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Physical Review Letters

Symmetries of the large-UHubbard model at half filling

Ling-Lie Chau, Ding-Wei Huang, Yue Yu

Published Apr 20, 1992• 2 citations
Publisher

Abstract

Symmetries of the Hubbard model in the large-U limit are discussed. WE show that the global SU(2) symmetry of the full even-lattice Hubbard model generated by Yang's \ensuremath{\eta}-pairing operators becomes a local symmetry in the large-U limit and is applicable to even-odd mixed lattices. This extends the well-known SU(2) local symmetry of Anderson, for which we construct the generators. The associated local gauge theory of the extended symmetry can be similarly formulated and has interesting physical implications: new mean-field theories with parity-changed pairings. We point out the importance of searching for the parity-dependent Aharanov-Bohm-type of effects related to these local gauge symmetries.

DOI
10.1103/physrevlett.68.2539

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Symmetries of the Hubbard model in the large-U limit are discussed. WE show that the global SU(2) symmetry of the full even-lattice Hubbard model generated by Yang's \ensuremath{\eta}-pairing operators becomes a local symmetry in the large-U limit and is applicable to even-odd mixed lattices. This extends the well-known SU(2) local symmetry of Anderson, for which we construct the generators. The associated local gauge theory of the extended symmetry can be similarly formulated and has interesting physical implications: new mean-field theories with parity-changed pairings. We point out the importance of searching for the parity-dependent Aharanov-Bohm-type of effects related to these local gauge symmetries.
We exhibit an additional SU(2) symmetry of the Hubbard model at half-filling, which suggests a simple physical manifestation of this symmetry in the presence of an applied magnetic field. The system governed by a negative-U Hubbard Hamiltonian will lose its superconductivity at an applied magnetic field well below the critical Hc{\mathit{H}}_{\mathit{c}}Hc​ associated with a conventional superconductor. Though departure from half-filling and the inclusion of a repulsive nearest-neighbor interaction break this symmetry, the above physical effect should still persist strongly and is indicative of a new type of SU(2) superconductor for the negative-U or attractive on-site Hubbard model.
We demonstrate explicitly the SU(2) gauge invariance of the large-UUU limit of the Hubbard model, i.e., the Heisenberg model written in terms of electron operators with a constraint of one particle per site. We use this result to demonstrate the equivalence of two apparently very different mean-field theories.
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 3He^{3}\mathrm{He}3He and the SU(2) Bogoliubov transformation in BCS theory.
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.