Chiral Separation Effect: Topological Transport

• The Chiral Separation Effect (CSE) is the generation of an axial current parallel to a magnetic field in a medium with a finite vector chemical potential, closely related to anomaly physics.
• It is derived from the lowest Landau level and expressed as a topological invariant of the Green function, ensuring an equilibrium, nondissipative transport response under chiral symmetry.
• Its universality is modified by factors such as mass generation, finite temperature, lattice regularization, and pairing, with significant implications in dense QCD matter, heavy-ion collisions, and Weyl materials.

Searching arXiv for recent and foundational papers on the Chiral Separation Effect to ground the article in cited literature. arxiv_search query="Chiral Separation Effect topological interactions lattice holography heavy ion Weyl semimetal" max_results=10 Chiral Separation Effect (CSE) is the generation of an axial, or chiral, current parallel to an external magnetic field in a medium with finite vector chemical potential. In a common normalization, one writes

$$j_5^i=\sigma_{\rm CSE}\,B^i, \qquad \sigma_{\rm CSE}^{\text{(free)}}=\frac{q\,N_c\,\mu}{2\pi^2},$$

while an equivalent per-Dirac-fermion form is $j_5^i=\frac{\mu}{2\pi^2}B^i$ in the massless limit. In QCD-oriented notation this is often written as

$\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$

The effect is treated as an equilibrium, nondissipative response, is closely tied to anomaly physics, and serves as a central building block in discussions of anomalous transport in dense QCD matter, heavy-ion collisions, and Weyl materials. At the same time, the extent to which its conductivity is “universal” depends on the regime: zero-temperature chirally symmetric systems admit a topological formulation, whereas mass generation, finite temperature, axial-charge relaxation, lattice regularization, or pairing can modify or even suppress the response (Zubkov et al., 2023, Khaidukov et al., 2017, Khaidukov et al., 2022).

1. Definition, constitutive relations, and physical mechanism

The CSE couples a vector chemical potential to an axial current in the presence of a magnetic field. In contrast, the Chiral Magnetic Effect (CME) couples an axial chemical potential to a vector current. In the conventions used in heavy-ion phenomenology,

$$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$

so the CME and CSE are naturally paired transport laws. The same distinction appears in relativistic and lattice treatments: the CME is an ordinary electric current along $\mathbf B$, whereas the CSE is an axial current along $\mathbf B$ (Belmont, 2014, Khaidukov et al., 2017).

A standard microscopic picture derives the CSE from the lowest Landau level. In a magnetic field, the lowest Landau level of a massless Dirac fermion is chiral, with one-dimensional propagation along $\mathbf B$. Filling states up to a vector chemical potential $\mu$ creates an imbalance of right- and left-moving modes and produces a net axial current. Using the lowest-Landau-level degeneracy $eB/(2\pi)$ per unit area and the one-dimensional density of states $1/(2\pi)$ per unit length yields the conventional coefficient $j_5^i=\frac{\mu}{2\pi^2}B^i$0 in the massless case (Khaidukov et al., 2017, Puhr et al., 2017).

The equilibrium status of the CSE is a defining conceptual feature. Several analyses emphasize that, unlike the CME, the CSE survives in equilibrium and is therefore a genuine equilibrium transport phenomenon. This distinction persists across continuum, lattice, and condensed-matter formulations, although the precise value of the conductivity may depend on whether one is in a chirally symmetric, gapless, zero-temperature regime or in a phase with masses, pairing, or strong finite-temperature effects (Zubkov et al., 2023, Khaidukov et al., 2017).

2. Topological and Green-function formulations

At $j_5^i=\frac{\mu}{2\pi^2}B^i$1, the CSE conductivity can be expressed as a topological invariant of the Green function. In one formulation, the averaged axial current obeys

$j_5^i=\frac{\mu}{2\pi^2}B^i$2

where $j_5^i=\frac{\mu}{2\pi^2}B^i$3 counts the number of chiral Dirac fermion species in the low-energy theory. In the interacting case, the same structure survives provided the renormalized axial current is used and chiral symmetry holds near the Fermi surface or Fermi point. The corresponding invariant is written in terms of the complete renormalized Green function and its inverse, integrated over a three-dimensional hypersurface $j_5^i=\frac{\mu}{2\pi^2}B^i$4 surrounding the singular manifold in momentum space (Zubkov et al., 2023).

