The Planar Thirring Model with K\"ahler-Dirac Fermions

TL;DR

This paper applies Kähler-Dirac formalism to three-dimensional fermions, deriving a continuum Thirring model with symmetries akin to lattice versions, and explores its implications for strongly interacting fixed points.

Contribution

It introduces a Kähler-Dirac-based continuum Thirring model that captures lattice symmetries and spin-taste entanglement, extending understanding of fermionic interactions in 3D.

Findings

The continuum theory exhibits U(N)×U(N) symmetry and parity invariance.

A generalized interaction term entangles spin and taste degrees of freedom.

Excluding scalar components yields a spin-1 fermion theory with specific polarization states.

Abstract

K\"ahler's geometric approach in which relativistic fermion fields are treated as differential forms is applied in three spacetime dimensions. It is shown that the resulting continuum theory is invariant under global U($N)\otimes$U($N)$ field transformations, and has a parity-invariant mass term, both symmetries shared in common with staggered lattice fermions. The formalism is used to construct a version of the Thirring model with contact interactions between conserved Noether currents. Under reasonable assumptions about field rescaling after quantum corrections, a more general interaction term is derived, sharing the same symmetries but now including terms which entangle spin and taste degrees of freedom, which exactly coincides with the leading terms in the staggered lattice Thirring model in the long-wavelength limit. Finally truncated versions of the theory are explored; it is found that excluding scalar and pseudoscalar components leads to a theory of six-component fermion fields describing particles with spin 1, with fermion and antifermion corresponding to states with definite circular polarisation. In the UV limit only transverse states with just four non-vanishing components propagate. Implications for the description of dynamics at a strongly interacting renormalisation-group fixed point are discussed.