VOLUME 68, NUMBER 2 PH YSICAL REVIEW LETTERS 13 JANUARY 1992

Exact Results for a Hubbard Chain with Long-Range Hopping

Florian Gebhard ' and Andrei E. Ruckenstein

Serin Physics Laboratory, Rutgers University, P.O. Box 849, Piscataway, A'em Jersey 08855-0849(Received 21 October 1991)

We give the exact spectrum and thermodynamics for a long-range hopping Hubbard chain with linear

dispersion. This model exhibits a Mott-Hubbard metal-insulator transition at half filling when the in-

teraction strength U equals the bandwidth W. The solution for U&& W also covers the corresponding t-Jmodel, which reduces to the spin model of Haldane and Shastry at half filling. We mention possible ex-

tensions of the model in one and higher dimensions.

PACS numbers: 7 l.30.+h, 05.30.Fk

There has been renewed interest in exactly solvablemodels of strongly correlated systems, especially follow-

ing the discovery of the high-T, . oxides in which strongcorrelations appear to play a key role. A few years ago,Haldane [I] and Shastry [2] independently introduced aspin- 2 Heisenberg model with long-range exchange,

L

H,„;,=(J/2) g, [d(l —m)] 'St S„, ,l&nt I

w here d(l —m) =(L/tr)sin[tr(! —m)/L] is the chord dis-tance between two lattice points (l,m) on a ring of lengthL. The ground state for the antiferromagnetic case(J & 0) has been identified [1,2] to be the paramagneticGutzwiller-projected Fermi sea (FS) with the number ofelectrons equal to the number of sites (half filling),)%'o) Pp p( FS) =Ht [I nt int 1]—) FS), where nt

=cI ~I is the number operator for electrons of spin o.~%'o) also turns out to be the ground state for a supersym-metric t-J model generalization of the Haldane-Shastrylong-range model away from half filling [3]. The exactground-state correlation functions are known from Ref.

[4]. In a remarkable Letter Haldane was also able toderive the full spectrum and thermodynamics of his mod-el [5]. In contrast to the solutions obtained by the Betheansatz, the structure of Haldane's spectrum suggestsextensions to higher dimensions. A specific scenario—involving a generalization of statistics —was recentlyoutlined in Ref. [6].

In this work we introduce a new Hubbard-type chainwith long-range hopping:

L L

H=T+UD= g t( et~„, +Urn(in/lcm [ o I [

with tt „,=it( —I )' "'[d(l —m)l ' =t„, t Since. it is

most convenient to work with even L, the specific form ofthe hopping matrix elements forces us to choose an-tiperiodic boundary conditions. Then the resulting dis-persion is linear, e(k) =tk, with k =(2tr/L)(m+ —,

' )(m = —L/2, . . . , L/2 —I). The itinerant model of Eq.(2) is constructed so that at half filling, in the Ulimit, we recover the Haldane-Shastry model. In particu-lar, the corresponding t-J Hamiltonian can be obtainedin the usual way [7]:

2 tl, n&n, m ~Hgj Pp o T+ g (St S~ 4 n(n~ ) g ~(&JEST )e(acn —wn —a em' Pp '0 r'

I~nr U I &n &m &I U ae'(3)

where nt =g nt At half. filling this~reduces to Eq. (I)with J=4t /U and an energy shift C—= (J/g)Z&~~-i[d(l—m)] =(J/24)(tr/L) L(L —I ) [8]. Note that H[Eq. (2)] has a definite parity, i.e., PHP =rtH, only forU=O (rt = —1), t =0 (rt=+ I), or in the limit of halffilling and U ~ (rt=+ I).

In this paper we use the exact diagonalization resultsfor small systems and analytical results in various limitsto conjecture an eAective Hamiltonian which recovers thefull spectrum and degeneracies of (2) for arbitraryfillings and interaction strengths This eflective H.amil-

tonian is then used to study the ground-state propertiesand thermodynamics of our model. We conclude by pro-posing an obvious extension of this type of long-rangehopping model which may be solvable even in higher di-mensions.

