Axial Hall Effect

• Axial Hall effect is a family of transverse transport phenomena characterized by an axial variable, such as chiral currents or pseudospin, driving the response in various materials.
• It encompasses distinct mechanisms including boundary-localized currents in Dirac semimetals, strain-induced valley responses, and universal conductivity ratios in holographic models.
• Key insights reveal that the phenomenon's manifestations depend on the physical context, influencing both experimental detection and theoretical modeling.

Searching arXiv for recent and foundational papers on the axial Hall effect and related usages of the term. {"query":"Axial Hall effect arXiv axial current Dirac semimetal chiral Hall effect Weyl semimetal", "max_results": 10} Axial Hall effect denotes a family of Hall-like transverse transport phenomena in which the transported quantity, the driving field, or the symmetry-breaking control parameter is “axial.” In the relativistic transport literature, the axial variable is the Noether current of chiral symmetry or the right-minus-left current; in Weyl and Dirac semimetals it can also arise from an axial gauge field that couples with opposite sign to opposite valleys; and in crystalline settings it may refer instead to a ferroaxial moment or to an axial pseudospin degree of freedom. The term therefore does not describe a single universal observable, but rather several distinct transverse responses that share an axial structure in their constitutive relations, symmetry properties, or microscopic origin (Okuma et al., 2016, Pu et al., 2014, Copetti et al., 2016, Araki, 2018, Hayami et al., 2023, Xu et al., 16 Sep 2025).

1. Terminology, definitions, and scope

In a massless Dirac-fermion system, the axial quantity is defined by chiral symmetry. The Lagrangian is invariant under the chiral rotation $\psi \to e^{i\gamma_5\theta}\psi$, and the corresponding Noether current is

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$

Here $\rho^{05}$ measures the imbalance of right- and left-handed fermions, and $j^{05}$ is the flow of that imbalance. In the massless approximation used there, this is a conserved current (Okuma et al., 2016).

In chiral plasma transport, the axial current is usually written in the right/left basis. One defines

$$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$

and the Hall conductivities

$$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$

In this usage, the axial Hall effect is a transverse axial current generated in a chiral medium at nonzero axial chemical potential $\mu_5$ (Pu et al., 2014).

In holographic Weyl-semimetal models, the relevant observable is the axial Hall conductivity $\sigma_A \equiv \sigma^5_{xy}$, extracted from a Kubo formula for the consistent axial current. In strained Dirac semimetals, by contrast, the axial object is the valley-odd gauge field $A_5$ and its curl $B_5$, which generate a valley or spin Hall flow with no net charge Hall current (Copetti et al., 2016, Araki, 2018). In ferroaxial metals and altermagnetic Lieb lattices, “axial” refers instead to crystalline axiality: a ferroaxial moment $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$0 in one case and an axial pseudospin or axial index in the other (Hayami et al., 2023, Xu et al., 16 Sep 2025).

Setting	Axial variable	Characteristic Hall response
Massless Dirac semimetal	$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$1	$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$2 near a boundary
Chiral plasma	$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$3, $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$4	$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$5 for $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$6
Holographic Weyl semimetal	$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$7	$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$8 after axial-field renormalization
Strained topological Dirac semimetal	$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$9, $\rho^{05}$0, $\rho^{05}$1	$\rho^{05}$2
Ferroaxial metal	ferroaxial moment $\rho^{05}$3	$\rho^{05}$4 at linear order in $\rho^{05}$5
Altermagnetic Lieb lattice	axial index $\rho^{05}$6	$\rho^{05}$7

2. Boundary axial Hall response in massless Dirac semimetals

A concrete axial Hall response was derived for a three-dimensional massless Dirac fermion model describing a Dirac semimetal. Starting from a quantum-kinetic equation for the density matrix and projecting onto the generators of the Clifford algebra, one obtains a closed diffusion system for eight local degrees of freedom: the charge density $\rho^{05}$8, the axial charge $\rho^{05}$9, the vector density $j^{05}$0, and the spin density $j^{05}$1. The diffusion equations are interdependent, and the applied electric field enters only in the combination $j^{05}$2, but through the $j^{05}$3- and $j^{05}$4-dependent couplings it induces spin and axial-charge dynamics (Okuma et al., 2016).

