., 2019, Shen et al., 2018, Vaidya et al., 30 May 2026, Wang et al., 2022).

1. Hermitian baseline and the scope of non-Hermitian generalization

The Hermitian reference problem remains the organizing template. In two dimensions, the standard Landau spectrum is $E_n=\hbar\omega_c(n+\tfrac12)$ for a charged particle in a uniform perpendicular magnetic field (Lizarraga et al., 18 Feb 2025). In Weyl systems under $B\hat z$, the higher Landau levels are

$$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$

while the zeroth level is a single chiral branch

$E_0(k_z,s)=s\,v_Fk_z,$

whose group velocity is fixed by the Weyl-node chirality $s=\pm1$ (Vaidya et al., 30 May 2026). For a pair of opposite-chirality Weyl nodes, the two zeroth modes counter-propagate, enforcing the vanishing of the net chirality current in accordance with the Nielsen–Ninomiya constraint (Vaidya et al., 30 May 2026).

Non-Hermitian generalizations preserve this Landau-level vocabulary but alter either the operator algebra, the nature of the gauge field, the spectral reality conditions, or the physical meaning of observability. In some constructions the magnetic field itself becomes complex; in others, non-Hermiticity enters through gain/loss gradients, imaginary Stark-like terms, disorder self-energies, or boundary-localized radiative loss (Montag et al., 22 May 2026, Xue et al., 2019, Lizarraga et al., 18 Feb 2025, Shen et al., 2018, Vaidya et al., 30 May 2026).

Mechanism	Hallmark of the Landau problem	Representative papers
Complex magnetic field	Equispaced complex Landau ladder with biorthogonal states	(Montag et al., 22 May 2026)
Gain/loss-generated pseudofield	Real Landau levels in a symmetry-unbroken regime	(Xue et al., 2019)
Exceptional-point Dirac/Weyl models	Real flat bands from non-Hermitian Landau quantization	(Zhang et al., 2019)
Imaginary linear perturbation	Complex Stark-shifted Landau levels labeled by $\lambda$	(Lizarraga et al., 18 Feb 2025)
Disorder self-energy	Complex Landau poles and spectrum collapse criteria	(Shen et al., 2018, Matsushita et al., 2020)
Axial field plus boundary loss	Observable chirality imbalance of Weyl zeroth levels	(Vaidya et al., 30 May 2026)
First-order non-Hermitian continuum models	Gaussian bound states with continuous complex spectrum	(Wang et al., 2022, Lin et al., 5 Sep 2025)

This range of mechanisms means that “non-Hermitian Landau levels” is not a single construction but a family of related quantization phenomena sharing Landau-type magnetic localization while differing sharply in spectral topology, boundary sensitivity, and physical interpretation.

2. Operator formulations and effective Hamiltonians

A direct extension of the continuum Landau problem is obtained by allowing the perpendicular magnetic field to be complex. In the symmetric gauge,

$$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$

the Hamiltonian retains the kinetic form

$$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$

but the cyclotron frequency $\omega_c=qB/(mc)$ is complex. The algebra is reorganized in terms of two pairs of ladder operators $(\hat a,\hat a^\S)$ and $B\hat z$0 satisfying $B\hat z$1, with the Hamiltonian reducing to

$B\hat z$2

(Montag et al., 22 May 2026). This is the most literal non-Hermitian analogue of the textbook Landau oscillator.

A second route replaces a physical magnetic field by an emergent pseudomagnetic gauge field induced by gain/loss modulation. In the non-Hermitian honeycomb model of "Non-Hermitian Dirac Cones" (Xue et al., 2019), a spatially varying onsite gain/loss parameter $B\hat z$3 shifts the Dirac points in momentum space. In a long-wavelength description this shift is reinterpreted as a valley-dependent vector potential $B\hat z$4, and a linear profile $B\hat z$5 generates a pseudo-magnetic field $B\hat z$6 for valley index $B\hat z$7 (Xue et al., 2019). The resulting continuum Dirac equation has precisely the Landau-quantized structure, but its field is induced by non-Hermiticity rather than by an external magnetic flux.

A third route keeps the electromagnetic field conventional but adds an explicitly non-Hermitian perturbation. In the two-dimensional Schrödinger problem with a constant perpendicular magnetic field and an electric field $B\hat z$8, the addition of

$B\hat z$9

produces the full Hamiltonian

$$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$0

which fails Hermiticity because of the $$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$1 term (Lizarraga et al., 18 Feb 2025). The exact solution requires a complex separation constant $$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$2, later identified as the eigenvalue of a commuting non-Hermitian symmetry operator $$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$3 (Lizarraga et al., 18 Feb 2025).

