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Acta Physica Polonica B

The Exact Diagonalization Technique for the Attractive Hubbard Model

Dorota Jakubczyk

Published Jan 1, 2013• 5 citations
Publisher

Abstract

This paper addresses the singular 1-soliton solution of the K(m, n) equation that is being considered with generalized evolution.The ansatz method will be used to extract the singular 1-soliton solution of this equation.A couple of constraint conditions will fall out in order for the singular soliton solutions to exist.Subsequently, the special cases of this equation will be studied.They are the KdV and the mKdV equations where the extended (G /G)-expansion method will be employed to extract a few nonlinear wave solutions.

DOI
10.5506/aphyspolb.44.1921

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Acta Physica Polonica B•Jan 1, 2014•0 citations
This paper addresses the singular 1-soliton solution of the K(m, n) equation that is being considered with generalized evolution.The ansatz method will be used to extract the singular 1-soliton solution of this equation.A couple of constraint conditions will fall out in order for the singular soliton solutions to exist.Subsequently, the special cases of this equation will be studied.They are the KdV and the mKdV equations where the extended (G /G)-expansion method will be employed to extract a few nonlinear wave solutions.
We constructed the one-particle spectral functions (diagonal and off-diagonal) which reproduce BCS for weak coupling and which take into account the effect of correlations on superconductivity in the attractive Hubbard model. The diagonal spectral function is composed of three peaks and the off-diagonal one is composed of two peaks. This ansatz satisfies the sum rules for the first six moments. Our solutions are valid for intermediate coupling, i.e., for U/t ≈ −4.0. Our set of analytical equations for the unknown variables is self-consistent and has been solved numerically in lowest order of the order parameter. As a result, we obtain that the presence of the third band, or upper Hubbard band, strongly renormalizes the two lower bands, making that the energy gap be k-dependent while the order parameter is pure s-wave. This shows that the order parameter and the gap are two different quantities.
We investigate the integrable structure and soliton dynamics of a coupled modified Korteweg-de Vries (cmKdV) system with a real symmetric coupling matrix. We introduce a vector reformulation of Hirota's bilinear formalism in which both the bilinear equations and their solutions are expressed directly at the vector level, rather than through a component-wise construction. This formulation preserves the intrinsic structure of the coupled system and provides a compact framework for multi-component nonlinear wave dynamics. Within this approach, we construct explicit one-, two-, and three-soliton solutions in closed vector form and recover the three-soliton condition directly at the vector level, confirming consistency with integrability. The method enables a unified treatment of focusing, defocusing, and mixed-sign regimes. In particular, for indefinite coupling, it reveals the existence of nontrivial vector ground states, leading to soliton solutions on non-zero backgrounds. These results highlight the structural advantages of the vector bilinear approach and open perspectives for the study of more general nonlinear excitations in multi-component integrable systems.