Tetrads in Solids: from Elasticity Theory to Topological Quantum Hall Systems and Weyl Fermions
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ontribution for the JETP special issue in honor of L.P. Pitaevskii’s 85th birthday
Tetrads in Solids: from Elasticity Theory to Topological Quantum Hall Systems and Weyl Fermions1 J. Nissinena,* and G. E. Volovika,b a
Low Temperature Laboratory, Aalto University, P.O. Box 15100, Aalto, FI-00076 Finland Institute for Theoretical Physics, Chernogolovka, Moscow oblast, 142432 Russia * e-mail: [email protected]
b Landau
Received April 9, 2018
Abstract—Theory of elasticity in topological insulators has many common features with relativistic quantum fields interacting with gravitational fields 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 3He-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. DOI: 10.1134/S1063776118110080
1. INTRODUCTION There are several sources of emergent tetrad gravity in solids. The tetrad field in particular emerges in Dirac and Weyl semimetals in the vicinity of nodes in the electron spectrum (the fermion tetrads or Weyl tetrads). A different set of tetrads (the elasticity tetrads) emerges in the theory of elasticity, see e.g. [1, 2]. Our consideration of the tetrad fields is based on approach formulated in two books by Landau and Lifshitz [2, 3] from their multi-volume “Course of Theoretical Physics.” Lev Petrovich Pitaevskii was not only one who had completed the course but later also the editor of the books in the series. Here we consider the elasticity tetrad fields in weak topological insulators and the fermion tetrads in Weyl fermion systems. More specifically, we discuss the role of the crystalline tetrads in the response of 3+1dimensional (D = 3 + 1) weak topological insulators with anomalous quantum Hall effect (AQHE) and Weyl semimetals/superfluids with the chiral/axial anomaly. For the topological AQHE response, we obtain a D = 2n + 2 = 6 mixed axial-gravitational anomaly, and consider its dimensional reduction to D = 2n + 1 = 5 and D = 2n = 4 anomalous actions, as 1 The article is published in the original.
well as the extension to driven Floquet-Bloch systems, which are expressed in terms of three or four integer topological invariants in the crystal 4-momentum space. We also note the possibility of an emergent fermion tetrad gravity that is different for left- and righthanded Weyl fermions. This is possible in condensed matter syste
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