The Hubbard chain: Lieb-Wu equations and norm of the eigenfunctions

Frank Göhmann, V. E. Korepin

Published Jan 1, 1999• 11 citations

Abstract

We argue that the square of the norm of the Hubbard wave function is proportional to the determinant of a matrix, which is obtained by linearization of the Lieb-Wu equations around a solution. This means that in the vicinity of a solution the Lieb-Wu equations are non-degenerate, if the corresponding wave function is non-zero. We further derive an action that generates the Lieb-Wu equations and express our determinant formula for the square of the norm in terms of the Hessian determinant of this action. PACS: 05.30.Fk; 71.10.Pm; 71.27.+a Keywords: Hubbard model; Bethe ansatz; Lieb-Wu equations; determinant representation

Related papers

The Bethe ansatz solution

Fabian H. L. Eßler, Holger Frahm, Frank Göhmann +2 more

The bare existence of this book is due to the amazing fact that the solution of the stationary Schrödinger equation for the one-dimensional Hubbard model can be reduced to a set of algebraic equations, which is tractable in the thermodynamic limit. These equations will be derived in this chapter. We will call them the Lieb-Wu equations to honour E. H. Lieb and F. Y. Wu, who first obtained them [298]. The derivation is based on a method, called the nested (coordinate) Bethe ansatz, which goes back to the seminal articles of C. N. Yang [493] and M. Gaudin [154], who generalized earlier work [60, 296] on exactly solvable models to models with internal degrees of freedom. The roots of the Lieb-Wu equations parameterize the eigenvalues and the eigenstates of the Hamiltonian of the one-dimensional Hubbard model. They encode the complete information about the model. These roots are not explicitly known in the general N -particle case. In the thermodynamic limit ( N →∞), however, only the distributions of the roots in the complex plane matter, and many physical quantities can be calculated as solutions of linear or non-linear Fredholm type integral equations. Moreover, it is sometimes possible to use the Lieb-Wu equations in an implicit way even for finite N , e.g., in the proof of the SO(4) highest weight properties of the eigenstates in Sections 3.D and 3.F of the appendix or in the calculation of their norm in Section 3.5.

Cambridge University Press eBooksFeb 7, 20051 citation

Bethe-ansatz studies of energy-level crossings in the one-dimensional Hubbard model

Akinori Nishino, Tetsuo Deguchi

Motivated by Heilmann and Lieb's work [Ann. N. V. Acad. Sci. 172, 583 (1971)]. We discuss energy level crossings for the one-dimensional Hubbard model through the Bethe ansatz, constructing explicitly the degenerate eigenstates at the crossing points. After showing the existence of solutions for the Lieb--Wu equations with one-down spin, we solve them numerically and construct Bethe-ansatz eigenstates. We thus verify all the level crossings in the spectral flows observed by the numerical diagonalization method with one down-spin. For each of the solutions we obtain its energy spectral flow along the interaction parameter U. Then, we observe that some of the energy level crossings can not be explained in terms of U-independent symmetries. Dynamical symmetries of the Hubbard model are fundamental for identifying each of the spectral lines at the level crossing points. We show that the Bethe-ansatz eigenstates which degenerate at the points have distinct sets of eigenvalues of the higher conserved operators. We also show a twofold permanent degeneracy in terms of the Bethe-ansatz wave function.

arXiv:cond-mat/0309024Physical review. B, Condensed matterAug 25, 20033 citations

Norm of Bethe Wave Function as a Determinant

V. E. Korepin

This is a historical note. Bethe Ansatz solvable models are considered, for example XXZ Heisenberg anti-ferromagnet and Bose gas with delta interaction. Periodic boundary conditions lead to Bethe equation. The square of the norm of Bethe wave function is equal to a determinant of linearized system of Bethe equations (determinant of matrix of second derivatives of Yang action). The proof was first published in Communications in Mathematical Physics, vol 86, page 391 in l982. Also domain wall boundary conditions for 6 vertex model were discovered in the same paper [see Appendix D]. These play an important role for algebraic combinatorics: alternating sign matrices, domino tiling and plane partition. Many publications are devoted to six vertex model with domain wall boundary conditions.

arXiv:0911.1881Nov 10, 20092 citations

Norms of Bethe Wave Functions

V. E. Korepin, N. M. Bogoliubov, Anatoli G. Izergin

Bethe Ansatz solvable models are considered, like XXZ Heisenberg\nanti-ferromagnet and Bose gas with delta interaction. Periodic boundary conditions lead to Bethe\nequation. The square of the norm of Bethe wave function is equal to a determinant of linearized\nsystem of Bethe equations (determinant of matrix of second derivatives of Yang action). The proof\nwas first published in Communications in Mathematical Physics, vol 86, page 391 in l982. Also\ndomain wall boundary conditions for 6 vertex model were discovered in the same paper [see\nAppendix D]. These play an important role for algebraic combinatorics: alternating sign matrices,\ndomino tiling and plane partition.

Cambridge University Press eBooksAug 19, 19930 citations

The Hubbard chain: Lieb–Wu equations and norm of the eigenfunctions

F. Göhmann, V.E. Korepin

arXiv:cond-mat/9908114Physics Letters ADec 1, 199924 citations

Algebraic Bethe ansatz for the gl(1|2) generalized model and Lieb-Wu equations

Frank Göhmann, Er Seel, Fachbereich Physik

We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang’s model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.

Jul 26, 201317 citations