Tetrads in Solids: from Elasticity Theory to Topological Quantum Hall Systems and Weyl Fermions

Abstract

Theory of elasticity in topological insulators has many common features with relativistic quantum fields interacting with gravitational field in the tetrad form. Here we discuss several issues in the effective topological (pseudo)electromagnetic response in three-dimensional weak crystalline topological insulators with no time-reversal symmetry that feature elasticity tetrads, including a mixed "axial-gravitational" anomaly. This response has some resemblance to "quasitopological" terms proposed for massless Weyl quasiparticles with separate, emergent fermion tetrads. As an example, we discuss the chiral/axial anomaly in superfluid 3 He-A. We demonstrate the principal difference between the elasticity tetrads and the Weyl fermion tetrads in the construction of the topological terms in the action. In particular, the topological action expressed in terms of the elasticity tetrads, cannot be expressed in terms of the Weyl fermion tetrads since in this case the gauge invariance is lost.PACS numbers:

Elasticity tetrads, mixed axial-gravitational anomalies, and ( 3+1 )-d quantum Hall effect

“…The CME for a time-periodic insulator with timelike E 0 µ = ω F δ 0 µ (with Floquet drive ω F ) and the temporal invariant N 0 = 0 was pointed out in Ref. 17. This can be extended to spatial deformations as well.…”

“…Here we show that the IQHE/AQHE in 3 + 1dimensional crystalline quantum Hall systems or topological insulators 13 is also described by CS term with mixed field content. Namely, the 3+1d CS term features elasticity tetrads [14][15][16][17] , which describe the geometry of elasticity theory, including crystals with dislocation defects. The density of dislocations corresponds to spatial torsion of the geometry, more familiar in gravitational theories, see e.g.…”

On Thermal Nieh-Yan Anomaly in Topological Weyl Materials

“…, which allows to have topological terms of the type E ∧ A ∧ dA with quantized dimensionless coefficients [21][22][23].…”

On DIMENSION OF TETRADS IN EffECTIVE GRAVITY

“…The dimensionless physics emerging in the frame of the Vladimirov-Diakonov dimensionful tetrads leads in particular to the new topological terms in action, since some of the dimensionless parameters appear to be the integer valued quantum numbers, which chacterize the topology of the quantum vacuum. This can be seen on example of the 3+1 dimensional quantum Hall effect in topological insulators 10,11,26 . When the Chern-Simons action is written in terms of the elasticity tetrads with [e A µ ] = 1/[l], its prefactor becomes dimensionless and universal, being expressed in terms of integer-valued momentum-space invariant.…”

“…This was first noticed by Diakonov 7 and Vladimirov and Diakonov (VD) 8,9 in the scenario, where tetrad fields emerge as bilinear combinations of the fermionic fields. Tetrads with dimension of inverse length emerge also in the model of the superplastic vacuum 10,11 .…”