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An Introduction to Hubbard Rings at U = ∞

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An Introduction to Hubbard Rings at U = ∞ An Introduction to Hubbard Rings at U = 1 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 = 1. 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 explicit results for some limiting cases are accessible to graduate level instruction and can give students some in- For nearly ¯fty years, the Hubbard model1 has been sight into many-body physics and some exercise in basic used to describe many-body e®ects in solids, capturing quantum mechanical principles for non-trivial systems. the dominant competition between the delocalizing ef- A common problem asked of students in an introduc- fects of the kinetic energy (with strength described by tory quantum mechanics course is to determine the en- a hopping energy t) and the localizing e®ects of the ergy and degeneracy of the ground state of a system. electron-electron repulsion (with strength described by In the present paper, we present a proof that the exact an on-site Coulomb energy U). Despite its simple form, ground state energy of the single band, one-dimensional it has provided signi¯cant insight into many-body prop- Hubbard model for the case that there are L lattice sites erties of solids such as metal-insulator transitions, high- (L · 1) with periodic boundary conditions and N elec- temperature superconductivity, and magnetic states2, trons (N · L) in the limit, U=t ´ u = 1 is µ ¶ largely because of the accessibility of its analytic and ¼N numerical solutions. The analytic understanding of the sin L Hubbard model stems primarily from the seminal work of Eg = ¡2t ³ ¼ ´ : (1) Lieb and Wu3,4 who derived a \Bethe anzatz" method5 sin for determining eigenvalues and eigenfunctions of the sin- L gle band, one-dimensional Hubbard model with L lattice This simple analytic form is helpful for analysis and nu- sites and N electrons, and obtained an explicit expres- merical studies of the one-dimensional Hubbard model, sion for the ground state energy for a half-¯lled system and our derivation of it provides insight into the nature in the thermodynamic limit (N = L ! 1). of the eigenstates of the model. The derivation of Eq. Typical textbooks for courses in condensed matter (1) is suitable for an introductory course on solids. We physics contain relatively little material about the Hub- also examine the degeneracy of the ground state for some bard model. For example, Ashcroft and Mermin6 have simple cases. 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 II. DERIVATION OF GROUND STATE molecule (the solution to which has been the subject of a ENERGY published paper7). Marder2 devotes a section of a chap- ter to the Hubbard model including a presentation of the Using second quantized notation, the Hamiltonian of mean ¯eld solution of an in¯nite one-dimensional sys- the Hubbard model is1 tem. At larger values of U=t, the mean ¯eld solution in- H = Hhop(t) + Hint(U) correctly predicts a ferromagnetic ground state and the X X X y ^ ^ section concludes with the sentence: \There is no better = ¡t cm,σcn,σ + U Nm;"Nm;#; (2) illustration of the di±culties involved in progressing sys- hm;ni σ=";# m tematically beyond the one-electron pictures of solids." y The more recent textbook by Snoke8 does not mention where cm,σ (cm,σ) creates (annihilates) an electron with the Hubbard model. spin σ in the Wannier state localized at lattice site m, ^ y Nevertheless, there has been substantial research de- and Nm,σ = cm,σcm,σ. The notation hm; ni denotes that voted to analyzing the mathematics and physics of the a sum is over nearest neighbor sites only. The creation Hubbard model, particularly in one-dimension. Two re- and annihilation operators that appear in Eq. (2) obey cent reviews9,10 summarize parts of the literature. Much the following relations (and their adjoints) h i of this literature is very complex, involving the enumer- y cm,σ; cm0,σ0 = ±m;m0 ±σ,σ0 and (3) ation of special symmetries and the analysis of compli- + cated nonlinear or combinatorial equations. On the other h i cm,σ; cm0,σ0 = 0; (4) hand, some of the basic ideas behind the analysis and + 2 h i where11 A; B ´ AB + ´BA. The disadvantage