An Introduction to Hubbard Rings at U = ∞ W. B. Hodge, N. A. W. Holzwarth and W. C. Kerr Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109-7507 (Dated: April 12, 2010) Using a straightforward extension of the analysis of Lieb and Wu, we derive a simple analytic form for the ground state energy of a one-dimensional Hubbard ring at U/t = ∞. This result is valid for an arbitrary number of lattice sites L and electrons N ≤ L. Furthermore, our analysis provides insight into the degeneracy and spin properties of the ground states in this limit. PACS numbers: I. INTRODUCTION For nearly fifty years, the Hubbard model 1 has been used to describe many-body effects in solids, capturing the dominant competition between the delocalizing ef- fects of the kinetic energy (with strength described by a hopping energy t) and the localizing effects of the electron-electron repulsion (with strength described by an on-site Coulomb energy U ). Despite its simple form, it has provided significant insight into many-body prop- erties of solids such as metal-insulator transitions, high- temperature superconductivity, and magnetic states 2 , largely because of the accessibility of its analytic and numerical solutions. The analytic understanding of the Hubbard model stems primarily from the seminal work of Lieb and Wu 3,4 who derived a “Bethe anzatz” method 5 for determining eigenvalues and eigenfunctions of the sin- gle band, one-dimensional Hubbard model with L lattice sites and N electrons, and obtained an explicit expres- sion for the ground state energy for a half-filled system in the thermodynamic limit (N = L →∞). Typical textbooks for courses in condensed matter physics contain relatively little material about the Hub- bard model. For example, Ashcroft and Mermin 6 have a one-paragraph qualitative description of the physics of the model plus an end-of-chapter problem involv- ing a two-site Hubbard model representing a hydrogen molecule (the solution to which has been the subject of a published paper 7 ). Marder 2 devotes a section of a chap- ter to the Hubbard model including a presentation of the mean field solution of an infinite one-dimensional sys- tem. At larger values of U/t, the mean field solution in- correctly predicts a ferromagnetic ground state and the section concludes with the sentence: “There is no better illustration of the difficulties involved in progressing sys- tematically beyond the one-electron pictures of solids.” The more recent textbook by Snoke 8 does not mention the Hubbard model. Nevertheless, there has been substantial research de- voted to analyzing the mathematics and physics of the Hubbard model, particularly in one-dimension. Two re- cent reviews 9,10 summarize parts of the literature. Much of this literature is very complex, involving the enumer- ation of special symmetries and the analysis of compli- cated nonlinear or combinatorial equations. On the other hand, some of the basic ideas behind the analysis and explicit results for some limiting cases are accessible to graduate level instruction and can give students some in- sight into many-body physics and some exercise in basic quantum mechanical principles for non-trivial systems. A common problem asked of students in an introduc- tory quantum mechanics course is to determine the en- ergy and degeneracy of the ground state of a system. In the present paper, we present a proof that the exact ground state energy of the single band, one-dimensional Hubbard model for the case that there are L lattice sites (L ≤∞) with periodic boundary conditions and N elec- trons (N ≤ L) in the limit, U/t ≡ u = ∞ is E g = −2t sin  πN L  sin  π L  . (1) This simple analytic form is helpful for analysis and nu- merical studies of the one-dimensional Hubbard model, and our derivation of it provides insight into the nature of the eigenstates of the model. The derivation of Eq. (1) is suitable for an introductory course on solids. We also examine the degeneracy of the ground state for some simple cases. II. DERIVATION OF GROUND STATE ENERGY Using second quantized notation, the Hamiltonian of the Hubbard model is 1 H = H hop (t)+ H int (U ) = −t  〈m,n〉  σ=↑,↓ c † m,σ c n,σ + U  m ˆ N m,↑ ˆ N m,↓ , (2) where c † m,σ (c m,σ ) creates (annihilates) an electron with spin σ in the Wannier state localized at lattice site m, and ˆ N m,σ = c † m,σ c m,σ . The notation 〈m, n〉 denotes that a sum is over nearest neighbor sites only. The creation and annihilation operators that appear in Eq. (2) obey the following relations (and their adjoints)  c m,σ ,c † m ′ ,σ ′  + = δ m,m ′ δ σ,σ ′ and (3)  c m,σ ,c m ′ ,σ ′  + =0, (4)