A fourth route arises in quasiparticle descriptions of disordered or interacting materials, where non-Hermiticity enters through the imaginary part of the self-energy. In the two-band formulation used for disorder-induced in-gap states, the effective Hamiltonian is

$$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$4

while in disordered Dirac systems one encounters

$$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$5

(Shen et al., 2018, Matsushita et al., 2020). In these formulations, Landau quantization acts on quasiparticle poles rather than on a closed Hermitian spectrum.

3. Spectral structures: real ladders, complex ladders, and collapse

For a complex perpendicular magnetic field in the symmetric gauge, the Landau ladder remains equidistant but moves into the complex plane:

$$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$6

Each level remains infinitely or macroscopically degenerate because the second ladder pair does not enter $$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$7, and the appropriate eigenbasis is biorthogonal rather than orthonormal (Montag et al., 22 May 2026). The non-Hermitian modification therefore changes spectral geometry without destroying the core oscillator structure.

In contrast, several non-Hermitian Landau problems retain a strictly real spectrum in a symmetry-unbroken regime. In "Non-Hermitian Dirac Cones" (Xue et al., 2019), the pseudo-magnetic-field Landau levels are

$$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$8

with the non-Hermitian parameter entering through the anisotropy factor $$E_{n,\pm}(k_z)=\pm\sqrt{2n\,eB\,v_F^2+(v_Fk_z)^2},\qquad n=1,2,\dots,$$9. For $E_0(k_z,s)=s\,v_Fk_z,$0, pseudo-Hermiticity and anti-PT symmetry keep all these energies real, while the zeroth level remains pinned at $E_0(k_z,s)=s\,v_Fk_z,$1 by particle-hole symmetry (Xue et al., 2019). In the exceptional Dirac/Weyl circuit models of Zhang and Franz, the exact Landau levels are likewise real,

$E_0(k_z,s)=s\,v_Fk_z,$2

despite $E_0(k_z,s)=s\,v_Fk_z,$3; the lowest level is sublattice polarized, and the partner at $E_0(k_z,s)=s\,v_Fk_z,$4 appears as an edge state under open boundaries (Zhang et al., 2019).

Other constructions yield genuinely complex Landau levels. In the linear non-Hermitian perturbation problem, the spectrum is

$E_0(k_z,s)=s\,v_Fk_z,$5

so the usual Landau ladder is shifted by a complex Stark-like correction $E_0(k_z,s)=s\,v_Fk_z,$6 whose real part is a Stark shift and whose imaginary part encodes decay or growth (Lizarraga et al., 18 Feb 2025). The exact eigenfunctions acquire the additional factor $E_0(k_z,s)=s\,v_Fk_z,$7 and a shifted polynomial $E_0(k_z,s)=s\,v_Fk_z,$8, explicitly deforming the standard Landau orbit structure (Lizarraga et al., 18 Feb 2025).

In non-Hermitian quasiparticle Hamiltonians for small-gap insulators, the Landau eigenvalues become complex poles. For $E_0(k_z,s)=s\,v_Fk_z,$9 one finds

$s=\pm1$0

with linewidths set by orbital-dependent scattering rates (Shen et al., 2018). This complexification directly modifies the density of states and the Lifshitz–Kosevich-like oscillatory amplitude, leading to the result that the effective mass can be controlled by scattering rates while the Dingle factor is controlled by the indirect gap over a wide parameter range (Shen et al., 2018).

A more singular phenomenon is spectrum collapse. In disordered Dirac Landau levels with spin-dependent scattering, the relevant energies are

$s=\pm1$1

and the gap between the $s=\pm1$2 levels collapses when

$s=\pm1$3

The collapse occurs in both weak and strong magnetic-field regimes, producing a reentrant behavior; in the strong-field regime, increasing $s=\pm1$4 can stabilize the collapse rather than suppress it (Matsushita et al., 2020). The collapse is tied to exceptional-point topology, specifically to a vortex texture and a winding number of the complex energy spectrum (Matsushita et al., 2020).

4. Weyl Landau levels, axial fields, and chirality imbalance

In Weyl systems, the most distinctive non-Hermitian effects concern the zeroth Landau level and the global chirality constraint. Under an axial field $s=\pm1$5, introduced through $s=\pm1$6 with $s=\pm1$7, the effective field at node $s=\pm1$8 is $s=\pm1$9, and the pseudo-Landau spectrum becomes

$\lambda$0

with zeroth level

$\lambda$1

For a pure axial field, $\lambda$2, this simplifies to

$\lambda$3

so both Weyl nodes co-propagate (Vaidya et al., 30 May 2026). In a finite lattice, the missing counter-propagating channel is not absent; it is transferred to boundary-localized Fermi-arc states (Vaidya et al., 30 May 2026).

