Odd Transport Phenomena

• Odd transport phenomena are nontraditional responses where fluxes do not align with driving gradients, arising from broken parity or time-reversal symmetry.
• They encompass Hall-like effects such as odd viscosity, thermal conductivity, and mobility observed in chiral fluids, topological semimetals, and nanoscale devices.
• Investigations quantify these effects via antisymmetric tensor coefficients and isolated transport anomalies like absolute negative mobility and reverse heat flow.

Odd transport phenomena are transport responses whose constitutive structure departs from ordinary longitudinal, dissipative diffusion or Ohmic conduction. In contemporary literature, the term covers several partially overlapping classes: parity-odd and often nondissipative currents generated by anomalies and topology; antisymmetric transport coefficients such as odd viscosity, odd diffusivity, odd thermal conductivity, and odd mobility in chiral fluids; Hall-like transverse responses in condensed matter; and broader anomalous or unconventional transport regimes in driven, active, and quantum many-body systems. A unifying feature is that fluxes need not align with their driving gradients, and in several cases the relevant coefficients are controlled by symmetry breaking, anomaly structure, or topology rather than by ordinary relaxation alone (Liao, 2014, Hargus et al., 2024, Zubkov et al., 2018, Bovet, 2015).

1. Symmetry structure and conceptual scope

The modern theory of odd transport is organized by discrete symmetries. In hydrodynamic and kinetic settings, odd coefficients are antisymmetric parts of linear-response tensors that become allowed when time-reversal symmetry and/or parity are broken. In two-dimensional chiral media, the Levi–Civita tensor permits constitutive laws in which a gradient along one direction generates a flux in the orthogonal direction; the resulting response is Hall-like and nondissipative in the sense that the odd sector does not contribute to viscous or diffusive entropy production (Hosaka et al., 2023, Jiao et al., 2024). In the language of constitutive matrices, one writes a decomposition into symmetric and antisymmetric pieces, with the symmetric part controlling bulk relaxation and the odd part generating transverse fluxes that are invisible to a purely relaxational description (Hargus et al., 2024).

This symmetry logic is not restricted to fluids. In Weyl semimetals and other topological electronic systems, parity-odd and time-reversal-odd responses appear as anomalous Hall currents, while anomaly-related longitudinal responses arise under parallel electric and magnetic fields (Hosur et al., 2013). In hot QCD matter, local P-odd and CP-odd domains created by topological fluctuations permit anomalous currents that vanish in global averages but survive in event-by-event correlations, a situation described as “environmental symmetry violation” (Liao, 2014). In chiral Brownian systems, the same logic is implemented at the particle level by an antisymmetric mobility tensor,

$\mu_{ij}=\mu_s\,\delta_{ij}+\mu_o\,\epsilon_{ij},$

which converts applied forces into both longitudinal and transverse drift components (Faedi et al., 20 Feb 2026).

The literature also uses “odd” or “anomalous” transport more broadly for counterintuitive regimes that violate ordinary monotonicity or diffusive scaling, such as superdiffusion, subdiffusion, negative mobility, reverse heat transport, and unconventional hydrodynamic relaxation at finite momentum and frequency. This broader usage does not always refer to antisymmetric tensors, but it shares the central theme that transport cannot be reduced to classical Brownian diffusion or simple longitudinal linear response (Hänggi et al., 2020, Zhang et al., 2024, Morettini et al., 14 Feb 2025, Bovet, 2015).

2. Anomaly-induced transport and momentum-space topology

In relativistic many-body systems, odd transport is tightly linked to topology and quantum anomalies. In the hot QCD fluid created by relativistic heavy-ion collisions, topological transitions between gluonic vacua carry winding number

$$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$

and the axial anomaly relates this topology to chirality through

$$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$

A local axial chemical potential $\mu_A$ then enables parity-odd transport responses such as the chiral magnetic effect (CME),

$$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$

the chiral separation effect (CSE),

$\vec j_A=\sigma_5\,\mu_V\,\vec B,$

and the chiral electric separation effect (CESE),

$\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$

These responses are coupled further into collective modes such as chiral magnetic waves and chiral electric waves (Liao, 2014).

