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Abstract
The paper is devoted to the memory of Igor E. Dzyaloshinsky. In our common paper Dzyaloshinskii and Volovick (1980), we discussed the elasticity theory described in terms of the gravitational field variables — the elasticity vielbein E_{μ}^{a}. They come from the phase fields, which describe the deformations of crystal planes. The important property of the elasticity vielbein E_{μ}^{a} is that in general they are not the square matrices. While the spacetime index μ takes the values μ=(0,1,2,3), in crystals the index a=(1,2,3), in vortex lattices a=(1,2), and in smectic liquid crystals there is only one phase field, a=1. These phase fields can be considered as the spin gauge fields, which are similar to the gauge fields in Standard Model (SM) or in Grand Unification (GUT). On the other hand, the rectangular vielbein e_{a}^{μ} may emerge in the vicinity of Dirac points in Dirac materials. In particular, in the planar phase of the spin–triplet superfluid ^{3}He the spacetime index μ=(0,1,2,3), while the spin index a takes values a=(0,1,2,3,4). Although these (4×5) vielbein describing the Dirac fermions are rectangular, the effective metric g^{μν} of Dirac quasiparticles remains (3+1)dimensional. All this suggests the possible extension of the Einstein–Cartan gravity by introducing the rectangular vielbein, where the spin fields belong to the higher groups, which may include SM or even GUT groups.
Original language  English 

Article number  168998 
Number of pages  8 
Journal  Annals of Physics 
Volume  447 
Early online date  2022 
DOIs  
Publication status  Published  Dec 2022 
MoE publication type  A1 Journal articlerefereed 
Keywords
 Dirac fermions
 Einstein–Cartan gravity
 Elasticity theory
 Superfluid He
 Tetrad
 Vielbein
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 1 Finished

TOPVAC: From Topological Matter to Relativistic Quantum Vacuum
Volovik, G., Nissinen, J., Eltsov, V., Rysti, J., Ikäheimo, A., Rantanen, R., Mäkinen, J., Zhang, K., Laurila, S., Volard, M. & Kamppinen, T.
01/10/2016 → 30/09/2022
Project: EU: ERC grants