Chiral Magnetic Skin Effect

• Chiral Magnetic Skin Effect is a phenomenon where chirality and magnetic bias induce unconventional boundary accumulation and altered electromagnetic penetration modes.
• In Weyl semimetals, incorporating the chiral magnetic current into Maxwell-Chern-Simons electrodynamics produces a CME-controlled skin depth and a capacitive low-frequency response.
• In non-Hermitian and magnonic systems, directional dissipative interactions lead to long-range edge and corner localization, offering robust platforms for photonic and quantum Hall applications.

The literature summarized here uses the expression chiral magnetic skin effect in at least two technical settings. In anomaly-driven transport, it denotes the unconventional electromagnetic skin effect that appears when a chiral magnetic current $\vec{J}_{ch}=\sigma_{ch}\vec{B}$ is incorporated into Maxwell-Chern-Simons electrodynamics of a Weyl semimetal, producing a chiral magnetic length, an altered skin depth, and magnetic-field-dependent plasma physics (Sa, 2016). In non-Hermitian and topological-wave settings, the same expression is used more broadly for boundary accumulation of chiral or magnetically biased excitations—such as magnons or quantum Hall edge states—when chirality, structured loss, or nonreciprocity generates a non-Hermitian skin effect rather than an ordinary extended boundary mode (Yu et al., 2022, Liu et al., 2023).

1. Anomaly-induced transport as the underlying chiral magnetic response

The anomaly-based foundation of the subject is the chiral magnetic effect (CME): in a medium with nonzero chirality, a magnetic field induces an electric current parallel to the field. In the hydrodynamical treatment of an ideal chiral fluid, the relevant macroscopic variables are a vector current $J^\mu$, an axial current $J_5^\mu$, chemical potentials $\mu$ and $\mu_5$, temperature $T$, and a fluid 4-velocity $u^\mu$ satisfying $u_\mu u^\mu=-1$. The hydrodynamic equations are

$$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$

so the vector current is conserved while the axial current is anomalous (Sadofyev et al., 2010).

Within that framework, the constitutive relations are expanded as

$T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$

$J^\mu$0

with first-derivative corrections parameterized by vorticity and magnetic-field terms. The entropy current is correspondingly modified to

$J^\mu$1

and the requirement $J^\mu$2 fixes the allowed transport coefficients. The magnetic contribution to the vector current is

$J^\mu$3

with

$J^\mu$4

For a single fermion species this reproduces

$J^\mu$5

This result is significant because it places the CME in a macroscopic hydrodynamic setting rather than treating it as only a microscopic single-particle effect. The leading coefficient agrees with the free-fermion expression at linear order in $J^\mu$6, and the paper interprets that agreement as a hydrodynamic non-renormalization theorem at first order in chemical potentials (Sadofyev et al., 2010). In the present context, this anomaly-controlled current is the ingredient that later enters electromagnetic skin-depth problems in Weyl semimetals.

2. Maxwell-Chern-Simons electrodynamics and the electrodynamic skin effect in Weyl semimetals

In the Weyl-semimetal formulation, the CME is tied to a finite energy splitting between the two Weyl nodes. The current is written as

$J^\mu$7

in contrast with the ordinary Ohmic contribution

$J^\mu$8

For the model discussed in the literature, the CME conductivity is

$J^\mu$9

so the node-energy splitting is the control parameter for the chiral magnetic response (Sa, 2016).

The electrodynamics is formulated through Maxwell equations augmented by the axion/Chern-Simons structure. In the simplified form used for the CME problem,

$J_5^\mu$0

$J_5^\mu$1

with anomaly equations

$J_5^\mu$2

and valley currents

$J_5^\mu$3

Combining these relations yields the wave-equation dispersion relation

$J_5^\mu$4

The skin-effect regime is obtained in the good-conductor and small-CME expansion

$J_5^\mu$5

For the relevant branch,

$J_5^\mu$6

so the skin depth is

$J_5^\mu$7

The distinctive length scale is the chiral magnetic length

$J_5^\mu$8

for which

$J_5^\mu$9

This replaces the standard Drude-controlled picture of metallic skin depth by a CME-controlled low-frequency penetration scale.

The same formalism predicts an unconventional field relation,

$\mu$0

so the magnetic and electric fields are phase shifted by approximately $\mu$1. The paper interprets this as a purely capacitive response in the low-frequency regime. At higher frequencies, the plasma frequency becomes magnetic-field dependent: $\mu$2 In the strong-field limit this gives $\mu$3. The same analysis suggests that Weyl semimetals may appear transparent in the optical or visible range because the estimated plasma frequency lies below optical frequencies (Sa, 2016). The stated regime of validity is the long-wavelength limit $\mu$4, with diffusion current neglected and the $\mu$5-dependent axion terms dropped.

