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Europhysics Letters (EPL)

Spin and Pairing Algebras and ODLRO in a Hubbard Model

M. Pernici

Published May 1, 1990• 53 citations
Publisher

Abstract

The states of a simple Hubbard model fall into representations of the su (2) spin algebra and of an su (2) pairing algebra. A half-filled band momentum eigenstate has off-diagonal long-range order (ODLRO) if it belongs to an irreducible representation with pairing spin η > ( N - s )/3, N being the number of lattice sites and s the spin of the representation.

DOI
10.1209/0295-5075/12/1/014

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The states of a simple Hubbard model fall into representations of the su (2) spin algebra and of an su (2) pairing algebra. A half-filled band momentum eigenstate has off-diagonal long-range order (ODLRO) if it belongs to an irreducible representation with pairing spin η > ( N - s )/3, N being the number of lattice sites and s the spin of the representation.
Yang's η pairing operator is generalized to explore off-diagonal long-range order in the Hubbard bilayer with an arbitrary chemical potential. With this operator and a constraint condition on annihilation and creation operators, we construct explicitly eigenstates which possess simultaneously three kinds of off-diagonal long-range order, i.e., the intralayer one and the interlayer one for on-site pairing, and that for interlayer nearest-neighbor pairing. As in the simple Hubbard model there is also an SO(4) symmetry, with the generators properly defined. A sufficient condition leads to at least one of the above three kinds of off-diagonal long-range order. A constraint relation among different kinds of off-diagonal long-range order is also given. There exists a triplet of collective modes if the U(1) symmetry of a subgroup is spontaneously broken.
We consider NNN-particle generalizations of $\ensuremath{\eta}$-pairing states in a chain of NNN-component fermions and show that these states are exact (high-energy) eigenstates of an extended SU(N)\mathrm{SU}(N)SU(N) Hubbard model. We compute the singlet correlation function of the states and find that its behavior is qualitatively different for even and odd NNN. When NNN is even, these states exhibit off-diagonal long-range order in NNN-particle reduced density matrix. On the other hand, when NNN is odd, the correlations decay exponentially with distance in the bulk, but end-to-end correlations do not vanish in the thermodynamic limit. Finally, we prove that these states are the unique ground states of suitably tailored Hamiltonians.
We show that in the translation invariant case and in the antiferromagnetic phase, the reduced density matrix ρ2 has no off-diagonal long-range order of on-site electron pairs for the single-band Hubbard model on a cubic lattice away from half filling at finite temperatures both for the positive coupling and for the negative coupling. In these cases the model can not give a mechanism for the superconductivity caused by the condensation of on-site electron pairs and the nearestneighbor electron pairs. PACS numbers: 71.10.+x, 74.20.-z, 05.30.-d. 1 It was proposed the possibility that the Hubbard model can give a mechanism to characterize high temperature superconductivity[1]. It has been shown that the η pairing states for the single band Hubbard model with constant hopping matrix element for nearest neighbor sites have ODLRO[2]. It has also been shown that in the ground state the model with negative coupling has off-diagonal long-range order (ODLRO) on some bipartite lattice[3]. However, as the temperature T> 0 whether the reduced density matrix ρ2 has ODLRO is inconclusive. The purpose of this work is to study the ODLRO of ρ2 in the single-band Hubbard model with constant hopping matrix element for nearest neighbor pairs on a cubic lattice at finite temperatures. In the translation invariant case we obtain an equation ( (10) in the following) which is satisfied by the long-range correlation functions of on-site electron pairs. From which we find that there is no ODLRO of on-site electron pairs. Then we consider the case with an antiferromagnetic background. In this case we will show that there is still no ODLRO of on-site electron pairs. The Hamiltonian of the model is H = −t ∑ (a † rar ′ + b † rb ∑