For a steady state in the half-space $j^{05}$5 with $j^{05}$6 and boundary conditions $j^{05}$7 at $j^{05}$8, the bulk solution has $j^{05}$9 aligned with the electric field and a boundary-localized spin accumulation. The explicit nonzero components are

$$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$0

$$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$1

with all other components vanishing. In the same steady state, the axial current follows from the continuity form of the axial-charge diffusion equation,

$$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$2

Since $$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$3 in the steady state,

$$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$4

The axial current therefore flows perpendicular to the electric current and is localized near the boundary over a distance of order $$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$5 (Okuma et al., 2016).

The microscopic origin is not the naïve spin-current operator. In that model the conventional spin-current operator $$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$6 is identically zero, so the usual Kubo formula would predict zero spin Hall conductivity. The nonzero transverse response instead emerges from the diffusion coupling among $$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$7, $$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$8, $$J_v^i = (J_R^i+J_L^i)/2, \qquad J_a^i = (J_R^i-J_L^i)/2,$$9, and $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$0, together with $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$1-independent vertex-correction terms generated by commutators of the $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$2-matrices in the gradient expansion. The extra drift $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$3 produces a bulk polarization $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$4, whose boundary curl drives $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$5, and hence $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$6 (Okuma et al., 2016).

This mechanism sharply distinguishes the effect from the conventional spin Hall effect. The current is a charge-neutral flow of chiral charge rather than a spin current, it is strictly conserved in the massless theory with no anomaly in the approximation used, and it is boundary-localized rather than a bulk transport coefficient in the ordinary sense (Okuma et al., 2016).

3. Chiral Hall effect in parity-odd media and anomaly-related Hall observables

In a parity-odd chiral medium at finite temperature, an axial Hall current is generated in the presence of orthogonal electric and magnetic fields provided there is a nonzero axial chemical potential $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$7. Taking $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$8 and $$\sigma_v^H=(\sigma_R^H+\sigma_L^H)/2, \qquad \sigma_a^H=(\sigma_R^H-\sigma_L^H)/2 \equiv \sigma_5^H.$$9, the transverse current is

$\mu_5$0

Microscopically, the effect is interaction-driven rather than anomaly-driven and exists only when $\mu_5$1. Its covariant constitutive form is

$\mu_5$2

and the corresponding Kubo formula is

$\mu_5$3

Under parity, $\mu_5$4 is even while $\mu_5$5 is odd, leading at small chemical potentials to the scaling expectations

$\mu_5$6

in weak fields, and to the explicit estimates

$\mu_5$7

with dimensionless $\mu_5$8's of order unity (Pu et al., 2014).

The same work also derived coupled diffusion-wave equations for vector and axial fluctuations. Their dispersion relation has two modes,

$\mu_5$9

and in the holographic Sakai–Sugimoto model the Hall part produces non-dissipative behavior of the “$\sigma_A \equiv \sigma^5_{xy}$0” mode at zero vector density. The resulting Hall density waves and chiral electric waves were proposed as a mechanism for rapidity-dependent charge asymmetries in asymmetric heavy-ion collisions such as Cu+Au, where large orthogonal $\sigma_A \equiv \sigma^5_{xy}$1 and $\sigma_A \equiv \sigma^5_{xy}$2 fields coexist with event-by-event $\sigma_A \equiv \sigma^5_{xy}$3 (Pu et al., 2014).

A distinct but related anomaly-associated Hall phenomenology was reported in TaAs. There, the axial anomaly among Weyl nodes gives

$\sigma_A \equiv \sigma^5_{xy}$4

Experimentally, a pronounced planar Hall effect with $\sigma_A \equiv \sigma^5_{xy}$5, an anomalous planar Hall signal odd in both $\sigma_A \equiv \sigma^5_{xy}$6 and $\sigma_A \equiv \sigma^5_{xy}$7, and a negative longitudinal magnetoresistivity were observed below the quantum limit. All three signatures are suppressed above the same critical field where quantum oscillations reveal a Fermi-surface reconstruction and hysteresis, and the data were interpreted as an “axial Hall effect” tied to the existence of ungapped Weyl nodes (Zhang et al., 2017).

4. Holographic Weyl semimetals and the universal $\sigma_A \equiv \sigma^5_{xy}$8 ratio

In the holographic Weyl-semimetal model, the bulk theory contains a vector gauge field $\sigma_A \equiv \sigma^5_{xy}$9, an axial gauge field $A_5$0, and a charged complex scalar $A_5$1, together with a Chern–Simons term chosen to reproduce the consistent AVV and AAA anomalies. The axial source is encoded in the ultraviolet boundary condition $A_5$2, while the infrared response is governed by the horizon value $A_5$3 (Copetti et al., 2016).