A parallel topological formulation appears in condensed matter. Using the Wigner transform of the two-point Green function and a gauge-invariant derivative expansion, odd and anomalous nondissipative currents can be expressed in terms of momentum-space topological invariants. The linear response takes the form

$$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$

with $\mathcal{M}_l$ a four-dimensional invariant built from $\mathcal{G}(p)$. In three-dimensional systems this yields anomalous Hall currents, while for a single massless Dirac fermion the axial CSE current becomes

$$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$0

Within the lattice-regularized equilibrium framework analyzed there, the equilibrium vector-current CME is absent because the corresponding invariant vanishes, whereas the CSE survives as a topological axial response (Zubkov et al., 2018).

Weyl semimetals provide the best-known condensed-matter realization of anomaly-rooted odd transport. Their Weyl nodes act as monopoles of Berry curvature, with Fermi-arc surface states connecting nodes of opposite chirality. The effective action with node-separation four-vector $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$1 generates the anomalous Hall and chiral magnetic currents

$$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$2

while parallel $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$3 pumps chiral charge and enhances longitudinal conductivity, producing negative magnetoresistance (Hosur et al., 2013). A plausible implication is that anomaly-induced transport forms a common language across high-energy and topological condensed-matter systems, even though the equilibrium status of particular currents, especially the CME, depends sensitively on regularization and limiting procedures (Zubkov et al., 2018, Hosur et al., 2013).

3. Odd hydrodynamics, odd viscosity, and reciprocal structure

In fluid mechanics, odd transport is most naturally expressed through constitutive laws for stress, heat, mass, or mobility. For incompressible flow with odd viscosity, the stress decomposes as

$$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$4

and in two dimensions the odd contribution can be written as

$$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$5

Because $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$6, odd viscosity is nondissipative and rotates stresses by $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$7 relative to strain-rate directions (Hosaka et al., 2023). In two-dimensional isotropic odd fluids, analogous constitutive laws arise for heat and mass fluxes,

$$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$8

so odd thermal conductivity and odd diffusivity accompany odd viscosity as transverse, reactive transport channels (Jiao et al., 2024).

The reciprocal structure of such media differs from that of ordinary Stokes flow. The classical Lorentz reciprocal theorem fails if one naively compares two flows with the same odd viscosity, because the antisymmetric part of the viscosity tensor violates the symmetry condition needed to cancel the bulk term. Reciprocity is restored by choosing the auxiliary problem with the sign of the odd viscosity flipped, so that $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$9 while $$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$0 (Hosaka et al., 2023). This generalization preserves a surface-only cross-work relation and leads to a sharp distinction between two propulsion classes: swimmers with prescribed surface velocity are unaffected by odd viscosity, whereas swimmers with prescribed active forces acquire transverse Hall-like propulsion. A torque dipole can therefore propel a body along the chirality axis, and in two dimensions the Hall angle is exactly $$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$1 with $$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$2 (Hosaka et al., 2023).

A broader statistical-mechanical formulation is supplied by the flux hypothesis, which generalizes Onsager’s regression hypothesis from conserved densities to fluxes. In Fourier space, it postulates that the average microscopic flux obeys the same constitutive law as the macroscopic one,

$$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$3

on intermediate timescales $$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$4. This yields Green–Kubo relations for both even and odd transport coefficients and clarifies the symmetry requirements: odd self-transport requires broken time-reversal symmetry, whereas odd cross-couplings can survive with time-reversal symmetry intact provided parity is broken (Hargus et al., 2024). This distinction becomes explicit in three-dimensional odd viscodiffusive fluids, where parity-breaking alone permits cross-couplings between concentration gradients and antisymmetric stress, or between vorticity and diffusive flux,

$$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$5

$$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$6

Because these couplings are reactive, they do not contribute to entropy production and can exist in passive equilibrium chiral liquids (Deshpande et al., 2024).

4. Chiral fluids, active matter, and particle-scale odd mobility

Odd transport in soft and active matter often appears first at the level of a mobility tensor rather than a stress tensor. In overdamped chiral fluids, the single-particle mobility may be written as

$$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$7

so the diffusion tensor becomes

$$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$8

For a driven tracer in a dilute interacting bath, the steady-state two-body Smoluchowski equation then predicts a rotated and eventually inverted density wake. For hard disks, the linear-response correction has the form

$$\partial_\mu J_A^\mu=\frac{g^2}{16\pi^2}F^a_{\mu\nu}\tilde F_a^{\mu\nu},\qquad N_R-N_L=2Q_w.$$9

with anisotropy coefficients

$\mu_A$0

As $\mu_A$1, the wake undergoes a $\mu_A$2 rotation: depletion forms ahead of the tracer and accumulation behind it, providing a unified microscopic mechanism for interaction-enhanced diffusion and absolute negative mobility (Faedi et al., 20 Feb 2026).

