Global SO(3) x SO(3) x U(1) symmetry of the Hubbard model on bipartite lattices

Journal Article · Sun Aug 15 00:00:00 EDT 2010 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2010.03.002· OSTI ID:21457140

Carmelo, J M.P., ^[1]; Ostlund, Stellan ^[2]; Sampaio, M J ^[1]

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1. GCEP-Centre of Physics, University of Minho, Campus Gualtar, P-4710-057 Braga (Portugal)
2. Goeteborgs Universitet, Gothenburg 41296 (Sweden)

In this paper the global symmetry of the Hubbard model on a bipartite lattice is found to be larger than SO(4). The model is one of the most studied many-particle quantum problems, yet except in one dimension it has no exact solution, so that there remain many open questions about its properties. Symmetry plays an important role in physics and often can be used to extract useful information on unsolved non-perturbative quantum problems. Specifically, here it is found that for on-site interaction U {ne} 0 the local SU(2) x SU(2) x U(1) gauge symmetry of the Hubbard model on a bipartite lattice with N{sub a}{sup D} sites and vanishing transfer integral t = 0 can be lifted to a global [SU(2) x SU(2) x U(1)]/Z{sub 2}{sup 2} = SO(3) x SO(3) x U(1) symmetry in the presence of the kinetic-energy hopping term of the Hamiltonian with t > 0. (Examples of a bipartite lattice are the D-dimensional cubic lattices of lattice constant a and edge length L = N{sub a}a for which D = 1, 2, 3,... in the number N{sub a}{sup D} of sites.) The generator of the new found hidden independent charge global U(1) symmetry, which is not related to the ordinary U(1) gauge subgroup of electromagnetism, is one half the rotated-electron number of singly occupied sites operator. Although addition of chemical-potential and magnetic-field operator terms to the model Hamiltonian lowers its symmetry, such terms commute with it. Therefore, its 4{sup N}{sub a}{sup D} energy eigenstates refer to representations of the new found global [SU(2) x SU(2) x U(1)]/Z{sub 2}{sup 2} = SO(3) x SO(3) x U(1) symmetry. Consistently, we find that for the Hubbard model on a bipartite lattice the number of independent representations of the group SO(3) x SO(3) x U(1) equals the Hilbert-space dimension 4{sup N}{sub a}{sup D}. It is confirmed elsewhere that the new found symmetry has important physical consequences.

• OSTI ID: 21457140
• Journal Information: Annals of Physics (New York), Vol. 325, Issue 8; Other Information: DOI: 10.1016/j.aop.2010.03.002; PII: S0003-4916(10)00054-0; Copyright (c) 2010 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
CUBIC LATTICES
EIGENSTATES
ELECTROMAGNETISM
ELECTRONS
EXACT SOLUTIONS
FIELD OPERATORS
GAUGE INVARIANCE
HAMILTONIANS
HILBERT SPACE
HUBBARD MODEL
KINETIC ENERGY
LATTICE PARAMETERS
MAGNETIC FIELDS
POTENTIALS
SO-3 GROUPS
SU-2 GROUPS
SYMMETRY
U-1 GROUPS
BANACH SPACE
CRYSTAL LATTICES
CRYSTAL MODELS
CRYSTAL STRUCTURE
ELEMENTARY PARTICLES
ENERGY
FERMIONS
INVARIANCE PRINCIPLES
LEPTONS
LIE GROUPS
MAGNETISM
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MATHEMATICAL SOLUTIONS
MATHEMATICAL SPACE
QUANTUM OPERATORS
SO GROUPS
SPACE
SU GROUPS
SYMMETRY GROUPS
U GROUPS