The key non-Hermitian ingredient in the photonic realization is radiative boundary loss. This is modeled by

$\lambda$4

Bulk pseudo-Landau states have negligible overlap with boundary sites and remain long-lived, whereas surface Fermi-arc states acquire large negative imaginary parts $\lambda$5 (Vaidya et al., 30 May 2026). In transmission spectroscopy, each mode contributes a Lorentzian with width $\lambda$6, so sufficiently lossy surface branches become too broad to resolve and effectively disappear from the observable spectrum. The experimentally visible zeroth-level sector then exhibits an apparent net chirality even though the full closed Hermitian accounting remains balanced (Vaidya et al., 30 May 2026).

This distinction between full-spectrum balance and observable-sector imbalance is central. A homogeneous synthetic magnetic field in the photonic Weyl system produces the expected pair of chirality-balanced zeroth Landau levels, while an axial field spatially separates the compensating channels into co-propagating bulk pseudo-Landau levels and boundary-localized surface states. Reducing the boundary loss by adding a partial Bragg reflector narrows the surface branch and restores it, where it hybridizes with the bulk zeroth level through an avoided crossing (Vaidya et al., 30 May 2026). The experiment therefore does not contradict the Nielsen–Ninomiya theorem; rather, it shows that non-Hermitian openness can hide one compensating sector from the long-lived measured spectrum (Vaidya et al., 30 May 2026).

A closely related surface phenomenon occurs in non-Hermitian Weyl semimetals with continuum Landau modes. Under a perpendicular magnetic field, the surface states can collapse onto one boundary and form a continuous family of localized modes. Only one surface hosts normalizable modes, yielding a net mode imbalance

$\lambda$7

and a surface current

$\lambda$8

identified with the non-Hermitian chiral magnetic effect (Lin et al., 5 Sep 2025).

5. Continuum Landau modes and the failure of discrete quantization

The most radical departure from Hermitian Landau quantization is the emergence of continuum Landau modes (CLMs): Gaussian-localized states with a continuous complex spectrum. In the two-dimensional non-Hermitian continuum model

$\lambda$9

exact eigenfunctions are Gaussian wavepackets

$$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$0

with

$$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$1

Normalizability requires $$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$2, but within that condition the energy is not quantized: varying $$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$3 and the center $$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$4 fills the complex energy plane (Wang et al., 2022). This directly overturns the Hermitian principle that bound states must have discrete energies (Wang et al., 2022).

In the non-Hermitian Weyl surface problem, the CLM Hamiltonian takes the form

$$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$5

with Gaussian eigenfunctions centered at $$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$6 and controlled by

$$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$7

Here normalizability requires $$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$8, so only one surface supports bound CLMs (Lin et al., 5 Sep 2025). Their localization length scales as

$$\mathbf A(\hat x,\hat y)=\tfrac12 B(\hat x\,\mathbf e_y-\hat y\,\mathbf e_x),\qquad B\in\mathbb C,$$9

matching the scaling required by the anomaly equation in that setting (Lin et al., 5 Sep 2025).

CLMs preserve one visual hallmark of Landau physics—Gaussian spatial localization—but destroy the usual equivalence between localization and discrete quantization. Their existence clarifies that non-Hermitian Landau physics is not limited to complexifying the standard ladder; it can also dissolve the ladder entirely while retaining magnetic localization.

6. Lattice, circuit, and photonic realizations

The topic has developed through several experimentally oriented platforms. In the one-dimensional Si/SiO$$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$0 multilayer synthetic Weyl semimetal, two synthetic momenta $$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$1 and $$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$2 are encoded through layer-thickness modulations, and transmission as a function of $$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$3 maps the Weyl bands (Vaidya et al., 30 May 2026). A homogeneous synthetic field is obtained by choosing $$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$4 to vary linearly with layer index, revealing two zeroth Landau branches with opposite slopes. An axial field is realized by modulating $$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$5, which yields co-propagating bulk zeroth pseudo-Landau levels; the hidden counter-propagating surface channel is recovered by adding a partial Bragg reflector that reduces boundary leakage (Vaidya et al., 30 May 2026).

Non-Hermitian electric circuits provide a second major realization. In the exceptional Dirac/Weyl circuit proposal, the imaginary term $$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$6 originates from connecting each $$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$7–$$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$8 pair by a resistor $$\hat H=\frac{1}{2m}\Bigl(\hat{\mathbf p}-\frac{q}{c}\mathbf A\Bigr)^2,$$9, with $\omega_c=qB/(mc)$0 (Zhang et al., 2019). A pseudo-magnetic field is implemented by grading inductances so that one hopping parameter varies linearly with position, and the resulting admittance matrix satisfies $\omega_c=qB/(mc)$1 (Zhang et al., 2019). The proposed measurements are impedance scans, voltage tomography, and edge-selective excitation. These access flat Landau bands, sublattice polarization of the zeroth level, edge states protected by a non-Hermitian energy-reflection symmetry, and the characteristic nodeless probability distribution that results from a complex displacement of the harmonic-oscillator orbit (Zhang et al., 2019).