The resulting transport coefficients are explicitly density dependent. In the hard-disk limit,

$\mu_A$3

Hence $\mu_A$4 becomes independent of density at $\mu_A$5, exceeds $\mu_A$6 for $\mu_A$7, and approaches $\mu_A$8 in the strong-chirality limit. If both tracer and host are driven, absolute negative mobility occurs when

$\mu_A$9

together with sufficiently large $$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$0, where $$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$1 is the drive ratio (Faedi et al., 20 Feb 2026). This suggests that odd mobility does not merely redirect forces; through contact constraints it qualitatively reorganizes microstructure and turns interactions from a drag mechanism into a propulsion mechanism.

A distinct particle-resolved mechanism operates in chiral active baths. In a rheological experiment with a passive tracer pulled through a bath of chiral bristle-bots, local collisions transfer chirality to the tracer, which displays circular trajectories and a systematic transverse drift under a constant pulling force. A reduced coarse-grained model takes the tracer to obey a chiral active Ornstein–Uhlenbeck dynamics with dry friction,

$$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$2

together with a precessing Ornstein–Uhlenbeck propulsion director. The crucial result is that nonlinear dry friction rectifies transferred chiral fluctuations into a macroscopic odd response; linear Stokes friction does not do so to the same extent (Goerlich et al., 24 May 2026).

At the mesoscale, odd fluids can be simulated directly by chiral stochastic rotation dynamics. In two dimensions, the collision step includes a fixed chirality angle $$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$3 that breaks time-reversal and parity while conserving mass, momentum, and energy. Kinetic theory then yields odd viscosity, odd thermal conductivity, and odd self-diffusion coefficients alongside their even counterparts, and simulations verify transverse mass accumulation in Poiseuille flow, tilted isotherms under a vertical temperature gradient, and tilted concentration fields under a density gradient (Jiao et al., 2024). The three-dimensional extension with cylindrical symmetry produces a full anisotropic odd viscosity tensor and Hall-like planar Poiseuille flows with transverse velocities or lateral density gradients depending on the forcing geometry (Jiao et al., 6 Aug 2025). A complementary hard-disk kinetic theory, in which chirality is injected solely by a collision-induced transverse impulse $$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$4, gives explicit odd coefficients

$$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$5

together with a nonzero torque density

$$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$6

thereby connecting odd transport directly to collisional orbital angular-momentum injection (Maire et al., 4 Mar 2026).

5. Nonequilibrium, anomalous, and unconventional transport regimes

In driven periodic systems, odd or anomalous transport need not be encoded by antisymmetric tensors; it can instead refer to transport that departs qualitatively from normal diffusion. For an inertial Brownian particle in a spatially asymmetric periodic potential,

$$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$7

directed transport without net bias requires broken spatial symmetry and suitable noise-assisted transitions between coexisting locked and running attractors. In the representative ratchet potential

$$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$8

the deterministic dynamics contains three coexisting velocity attractors $$\vec j_V=\sigma_5\,\mu_A\,\vec B,\qquad \sigma_5=\frac{N_c e}{2\pi^2},$$9, $\vec j_A=\sigma_5\,\mu_V\,\vec B,$0, and $\vec j_A=\sigma_5\,\mu_V\,\vec B,$1. As a result, the mean-square displacement exhibits a long sequence of early superdiffusion, intermediate subdiffusion, and only asymptotically normal diffusion. For $\vec j_A=\sigma_5\,\mu_V\,\vec B,$2, $\vec j_A=\sigma_5\,\mu_V\,\vec B,$3, $\vec j_A=\sigma_5\,\mu_V\,\vec B,$4, $\vec j_A=\sigma_5\,\mu_V\,\vec B,$5, and $\vec j_A=\sigma_5\,\mu_V\,\vec B,$6, the superdiffusive stage persists up to $\vec j_A=\sigma_5\,\mu_V\,\vec B,$7, while subdiffusion can last until $\vec j_A=\sigma_5\,\mu_V\,\vec B,$8–$\vec j_A=\sigma_5\,\mu_V\,\vec B,$9 (Hänggi et al., 2020).

