Non-Hermitian Aharonov-Bohm Effect

• Non-Hermitian Aharonov-Bohm effect is characterized by the interplay of magnetic flux interference with gain, loss, and complex gauge fields.
• It demonstrates gauge-invariant quantum transport in systems with complex dot energies and nonreciprocal hopping, revealing asymmetric Fano profiles.
• Experimental platforms like photonic lattices and electrical circuits exploit these effects to explore exceptional points, AB caging, and the non-Hermitian skin effect.

The non-Hermitian Aharonov-Bohm effect is the extension of Aharonov-Bohm (AB) physics to non-Hermitian systems, where magnetic-flux-induced interference coexists with gain, loss, nonreciprocal hopping, exceptional points, or imaginary gauge fields. In conventional AB settings, a charged particle acquires a phase shift when encircling magnetic flux, and conductance oscillates as a function of flux. In non-Hermitian settings, that basic interference structure persists in some observables, but the response can also involve amplification or decay of wave-function norm, complex orbital magnetic moments, coalesced flat bands, and boundary accumulation associated with the non-Hermitian skin effect (NHSE) (Zeng et al., 2016, Ozawa et al., 2023, Alon et al., 5 Jun 2025, Zhang et al., 2020, Li et al., 2024).

1. Conceptual scope and gauge structure

The term encompasses several distinct but related phenomena. In mesoscopic transport, non-Hermiticity is introduced by assigning complex energies to quantum dots embedded in an AB ring, with the imaginary parts representing physical gain or loss. In lattice systems with an imaginary magnetic field, the AB effect no longer appears only as a phase; the norm of a transported wave packet changes after a closed loop. In semiclassical band theory, a complex orbital magnetic moment acquires an imaginary component tied to gain or loss. In flat-band photonic and rhombic lattices, synthetic flux and non-Hermitian couplings generate AB caging, coalesced flat bands, or disorder-enabled NHSE (Zeng et al., 2016, Ozawa et al., 2023).

It is sometimes assumed that non-Hermiticity simply destroys the usual AB gauge freedom. The available analyses are more specific. In the non-Hermitian AB ring, the transmission probability is independent of how the flux phase factor is allocated among tunnel couplings, as in the Hermitian case. In the lattice with an imaginary magnetic field, the Landau and symmetric gauges are related by a generalized gauge transformation that changes amplitudes as well as phases, and the total norm change after a closed loop is gauge invariant even though intermediate amplitudes are gauge dependent (Zeng et al., 2016, Ozawa et al., 2023).

Setting	Non-Hermitian ingredient	AB manifestation
AB ring with two quantum dots	Complex dot energies $E_{u,d}=\epsilon_{u,d}+i\gamma_{u,d}$	Gauge-invariant transmission; Fano conductance profiles
Two-dimensional lattice	Imaginary magnetic field $B=i\mathcal{B}$	Closed-loop norm amplification or decay
Non-Hermitian periodic band	Complex orbital magnetic moment	Magnetic-field-tunable gain/loss response
Photonic or rhombic flat-band lattice	Exceptional points, nonreciprocal hopping, disorder	AB caging, coalesced flat bands, or NHSE

2. Mesoscopic transport in a non-Hermitian AB ring

A minimal transport realization is an AB ring threaded by magnetic flux $\phi$, with one quantum dot in each arm and two one-dimensional leads. The quantum-dot levels are complex,

$$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$

where $\gamma_{u/d}>0$ models gain and $\gamma_{u/d}<0$ models loss. The Hamiltonian is

$$\mathcal{H}=\mathcal{H}_{DQD}+\mathcal{H}_{Leads}+\mathcal{H}_T,$$

with

$$\mathcal{H}_{DQD}=E_u f_u^\dagger f_u+E_d f_d^\dagger f_d,$$

$$\mathcal{H}_{Leads}=-t_0\sum_j(c_j^\dagger c_{j+1}+h.c.),$$

$$\mathcal{H}_T=-\sum_{n=u,d}(t_{nL}c_{-1}^\dagger f_n+t_{nR}c_1^\dagger f_n+h.c.).$$

Two canonical allocations of the AB phase were analyzed. In the symmetric allocation,

$B=i\mathcal{B}$0

and in the asymmetric allocation,

$B=i\mathcal{B}$1

The symmetric choice is $B=i\mathcal{B}$2-symmetric if the imaginary parts of $B=i\mathcal{B}$3 and $B=i\mathcal{B}$4 are equal in magnitude and opposite in sign; the asymmetric choice is $B=i\mathcal{B}$5-asymmetric but physically allowed (Zeng et al., 2016).

