The Planar Thirring Model with Kähler-Dirac Fermions

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(N)⊗U(N) 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.

A few solvable two-dimensional quantum field theories (QFT)

The chapter is devoted to several two-dimensional quantum field theories (QFT), whose properties can be determined by non-perturbative methods. The Schwinger model, a model of two-dimensional quantum electrodynamics (QED) with massless fermions, illustrates the properties of confinement, spontaneous chiral symmetry breaking, asymptotic freedom and anomalies, properties one also expects in particle physics from quantum chromodynamics. The equivalence between the massive Thirring model, a fermion model with current–current interaction, and the sine-Gordon model is derived, using the bozonisation technique. The bosonization technique, based on an identity for Cauchy determinants, establishes relations, specific to two dimensions, between fermion and boson local field theories. Several generalized Thirring model are also discussed. In the discussion of the O(N) non-linear σ-model, it has been noticed that the Abelian case N = 2 is special, because the renormalization group (RG) β-function vanishes in two dimensions. The corresponding O(2) invariant spin model is especially interesting: it provides an example of the celebrated Kosterlitz–Thouless (KT) phase transition and will be studied elsewhere. This chapter also provides the necessary technical background for such an investigation.

Duality between Dirac fermions in curved spacetime and optical solitons in non-linear Schrodinger model: magic of $$1+1$$-dimensional bosonization

Bosonization in curved spacetime maps massive Thirring model (self-interacting Dirac fermions) to a generalized Sine–Gordon model, both living in $$1+1$$1+1-dimensional curved spacetime. Applying this duality we have shown that the Thirring model fermion, in non-relativistic limit, gets identified with the soliton of non-linear Scrodinger model with Kerr form of non-linearity. We discuss one particular optical soliton in the latter model and relate it with the Thirring model fermion.