The renormalized axial current used in this framework is

$j_5^i=\frac{\mu}{2\pi^2}B^i$5

with $j_5^i=\frac{\mu}{2\pi^2}B^i$6. This definition is essential because the axial current is not a conserved Noether current in the interacting theory. Under exact chiral symmetry near the Fermi manifold and in the absence of gap opening, the conductivity remains equal to the noninteracting value per Dirac fermion, $j_5^i=\frac{\mu}{2\pi^2}B^i$7, even after interactions are included through the full Green function (Zubkov et al., 2023).

A closely related extension exists for spatially non-homogeneous systems. There, the CSE conductivity is expressed through a Wigner–Weyl invariant involving the Wigner-transformed Green function $j_5^i=\frac{\mu}{2\pi^2}B^i$8, the Weyl symbol $j_5^i=\frac{\mu}{2\pi^2}B^i$9, and Moyal star products. In weakly inhomogeneous situations this reduces to an ordinary-product expression, but the central conclusion is unchanged: the bulk CSE coefficient is controlled by a topological invariant, robust to smooth inhomogeneity that does not break low-energy chiral symmetry or move the relevant singularities across the integration surface (Suleymanov et al., 2020).

In magnetic Weyl semimetals, the same logic yields

$\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$0

with $\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$1 the number of Weyl-node pairs. The invariant can again be written in terms of the full interacting or disordered Green function, so the condensed-matter version of the CSE is likewise topological under the stated assumptions (Zubkov, 2023).

3. Regularization, anomaly, and the limits of universality

Although the CSE is anomaly-related, its conductivity need not inherit the anomaly’s full immutability in every regime. A real-time point-splitting analysis in QED separates the axial current into a UV-dominated divergent part, which generates the chiral anomaly, and an IR-dominated convergent part, which produces the CSE. In that treatment the zero-temperature result becomes

$\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$2

so the response vanishes when $\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$3. This establishes that, in that setting, the CSE is sensitive to infrared physics such as mass generation, whereas the anomaly coefficient is fixed by UV structure and Lorentz symmetry (Khaidukov et al., 2022).

Lattice regularization makes this issue especially sharp. For Wilson fermions and conventional overlap fermions, which eliminate doublers at the price of breaking exact anticommutation with $\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$4, the lattice computation reproduces the conventional CSE in the massless limit. By contrast, for naive fermions with exact chiral symmetry $\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$5, the contributions of the 16 doublers cancel, and the net CSE vanishes. A modified overlap regularization with exact chiral symmetry exhibits the same cancellation through poles and zeros of the Green function (Khaidukov et al., 2017).

These lattice results do not imply that every chirally symmetric lattice study must find zero CSE. In quenched finite-density QCD with overlap fermions satisfying the Ginsparg–Wilson relation and a conserved covariant axial current, the measured transport coefficient agrees with the universal free value within small statistical errors in both confined and deconfined phases. The key distinction is that the overlap formulation used there does not enforce the strict propagator anticommutation $\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$6 of the naive lattice theory; instead it preserves lattice chiral symmetry through the Ginsparg–Wilson relation while allowing a nonvanishing conserved axial current operator (Puhr et al., 2017).

A separate source of non-universality appears in interacting QCD because the axial current is not conserved even at vanishing quark mass: the gluonic axial anomaly induces axial-charge relaxation. In a holographic V-QCD analysis, this leads to radiative corrections to the CSE conductivity and a nontrivial dependence on temperature and density. The resulting picture is not in contradiction with the topological $\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$7 formulation; rather, it applies to a regime where axial-charge relaxation, finite temperature, and the consistent-current convention materially alter the transport coefficient (Gallegos et al., 2024).

