Nieh–Yan Anomaly in Field Theory

• The Nieh–Yan anomaly is a torsion-induced chiral anomaly in four dimensions, characterized by its dependence on spacetime torsion and a dimensionful coefficient.
• It is derived via methods like spectral flow analysis and holographic dual models, which reveal distinct thermal and nonthermal behaviors influenced by ultraviolet regularization.
• Experimental realizations in Weyl semimetals and chiral superfluids demonstrate its practical impact on anomalous transport and thermal Hall effects.

The Nieh–Yan anomaly is a torsion-induced contribution to the chiral (axial) anomaly of Dirac or Weyl fermions in four-dimensional spacetime. Distinct from the standard Adler–Bell–Jackiw (ABJ) anomaly, which links the nonconservation of the axial current to gauge and gravitational field strengths, the Nieh–Yan (NY) sector involves spacetime torsion and enters the divergence of the axial current through a specific four-form. The anomaly has significant implications for both high-energy field theory and condensed-matter systems, particularly in settings where emergent torsion arises dynamically, such as Weyl semimetals, chiral superfluids, and certain holographic models.

1. Definition and Structure of the Nieh–Yan Term

The NY four-form is built from the torsion and curvature two-forms associated with a vierbein $e^a$ and spin connection $\omega^a{}_b$. In differential form notation, the Nieh–Yan invariant reads

$$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$

where $T^a = de^a + \omega^a{}_b \wedge e^b$ is the torsion and $$R_{ab} = d\omega_{ab} + \omega_{ac} \wedge \omega^c{}_b$$ is the curvature two-form. In component language, the anomaly in the divergence of the axial current $J_{1,5}^\mu = \bar\psi \gamma^\mu \gamma_5 \psi$ takes the form

$$\partial_\mu J_{1,5}^\mu = -i\,c_{\rm NY} \, \epsilon^{\mu\nu\lambda\rho} \left( \eta_{\alpha\beta}\,T^\alpha_{\mu\nu}\,T^\beta_{\lambda\rho} - \frac12 R_{\mu\nu\lambda\rho} \right)$$

where $c_{\rm NY}$ is a model-dependent anomaly coefficient, $T^\alpha_{\mu\nu}$ is the torsion tensor, and $R_{\mu\nu\lambda\rho}$ the Riemann curvature (Hoyos et al., 2024, Erdmenger et al., 2024, Rasulian et al., 2023, Huang et al., 2019).

The NY term is exact and locally a total derivative, but it yields physical contributions in settings with nontrivial topology, singularities, boundaries, or in thermal (finite-temperature) or finite-density backgrounds.

2. Physical Origin, Differential-Geometric Context, and Distinction from Standard Chiral Anomalies

In the Cartan formulation, torsion couples to the totally antisymmetric spin current, $\omega^a{}_b$0. The standard gauge and gravitational chiral anomalies involve $\omega^a{}_b$1 and $\omega^a{}_b$2—with dimensionless, quantized coefficients—derived from topological classes via the descent formalism.

In sharp contrast, the NY anomaly arises through quadratic torsion terms, is non-topological in four dimensions, and is associated with a dimensionful coefficient (e.g., $\omega^a{}_b$3 for a momentum-space cutoff, or $\omega^a{}_b$4 at finite temperature). This term does not descend from a gauge-invariant local functional and is sensitive to the ultraviolet regularization or infrared boundary data in its nonthermal version (Hoyos et al., 2024, Nissinen, 2019, Rasulian et al., 2023).

Geometrically, $\omega^a{}_b$5 measures the "failure" of infinitesimal parallelograms to close due to torsion, corrected by the addition of a curvature piece, and constitutes the unique torsion-invariant four-form without derivatives of torsion.