3. Long-range chiral magnon coupling and non-Hermitian skin accumulation in magnetic arrays

A distinct usage of the phrase arises in non-Hermitian magnonics. In a periodic array of magnetic nanowires deposited on a thin magnetic film, the nanowire Kittel modes do not couple directly because $\mu$6; instead, they interact through traveling spin waves in the film. The linearized Hamiltonian is

$\mu$7

with film-spin-wave dispersion

$\mu$8

The coupling

$\mu$9

is chiral when the film and wire magnetizations are parallel or antiparallel: for $\mu_5$0, only right-going spin waves couple and $\mu_5$1; for $\mu_5$2, only left-going spin waves couple and $\mu_5$3 (Yu et al., 2022).

After the film modes are integrated out in the Markov approximation, the mediated interaction becomes long-ranged and exponentially attenuated because damping shifts the pole to a complex wave vector,

$\mu_5$4

The directional rates

$\mu_5$5

encode the asymmetry. The effective Hamiltonian is

$\mu_5$6

which the paper describes as a long-range generalization of the Hatano-Nelson model.

The physical mechanism has three components: chirality makes the coupling directional, the propagation phase $\mu_5$7 introduces interference, and damping suppresses long-distance destructive interference. A central point is that coherent propagation by itself can prevent accumulation, so sufficiently strong damping is required before the non-Hermitian skin effect becomes robust. The numerical criterion is direct: if $\mu_5$8, all modes localize at the right edge; if $\mu_5$9, all modes localize at the left edge; if chirality vanishes or damping is weak, skin localization disappears (Yu et al., 2022).

The paper also gives a concrete microwave-detection proposal. A local stripline driving one wire with a field as small as

$T$0

at $T$1 GHz excites a large steady-state response in the skin-localized mode, with a reported magnetization deviation in CoFeB wires of about

$T$2

and a precession cone angle of about

$T$3

The response appears at the opposite edge of the array, and the effect is reported to remain robust against random shifts in Kittel frequencies up to $T$4. In this magnonic setting, the phrase chiral magnetic skin effect denotes edge accumulation of magnetic excitation produced by directional, damped spin-wave mediation rather than by the anomaly-driven CME of Weyl media.

4. Localization of chiral edge states by loss in quantum Hall photonics

A further extension appears in lossy quantum Hall photonics, where chiral edge states of a gyromagnetic photonic crystal remain topological yet become spatially localized to part of the boundary or to a corner. The platform is a square-lattice gyromagnetic photonic crystal with three gyromagnetic rods per unit cell, operated in the TM mode inside a parallel-plate microwave waveguide. A static magnetic field of $T$5 along $T$6 breaks time-reversal symmetry and places the system in a quantum Hall photonic phase. Structured microwave-absorbing materials then introduce loss in a directionally asymmetric way (Liu et al., 2023).

The crucial distinction is that the bulk topology remains characterized by nonzero Chern numbers,

$T$7

while the edge spectrum acquires point-gap winding in the complex-frequency plane. For direction $T$8 and gap $T$9, the winding number is

$u^\mu$0

Nonzero $u^\mu$1 drives a non-Hermitian skin effect of the edge modes. The paper therefore proposes a generalized bulk-boundary correspondence

$u^\mu$2

where $u^\mu$3 determines how many chiral edge modes exist and $u^\mu$4 determine where they localize.

Two loss geometries are analyzed. When mirror symmetry is broken only along $u^\mu$5, the system exhibits single-edge localization under double open boundaries: $u^\mu$6 Gap 2 modes localize on the right edge, while gap 3 modes localize on the left edge. When mirror symmetry is broken along both $u^\mu$7 and $u^\mu$8, localization becomes corner-like: $u^\mu$9 Gap 2 modes localize at the upper-right corner and gap 3 modes at the lower-left corner. The paper emphasizes that this is not conventional disorder-induced localization; it is a non-Hermitian skin effect generated by complex spectral winding, while the Chern topology remains intact (Liu et al., 2023).

The experimental observation is based on pump-probe microwave measurements with a source antenna placed at each corner in turn and a detection antenna mapping the field distribution. Frequencies inside the second and third band gaps are probed at $u_\mu u^\mu=-1$0 GHz and $u_\mu u^\mu=-1$1 GHz, respectively. Because loss produces trivial attenuation along the propagation path, the measured fields are processed with an artificial exponential compensation factor based on fitted decay constants; after compensation, the fields accumulate on the predicted edges or corners, independent of source location. The paper further argues that these skin modes are more robust to local defects and disorder than earlier non-Hermitian skin-effect realizations because forward and backward transport are spatially separated on opposite edges by the chiral quantum Hall structure.

