Preliminary physical consequences of the Hubbard model on a square lattice extended global SO(3) × SO(3) × U(1) symmetry
By: José M. P. Carmelo
From: Univ. Minho
At: Instituto de Investigação Interdisciplinar, Anfiteatro
[2011-06-09]
A trivial result is that at onsite repulsion U = 0 the global symmetry of the half-filled Hubbard model on a bipartite lattice is O(4) = SO(4) × Z2. Here the factor Z2 refers to the particle-hole transformation on a single spin under which the model Hamiltonian is not invariant for U /= 0. C. N. Yang and S. C. Zhang considered the most natural possibility that the SO(4) symmetry inherited from the U /= 0 Hamiltonian O(4) = SO(4) × Z2 symmetry was the model global symmetry for U > 0.[1] However, a recent study of the problem by the author and collaborators [2] revealed an exact extra hidden global U(1) symmetry emerging for U /= 0 in addition to SO(4), so that the model global symmetry is [SO(4) × U(1)]/Z2 = SO(3) × SO(3) × U(1) = [SU(2) × SU(2) × U(1)]/Z22 . The factor 1/Z22 in SO(3) × SO(3) × U(1) = [SU(2) × SU(2) × U(1)]/Z22 imposes that both [Sc + Sc] and [Se + Sc] are integer numbers. Here Se, Ss, and Sc are the e-spin, the spin, and the eigenvalue of the generator of the new global U(1) symmetry, respectively. The latter is found in [2] to be one half the number of rotated-electron singly occupied sites. The extra hidden global U(1) symmetry is related to the U /= 0 local SU(2) × SU(2) × U(1) gauge symmetry of the Hubbard model on a bipartite lattice with transfer integral t = 0.[3] Such a local SU(2) × SU(2) × U(1) gauge symmetry becomes for finite U and t a group of permissible unitary transformations. Rather than the ordinary U(1) gauge subgroup of electromagnetism, for finite U/t here U(1) refers to a “nonlinear†transformation.[3] Since the chemical-potential and magnetic-field operator terms commute with the Hamiltonian, for all densities its energy eigenstates refer to representations of the new found global SO(3)×SO(3)×U(1) = [SO(4)×U(1)]/Z2 symmetry, which is expected to have important physical consequences. In addition to introducing the new-found extended global symmetry, in this talk some preliminary physical consequences are reported for the Hubbard model on the bipartite square lattice.[4]
1. C. N. Yang and S. C. Zhang, Mod. Phys. Lett. B 4, 758 (1990); S. C. Zhang, Phys. Rev. Lett. 65, 120 (1990).
2. J. M. P. Carmelo, Stellan ¨Ostlund, and M. J. Sampaio, Ann. Phys. 325, 1550 (2010).
3. Stellan ¨Ostlund and Eugene Mele, Phys. Rev. B 44, 12413 (1991).
4. J. M. P. Carmelo, Nucl. Phys. B 824, 452 (2010); Nucl. Phys. B 840, 553 (2010).