The electric and axial Hall conductivities are defined by zero-momentum Kubo formulae,

$A_5$4

with $A_5$5 and $A_5$6. At zero frequency and momentum, the vector radial current is conserved, which yields the horizon formula

$A_5$7

The axial Hall conductivity requires solving coupled axial and metric fluctuations, because the axial gauge perturbations mix with $A_5$8 and $A_5$9 (Copetti et al., 2016).

The anomaly algebra predicts that the axial Hall conductivity should be one third of the electric Hall conductivity. Naïvely, numerical data do not satisfy $B_5$0. The resolution is a nontrivial renormalization of the external axial gauge field: the ultraviolet source $B_5$1 is screened by the scalar sector, and only the infrared fraction

$B_5$2

reaches the low-energy theory. After this renormalization, the conductivities are

$B_5$3

where $B_5$4. Numerical data across a wide range of $B_5$5 confirm that once the computed $B_5$6 is rescaled by $B_5$7, the ratio $B_5$8 is one third to high precision and independent of the state (Copetti et al., 2016).

The same model exhibits a zero-temperature quantum phase transition as the ultraviolet ratio $B_5$9 crosses a critical value. For $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$00, the system is in a topological phase with nonzero $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$01 and $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$02; for $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$03, both vanish in the trivial phase. The ratio $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$04 persists throughout the topological phase, and an analogous transition appears in a top-down type-IIB construction (Copetti et al., 2016).

5. Axial gauge fields from strain and valley-odd Hall transport

In topological Dirac semimetals such as Na$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$05Bi and Cd$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$06As$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$07, smooth lattice deformations act as an axial gauge field that shifts the two Dirac cones in opposite directions. Near the two Dirac points, the low-energy Hamiltonian is

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$08

and the strain-induced vector potential can be written as

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$09

Examples include a screw dislocation along $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$10, for which $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$11 and $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$12, as well as torsion or bending geometries that can generate macroscopically uniform $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$13 up to $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$14–$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$15 (Araki, 2018).

In semiclassical chiral kinetic theory, the valley index $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$16 experiences the effective fields

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$17

For the geometry $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$18, $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$19, an electric field together with $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$20 produces a Hall response that is odd in valley. Summing over spin and valley with weight $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$21 gives the axial current

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$22

In component form,

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$23

The ordinary Hall deflections of the two valleys therefore cancel in charge but add in the axial channel, yielding a pure valley current. Because spin and valley are locked in Na$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$24Bi and Cd$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$25As$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$26, the same response is also a spin Hall effect linear in $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$27 and $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$28 (Araki, 2018).

The same study further showed that a strained topological Dirac semimetal can exhibit a nonlinear spin Hall effect quadratic in the electric field. That effect arises as the cross effect between the regular Hall effect driven by the axial magnetic field and the anomalous Hall effect coming from the momentum-space topology. The linear axial Hall current and the quadratic spin Hall current therefore belong to the same strain-engineered transport framework (Araki, 2018).

6. Crystalline axiality: ferroaxial metals and altermagnetic Lieb lattices

In metallic ferroaxial systems, the axial quantity is a ferroaxial moment $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$29, an axial and time-even vector. The magnetoconductivity tensor may be expanded as

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$30

Here $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$31 gives the conventional Hall effect, while $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$32 is nonzero only when both a ferroaxial moment and an external magnetic field are present. For the tetragonal point group $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$33, symmetry lowering from $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$34 activates an electric toroidal dipole $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$35 with the same $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$36 symmetry as the ferroaxial order, and its off-diagonal contribution appears in the antisymmetric Hall sector. The resulting unconventional Hall response is odd in $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$37, vanishes if the crystalline-electric-field hybridization $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$38 is zero, and is controlled by the symmetry-breaking term $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$39 that hybridizes $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$40 orbitals (Hayami et al., 2023).

A central conclusion of that analysis is that relativistic spin-orbit coupling is not required. The crystalline electric field arising from the symmetry reduction from $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$41 to $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$42 is essential for the ferroaxial-related magnetotransport, whereas numerical results at $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$43 still show the axial Hall effect. Candidate materials include Ca$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$44Ir$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$45O$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$46, Co$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$47Nb$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$48O$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$49, RbFe(MoO$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$50)$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$51, and NiTiO$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$52. For Ca$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$53Ir$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$54O$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$55, density-functional tight-binding parameters suggest $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$56 eV and $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$57, with an in-plane geometry $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$58, $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$59, and a transverse voltage measured along $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$60 or $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$61 (Hayami et al., 2023).