3. Microscopic Mechanisms, Holographic Reinterpretation, and the Dimensional Ladder

Several derivations of the NY anomaly exist:

• Spectral Flow and Torsional Landau Levels: In a background with uniform "torsional magnetic field" $\omega^a{}_b$6, Dirac fermions exhibit Landau quantization in the torsional field. All torsional Landau levels except the lowest cancel pairwise in the chiral charge. The surviving lowest modes reduce the effective theory to $\omega^a{}_b$7 dimensions and the anomaly coefficient is shown to be proportional to the $\omega^a{}_b$8-dimensional central charge, especially at finite $\omega^a{}_b$9 (Nissinen et al., 2019, Huang et al., 2019, Huang et al., 2019).
• Callan–Harvey Descent: The anomaly arises as a boundary inflow effect from a five-dimensional parity-odd effective action, linking the $$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$0D NY term to the $$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$1D Hughes–Leigh–Fradkin (HLF) action and the $$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$2D chiral energy-momentum anomaly via integration over momentum space slices (Huang et al., 2019).
• Holographic Duals: In asymptotically AdS$$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$3 spacetimes, each fermion bilinear current is dual to a $$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$4-form field in the bulk. In the absence of torsion, Hodge duality links the various bilinears, but when torsion is present, Hodge duality relations break down, and the non-conservation of the axial current arises not from explicit $$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$5 breaking, but from the failure of typically-exact Hodge duality relations. This holographically realizes the NY anomaly and predicts the $$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$6-scaling of its coefficient at finite temperature (Hoyos et al., 2024).

4. Universal Versus Non-Universal (Thermal and Non-Thermal) Nieh–Yan Anomaly

Zero-Temperature (Non-Thermal) NY Anomaly

In the zero-temperature limit, the NY anomaly enters the divergence of the axial current with a coefficient $$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$7 proportional to a dimensionful UV cutoff $$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$8: $$\mathcal{N}_{\rm NY} = T^a \wedge T_a - e^a \wedge e^b \wedge R_{ab} = d(e^a \wedge T_a)$$9 This coefficient is non-universal, scheme-dependent, and traceable to the depth of the Dirac sea or band-width in condensed-matter realizations, and can always be removed or altered by renormalization conditions (Erdmenger et al., 2024, Rasulian et al., 2023, Huang et al., 2019, Nissinen, 2019).

Thermal Nieh–Yan Anomaly

At finite temperature, the anomaly coefficient becomes universal and dimensionless: $T^a = de^a + \omega^a{}_b \wedge e^b$0 with $T^a = de^a + \omega^a{}_b \wedge e^b$1 for a single chiral fermion, independent of the UV regulator, and fixed by geometry, topology, and fermion number. This result is derived via thermal field theory computations, spectral flow calculations in a torsional field, and holographic models (Hoyos et al., 2024, Nissinen et al., 2019, Nissinen et al., 2019, Huang et al., 2019).

The appearance of $T^a = de^a + \omega^a{}_b \wedge e^b$2—the only available IR scale for $T^a = de^a + \omega^a{}_b \wedge e^b$3 (gap or linearity scale)—demonstrates a sharp separation between thermal and non-thermal NY responses.

5. Boundaries, Regularization, and Observable Content

The status of the NY anomaly critically depends on boundary conditions and the global structure of the spacetime manifold:

• Closed, Smooth Manifolds (No Boundaries): On such backgrounds, the global integral of the NY four-form vanishes if the Dirac index is well-defined, as required by the index theorem. The heat-kernel approach shows that the $T^a = de^a + \omega^a{}_b \wedge e^b$4-dependent part in the regularized index expansion must be absent for the index to be regulator-independent (Erdmenger et al., 2024, Rasulian et al., 2023). In these cases, the NY contribution can be removed by a local chiral transformation ("rotated away") and does not influence the quantum anomaly.
• Singularities, Boundaries, or Noncompactness: The NY anomaly can be nonzero when the manifold has boundaries, singular vierbein configurations, or fails to be globally regular. For instance, lattice dislocations, domain walls, or spatial infinity in condensed-matter systems act as effective boundaries supporting nontrivial NY terms, which may generate boundary Chern–Simons actions and induce anomalous transport (e.g., torsional Hall effects) (Erdmenger et al., 2024, Rasulian et al., 2023).
• Removal via Counterterms: In nonthermal settings, the NY term can always be absorbed into a renormalization of the torsion coupling in the action (e.g., $T^a = de^a + \omega^a{}_b \wedge e^b$5 counterterms). Only thermal NY terms, depending on periodicity in Euclidean time, are robust against such counterterm redefinitions (Erdmenger et al., 2024, Nissinen et al., 2019).

