Chiral Transport in Metric-Affine Geometries

The gist The formal study of fluids in nontrivial background geometries has found interesting applications to condensed matter systems and this paper provides a detailed analysis of chiral transport driven by background spacetime nonmetricity in relativistic fermionic fluids at equilibrium.

Coupling fermionic matter to nonmetricity

Nonmetricity measures the incompatibility of a connection with the metric, defined as ∇µgαβ ≠ 0. In the Cartan formalism, this is encoded in the one-form nonmetricity tensor Qab ≡ −Dηab = 2ω(ab). The nonmetricity tensor (2.1) can be decomposed into its trace and shear (traceless) pieces, Qab ≡ Q−→ab + Qηab. A second vector field can be defined from the components of the nonmetricity tensor by Qb a ≡ eb⌋Qb a = eb⌋Q−→b a + ea⌋Q. Under Weyl transformations, the components of the nonmetricity tensor change as Q −→ Q − 2dσ, Q−→ab −→ Q−→ab.

Nonmetricity invariant

A nonmetricity invariant is constructed analogous to the torsional Nieh-Yan invariant in terms of which the anomaly polynomial is written. This invariant, I Q 4 ≡ 1/4 DQab ∧ DQab − 1/2 Qca ∧ R[ab] ∧ Qb c, can be regarded as a kind of nonmetricity counterpart to the Nieh-Yan invariant and is Weyl invariant. This quantity can be expressed as I Q 4 = 8π/2 h p1(Re) − p1(R) i, where p1(R) and p1(Re) are, respectively, the first Pontrjagin classes of the full connection ω a b and the metric-compatible, Weyl invariant connection ωe a b introduced above.

Descent analysis

The anomaly polynomial is written as P6(FA, IQ 4) = cW FA ∧ I Q 4, where FA = dA is the field strength of the external gauge field coupling to the axial-vector fermion current and cW is a pure number. The associated Chern-Simons five-form ω 0 5(A, IQ 4) = cW A ∧ I Q 4. The consistent anomaly for the axial-vector current coupling to A δγω 0 5 = dω1/4 =⇒ ω1/4 = cW γIQ 4. The covariant anomaly is then given by d⟨⋆J5⟩cov = −cW I Q 4.

Chiral current constitutive relation

The equilibrium partition function is obtained by integrating eq. (5.8) over the five-dimensional bulk Euclidean space M5. The covariant axial-vector current is obtained from the boundary contribution of the functional variation of bulk partition function with respect to A. This expression reveals the existence of two sources of chiral charge transport induced by nonmetricity, respectively driven by the Weyl magnetic field and the fluid’s vorticity.

Discussion

The main result is the existence of magnetic and vortical nonmetricity-driven chiral separation effects that, in view of the results of refs. [17,18], might be of physical relevance for the context of Weyl semimetals with point defects. Both effects disappear for static geometries with Q0 = 0. The corresponding transport coefficients of these nonmetricity magnetic and chiral separation effects depend on the nonmetricity chemical potential, determined by the time component of the Weyl gauge field.

How it works

The equilibrium partition function is computed from the Chern-Simons five-form (4.2) by using transgression methods. The constitutive relation for the chiral coupling of ref. [20] can be obtained from the one for the model proposed in [17] by an appropriate replacement of the Weyl gauge field. The final expression for the covariant axial-vector current is ⟨⋆J5⟩cov = −cW W3. This expression reveals the existence of two sources of chiral charge transport induced by nonmetricity, respectively driven by the Weyl magnetic field and the fluid’s vorticity. The corresponding transport coefficients of these nonmetricity magnetic and chiral separation effects depend on the nonmetricity chemical potential, determined by the time component of the Weyl gauge field.

How it works

The constitutive relation for the covariant axial-vector current, obtained by functional differentiation of the bulk action with respect to BA, reads ⟨⋆J5⟩cov = 2cW u ∧ µQBQ + µ/2 Qω. This expression reveals the existence of two sources of chiral charge transport induced by nonmetricity, respectively driven by the Weyl magnetic field and the fluid’s vorticity. The corresponding transport coefficients of these nonmetricity magnetic and chiral separation effects depend on the nonmetricity chemical potential, determined by the time component of the Weyl gauge field. The constant cW is given by cW = −e 2/4π/2, with e the electron charge.