The transmission amplitudes for the two allocations differ only by an overall phase,

$B=i\mathcal{B}$6

so the transmission probability

$B=i\mathcal{B}$7

is independent of the way the AB phase is split among the tunnelings. This establishes that the gauge-invariance of transport under AB phase reallocation survives the inclusion of gain and loss. The conductance is

$B=i\mathcal{B}$8

The environmental broadening of the two channels is controlled by $B=i\mathcal{B}$9 and $\phi$0, with $\phi$1. When $\phi$2, the two channels acquire different broadenings; the broader channel acts as a background and the narrower channel as a discrete resonance, producing an asymmetric Fano profile in the conductance spectrum. In this formulation, gain and loss directly tune the Fano lineshape. For large gain, the conductance can exceed the usual quantum maximal value $\phi$3, evidencing the non-conservation of probability inherent in non-Hermitian physics (Zeng et al., 2016).

3. Imaginary magnetic fields and norm-changing AB transport

A distinct realization arises in a two-dimensional square lattice with a uniform imaginary magnetic field. The hopping obeys a generalized Peierls substitution,

$\phi$4

with gauge-dependent phases $\phi$5. For a real magnetic field the Peierls factors are unitary. For $\phi$6, they acquire modulus different from unity, making the model non-Hermitian and generating non-reciprocal, exponentially growing or decaying hopping amplitudes as a function of position (Ozawa et al., 2023).

The paper considers both the Landau gauge,

$\phi$7

and the symmetric gauge,

$\phi$8

For imaginary $\phi$9, the two gauges are connected not by a unitary phase rotation but by a diagonal similarity transformation,

$$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$0

This generalized gauge transformation changes not only the phase but also the amplitude of the wave function (Ozawa et al., 2023).

In this setting the non-Hermitian AB effect is most directly visible in adiabatic wave-packet transport. For a closed path $$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$1, the wave function transforms as

$$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$2

Because the vector potential is purely imaginary, the factor $$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$3 is real. For a uniform field, the norm change depends on the enclosed area: $$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$4 Reversing the orientation of the loop reverses amplification and decay. The total change of norm is a gauge-invariant observable determined by the imaginary magnetic flux enclosed by the path, whereas intermediate values of the norm along the path are gauge dependent (Ozawa et al., 2023).

The same model also exhibits spectral behavior absent in Hermitian lattice models with real magnetic fields. The energy spectrum does not converge as the lattice size is made larger because of intrinsic nonperiodicity, but it does converge when one fixes the length of one side and makes the other side longer. That asymptotic behavior is interpreted within non-Bloch band theory. Proposed experimental platforms include mechanical metamaterials, electrical circuits, photonic lattices, and Floquet-engineered systems (Ozawa et al., 2023).

4. Semiclassical formulation and the complex magnetic moment

A more general formulation appears in the semiclassical theory of electrons in a non-Hermitian periodic system subject to perturbations varying slowly in space and time. The theory uses left and right Bloch eigenstates, $$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$5 and $$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$6, together with the corresponding Berry connections $$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$7 and $$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$8. For a single-band wavepacket,

$$E_u=\epsilon_u+i\gamma_u,\qquad E_d=\epsilon_d+i\gamma_d,$$9

the effective energy is

$\gamma_{u/d}>0$0

where

$\gamma_{u/d}>0$1

is the non-Hermitian projection operator onto the band (Alon et al., 5 Jun 2025).

For a uniform external magnetic field, the wavepacket energy acquires a Zeeman-like contribution that defines a complex orbital magnetic moment. In the notation of the paper,

$\gamma_{u/d}>0$2

The last term is the generalized orbital magnetic moment. Its real part produces the conventional orbital magnetization energy shift, while its imaginary part shifts the imaginary part of the energy and therefore the amplification or decay rate of the wavepacket (Alon et al., 5 Jun 2025).

