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arXiv:cond-mat/9712228 · Journal of Physics A Mathematical and General

Coherent pairing states for the Hubbard model

A. I. Solomon, K. A. Penson

Published May 8, 1998• 6 citations
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Abstract

We consider the Hubbard model and its extensions on bipartite lattices. We define a dynamical group based on the -pairing operators introduced by Yang, and define coherent pairing states, which are combinations of eigenfunctions of -operators. These states permit exact calculations of numerous physical properties of the system, including energy, various fluctuations and correlation functions, including pairing off-diagonal long-range order to all orders. This approach is complementary to BCS, in that these are superconducting coherent states associated with the exact model, although they are not eigenstates of the Hamiltonian.

DOI
10.1088/0305-4470/31/18/004

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We consider the Hubbard model and its extensions on bipartite lattices. We define a dynamical group based on the -pairing operators introduced by Yang, and define coherent pairing states, which are combinations of eigenfunctions of -operators. These states permit exact calculations of numerous physical properties of the system, including energy, various fluctuations and correlation functions, including pairing off-diagonal long-range order to all orders. This approach is complementary to BCS, in that these are superconducting coherent states associated with the exact model, although they are not eigenstates of the Hamiltonian.
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.
The ground-state properties of the Hubbard model with attractive local pairing interactions are investigated in the framework of lattice density-functional theory. A remarkable correlation is revealed between the interaction-energy functional $W[\mathbit{\ensuremath{\eta}}]$ corresponding to the Bloch-state occupation-number distribution ${\ensuremath{\eta}}_{\ensuremath{\kappa}\ensuremath{\sigma}}$ and the entropy $S[\mathbit{\ensuremath{\eta}}]$ of a system of noninteracting fermions having the same ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$. The relation between $W[\mathbit{\ensuremath{\eta}}]$ and $S[\mathbit{\ensuremath{\eta}}]$ is shown to be approximately linear for a wide range of ground-state representable occupation-number distributions ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$. Taking advantage of this statistical analogy, a simple explicit ansatz for $W[\mathbit{\ensuremath{\eta}}]$ of the attractive Hubbard model is proposed, which can be applied to arbitrary periodic systems. The accuracy of this approximation is demonstrated by calculating the main ground-state properties of the model on several 1D and 2D bipartite and nonbipartite lattices and by comparing the results with exact diagonalizations.
We present a new model describing strongly correlated electrons on a general d-dimensional lattice. It is an extended Hubbard model and it contains the t-J model as a special case. The model naturally describes local electron pairs, which can move coherently at arbitrary momentum. By using an \ensuremath{\eta}-pairing mechanism we can construct eigenstates of the Hamiltonian with off-diagonal long-range order. In the attractive case the exact ground state is superconducting in any number of dimensions. On a one-dimensional lattice, the model is exactly solvable by Bethe ansatz.
ηηη-pairing symmetry generalizes the pairing mechanisms in superconductivity but is broken in the presence of interlayer interactions. In this work, we extend this approach to triplet pairs. We propose interlayer triplet-pairing operators for the multi-layer extended Hubbard model. We find that a set of exact condensate-pair eigenstates can be constructed, which exhibit off-diagonal long-range order. In contrast to the ηηη-pairing mechanism, this originates from restricted spectrum generating algebra and is only available for bilayer and trilayer systems in the presence of interlayer Hubbard interactions. Nevertheless, the system also retains the original on-site ηηη-pairing symmetry in the absence of interlayer interactions. Consequently, both singlet and triplet pairs coexist in the eigenstates of the multi-layer Hubbard model. We employ quench dynamics to demonstrate the results through numerical simulations. Our findings open avenues for the study of exact condensate-pair states in strongly correlated systems.
We extend previous studies of the (BCS) canonical approach for the attractive Hubbard model. A derivation of the BCS formulation is presented for both the Hubbard model and a simpler reduced Hamiltonian. Using direct diagonalization, exact one- and two-dimensional solutions for both Hamiltonians are compared to BCS variational calculations. Approximate and exact ground state energies and energy gaps are computed for different electron number systems as well as correlation observables not previously predicted. Reproducing published one-dimensional findings, the BCS approach is an excellent approximation for the Hubbard ground state energy but not the energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wave functions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. Even for weak coupling pronounced differences are found between the BCS and Hubbard on-site correlators. Significantly, in the extreme strong coupling limit the Hubbard model exhibits no dimensional crossover, or density induced transitions, to long-range order in two dimensions. However, for the reduced Hamiltonian the BCS approach is an excellent approximation for all observables in both one and two dimensions.