We consider density-density correlations in the one-dimensional Hubbard model
at half filling. On intuitive grounds one might expect them to exhibit an
exponential decay. However, as has been noted recently, this is not obvious
from the Bethe Ansatz/conformal field theory (BA/CFT) approach. We show that by
supplementing the BA/CFT analysis with simple symmetry arguments one can easily
prove that correlations of the lattice density operators decay exponentially.Comment: 3 pages, RevTe

A. A. Ovchinnikov

C. A. Stafford

C. N. Yang

E. H. Lieb

F. H. L. Eßler

F. Woynarovich

Fabian H. L. Eßler

H. Frahm

Holger Frahm

J. Voit

O. J. Heilmann

P. Azaria

R. Assaraf

V. J. Emery

English

arXiv

We consider density-density correlations in the one-dimensional Hubbard model at half filling. On intuitive grounds one might expect them to exhibit an exponential decay. However, as has been noted recently, this is not obvious from the Bethe Ansatz/conformal field theory (BA/CFT) approach. We show that by supplementing the BA/CFT analysis with simple symmetry arguments one can easily prove that correlations of the lattice density operators decay exponentially

Eßler, F.H.L.

Frahm, Holger

Institutionelles Repositorium der Leibniz Universität Hannover

PHYSICAL REVIEW B 15 SEPTEMBER 1999-IIVOLUME 60, NUMBER 12Density correlations in the half-filled Hubbard modelFabian H. L. EßlerDepartment of Physics, Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, United KingdomHolger FrahmInstitut fu¨r Theoretische Physik, Universita¨t Hannover, D-30167 Hannover, Germany~Received 4 March 1999!We consider density-density correlations in the one-dimensional Hubbard model at half filling. On intuitivegrounds one might expect them to exhibit an exponential decay. However, as has been noted recently, this isnot obvious from the Bethe Ansatz/conformal field theory ~BA/CFT! approach. We show that by supplement-ing the BA/CFT analysis with simple symmetry arguments one can easily prove that correlations of the latticedensity operators decay exponentially. @S0163-1829~99!12135-0#Recently, density correlation functions have been studiedfor various multicomponent one-dimensional electron sys-tems within the framework of the bosonization approach.1,2An important issue raised in these works is the questionwhether or not these correlators exhibit a power-law behav-ior at large distances in the presence of a Mott-Hubbard gap~see also Refs. 3 and 4!. For the particular case of the halffilled, one-dimensional Hubbard model a bosonization analy-sis yields the expected exponential decay, which at first sightappears to be at odds with the results obtained from theBethe Ansatz5 ~see, e.g., Ref. 20 of Ref. 1!.In one spatial dimension, the asymptotic behavior of cor-relation functions can be analyzed by means of the Bethe-Ansatz/conformal field theory ~BA/CFT! approach. Further-more, all possible critical dimensions can be obtained fromthe exact solution.5 Since the expansion of the lattice opera-tors in terms of the conformal fields is not known explicitly,the resulting expressions contain unknown amplitudes. Inparticular, the leading term in the BA/CFT expression for thecorrelation function under question could actually vanish.This is the origin of the controversy mentioned above. Incertain cases, including the one discussed here, it is possibleto resolve this issue by employing symmetry considerationsof the underlying microscopic model.In this note, we are interested in connected correlationfunctions of electron densities at half filling. Using the selec-tion rules of conservation of numbers of electrons with spinup and down, one can show from the exact solution ~special-ized to the case of zero-magnetic field! thatGnn~ t ,x !5^@n~ t ,x !2^n&#@n~0,0!2^n&#&→A1cos~px/a0!ux1ivtu 1A2x22v2t2~x21v2t2!21 , ~1!where v is the spin velocity, a0 is the lattice spacing and A1,2are unspecified amplitudes that depend on the physical pa-rameters of the problem, i.e., interaction strength and in thegeneral case magnetic field. Note that in Eq. ~1! we haveomitted similar, subdominant terms.We will now show that by using the full set of selectionrules for the half-filled Hubbard model, one can establishPRB 600163-1829/99/60~12!/8540~3!/$15.00that the amplitudes of all contributions to Eq. ~1!, that decayalgebraically, are identically zero and that the correlator thusdecays exponentially.The Hamiltonian of the half-filled Hubbard model is ofthe formH~U !52(^i j&$cis† c js1H.c.