A more recent circuit experiment targets high-order Landau modes. In a $\omega_c=qB/(mc)$2 honeycomb network printed on a PCB, inhomogeneous coupling realizes a pseudomagnetic field, a linear onsite gradient realizes a pseudoelectric field, and non-reciprocal coupling produces an imaginary momentum (Wang et al., 15 Apr 2026). A Vector Network Analyzer measures the multiport $\omega_c=qB/(mc)$3-matrix and reconstructs the admittance Laplacian, while direct voltage measurements map eigenmode profiles (Wang et al., 15 Apr 2026). With only the pseudomagnetic field, high-order modes remain degenerate and delocalized in $\omega_c=qB/(mc)$4; with an added pseudoelectric field, the degeneracy is lifted; with added non-reciprocity, each high-order mode collapses to a single peak in $\omega_c=qB/(mc)$5, yielding the reported “multi-frequency single-peak localization” (Wang et al., 15 Apr 2026).

On the lattice-theory side, a non-Hermitian Harper–Hofstadter model with complex Peierls phases reproduces the continuum theory of complex magnetic fields (Montag et al., 22 May 2026). Exact diagonalization on a finite square lattice yields clustered eigenvalues

$\omega_c=qB/(mc)$6

with near-flat bands and finite-size splitting in the imaginary part caused by non-reciprocal boundary amplification (Montag et al., 22 May 2026). Gaussian wave-packet dynamics follow semiclassical equations with a complex Lorentz force, producing logarithmic spirals and drifting skipping orbits (Montag et al., 22 May 2026).

7. Conceptual issues, invariants, and common misconceptions

A recurring misconception is that non-Hermiticity necessarily makes Landau levels complex. Several constructions demonstrate the opposite. The gain/loss-induced pseudo-magnetic Landau levels in the symmetry-unbroken regime remain strictly real because pseudo-Hermiticity and anti-PT symmetry stabilize diabolic rather than exceptional Dirac points (Xue et al., 2019). The exceptional-point circuit models also yield purely real Landau levels even though the zero-field system contains exceptional points and rings (Zhang et al., 2019). Non-Hermitian Landau quantization therefore does not reduce to “complex energies under a magnetic field.”

A second misconception is that any apparent chirality imbalance in Weyl Landau levels violates lattice no-go theorems. In the photonic Weyl experiment, the imbalance is explicitly an observable-sector effect produced by selective boundary loss: the compensating channel is present in the full problem but is too short-lived to appear as a resolvable spectral feature (Vaidya et al., 30 May 2026). This is a relaxation of the chirality constraint in the long-lived spectrum, not a literal violation of the Nielsen–Ninomiya theorem (Vaidya et al., 30 May 2026).

A third issue concerns gauge choice. When the magnetic field is complex, gauge transformations are non-unitary:

$\omega_c=qB/(mc)$7

As a result, changing gauge can affect the normalizability of the transformed right and left eigenstates; in the symmetric-gauge construction, square integrability is maintained provided $\omega_c=qB/(mc)$8, whereas in the Landau gauge one may need the stronger condition $\omega_c=qB/(mc)$9 (Montag et al., 22 May 2026). Non-Hermitian gauge transformations thus have physical consequences absent in the Hermitian theory.

A fourth issue concerns the relation between non-Hermitian Landau physics and the non-Hermitian skin effect. Magnetic confinement can suppress skin accumulation in some models: the exceptional circuit Landau levels were identified precisely as a setting that can avoid the non-Hermitian skin effect (Zhang et al., 2019). In other settings the two phenomena coexist. Under full open boundaries in the non-Hermitian Weyl model with continuum Landau surface states, some point-gap eigenstates remain CLMs on the bottom face while the remainder are $(\hat a,\hat a^\S)$0-skin modes (Lin et al., 5 Sep 2025).

Finally, topological response can remain unexpectedly rigid. In the two-dimensional Schrödinger problem with the imaginary linear Stark term, the lowest-Landau-level Hall conductivity is

$(\hat a,\hat a^\S)$1

exactly the quantized value despite the non-Hermitian deformation (Lizarraga et al., 18 Feb 2025). By contrast, in disordered small-gap insulators and Dirac systems, non-Hermiticity strongly reshapes oscillatory observables: the effective mass governing thermal damping can be set by scattering rates, while spectrum collapse is controlled by the competition between Landau splitting and spin-dependent scattering (Shen et al., 2018, Matsushita et al., 2020). Taken together, these results show that non-Hermitian Landau levels form a heterogeneous but coherent field in which the central questions are no longer limited to quantization alone, but extend to spectral reality, biorthogonality, observability, boundary selectivity, and topological response.