In altermagnetic Lieb lattices, “axial” denotes a hidden topological degree of freedom associated with the two edge sublattices. The axial index is

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$62

and in the unstrained lattice the combined $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$63 symmetry enforces a double degeneracy between the $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$64-chain and $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$65-chain sectors. Under uniaxial strain, a piezomagnetic response produces $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$66 and $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$67, breaks the degeneracy, and permits an axial Hall conductivity

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$68

Dresselhaus spin-orbit coupling mixes the spin sectors near $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$69, and the strain-induced gap-opening there creates sharply localized Berry-curvature hotspots (Xu et al., 16 Sep 2025).

A first-principles case study for Mn$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$70WS$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$71 reported a piezomagnetic moment $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$72 per cell under $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$73 strain, a narrow gap $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$74 meV at $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$75, Berry curvature peaks of order $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$76 Å$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$77, and an intrinsic axial Hall conductivity

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$78

for hole doping $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$79 meV below $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$80. That conductivity remains essentially constant, to within a few percent, from $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$81 to $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$82 strain, and multilayers show an odd–even modulation in charge and spin Hall responses: odd $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$83 gives nonzero charge axial Hall conductivity and zero spin response, while even $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$84 gives the opposite (Xu et al., 16 Sep 2025).

7. Detection schemes, broader manifestations, and conceptual distinctions

Proposed and realized signatures differ substantially because the underlying axial variable differs. For the boundary axial current in a Dirac semimetal, the predicted probes include nonlocal transport, magneto-optical Kerr or Faraday rotation, interface torque on an adjacent ferromagnet, and valley-resolved ARPES or photocurrent. In high-mobility Dirac semimetals such as TlBi(S$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$85Se$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$86)$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$87 or Cd$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$88As$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$89, using $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$90 s, $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$91 m/s, and $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$92, the estimates are $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$93 nm and $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$94, which were described as well within experimental reach (Okuma et al., 2016).

In chiral plasma physics, the observables are collective and event-averaged rather than boundary-local transport coefficients. The chiral Hall effect and Hall density waves were proposed to yield rapidity-dependent charge asymmetries and differences in flow harmonics $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$95 in asymmetric heavy-ion collisions. In Weyl semimetals such as TaAs, the experimentally accessible quantities are instead a giant Hall angle, planar Hall oscillations with $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$96 periodicity, an anomalous planar Hall component odd in field, and negative longitudinal magnetoresistivity below the quantum-limit field (Pu et al., 2014, Zhang et al., 2017).

The phrase also appears in a non-quantum magneto-thermal context. In a conducting cylinder with a uniform axial magnetic field and a purely radial temperature gradient, the Hall current is azimuthal,

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$97

and Ampère’s law then generates an induced axial magnetic field $$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$98 obeying

$$j^{05}_\mu \equiv (\rho^{05}, j^{05}), \qquad \rho^{05}(x)=\bar\psi(x)\gamma_5\psi(x), \qquad j^{05}(x)=\bar\psi(x)\gamma_5\gamma\psi(x).$$99

The induced field opposes the applied one and can strongly suppress it when the thermodiffusion electromotive force is large. This usage concerns an axial magnetic-field response generated by a Hall current, rather than a Hall current carried by an axial charge or an axial pseudospin (Bisnovatyi-Kogan et al., 2023).

Taken together, these results establish several non-equivalent meanings of axial Hall effect. In some works it is a conserved Noether current localized near a boundary; in others it is a bulk axial conductivity $\rho^{05}$00; in holography it is fixed by anomaly coefficients and infrared screening; in strain engineering it is a valley-odd Hall response to an axial magnetic field; and in crystalline transport it can be controlled by a ferroaxial moment or by an axial topological index (Okuma et al., 2016, Pu et al., 2014, Copetti et al., 2016, Araki, 2018, Hayami et al., 2023, Xu et al., 16 Sep 2025). A common misconception is therefore to treat the expression as if it named a single mechanism. The literature instead supports a narrower statement: the unifying feature is a transverse response whose constitutive law is odd in an axial variable, while the carrier, symmetry class, conservation law, and experimental signature depend on the physical setting.