The spectrum of the kinetic-energy operator T for afixed total momentum Q [Q =(T/t)mod2x] consists of

highly degenerate equidistant levels separated by 2nt.To see this, let the momentum transferred be q & 0 andconsider the two-particle scattering process (k,p) (k+q,p —q). The initial and scattered states have thesame kinetic energy T if (i) k+q & tr and p —

q & —tr or(ii) k+q & tr and p —

q ( —n. In the remaining cases Tchanges by ~ 2xt.

To gain some intuition we analyze the L states with

S =L/2 —I (single spin flip). There are L states with

T=Q =0; of these, the L —I states with S =L/2 —I have

double occupancy D = I, and the remaining S =L/2 statehas D=O. For fixed total momentum Q=(2n/L)mti&0one finds (L —1)(L —2) simultaneous eigenstates of Tand D with D =1, of which (L —I )(L —

~mt'~—1) have

kinetic energy T=tQ, and (L —1)(~mg~ —I) have T=t (Q —2nl, g) [where Et' =sgn(Q)]. The 2(L —1)remaining Q&0 states can be chosen as eigenstates of T

244 1992 The American Physical Society

VOLUME 68, NUMBER 2 PHYSICAL REVIEW LETTERS 13 JANUARY 1992

and are superpositions between D 0 and D=l chargestates . These bonding and antibonding states [(L —I)doublets] for the spin flip and the hole are obtained fromthe diagonalization of the 2x2 matrix for each Q, with

diagonal elements tQ+U)mgllL and t(Q —2/rkty)+U(I—[mq(/L), and off-diagonal elements U[((mg(/L)(I—[m, [/L)] '/'.

In what follows it wi11 be useful to represent the Hil-bert space of the above example pictorially: For this pur-pose we introduce a "quasimomentum" space with "coor-dinates" 4' which, as the physical momenta, are given

by R=(2/r/L)(my&+ 2 ) (my&= L/2,—. . . , L/2 —1). A

particular state will be depicted by filling the R stateswith spin-carrying (f, J) and charge-carrying (x,o) bo-sons representing spin up and down, and doubly occupiedand empty 8' sites. In addition, a completeness con-straint that each 8' site is occupied by one and only oneboson must be fulfilled. For example, there are L stateswhich we would represent by f

. f J f f which in-

volve a single spin flip with D=O. However, from oursolution above we learn that there is only one eigenstatewith D =0, namely, the state with S=L/2 and totalmomentum Q =0. We will arbitrarily select the pictorialrepresentation of this state to be J f f, i.e., the down

spin occupies the first position on the R chain. Allconfigurations which involve an up spin to the immediateleft of a down spin —and there are L —

1 such con-figurations in the single-spin-flip case—must then be-long to the L —

1 doublets. We depict these configura-tions by "boxing" the nearest-neighbor up-down pair,f . f(f f)«f f. Similarly, the L(L —1) states withD I could be represented as f

. f x f' fof

Of these, L —1 are part of the doublets while the remain-

ing (L —1) are simultaneous eigenstates of T and D.We then choose to represent the former by boxing aneighboring pair of doubly occupied and empty siteswhenever the doubly occupied site occurs to the left of the

empty one: f f (xo) y& f . . f. The L —I doubletsare then identified with the 2(L —I ) states involving thesuperposition (f j ) y& and (xo) y& with R'=R.

We now assign physical momenta and kinetic energies

to the various configurations. For configurations without

boxes the physical momentum of the singly occupied 8'sites is identified with the % momentum awhile doubly oc-cupied and empty R states carry zero momentum. Thesestates can then be described in terms of an effective Ham-

iltonian diagonal in the occupation-number representation for our set of boson operators for spin-carrying($ y& ), and charge-carrying (ey&, dy&) quasiparticles:

A I

Ho = g [2 tR[(Sy&1$y& f+Sy&1$y&f)—s& R&x

(d „'d,—+e,'e„)]+Ud,'d, ],(4)

where we made use of the completeness constraint

g~/& $/& +d/, d«+e«e/, =1 for each R T. h.e primeon the sum is a reminder of the fact that the L —

1 dou-

blets are not represented by Hp .For boxed configurations, (f ))y& and (xo)y& are as-

signed momenta yrA, y& and —/rXy&, respectively. The mix-

ing of the boxed configurations due to the Hubbard in-

teraction can be read off from the solution of our single-spin-flip problem. We use the identity mg/L =[2&,

y&—(2%'+5)]/4/r with 6=2/r/L, so that the second part ofthe efl'ective Hamiltonian reads

