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Perturbation expansion of the ground‐state energy for the one‐dimensional cyclic Hubbard system in the Hückel limit
Author(s) -
Takahashi M.,
Bracken P.,
Čížek J.,
Paldus J.
Publication year - 1995
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560530502
Subject(s) - ground state , coupling constant , physics , algebraic number , mathematical physics , algebraic equation , perturbation (astronomy) , transcendental equation , hubbard model , quantum mechanics , quantum electrodynamics , mathematical analysis , mathematics , differential equation , nonlinear system , superconductivity
The perturbation expansion coefficients for the ground‐state energy of the half‐filled one‐dimensional Hubbard model with N = 4ν + 2, (ν = 1, 2,…) sites and satisfying cyclic boundary conditions were calculated in the Hückel limit ( U /β → 0), where β designates the one‐electron hopping or resonance integral, and U , the two‐electron on‐site Coulomb integral. This was achieved by solving—order by order—the Lieb–Wu equations, a system of transcendental equations that determines the set of quasi‐momenta { k i } and spin variables {τ α } for this model. The exact values for these quantities were found for the N = 6 member ring up to the 20th order in terms of the coupling constant B = U /2β, as well as numerically for 10 ⩽ N ⩽ 50, and the N = 6 Lieb–Wu system was reduced to a system of sextic algebraic equations. These results are compared with those of the infinite system derived analytically by Misurkin and Ovchinnikov [Teor. Mat. Fiz. 11 , 127 (1972)]. It is further shown how the results of this article can be used for the interpolation by the root of a polynomial. It is demonstrated that a polynomial of relatively small degree provides a very good approximation for the energy in the intermediate coupling region. © 1995 John Wiley & Sons, Inc.