6. Experimental and Phenomenological Manifestations

The NY anomaly is relevant in systems where effective (emergent) torsion is present:

• Weyl Semimetals and Chiral Superfluids: In crystalline Weyl materials or the $T^a = de^a + \omega^a{}_b \wedge e^b$6He-A phase, strain, dislocations, or superfluid textures act as sources of emergent torsion. The thermal NY anomaly predicts a $T^a = de^a + \omega^a{}_b \wedge e^b$7 scaling of torsional axial conductivity, universal per Weyl node. Experimental consequences include: thermal generation of axial currents, torsional contributions to thermal Hall responses, and quantized thermal Hall conductance proportional to the $T^a = de^a + \omega^a{}_b \wedge e^b$8D central charge per node pair (Nissinen et al., 2019, Nissinen et al., 2019, Huang et al., 2019, Hoyos et al., 2024).
• Kramers–Weyl Semimetals: Electron–phonon couplings in nonmagnetic chiral crystals yield phonon-induced torsion fields for Kramers–Weyl fermions. Integrating out the fermions produces the NY term in the phonon action, giving rise to phonon helicity (circular polarization of acoustic phonons) and a net phonon angular momentum under temperature gradients, with a temperature dependence carrying both cutoff-sensitive and universal NY contributions (Liu, 2021).
• Torsional Hall Effects: Boundary-induced NY contributions lead to a boundary Chern–Simons action for the axial torsion field, producing torsional anomalous Hall responses at material surfaces or interfaces, with the Hall coefficient determined by boundary jumps or domain-wall structure (Erdmenger et al., 2024).

A selection of key observable predictions is organized in the following table:

System	Observable Effect	NY Signature
Weyl semimetal (bulk)	Anomalous thermal Hall conductivity	$T^a = de^a + \omega^a{}_b \wedge e^b$9 scaling, central charge factor
$$R_{ab} = d\omega_{ab} + \omega_{ac} \wedge \omega^c{}_b$$0He-A superfluid	Torsional correction to linear momentum flow	$$R_{ab} = d\omega_{ab} + \omega_{ac} \wedge \omega^c{}_b$$1 scaling of texture-induced flow
Chiral crystal (phonons)	Phonon helicity, angular momentum response	$$R_{ab} = d\omega_{ab} + \omega_{ac} \wedge \omega^c{}_b$$2 and $$R_{ab} = d\omega_{ab} + \omega_{ac} \wedge \omega^c{}_b$$3 dependences, change of sign with $$R_{ab} = d\omega_{ab} + \omega_{ac} \wedge \omega^c{}_b$$4
System with boundary/domain	Boundary axial (Hall) current	Boundary Chern–Simons NY term

Thermal NY effects are thus accessible via high-precision measurements of thermal/axial conductivities and chiral transport under engineered strain, texture, or boundary conditions.

7. Outlook and Open Issues

The status of the NY anomaly in the generic quantum field theory remains subtle. On smooth, closed manifolds it is a total derivative without physical anomaly content; its UV-sensitive coefficient in the nonthermal case implies that its magnitude is scheme-dependent and not topologically protected. Only in the presence of infrared cutoffs (temperature, cosmological constant), boundaries, singularities, or emergent torsional dynamics—as in condensed matter or holography—does the NY anomaly become quantifiable and phenomenologically relevant.

Recent research emphasizes the universality of the thermal NY anomaly, its confirmation via spectral flow and dimensional reduction arguments, and its precise experimental manifestations, contrasting sharply with the regulator-dependence or removability of the nonthermal term. These developments clarify long-standing debates about the NY term’s role in chiral anomalies and identify clear physical signatures in transport, dynamic responses, and topological phenomena in both high-energy and condensed-matter platforms (Hoyos et al., 2024, Nissinen et al., 2019, Nissinen et al., 2019, Huang et al., 2019, Erdmenger et al., 2024, Huang et al., 2019, Rasulian et al., 2023, Liu, 2021, Nissinen, 2019).