—tI & '/1'&n —h

U U A$

A——ky& ytt — (28+6) d/&dt&e/&+&e«+/,

2 4z

U U Af A At A+ +A, y& ytt (2.8+6,) $ y& 1$y& 1$ y&+g ($ y&+g )2 4z

A+ [(2/r) —(2.8+6) ]' [d/&e/&+&$/& tsy&+z J+H.c.] '.4x

(5)

The total effective Hamiltonian is given by H' Ho+HeA'

A

We now conjecture that H' is indeed correct in allspin sectors, for all fillings and arbitrary interactionstrengths. For two particles on a lattice of size L this canbe checked by applying a particle-hole transformationand turning the problem into the single-spin-flip case.We have also confirmed our conjecture for lattices of sizeL =2, 4, and 6 at half filling where the dimension of theHilbert space is 6, 70, and 924, respectively. Furtherconfirmation comes from agreement with (i) the small-Uexpansion at zero temperature, (ii) the low-density ex-pansion at zero temperature, (iii) the high-temperatureexpansion, and (iv) all of Haldane's results [5] for the

large-U limit at half filling (see below).The effective Hamiltonian can be easily brought into

an occupation-number form by diagonalizing the 2x2matrices describing the independent doublets for each R.Including an external magnetic field (Po) and chemicalpotentials p =p —age/&&0/2 we may write the resultingeffective Hamiltonian as

AH' = h'~ n'j& + h q&n ~+ h'q&n'q&a

+Jy&(n« /, n/&—n /, &, in «J)'''

with h /&=t R/2 p, h /&

= t R. /2 2p+U, h «

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VOLUME 68, NUMBER 2 PH YSICAL R EVI EW LETTERS 13 JANUARY 1992

= —t /t/2, and

Jy, =It(2R —4) —U+ [(2trt) +U —2tU(2'8 —6)] ' ]/2~ 0.Here, the obvious notations n j&

=sq& ~/~, etc. , have been used. To be precise, we also restrict ourselves to t ~ 0 and

U~ —2trt in which case we may identify 8„„„—6—:R .„„because Jt, vanishes for A'= R;„=—[(L —I)/2]A.F'or an even particle number, A ~ I, the ground state in zero magnetic field is obtained by only filling R states with

.~ tq Is tq+~ i pairs from /t = —tt to 8 = S =n(2n —I ). The ground-state energy per lattice site is found to be

4 24WUeo(n ~ I ) = — [(W+U) ' —[(

where W =2zt is the bandwidth. Using particle-holesymmetry we obtain eo(n ~ I ) =eo(2 n—)+U(n —I ),and, correspondingly, for the chemical potential at zerotemperature, p(n & I ) =t)eo(n ( I )/tin, p(n & I ) =U—p(2 n). A—t half filling we find for the left derivativeof the ground-state energy density p-(n = I ) =(W+U—

I W —UI)/4, while the right derivative is p ~(n = I ) =U—p —(n = I ). For U ~ W the chemical potential is con-tinuous (p =U/2), while for U & W, there is a gap4p(n = I ) =U —W, i.e., our model exhibits a Mott-Hubbard metal insulator transition at half filling for afinite value of the interaction, U = W.

We note that, in the limit of small U and all n, or smalln and all U, the ground-state energy (7) agrees with theappropriate limits derived from the Gutzwiller wave func-tion, I V~) =+t [I —(I g)nt—I n(1] I

FS) [9]. It can bechecked explicitly from the appropriate perturbationtheory that this wave function is indeed the ground-statewave function, but this is true only in these special limits.In addition, given our eAective Hamiltonian, it can beseen that the Gutzwiller-projected Fermi sea also be-comes the ground state of the t -J model (3) forJ =4t /U((t.