The same model shows a non-monotonic temperature dependence of the asymptotic diffusion coefficient: $\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$0 has a local maximum near $\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$1, a pronounced minimum near $\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$2, and then grows roughly linearly at high $\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$3. The physical mechanism is the competition between thermal activation, inertial phase-locking into a preferred running state, and high-temperature noise broadening (Hänggi et al., 2020). A plausible implication is that some anomalous transport phenomena are best understood not as symmetry-protected coefficients but as long-lived transient consequences of multistability and nonequilibrium state selection.

At the level of stochastic transport theory, non-diffusive models generalize Brownian motion by relaxing Gaussianity, Markovianity, or locality. In continuous-time random walks and fractional Lévy motion, anomalous transport is characterized by nonlinear mean-square-displacement scaling,

$\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$4

with $\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$5 for subdiffusion, $\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$6 for superdiffusion, and $\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$7 for ballistic motion. Time-fractional and space-fractional diffusion equations encode memory and long jumps through Caputo derivatives and Riesz fractional Laplacians, respectively (Bovet, 2015). Here the term “odd transport” is a looser one; it signifies a departure from classical diffusion rather than an antisymmetric Hall-like coefficient.

An unconventional quantum many-body variant appears in a spin-1 scar model supporting an exact tower of quantum many-body scars. For coherent initial states aligned with the scar manifold, local autocorrelators of the operator $\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$8 obey an oscillatory hydrodynamic form

$\vec j_A=\chi_e\,\mu_V\,\mu_A\,\vec E.$9

with numerical evidence for $$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$0 in the accessible time window. The key point is that the relevant slow mode is not a standard conserved density at $$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$1 but a symmetry restricted to the scar subspace at finite momentum and frequency (Morettini et al., 14 Feb 2025). This is transport that is “odd” relative to conventional hydrodynamics rather than parity-odd in the tensorial sense.

6. Mesoscopic, nonlinear, and superconducting manifestations

At the nanoscale, odd or counterintuitive transport can arise purely from nonequilibrium energy filtering. In a single-level quantum dot with sequential tunneling rates

$$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$2

the steady particle current is

$$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$3

and the heat currents are

$$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$4

Under combined temperature and voltage bias, the system exhibits anomalous thermodiffusion, absolute negative mobility, and reverse heat transport in distinct energy windows organized by the reversible energy level

$$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$5

These regimes do not violate the second law because

$$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$6

with equality only at $$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$7 (Zhang et al., 2024). This suggests a useful distinction between reactive odd transport and nonlinear nonequilibrium transport: both produce transverse or counterdirected responses, but only the former is fundamentally tied to antisymmetric constitutive tensors.

Another nonlinear setting is provided by odd-parity magnetic multipole systems with $$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$8 symmetry. There the ordinary Berry-curvature dipole vanishes at zero field because $$j^{(1)k}(R)=-\frac{1}{4\pi^2}\epsilon^{ijkl}\mathcal{M}_l A_{ij}(R),$$9 enforces $\mathcal{M}_l$0, yet parity violation allows a dominant second-order Drude response,

$\mathcal{M}_l$1

In the model for BaMn$\mathcal{M}_l$2As$\mathcal{M}_l$3, the zero-field nonlinear Hall effect is generated by asymmetric dispersion rather than by a Berry-curvature dipole, and symmetry permits

$\mathcal{M}_l$4

When an external magnetic field breaks $\mathcal{M}_l$5, a magnetic Berry-curvature dipole appears and produces additional nonlinear Hall components with distinct one-fold angular dependence (Watanabe et al., 2020).

Odd transport also enters superconducting electronics through hidden altermagnetism. In Josephson junctions based on CsV$\mathcal{M}_l$6Te$\mathcal{M}_l$7O-family materials, monolayer planar junctions exhibit a fully spin-polarized supercurrent with strong directional anisotropy, because quasi-one-dimensional spin-polarized bands permit equal-spin triplet transport only in one spin channel along a given crystalline axis. In multilayers, an altermagnetic even–odd effect appears: spin-polarized supercurrents persist only in odd-layer planar junctions and cancel exactly in even layers. In vertical junctions, odd-layer barriers enhance equal-spin triplet transport while even layers favor opposite-spin transport, yielding a robust period-two oscillation of the total supercurrent with layer number (Li et al., 16 Feb 2026). Here “odd transport” refers simultaneously to a layer-parity effect and to a spin-selective transport response rooted in hidden altermagnetic symmetry.