How it works

The two chiral separation effects survive in the limit of vanishing nonmetricity, when they are driven by the background’s edge torsion T alone. This is in addition to chiral transport effects resulting from the minimal chiral coupling of microscopic fermions to the screw torsion gauge field Sa, that were investigated in [25]. The comparison between the actions (2.19) and (2.15) shows that the results for the latter model can be found from our analysis by replacing Q → 1/2 βTeaθ a. The corresponding constitutive relation for the chiral current is obtained from (5.21) by substituting µQ −→ β/4 2µT + µQb − 4µQ, BQ −→ β/4 2BT + BQb − 4BQ. This expression reveals the existence of two sources of chiral charge transport induced by nonmetricity, respectively driven by the Weyl magnetic field and the fluid’s vorticity. The corresponding transport coefficients of these nonmetricity magnetic and chiral separation effects depend on the nonmetricity chemical potential, determined by the time component of the Weyl gauge field. The constant cW is given by cW = −e 2/4π/2, with e the electron charge.

How it works

The bulk and boundary equilibrium partition functions, (5.11) and (5.12), as well as the constitutive relation of the chiral current (5.16), are invariant under time-independent Weyl transformation. The constitutive relation for the covariant axial-vector current, obtained by functional differentiation of the bulk action with respect to BA, reads ⟨⋆J5⟩cov = 2cW u ∧ µQBQ + µ/2 Qω. This expression reveals the existence of two sources of chiral charge transport induced by nonmetricity, respectively driven by the Weyl magnetic field and the fluid’s vorticity. The corresponding transport coefficients of these nonmetricity magnetic and chiral separation effects depend on the nonmetricity chemical potential, determined by the time component of the Weyl gauge field. The constant cW is given by cW = −e 2/4π/2, with e the electron charge <ref:2606.

1. Improved understanding of chiral transport in Weyl semimetals by modeling lattice point defects using nonmetricity as a background field. This allows AI systems to predict anomalous transport signatures in materials exhibiting Weyl physics with structural imperfections.

2. Development of a framework for calculating the equilibrium partition function for fermionic fluids coupled to nonmetricity using transgression techniques, as described by the equilibrium partition function obtained using transgression techniques. This capability enables AI models to simulate thermodynamic properties of exotic matter under non-Riemannian conditions.

3. Implementation of a descent analysis method that encodes the dependence of the anomaly polynomial on the nonmetricity tensor in a Weyl invariant four-form. This allows AI to systematically derive physically meaningful transport laws from fundamental geometric invariants, such as a nonmetricity invariant is constructed, analogous to the torsional Nieh-Yan invariant.

4. Ability to determine if chiral separation effects are mediated by fluid vorticity and the Weyl magnetic field by evaluating the axial-vector current constitutive relation, which shows these effects are driven by the fluid’s vorticity and the magnetic Weyl field.

5. Modeling of nonminimal couplings of fermionic matter to metric-affine geometries, allowing AI to analyze actions proposed in literature like [20] where fermions couple chirally to both nonmetricity gauge fields, Qa and Qba, as well as edge torsion Ta.

6. Prediction of transport coefficients for nonmetricity magnetic and chiral separation effects by using the derived constitutive relation: ⟨⋆J5⟩cov = 2cW u ∧ µQBQ + µ2Qω. This enables AI to quantify how specific geometric background parameters influence charge flow.

Anomalous transport in equilibrium fermionic fluids chirally coupled to background Weyl-type nonmetricity is studied. A formal descent analysis is carried out in which the dependence of the anomaly polynomial on the nonmetricity tensor is encoded in a Weyl invariant four-form. The constitutive relation of the axial-vector current is evaluated from the equilibrium partition function obtained using transgression techniques, showing the existence of nonmetricity-mediated chiral separation effects driven by the fluid's vorticity and the Weyl magnetic field. A second nonminimal coupling of fermionic matter to metric-affine geometries proposed in the literature is also discussed.