It is also interesting to consider the t-J model which is

obtained by replacing 4t /U by J in the Hamiltonian (3)and then treating J as an independent parameter. Withthe help of the large-U limit of our eA'ective Hamiltonian

W+U)' —4WUn] j,

we obtain the ground-state energy as eo (J6 J,. )= —Wn(l —n)/2 —Jtr n (3 —2n)/12 and eoj(J ~ J,. )= —nW /2Jtr J—tr n(3 —n )/24, where J„=2W/tr (I

n) —At. half filling we can compare our results withthose of Haldane [5]. Since in the large-U limit st—=s tt the antiferromagnetic ground state is given by L/2(t J ) pairs, a state which is represented by L/2+ I empty"orbitals" in Ref. [5]. For finite L the corresponding en-

ergy is given by

Eo =C+g Jtt= —(J/24)(tr/L) L(L +5),where the prime on the sum indicates that only everysecond 8 had to be included in the sum [10].

The resulting spectrum consists of two fundamental ex-citations: "spinons" [5,6] as in the Haldane-Shastrymodel, and "holons, " which are already present in the t

model (U=~, i.e., J—=0). This charge-spin separation[I I] is already evident from the effective Hamiltonian forU ~, which can be explicitly written as an itinerantand a spin part

~dr ~'ir+ H"'IJ f J

Here, J q,= (J/4) (tr- —'A'-). A "spinon" for J ~ J, cor-

responds to a broken spin pair in an otherwise unchangedhole background. It can be represented graphically as

(tl) I (tl) (t~) 'I (tl) . (tl)l.t. „o . o.The excitations are always created in pairs, and their sep-aration 8' —R' has to be an even multiple of h. It is thenconvenient to rescale 4' 2%' to make contact withHaldane's Eq. (17); in the infinite volume limit we obtainfor a single spinon ~;e„,„(R)=(J/2)[(tr/2) —R ] with—tr/2 S %' ~ tr(2n I )/2. Th—e spinon velocity at= —tr/2 is v,v;„,„=Jtr/2. For J & J, the lowest-lying spinexcitations are two holes at R =R —2zn and—2xn+h, and two spins at 4', R' which can have arbi-trary separation. Here, R = tr [—J,/J+ n (I +J,/J) ].The dispersion for a single spinon is then given by

, ev(R) =t%' —t(%' 2trn)+ JzF z „/2 with— —tr—2zn. The spinon spectrum opens a gap at

J=J, , i.e., the spin Auid becomes "incompressible" lead-

ing to the vanishing of the spin susceptibility above J,.At half filling we have R =x and the spinon dispersionacquires another linear part around R =tr/2 with velocity

v', v;„,„=—v,'~;„,„=—Jtr/2. We further notice that we

obtain two spinons for every broken ( t f ) pair, so that onehas L/2+1 —M orbitals for 2M spinons (M=O, I,. . . ,L/2). Spinons between unbroken (t j ) pairs occupythe corresponding orbitals. This construction provides

the link to Haldane's construction of the spinon states[I 2].

The other excitation, the holon, involves a hole ofmomentum R which is surrounded by unbroken spin

pairs. As for the spinon, the allowed 4' values are spaced

by 4tr/L and we again rescale R 2%'. Then the excita-tion energies are

X~—2X2

[W+ 4 Jx(R +2%)],

V(F/2 —nn ~ R ~ RF/2.

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VOLUME 6S, NUMBER 2 P H YSICAL REVI EW LETTERS 13 &ANUAzv 1992

For J=0 the above dispersion agrees with that of a gas ofspinless fermions with linear dispersion and Fermimomentum Jf tr(2n —I ). For J~ J„ the holon veloci-

ty at % =% /2 is vh ~„=(—I/tr)[W+Jtr (2n —I)/2].

For J & J, we find t~o linear excitations, the first atR =R /2 with velocity ch„~,„=—nJtr/2, and the secondat 4' =yf /2 —trn with velocity v'h, ~,„=nJtr/2

The model (6) belongs to the class of one-dimensional

Ashkin-Teller models [13] in external fields. Its free-

energy density in the thermodynamic limit can be easilyfound by the use of the transfer-matrix method. Wedefine S t, =exp( —Ph'tt ), D tt =exp( —Ph "tt), Ett=exp( ph'—tt ), X tt =S tt I

+S tt I +D tt+ E tt, and P tt=exp( —PJ tt), and obtain

f ——„ ln( 2 Xyt+ 2 [Xvt 4[ST—ISvt 1(1 —Pvt ')+DgErt(1 —Prt)]] 't ) .

We checked this expression for high temperatures to firstorder in P.