4. Dense QCD, finite temperature, pairing, and impurity effects

In chirally symmetric, gapless QCD matter at $\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$8 and sufficiently large $\vec J_A=\frac{N_c e}{2\pi^2}\mu_V\vec B.$9, the topological formulation predicts

$$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$0

summing over flavors with masses below $$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$1. The analysis explicitly states that this requires restored chiral symmetry and the absence of color superconductivity, because pairing opens gaps and changes the topology of the Green function. A possible realization is cold dense quark matter inside neutron stars (Zubkov et al., 2023).

At finite temperature, nonperturbative effects can suppress the CSE substantially. Using Simonov’s field correlator method, one study models the dominant finite-$$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$2 effect as a thermal mass $$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$3 generated by color-magnetic confinement and finds that the topological value is recovered only when $$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$4 is much larger than the thermal mass and the Polyakov-loop potential. With the nonperturbative correction included, the screened mass scale near $$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$5 is $$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$6 MeV, and the conductivity is roughly suppressed by a factor $$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$7 in the RHIC region; the topological value is approached only at very high temperatures, around the electroweak scale $$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$8 GeV (Zubkov et al., 2023).

Lattice results at finite density show that the phase structure matters. In quenched SU(3) finite-density QCD, the CSE transport coefficient agrees with the universal anomaly value in both confinement and deconfinement phases within small statistical errors, and this agreement led to the suggestion that the CSE can be used to determine the axial-current renormalization factor $$\vec{J}_V=\frac{N_c e}{2\pi^2}\mu_A\vec B, \qquad \vec{J}_A=\frac{N_c e}{2\pi^2}\mu_V\vec B,$$9 for other lattice discretizations. In two-color QCD with dynamical quarks, however, the CSE is close to the free-quark result in the high-temperature quark–gluon plasma phase but is gradually suppressed in the confined, chirally broken regime for $\mathbf B$0. In that regime the data are described more effectively by $\mathbf B$1 than by the free-theory proportionality to $\mathbf B$2 (Puhr et al., 2017, Buividovich et al., 2020).

Pairing can suppress the conductivity even more strongly. In two-color dense QCD with a diquark condensate carrying electric charge $\mathbf B$3, the condensate produces electromagnetic Meissner effects. At one loop, destructive interference in the particle–hole channel suppresses the static, long-wavelength CSE conductivity to one third of the normal-phase value in the chiral limit at $\mathbf B$4: $\mathbf B$5 The same analysis shows that the Nambu–Goldstone modes required by the Ward–Takahashi identities are longitudinal and do not contribute to the bulk CSE conductivity in linear response (Suenaga et al., 2021).

Quark masses and heavy impurities provide additional deformations. In a D3/D7 holographic study, the quark-mass correction to the CSE is related to static correlators of the pseudoscalar condensate, quark number density, and quark condensate; in the small-mass regime, the correction has the structure $\mathbf B$6, and beyond small momentum the model exhibits a normalizable mode that may lead to a spiral phase (Guo et al., 2016). By contrast, in a heavy-impurity Kondo model, the Kondo condensate enhances rather than suppresses the CSE: both static and dynamical conductivities increase, and in the dynamical limit the enhancement can be approximately a factor of three for $\mathbf B$7 GeV and $\mathbf B$8 GeV (Suenaga et al., 2020).

5. Heavy-ion collisions and the Chiral Magnetic Wave

In heavy-ion phenomenology, the CSE is studied primarily through its coupling to the CME. A finite vector chemical potential drives an axial current via the CSE, spatially separating right- and left-handed densities along $\mathbf B$9; the resulting axial imbalance then drives CME vector currents away from the reaction plane. In the chiral basis $\mathbf B$0, $\mathbf B$1, this coupled transport gives a collective excitation, the Chiral Magnetic Wave (CMW), which carries an electric quadrupole moment of fixed sign from event to event (Belmont, 2014).

