From elasticity tetrads to rectangular vielbein

G. E. Volovik*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)
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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 Eμa. They come from the phase fields, which describe the deformations of crystal planes. The important property of the elasticity vielbein Eμa 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 eaμ may emerge in the vicinity of Dirac points in Dirac materials. In particular, in the planar phase of the spin–triplet superfluid 3He 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 gμν 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 languageEnglish
Article number168998
Number of pages8
JournalAnnals of Physics
Early online date2022
Publication statusPublished - Dec 2022
MoE publication typeA1 Journal article-refereed


  • Dirac fermions
  • Einstein–Cartan gravity
  • Elasticity theory
  • Superfluid He
  • Tetrad
  • Vielbein


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