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International Journal of Modern Physics B

EXTENDED HUBBARD HAMILTONIAN WITH (SUPER)SYMMETRIES: ADDITIVE POLYNOMIAL R-MATRIX FOR SOME INTEGRABLE CASES

Fabrizio Dolcini, Arianna Montorsi

Published Oct 10, 1999• 2 citations
Publisher

Abstract

We propose a constructive method to prove the integrability of a given physical Hamiltonian in one dimension, which amounts to looking for appropriate polynomial R-matrices, solutions of GYBE, whose first coefficient in the power expansion with respect to the spectral parameter is the Hamiltonian itself. The method is applied to the extended Hubbard Hamiltonian, in particular to the cases in which it exhibits so(4) or gl(2,1) symmetries. We show that in the latter case the R-matrices are at most polynomial of second degree, whose coefficients are nothing but the Hamiltonian, the identity and the permutation operator. In this way, all known completely integrable cases are recovered. Also, the method allows to recognize that the possible integrability of the most general gl(2,1) invariant Hamiltonian depends on the existence of a non-additive R-matrix.

DOI
10.1142/s0217979299002782

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We propose a constructive method to prove the integrability of a given physical Hamiltonian in one dimension, which amounts to looking for appropriate polynomial R-matrices, solutions of GYBE, whose first coefficient in the power expansion with respect to the spectral parameter is the Hamiltonian itself. The method is applied to the extended Hubbard Hamiltonian, in particular to the cases in which it exhibits so(4) or gl(2,1) symmetries. We show that in the latter case the R-matrices are at most polynomial of second degree, whose coefficients are nothing but the Hamiltonian, the identity and the permutation operator. In this way, all known completely integrable cases are recovered. Also, the method allows to recognize that the possible integrability of the most general gl(2,1) invariant Hamiltonian depends on the existence of a non-additive R-matrix.
We consider the most general form of extended Hubbard Hamiltonian conserving the total spin and number of electrons, and find all the 1-dimensional completely integrable models which can be derived from first degree polynomial solution of the Yang–Baxter equation. It is shown that such models are 96. They are identified with the 16-dimensional representations of the class of solutions of symmetric group relations acting as generalized permutators. As particular examples, the EKS and some other known models are obtained. A method for determining the physical features of the above models is outlined.
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Abstract We present a general formula for constructing R-matrices with non-additive spectral parameters associated with a type-I quantum superalgebra. The spectral parameters originate from two one-parameter families of inequivalent finite-dimensional irreducible representations of the quantum superalgebra upon which the R-matrix acts. Applying to the quantum superalgebra Uq(gl(2|1)), we obtain the explicit expression for the Uq(gl(2|1))− invariant R-matrix which is of non-difference form in spectral parameters. Using this R­ matrix we derive a new two-parameter integrable model of strongly correlated electrons with pure imaginary pair hopping terms.
We consider the most general form of extended Hubbard Hamiltonian conserving the total spin and number of electrons, and find all the 1-dimensional completely integrable models which can be derived from first degree polynomial solution of the Yang-Baxter equation. It is shown that such models are 96. They are identified with the 16-dimensional representation of a new class of solutions of symmetric group relations, acting as generalized permutators. As particular examples, the EKS and some other known models are obtained.
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The one-dimensional Hubbard model is known to possess an extended su(2) symmetry and to be integrable. I indroduce an integrable model with an extended su(n) symmetry. This model contains the usual su(2) Hubbard model and has a set of features that make it the su(n) generalization of the Hubbard model. Complete integrability is shown by introducing the L-matrix and showing that the transfer matrix commutes with the hamiltonian. While the model is integrable in one dimension, it provides a generalization of the Hubbard hamiltonian in any dimension.