The same broad theme appears in colloidal phoresis in odd fluids. Under a concentration or temperature gradient, a colloid in a two-dimensional odd fluid experiences a phoretic force

$\mathcal{M}_l$8

where $\mathcal{M}_l$9 is either $\mathcal{G}(p)$0 or $\mathcal{G}(p)$1. Mesoscale simulations of a chiral stochastic-rotation fluid directly realize odd diffusiophoresis and odd thermophoresis. For a chirality angle $\mathcal{G}(p)$2, the measured diffusiophoretic coefficients are $\mathcal{G}(p)$3 and $\mathcal{G}(p)$4, while the thermophoretic coefficients are $\mathcal{G}(p)$5 and $\mathcal{G}(p)$6; reversing chirality flips the sign of the odd part (Jiao et al., 30 Mar 2026). A plausible implication is that odd transport offers a direct route to lateral manipulation and size-dependent sorting in microfluidic environments.

7. Observables, diagnostics, and open problems

Because odd transport is often encoded in transverse fluxes or event-by-event signatures rather than in simple scalar conductivities, its diagnostics are unusually geometry dependent. In heavy-ion collisions, the CME is searched for through charge-dependent azimuthal correlators such as

$\mathcal{G}(p)$7

and through charge-dependent elliptic-flow splitting

$\mathcal{G}(p)$8

Signal dilution by the decorrelation between the magnetic-field direction and participant planes is quantified by

$\mathcal{G}(p)$9

which peaks at intermediate centrality (Liao, 2014). In lattice QCD, the corresponding axial response is extracted nonperturbatively from the slope of $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$00 versus $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$01, yielding the CSE conductivity. In full QCD with $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$02 flavors at physical masses, this conductivity approaches the free-theory value above $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$03 and is strongly suppressed below $$Q_w=\frac{g^2}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F_a^{\mu\nu},$$04 (Brandt et al., 2022).

In odd-fluid and active-matter contexts, diagnostics often rely on direct field reconstruction. Hydrodynamic simulations and experiments identify odd viscosity through transverse stresses, tilted isotherms, tilted density profiles, antisymmetric force–velocity relations, or counter-rotating marginal probability currents (Hosaka et al., 2023, Jiao et al., 2024, Abdoli et al., 25 Jun 2026). In chiral tracer problems, pair-correlation wakes, Hall angles, and drift reversal thresholds provide discriminating observables (Faedi et al., 20 Feb 2026, Goerlich et al., 24 May 2026). In kinetic theories of chiral disks, the antisymmetric homogeneous stress and torque density are especially direct observables because they arise even before gradient corrections (Maire et al., 4 Mar 2026).

Across these fields, three broad challenges recur. One is disentangling genuine odd response from ordinary backgrounds or geometric artifacts, such as elliptic-flow backgrounds in heavy-ion data, wall-induced currents in active matter, or nonlinear but parity-even transport in mesoscopic electronics (Liao, 2014, Zhang et al., 2024). A second is extending dilute or weak-coupling theories into dense, strongly interacting, or finite-frequency regimes, where additional tensor structures, collective modes, and memory effects appear (Deshpande et al., 2024, Jiao et al., 6 Aug 2025). A third is terminological: the same phrase, “odd transport,” is used both for antisymmetric constitutive responses and for broader unconventional transport phenomena such as anomalous diffusion or negative mobility (Hänggi et al., 2020, Bovet, 2015). This suggests that the most precise usage is contextual. In hydrodynamics and linear response, “odd” refers to antisymmetric, Hall-like constitutive sectors; in nonequilibrium statistical physics, it often designates transport that is anomalous, non-diffusive, or counterdirected relative to naive expectations.

Taken together, the literature shows that odd transport phenomena are not a single mechanism but a family of symmetry-governed and nonequilibrium-governed responses spanning quantum anomalies, topological matter, chiral hydrodynamics, active tracer dynamics, ratchet diffusion, nanoscale thermodynamics, and superconducting spin transport. What unifies them is the breakdown of the default assumption that transport must be purely longitudinal, diffusive, and relaxational.