of using Bloch states as a basis for ´ solving the Hubbard Hamiltonian in Eq. (2) is that they In this paper, we assume that the system described complicate the two-particle interaction term H ; by in- by Eq. (2) is one-dimensional and has periodic boundary int troducing this interaction term, the model is no longer conditions (such a system is often referred to as a Hubbard one of independent electrons. In their study, Lieb and ring). These two conditions ensure that each lattice site Wu considered solutions to the one-dimensional Hub- has exactly two nearest neighbor sites. The site index bard model for a speci¯ed electron spin distribution in Eq. (2) takes values 1 · m · L, with indices m and with N and N indicating the total number of up and m+L being equivalent. It can be shown (see Problems 1 " # down z-component spin orientations, respectively, where and 2) that the Hubbard Hamiltonian obeys the relations N = N + N . Applying the Bethe ansatz5, they found h i " # that the total energy eigenvalues have a similar form to H; S2 = 0; (5) those given in Eq. (12) for independent electrons3 h i¡ H;Sz = 0; (6) XN ¡ h i E = ¡2t cos kj: (13) H;S§ = 0; (7) j=1 ¡ The so-called charge momenta k , however, do not have where S2 and S are the operators for total spin and j z the simple form 2¼º =L given in Eq. (12) for indepen- z-component of spin, and S and S are the spin rais- j + ¡ dent electrons. To ¯nd the charge momenta requires solv- ing and lowering operators. Due to Eqs. (5) and (6), we ing the set of coupled nonlinear equations (the Lieb-Wu choose the energy eigenstates of the Hubbard Hamilto- 2 equations) nian to be also simultaneous eigenstates of S and Sz N · ¸ with quantum numbers S and MS, respectively. X# ¡1 4 The one-particle hopping term Hhop favors delocalized Lk = 2¼I + 2 tan (sin k ¡ ¸ ) ; (14) j j u j ¯ states, an observation perhaps not immediately obvious ¯=1 in Eq. (2) since it has been written in the basis of local- ized Wannier states. We can rewrite this term in a more and · ¸ revealing manner by noting that localized Wannier states XN 4 and delocalized Bloch states are related by the Fourier 2 tan¡1 (¸ ¡ sin k ) = 2¼J u ® j ® transforms j=1 N · ¸ 1 XL X# 2 c = p e+2¼mºi=Lc~ and (8) + 2 tan¡1 (¸ ¡ ¸ ) : (15) m,σ º,σ u ® ¯ L º=1 ¯=1 XL 1 ¡2¼mºi=L Henceforth, we assume that N · L and N"¡N# = 0 (+1) c~º,σ = p e cm,σ; (9) if N is even (odd). The parameter Ij, where 1 · j · N, L m=1 is an integer (half-odd-integer) if N# is even (odd). The y wherec ~º,σ (~cº,σ) creates (annihilates) an electron with parameter J®, where 1 · ® · N#, is an integer (half-odd- 0 spin σ in the Bloch state with quantum number º, and integer) if N" = N ¡ N# is odd (even). The ¸ s are a set obeys fermion relations analogous to Eqs. (3) and (4). of ordered, unequal real numbers ¸1 < ¸2 < ::: < ¸N# . Using Eq. (8), the Fourier transform of Hhop is (see Prob- Details of the derivation of these equations are given in lem 3) Yang's examination of a one-dimensional system with delta function interaction12 as well as a more recent re- L µ ¶ X X 2¼º view of the Hubbard model by Lieb and Wu4. While the H = ¡2t cos N~ ; (10) hop L º,σ derivation of these equations is quite challenging, Kar- º=1 σ=";# bach et al.13,14 provide an excellent introduction to the ~ y Bethe ansatz for the Heisenberg model. where Nº,σ =c ~º,σc~º,σ. In cases where only this term in the Hubbard model is important (such as when U = 0 or In Figure 1, some of the lowest energy eigenvalues of when all the electrons' spins are aligned), the Bloch states the Hubbard Hamiltonian are plotted as a function of u serve as a natural basis. In those cases, Eq. (2) reduces for the case N = L = 6. Even for such a small system, to a model of independent electrons on a periodic lattice, the low-energy spectrum is quite complex for ¯nite u. and it is easy to verify that any N-electron Bloch state However, in the in¯nite-u limit, disparate energy curves asymptotically approach the same limiting energy level. y y jº1; σ1; ¢ ¢ ¢ ; ºN ; σN i =c ~º1,σ1 ¢ ¢ ¢ c~ºN ,σN j0i; (11) This limit, which represents a highly correlated system, simpli¯es the mathematical properties of the Hubbard is an eigenstate of H with energy hop model considerably.
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