Vielbein with mixed dimensions and gravitational global monopole in the planar phase of superfluid $^3$He

TL;DR

This paper explores the effects of Dirac monopoles in the planar phase of superfluid helium-3, revealing a conical effective metric and introducing a novel vielbein structure distinct from traditional tetrad fields.

Contribution

It demonstrates that Dirac monopoles induce a conical metric in superfluid helium-3 and introduces a new 4x5 vielbein matrix as the primary variable, differing from standard tetrad gravity.

Findings

Dirac monopoles produce a conical effective metric.

The primary variable is a 4x5 vielbein matrix, not a 4x4 tetrad.

The effective metric resembles that of a global monopole in general relativity.

Abstract

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 $4\times 5$ matrix, as distinct from the conventional $4\times 4$ matrix of the tetrad field in tetrad gravity.