Functional Bosonization of Nonrelativistic Fermions in 2+1 Dimensions

Abstract

We analyze the universality of the bosonization rules in non-relativistic fermionic systems in (2 + 1)d. We show that, in the case of linear fermionic dispersion relations, a general fermionic theory can be mapped into a gauge theory in such a way that the fermionic density maps into a magnetic flux and the fermionic current maps into a transverse electric field. These are universal rules in the sense that they remain valid whatever the interaction considered. We also show that these rules are universal in the case of nonlinear dispersion relations provided we consider only density-density interactions. We apply the functional bosonization formalism to a non-relativistic and non-local massive Thirring-like model and evaluate the spectrum of collective excitations in several limits. In the large mass limit, we are able to exactly calculate this spectrum for arbitrary density-density and current-current interactions. We also analyze the massless case and show that it has no collective excitations for any density-density potential in the Gaussian approximation. Moreover, the presence of current interactions may induce a gapless mode with a linear dispersion relation.

Effective field theories for topological insulators by functional bosonization

“…This approach naturally is only meaningful for massive theories. In the case of massive Dirac fermions in D = 2 + 1 dimensions it has been shown [51][52][53][54][55][56] that the correlation functions of the conserved fermionic currents are the same as the correlation functions of a dual topological Chern-Simons gauge theory in the low energy regime (only!). A mapping of the correlators of other fermion bilinears can also be determined but, again, only in the low-energy and long wavelength regime.…”

Universal transverse conductance between quantum Hall regions and (2+1) D bosonization

“…Eqs. (34) and (52)), has the same form of the localized interaction term. Then, when looking at universal behavior, the exact form of this unknown term is irrelevant, all we need is that this term be localized outside the perfect Hall regions.…”

“…In Ref. 34 we have shown that for the nonrelativistic interactions consider in this model the current bosonization rules,…”

“…The nonrelativistic fermionic determinant is a very complicated object and no exact analytic result is known for the general case. In the gaussian approximation, when B ext ≈ 0, the spectrum is gapless and there is no signal of topology in the structure of the determinant 34 .…”

Bosonization and effective vector-field theory of the fractional quantum Hall effect

“…Let us first briefly review this procedure, known as functional bosonization. [17][18][19][20] The gauge invariance of the action…”