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Physical Review Letters

Symmetries and canonical transformations of the Hubbard model on bipartite lattices

Stellan OIUMLstlund

Published Sep 14, 1992• 20 citations
Publisher

Abstract

Recently an exact SU(2)\ensuremath{\bigotimes}SU(2) symmetry for the half-filled Hubbard model has been elucidated but has not yet been properly incorporated in many analyses of this model. We compute the irreducible representations of the symmetry group, a necessary step for any consistent mean-field analysis. A proper mean-field theory valid for both negative- and positive-U Hubbard models is then presented. A by-product of the description is a systematic enumeration of the Lie group SU(8) of unitary canonical transformations that is a direct generalization of the SU(4) transformation in the theory of superfluid $^{3}\mathrm{He}$ and the SU(2) Bogoliubov transformation in BCS theory.

DOI
10.1103/physrevlett.69.1695

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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.
By permitting canonical transformations that are nonlinear in fermion creation and annihilation operators, we show that the space of canonical transformations of ordinary spin-1/2 operators local to a point in space is SU(2)\ensuremath{\bigotimes}SU(2)\ensuremath{\bigotimes}U(1)\ensuremath{\bigotimes}Z2{\mathit{Z}}_{2}Z2​. We identify those subgroups that form local and global gauge symmetries of the Hubbard-Heisenberg model on and off half filling. Our systematic method of generating all ocal canonical transformations enables us to discover a ``nonlinear'' local U(1) gauge symmetry of the Heisenberg-Hubbard model that remains a local symmetry away from half filling. The paper presents this group together with all other known canonical transformations in a unified framework.
The two-dimensional extended Hubbard model that includes a nearest-neighbor Heisenberg interaction is studied using a mean-field theory where quasiparticles are defined by an U(8) group of canonical transformations permitting both broken gauge, spin, and sublattice symmetry. The theory is further extended to incorporate a possible twist in the spin-quantization axis, so that the competition between superconductivity, charge density waves, and N\'eel and spiral antiferromagnetic order can be monitored within one single theory. Our results for positive Hubbard U and Heisenberg exchange J suggest that antiferromagnetic ordering dominates close to half-filling, while spiral states and d-wave superconductivity compete when doping is introduced. For moderate values of J, we find a phase diagram where a phase transition occurs from an antiferromagnet to a d-wave superconductor as doping is increased. A narrow region of (s+id)-wave superconductor is found for some values of J and U.
The one band Hubbard model on a bi-partite lattice possess a global SO(4) symmetry group which is a direct product of the spin SU(2) and the pseudospin SU(2) symmetry group. Recently, the group theorectical consequence of this symmetry property has been investigated systematically. It is shown that this symmetry property can be used to reduce the complexity of diagonalizing the Hubbard Hamiltonian, and classify all the eigenstates according to the representation of the SO(4) group. It also leads to an exact one-to-one corespondence of the eigenstates at half-filling to that away from half-filling. Most importantly however, it is shown that this symmetry property leads to a direct experimental prediction for the high Tc superconductivity. If the high Tc superconductivity is correctly described by the Hubbard model, then the SO(4) symmetry dictates the existence of a collective mode that only appears below the superconducting transition temperature. Such a mode can be detected in an electron energy loss experiment as a sharp resonance feature. The location of the peak measure the parameters in the Hubbard model, the intensity is proportional to the superconducting order parameter and the width measures the departure of the Hubbard model from reality.