Abstract
In this work, we present a proof of the existence of real and ordered solutions to the generalized Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U>0 to U=∞ is also shown. We use this continuity property, combined with the proof that the norm of the wavefunction obtained with the generalized Bethe Ansatz is not zero, to prove that the solution gives us the ground state of the finite system, as assumed by Lieb and Wu. Lastly, for the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the Lieb-Wu solution of the 1D Hubbard model.
Similar content being viewed by others
References
Bethe, H.A.: Zur Theorie der Metalle: I. Eigenwerte und Eigenfunktionen der linearen Atom Kette. Zeits. f. Physik 71, 205–226 (1931)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. New York: Springer-Verlag, USA, 1992
Essler, F.H.L., Korepin, V.E., Schoutens, K.: Completeness of the SO(4) extended Bethe Ansatz for the one-dimensional Hubbard model. Nucl. Phys. B 384, 431–458 (1992)
Gaudin, M.: Travaux de Michel Gaudin: Modèles exactement résolus. Paris Cambridge, MA: Les Éditions de Physique, 1995
Göhmann, F., Korepin, V.E.: The Hubbard chain: Lieb-Wu equations and norm of the eigenfunctions. Phys. Lett. A 263, 293–298 (1999)
Guillemin, V., Pollack, A.: Differential Topology. Englewood Cliffs, NJ: Prentice-Hall Inc., 1974
Gottlieb, D.H., Samaranayake, G.: The index of discontinuous vector fields. New York J. Math. 1, 130–148 (1995)
Hubbard, J.: Electron correlation in narrow energy bands. Proc. Roy. Soc. (London) A 276, 238–257 (1963)
Lieb, E.H., Loss, M.: Analysis 2nd edition. Providence RI: Am. Math. Soc., 2001
Lieb, E.H., Wu, F.Y.: Absence of Mott transition in an exact solution of the short-range one-band model in one dimension. Phys. Rev. Lett. 20, 1445–1448 (1968)
Lieb, E.H., Wu, F.Y.: The one-dimensional Hubbard model: A reminiscence. Physica A 321, 1–27 (2003)
Woynarovich, F.: Excitations with complex wavenumbers in a Hubbard chain: I. States with one pair of complex wavenumbers. J. Phys. C 15, 85–96 (1982)
Yang, C.N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19, 1312–1314 (1967)
Yang, C.N., Yang, C.P.: One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe’s hypothesis for ground state in a finite system. Phys. Rev. 150, 321–327 (1966)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Rights and permissions
About this article
Cite this article
Goldbaum, P. Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model: Finite Lattice and Thermodynamic Limit. Commun. Math. Phys. 258, 317–337 (2005). https://doi.org/10.1007/s00220-005-1357-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-005-1357-y