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Modern Physics Letters B

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

Charilaos Aneziris, A. P. Balachandran, Ajit Mohan Srivastava

Published Apr 10, 1990• 1 citation

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.

DOI: 10.1142/s0217984990000568

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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.

At half-filling the repulsive Hubbard model is in a Mott insulating phase. The charge degrees of freedom are gapped, whereas the spin degrees of freedom remain gapless. At low energies the spin sector is actually scale invariant (apart from logarithmic corrections) and Conformal Field Theory (CFT) methods may be applied to determine the low-energy behaviour of correlation functions involving only the spin sector. On the other hand, the charge sector is not scale invariant and CFT does not provide any information for correlators involving the charge degrees of freedom. In this chapter we will show that there exists a particular continuum limit of the half filled Hubbard model, in which it is possible to calculate dynamical correlation functions by means of methods of integrable quantum field theory. We first construct a Lorentz invariant scaling limit starting from the results for the excitation spectrum and the S-matrix discussed in Chapter 7. This scaling limit is identified as the SU(2) Thirring model, which is an integrable relativistic quantum field theory. Next we discuss a continuum limit , which is obtained directly from the Hubbard Hamiltonian and describes the vicinity of the scaling limit. Construction of the scaling limit The simplest way of constructing the scaling limit is to start with the results for the dispersions of the elementary excitations and the S-matrix derived in Chapter 7 and then look for a particular limit in which Lorentz invariance is recovered.

Abstract We discuss the one-dimensional Hubbard model, on finite sites spin chain, in context of the action of the direct product of two unitary groups SU(2)×SU(2). The symmetry revealed by this group is applicable in the procedure of exact diagonalization of the Hubbard Hamiltonian. This result combined with the translational symmetry, given as the basis of wavelets of the appropriate Fourier transforms, provides, besides the energy, additional conserved quantities, which are presented in the case of a half-filled, four sites spin chain. Since we are dealing with four elementary excitations, two quasiparticles called “spinons”, which carry spin, and two other called “holon” and “antyholon”, which carry charge, the usual spin-SU(2) algebra for spinons and the so called pseudospin-SU(2) algebra for holons and antiholons, provide four additional quantum numbers.