Vielbein with Mixed Dimensions and Gravitational Global Monopole in the Planar Phase of Superfluid 3 He
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Vielbein with Mixed Dimensions and Gravitational Global Monopole in the Planar Phase of Superfluid 3 He G. E. Volovik 1) Low Temperature Laboratory, Aalto University, School of Science and Technology, P.O. Box 15100, FI-00076 Aalto, Finland Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia Submitted 20 September 2020 Resubmitted 20 September 2020 Accepted 20 September 2020
The planar phase of superfluid 3 He has Dirac points in momentum space and the analog of Dirac monopole in the real space. Here we discuss the combined effect of Dirac point and Dirac monopole. It is shown that in the presence of the monopole the effective metric acting on Dirac fermions corresponds to the metric produced by the global monopole in general relativity: it is the conical metric. Another consequence is that the primary variable, which gives rise to the effective metric, is the unusual vielbein field in the form of the 4 × 5 matrix, as distinct from the conventional 4 × 4 matrix of the tetrad field in tetrad gravity. DOI: 10.1134/S0021364020200035
where σ α are the Pauli matrices for spin and Aαi is the 3 × 3 complex matrix of the order parameter, see the book [1]. In the planar phase the particular representative of the order parameter is: Aαi = c⊥ eiΦ δαi − ˆlα ˆli , (2)
I. Introduction. The planar phase is one of the possible superfluid phases of liquid 3 He [1]. It may exist in some region of the phase diagram of superfluid 3 He confined in aerogels [2]. The planar phase has two Dirac points in the quasiparticle spectrum, which are supported by combined action of topology and some special symmetry, see e.g. [3]. The quasiparticles in the planar phase with fixed spin behave as Weyl fermions. Similar to the chiral superfluid 3 He-A, they experience the effective gravity and gauge field produced by the deformation of the order parameter. But there is the following important difference. In 3 He-A, the spin-up and spin-down fermions have the same chirality, while in the planar phase the spin-up and spin-down fermions have the opposite chirality. As a result the Weyl fermions in planar phase form the massless Dirac fermions, see [4]. Here we study the planar phase fermions in the presence of the topological defect – the hedgehog. The effective gravity produced by the hedgehog appears to be similar to the gravitational effect of the global monopole in general relativity: it gives rise to the conical space [5–12]. Another consequence of the hedgehog is that the vielbein, which describes the effective gravity, is the 4×5 matrix, as distinct from the conventional 4 × 4 matrix in the tetrad formalism of general relativity. II. Weyl-Dirac points and 4 × 5 vielbein. In the general spin triplet p-wave pairing state the 2 × 2 matrix of the gap function is: ˆ = Aiα σ α pi , Δ 1) e-mail:
where Φ is the phase of the order parameter and ˆl is the unit vector. All the other degenerate states of the planar phase are obtained by spin, orbital and phase rotations of the group G = SO(3)S × SO(3)L × U (1) (here we ignore the discrete symmetry
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