Effective Lagrangian for the macroscopic motion of Weyl fermions in $^3$He-A

TL;DR

This paper develops an effective Lagrangian framework for describing the macroscopic motion of the normal component of superfluid $^3$He-A, incorporating emergent relativistic fermions, torsion, and anomalies, with applications to vortex rotation.

Contribution

It introduces a novel effective Lagrangian formulation for the normal component's motion in superfluid $^3$He-A, including torsion and anomaly effects, using a matrix-valued vierbein approach.

Findings

Explicit form of the effective Lagrangian for emergent relativistic fermions.

Constraints on superfluid background and normal flow derived from thermodynamic equilibrium.

Analysis of thermodynamic quantities during macroscopic rotation around vortices.

Abstract

We consider macroscopic motion of the normal component of superfluid $^3$He - A in global thermodynamic equilibrium within the context of the Zubarev statistical operator method. We formulate the corresponding effective theory in the language of the functional integral. The effective Lagrangian comprising macroscopic motion of fermionic excitations is calculated explicitly for the emergent relativistic fermions of the superfluid $^3$He - A phase immersed in a non-trivial bosonic background due to a space and time dependent matrix-valued vierbein featuring nonzero torsion as well as the Nieh-Yan anomaly. We do not consider the dynamics of the superfluid component itself and thereby its backreaction effects due to normal component macroscopic flow. It is being treated as an external background within which the emergent relativistic fermions of the normal component move. The matrix-valued vierbein formulation comprises an additional two dimensional internal spin space for the two axially charged Weyl fermions living at the Fermi points which may be replaced by one featuring a Dirac fermion doublet with a real valued vierbein, an axial Abelian gauge field and a spin connection gauge field mixing the Dirac and internal spin spaces. We carry out this change of description in detail and determine the constraints on the superfluid background as well as the the normal component motion as determined from the Zubarev statistical operator formalism in global thermodynamic equilibrium. As an application of the developed theory we consider macroscopic rotation around the axis of pure integer mass vortices. The corresponding thermodynamic quantities of the normal component are analyzed. Our formulation incorporates both superfluid background flow and macroscopic motion flow of the normal component and thereby enables an analysis of their interrelation.