Hubbard model

introduction

• The Hubbard model describes hopping electrons on a lattice
• 1968 Lieb and Wu
  • application of Bethe ansatz
• 1972 Takahasi
  • string hypothesis
  • replace the Lieb-Wu equations by simpler ones
  • proceeded to drive a set of non-linear integral equations known as thermodynamic Bethe ansatz equations
• algebraic Bethe ansatz for the Hubbard model

Lieb-Wu equations

• describing Eigenstates of the Hubbard Hamiltonian
• Bethe ansatz equation

\[\exp(ik_jL)=\prod_{l=1}^{M}\frac{\lambda_{l}-\sin k_j-i u}{\lambda_{l}-\sin k_j+i u},\,j=1,\cdots, N\] \[\prod_{j=1}^{N}\frac{\lambda_{l}-\sin k_j-i u}{\lambda_{l}-\sin k_j+i u}=\prod_{m=1,m\neq l}^{M}\frac{\lambda_{l}-\lambda_{m}-2i u}{\lambda_{l}-\lambda_{m}+2i u},\,l=1,\cdots, M\]

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

• Bethe ansatz
• Bose-Hubbard Model

encyclopedia

• http://en.wikipedia.org/wiki/Hubbard_model

books

• The One-Dimensional Hubbard Model

articles

• Popkov, Vladislav, and Tomaz Prosen. “Infinitely Dimensional Lax Structure for One-Dimensional Hubbard Model.” arXiv:1501.02230 [cond-Mat, Physics:math-Ph, Physics:nlin], January 9, 2015. http://arxiv.org/abs/1501.02230.
• Wadati, Miki, Eugenio Olmedilla, and Yasuhiro Akutsu. “Lax Pair for the One-Dimensional Hubbard Model.” Journal of the Physical Society of Japan 56, no. 4 (April 15, 1987): 1340–47. doi:10.1143/JPSJ.56.1340.