THE HALF-FILLED HUBBARD MODEL AND THE 2 + 1 DIMENSIONAL DIRAC LAGRANGIAN

Abstract

There is a proposal that a 2 + 1 dimensional Lagrangian for an appropriate number of SU(2) doublets of Dirac fields with couplings to an SU(2) gauge field describes the continuum limit of the strongly coupled half-filled Hubbard model. In this paper, we compare the symmetries of these two models and show that the Dirac Lagrangian does have all the symmetries of the latter. In particular, the SU(2) spin rotation symmetry of the latter is shown to be realized as a Pauli-Gürsey symmetry of the Dirac system. The occurrence of the Pauli-Gürsey symmetry in a class of 2 + 1 dimensional Dirac Lagrangians more general than those with SU(2) gauge couplings is pointed out. The Dirac Lagrangian with SU(2) gauge invariance has more symmetries than the half-filled Hubbard system, being in particular Lorentz invariant. Starting with a general ansatz for the continuum Lagrangian and systematically requiring that it fufills the wanted symmetries of the half-filled Hubbard model, a one-parameter family of possible continuum approximations to this model are derived. The Dirac system corresponds to a special value of this parameter. Except for this special case, these approximations are not Lorentz invariant, but they have rotational invariance.