The standard experimental parametrization uses the event charge asymmetry

$\mathbf B$2

and the expected CMW-induced elliptic-flow splitting

$\mathbf B$3

The ALICE analysis at $\mathbf B$4 TeV instead employed an efficiency-independent three-particle correlator,

$\mathbf B$5

together with charge-independent and charge-dependent subtraction schemes designed to remove trivial factorization and reduce local-charge-conservation backgrounds (Belmont, 2014).

The second-harmonic correlator increases substantially toward peripheral collisions. After charge-dependent subtraction, both its magnitude and its $\mathbf B$6 range are substantially reduced, consistent with the removal of a large local-charge-conservation component, yet a residual $\mathbf B$7 signal remains. The third-harmonic correlator is largely removed by the same subtraction. ALICE therefore concluded that the residual second-harmonic correlation could be a mixture of remaining local-charge-conservation backgrounds and possible CMW-related $\mathbf B$8-violating correlations, but that the data do not establish a definitive observation of the CMW and hence do not establish the presence of the CSE in heavy-ion collisions (Belmont, 2014).

Real-time lattice simulations support the qualitative CME–CSE mechanism. In simulations with dynamical fermions coupled to non-Abelian $\mathbf B$9 and Abelian $\mathbf B$0 gauge fields, a sphaleron transition in the presence of an external magnetic field generates axial charge through the non-Abelian anomaly; the resulting CME vector current produces a vector-charge dipole, which then sources an axial current through the CSE. In the strong-field, non-diffusive limit this yields a propagating chiral magnetic wave with $\mathbf B$1. The same study found strong mass dependence: for $\mathbf B$2, the maximal axial charge imbalance is reduced by a factor of two, and for $\mathbf B$3 the induced vector current even reverses direction immediately after the sphaleron transition (Mueller et al., 2016).

6. Weyl materials, non-homogeneous systems, and higher-spin generalizations

Weyl semimetals realize the CSE as an axial current of right-handed minus left-handed quasiparticles. In the low-temperature regime, the same topological expression as in relativistic field theory applies, provided one works near the Weyl nodes and uses the full Green function. For one Dirac fermion, the predicted value is again $\mathbf B$4, and in magnetic Weyl semimetals with $\mathbf B$5 pairs of Weyl nodes one has

$\mathbf B$6

in natural units. This robustness extends to smooth inhomogeneity and to interaction or disorder effects, as long as the poles of the Green function at the nodes remain intact (Zubkov et al., 2023, Suleymanov et al., 2020, Zubkov, 2023).

A particularly developed condensed-matter picture combines the bulk CSE current with surface Fermi-arc transport. The axial current along $\mathbf B$7 is accompanied by momentum-space flow along the Fermi arcs; together they form closed Weyl orbits. These orbits admit the semiclassical quantization

$\mathbf B$8

where $\mathbf B$9 is the sample thickness, $\mu$0 the momentum-space separation between opposite-chirality nodes, and $\mu$1 the field tilt. The proposal advanced in this context is that Weyl-orbit contributions to Hall conductance offer an experimental probe of the CSE complementary to quantum oscillations (Zubkov, 2023).

The topological dependence of the CSE on node charge extends beyond simple Weyl fermions. For a Rarita–Schwinger–Weyl semimetal with a four-fold crossing and quasispin $\mu$2, the node Chern number is $\mu$3 and the conductivity becomes

$\mu$4

For Adler’s relativistic spin-$\mu$5 model, the CSE conductivity is five times the massless Dirac value, while in the four-fold condensed-matter realization it is four times the Weyl value. More generally, the conductivity is proportional to the Chern number of the Fermi point and is topologically protected (Khaidukov et al., 2020).

These developments indicate that the CSE is best regarded not as a single formula with universal applicability in every medium, but as a family of anomaly-related axial transport responses whose precise coefficient is fixed by topology only when the low-energy spectrum remains chiral, gapless, and sufficiently close to equilibrium, and whose deviations encode masses, thermal screening, axial relaxation, pairing, or impurity-induced hybridization.