5. Symmetry constraints, generalized Brillouin zones, and observables

The broader non-Hermitian literature clarifies when dissipation can and cannot produce skin localization. For a one-dimensional Hermitian system coupled locally to a bath, integrating out the bath yields an effective non-Hermitian Bloch Hamiltonian

$u_\mu u^\mu=-1$2

where the dissipative term is onsite and reciprocal rather than explicitly nonreciprocal (Yi et al., 2020). Skin modes are then analyzed through the generalized Brillouin zone (GBZ), defined by the complex-momentum roots $u_\mu u^\mu=-1$3 of

$u_\mu u^\mu=-1$4

They appear when the GBZ differs from the unit circle.

The paper establishes a no-go theorem: if the underlying Hermitian Hamiltonian has spinless time-reversal symmetry

$u_\mu u^\mu=-1$5

onsite dissipation cannot induce skin modes. In that case the symmetry constraints force the root pairing $u_\mu u^\mu=-1$6, which preserves the unit circle. By contrast, if the underlying system has spinful time-reversal symmetry

$u_\mu u^\mu=-1$7

skin modes may occur under specific inversion-symmetry conditions. The paper lists three representative possibilities: the Hermitian system breaks inversion symmetry; or it preserves inversion symmetry but $u_\mu u^\mu=-1$8; or it preserves inversion with $u_\mu u^\mu=-1$9 but $$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$0. The spinful case allows a $$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$1 skin effect in which Kramers partners localize on opposite boundaries (Yi et al., 2020).

The same work also addresses detection. It states explicitly that the local density of states is not a good detector of skin modes, because in the thermodynamic limit the right and left eigenfunctions contribute factors proportional to $$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$2 and $$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$3, whose product largely cancels. Instead, the proposed observable is the chiral tunneling effect, defined through

$$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$4

Comparing $$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$5 and $$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$6 reveals a strong directional asymmetry once skin modes appear: tunneling is enhanced toward the boundary where the skin modes accumulate and suppressed in the opposite direction. The asymmetry scales as

$$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$7

so it grows exponentially with system size and the localization parameter. This supplies a general diagnostic framework for dissipation-induced chiral skin phenomena, including magnetic or photonic realizations.

6. Flux-threaded, interacting extensions and the scope of the term

Recent work on a nonreciprocal bosonic two-leg ladder with artificial magnetic flux shows that the boundary-accumulation problem extends beyond linear eigenmode localization to nonlinear mean-field dynamics. The model combines artificial magnetic flux $$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$8, nonreciprocal hopping $$\partial_\mu T^{\mu\nu}=F^{\nu\lambda}J_\lambda,\qquad \partial_\mu J^\mu=0,\qquad \partial_\mu J_5^\mu=C\,E_\mu B^\mu,$$9, and interaction $T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$0: $T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$1 In the Hermitian limit $T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$2, flux produces chiral motion, but increasing $T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$3 drives nonlinear self-trapping and breaks that chiral dynamics (Chen et al., 2024).

With nonreciprocity present, the paper finds a transition between chiral dynamics and antichiral dynamics. Here antichiral motion means that the currents on both legs flow in the same direction, because the skin currents under open boundary conditions dominate the flux-induced opposite-leg bias. Skin accumulation is quantified by the time-averaged centers of mass

$T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$4

and the paper reports that for $T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$5, skin dynamics appears on both legs and is almost independent of $T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$6. The associated Bloch winding number

$T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$7

is nonzero whenever $T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$8, zero only for $T^{\mu\nu}=w\,u^\mu u^\nu + p\,g^{\mu\nu}+\cdots,$9, and essentially independent of flux. The explicit conclusion is that, in this ladder model, flux does not suppress the non-Hermitian skin effect (Chen et al., 2024).

The same model exhibits trap-skin dynamics, defined as the coexistence of self-trapping and skin accumulation. The diagnostics

$J^\mu$00

organize the parameter plane into regions of skin dynamics, self-trapping dynamics, and trap-skin dynamics. This broadens the conceptual scope of the subject. A plausible implication is that the phrase chiral magnetic skin effect now functions as an umbrella term for several boundary-accumulation phenomena in which chirality, magnetic bias, flux, or anomaly physics reshapes either electromagnetic penetration or the localization of collective modes. The common structure is not a single microscopic mechanism, but the coupling of chirality-related transport to boundary-selective accumulation.