The Planar Thirring Model with Kähler-Dirac Fermions

• Main Author: Simon Hands
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Symmetry
• Subjects: interacting fermions; field theories in dimensions other than four; staggered lattice fermions; renormalisation group fixed point
• Online Access: https://www.mdpi.com/2073-8994/13/8/1523

Kähler’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<inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>N</mi><mo>)</mo><mo>⊗</mo></mrow></semantics></math></inline-formula>U<inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula> field transformations and has a parity-invariant mass term, which are 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.