The operator that first appears in the magnetic Hamiltonian is

$\gamma_{u/d}>0$3

Because $\gamma_{u/d}>0$4, this operator is generally not Hermitian. A physically meaningful angular momentum is instead obtained from the non-Hermitian Ehrenfest theorem via the Hermitian velocity operator

$\gamma_{u/d}>0$5

which gives

$\gamma_{u/d}>0$6

The relation

$\gamma_{u/d}>0$7

identifies the real part of $\gamma_{u/d}>0$8 as the physical angular momentum. The imaginary part is an “imaginary angular momentum” that generates dissipation proportional to the AB phase gained by rotating about the wavepacket center, i.e. proportional to the total magnetic flux through the wavepacket. In this semiclassical framework, the non-Hermitian AB effect is therefore the conversion of AB phase accumulation into gain or loss (Alon et al., 5 Jun 2025).

5. Exceptional points and non-Hermitian AB caging

AB caging is the flat-band limit of flux-induced destructive interference, where arbitrary excitations remain localized and do not spread. A non-Hermitian extension was proposed in photonic crystals and coupled waveguides by combining synthetic magnetic flux with gain and loss, or by exploiting the unidirectionality of an exceptional point (EP). In this setting, the spectrum of the non-Hermitian AB cage is entirely constituted by coalesced flat bands, rather than distinct orthogonal flat bands as in the Hermitian case (Zhang et al., 2020).

The model is a one-dimensional chain of resonators or waveguides with reciprocal hopping $\gamma_{u/d}>0$9, asymmetric couplings $\gamma_{u/d}<0$0, and alternating on-site gain/loss $\gamma_{u/d}<0$1. At the EP, when $\gamma_{u/d}<0$2, one of the asymmetric inter-cell couplings vanishes and the coupling becomes completely unidirectional. This produces a non-Hermitian caging mechanism additional to ordinary flux-induced interference: excitations cannot tunnel back out once they enter inward-coupled cells. A second mechanism uses destructive interference of synthetic magnetic flux together with gain and loss at the EP, such that the flat-band eigenstates survive as coalesced states (Zhang et al., 2020).

The resulting dynamics remain localized but are no longer unitary. Arbitrary light excitation is still confined, although the localization area may alter. Because the Hamiltonian is defective at the EP, localized excitation can be static, oscillatory, or grow polynomially with time. When the initial state overlaps generalized eigenstates, the intensity can grow quadratically. The same basic phenomenon can be realized not only in active structures with true gain and loss but also in passive photonic crystals, including coupled waveguides, by engineering losses so that the non-Hermitian AB cage is reproduced after shifting the imaginary part of the energy (Zhang et al., 2020).

6. Flat bands, disorder, and the non-Hermitian skin effect

A further development appears in the one-dimensional rhombic lattice with three sublattices per unit cell, magnetic flux $\gamma_{u/d}<0$3 in each plaquette, nonreciprocal hopping, and disorder. In the clean Hermitian limit, applying flux $\gamma_{u/d}<0$4 renders all bands perfectly flat and creates an AB cage, so that eigenstates are compactly localized. Nonreciprocal hopping introduces point-gap topology in the complex spectrum under periodic boundary conditions, and under open boundary conditions this topology is associated with the NHSE, in which an extensive number of eigenstates accumulate at a boundary (Li et al., 2024).

The point-gap topology is characterized by the winding number

$\gamma_{u/d}<0$5

In the presence of the AB cage, destructive interference suppresses the skin effect. Disorder can remove that protection, but the outcome depends sharply on the disorder structure. For anti-symmetric disorder, $\gamma_{u/d}<0$6, nonreciprocity drastically changes the localization physics. With Bernoulli anti-symmetric disorder, even very small nonreciprocity makes bulk eigenstates completely delocalized and collapsed to the boundary; the NHSE persists regardless of disorder strength. With random anti-symmetric disorder, nonreciprocal hopping can also induce NHSE and complete delocalization, although for large disorder a coexistence of localized bulk states and delocalized skin states can occur before reentrant skin behavior dominates. By contrast, symmetric disorder does not break the AB cage in the same way, and the NHSE does not emerge (Li et al., 2024).

These results sharpen the meaning of the non-Hermitian AB effect in flat-band systems. In one limit, non-Hermiticity and flux cooperate to preserve caging through coalesced flat bands at an EP. In another, nonreciprocity, point-gap topology, and correlated disorder destroy the cage and drive boundary accumulation. The proposed experimental platform is the electrical circuit, where capacitors and inductors implement hoppings and onsite terms, negative impedance converters realize nonreciprocal hopping, correlated disorder is introduced through randomized elements, and the flux $\gamma_{u/d}<0$7 is emulated by crossing wiring (Li et al., 2024).