At half filling we have p =U/2 due to particle-holesymmetry. For U and finite external magnetic fieldwe can therefore neglect the terms with Dq& and EI&, theeffective Hamiltonian (6) can then be transformed intoan Ising model. The corresponding expressions for theentropy density at vanishing external magnetic field, andthe static magnetic susceptibility can be easily derivedand are found to completely agree with Eqs. (15) and(16) of Ref. [5].

The reason for the integrability of the model is relatedto the relatively simple algebra satisfied by the kinetic-and potential-energy operators and their highly degen-erate spectrum. It is not difficult to see how to systemati-cally construct long-range hopping models which preservethese features in d» 1. We start from the observationthat nearest-neighbor hopping on a cluster of size R givesR values for the kinetic energy, e(k„), with n = I, . . . , R.A long-range hopping model on the lattice can then bedefined by choosing a dispersion relation e(k) which is

limited piecewise to the constant values e(k„) in regionsaround k„. Such a finite cluster model should lead to aneA'ective Hamiltonian in an occupation-number basis,describing a classical model in the same number of di-mensions. The latter, although rarely completely solv-

able, can be investigated by a variety of techniques ofclassical statistical mechanics. We expect that, in thisnew class of models, the metallic ground states are Fermiliquids, like those obtained from variational Gutzwillerwave functions [4), and one might as well call suchphases "Gutzwiller liquids.

"In this Letter we have studied a new itinerant one-

dimensional model with long-range hopping and Hubbardinteraction. In spite of its simplicity this model displays ametal-to-insulator transition at half filling for a finite in-

teraction strength. It remains to justify the conjectures ofthis paper by a direct algebraic approach.

We would like to thank Prasenjit Saha (University ofToronto) for his substantial numerical support. We alsothank the International Centre for Theoretical Physics,Trieste, Italy, and the Aspen Center for Physics wherepart of this work was carried out. Support throughNATO (F.G.) and the Sloan Foundation (A.E.R.) is

gratefully acknowledged.iVote added R..—Shankar informed us that he has

checked the validity of our ground-state energy, Eq. (7),by perturbation theory up to third order in U for all

fillings. He also observed that at half filling et'(n) has anonanalyticity of the form (I n) ~I

——n~ for all values

of the coupling constant.

" Address after l March l992: Philipps Universitat Mar-

burg, Fachbereich Physik, Renthof 5, W-3550 Mar-burg/Lahn, Germany.

[I] F. D. M. Haldane, Phys. Rev. Lett. 60, 635 (1988).[2] B. S. Shastry, Phys. Rev. Lett. 60, 639 (1988).[31 Y. Kuramoto and H. Yokoyama, Phys. Rev. Lett. 67,

1338 (1991).[4) W. Metzner and D. Vollhardt, Phys. Rev. Lett. 59, 121

(1987); F. Gebhard and D. Vollhardt, Phys. Rev. Lett.59, 1472 (1987).

[5] F. D. M. Haldane, Phys. Rev. Lett. 66, 1529 (1991).[61 F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991).[71 P. W. Anderson, in Solid State Physics: Adt ances in

Research and App/ications, edited by F. Seitz and D.Turnbull (Academic, New York, 1963) Vol. 14, pp.99-2l4; 3. E. Schrieffer and P. A. %'olA, Phys. Rev. 149,491 (1966).

[8] E. R. Hansen, A Table of Series and Products (Pren-tice-Hall, Englewood Cliffs, NJ, 1975), No. 24. 1.2, p.260.

[9] M. C. Gutzwiller, Phys. Rev. Lett. IO, 159 (1963).[10] We checked our result for L =2; we therefore are in

disagreement with Eqs. (5) and (9) of Ref. [5].[I I] S. A. Kivelson, D. S. Rokshar, and 3. P. Shetna, Phys.

Rev. B 35, 8865 (1987); Z. Zou and P. W. Anderson,Phys. Rev. B 37, 627 (1988).

[12] For example, in our two-spinon configuration,I

U(l 1)u(l 1)U(l l) u(l 1)u,Haldane's orbitals are denoted by U, and thus, in

Haldane's language (the occupation representation forthese orbitals}, it translates into

UUUUU

[13]3. Ashkin and E. Teller, Phys. Rev. 64, 178 (1943).