%1U(iS ni↑2 12 D S ni↓2 12 D .~2!Here cis ,cis† are canonical annihilation and creation opera-tors for electrons with spin s on site i, ni5(snis5(scis† cis is the number operator density for electrons.In addition to the SU(2) invariance in the spin degrees offreedom the Hubbard Hamiltonian on a bipartite lattice com-mutes with the generators of a second SU(2) related toparticle-hole symmetry.6,7 In the one-dimensional case, thelocal generators of this h-pairing SU(2) areh j15~21 ! jc j↑c j↓ ,~3!h j25~h j1!†, h jz5 12 ~12n j!,satisfying commutation relations @h j6,h jz#56h j6 and@h j1,h j2#52h jz. Combining the two symmetries one obtainsthe well-known SU(2) ^ SU(2)/Z25SO(4)-symmetry ofthe Hubbard model.At half filling, the ground state uV& is an SO(4) singlet.8,9Denoting by n(t ,x) the density operator at x5a0 j and time twe want to determine the asymptotic behavior for j→‘ ofthe connected correlation functionGnn~ t ,x !54^hz~ t ,x !hz~0,0!&54(n^Vuhz~ t ,x !un&^nuhz~0,0!uV&54(nexp@ i~ent2knx !#u^nuhz~0,0#uV&u2. ~4!Hence, the matrix elements8540 ©1999 The American Physical SocietyPRB 60 8541DENSITY CORRELATIONS IN THE HALF-FILLED . . .^nuh1z uV& ~5!determine which intermediate states will contribute to theasymptotic behavior of Eq. ~4!.In the BA/CFT approach to correlation functions in inte-grable models one usually proceeds as follows: Firstly, onecalculates the finite-size scaling behavior of the energies enand momenta kn of the low-lying states entering the expres-sion ~4!. Secondly, one uses the quantum numbers of theoperator entering the matrix element to eliminate ‘‘incompat-ible’’ intermediate states. Finally, the CFT relation betweenfinite-size energies and scaling dimensions is used to obtainthe asymptotic ~power law! behavior of correlation functionsby summing over the remaining low-lying intermediatestates.In Ref. 5 only particle numbers for spin-s electrons wereused in the second step. This corresponds to taking into ac-count only quantum numbers associated with the Cartan gen-erators Sz and hz but not those with the total spins SW 2 andhW 2. As a result, one obtains Eq. ~1!, which seems to indicatethat Gnn decays algebraically for large distances in spite ofthe presence of a charge gap at half filling. We will nowshow, by considering the full set of SO(4) quantum num-bers, that only intermediate states with a gap contribute toEq. ~5! and the density correlations decay exponentially.Let us calculate the total h-spin quantum number of thestate h1z uV&: using the commutation relations between theh-pairing operators @and the fact that uV& is a singlet underthe h-pairing SU~2!# one easily obtainshW 2~h1z uV&)5@hW 2,h1z #uV&52~h1z uV&), ~6!showing that this state is a triplet of the h-pairing SU(2). Asa consequence, only intermediate states that are h-pairingtriplets can contribute to the correlation function ~4!.As is well known only pure spin excitations are gapless inthe repulsive half-filled Hubbard model. However, these areall singlets of the h-pairing SU(2).9 Therefore the corre-sponding matrix elements ~5! vanish identically and cannotcontribute in Gnn . We conclude that the lowest energy in-termediate states with nonzero matrix elements ~5! areholon-antiholon scattering states with energy above theMott-Hubbard gap. As a result the density-density correla-tion function exhibits exponential decay at large distancesfor any positive UGnn~ t ,x !→exp~2ax !, x→‘ . ~7!The determination of a(U) is an interesting open problem.On general grounds we expect it to be proportional to theholon gap, which is given by10,11 ~see also Ref. 12!D~U !5U2 2214 (n51‘~21 !n@A11~nU/2!22nU/2#;4pAU exp~22p/U !, for U→0. ~8!If one considers correlation functions of ‘‘point-split’’densities such as N j(ps)5(sc js† c j11,s the above symmetryargument does not imply the vanishing of, e.g., the matrixelement ^STuN (ps)(0,0)u0&, where uST& denotes a spin-tripletexcitation.13 Given that symmetry does not force such matrixelements to vanish we expect them to be nonzero, which thenimmediately implies an algebraic decay like in Eq. ~1!.We emphasize, that to establish the exponential decay ofGnn in the above way it is essential that: ~1! There is an exactsymmetry in the charge sector of the microscopic Hamil-tonian ~2!. In general this symmetry may not be obvious ~asis the case for the Hubbard model!. ~2! The ground state is asinglet of the corresponding algebra. Note that the aboveconsiderations still hold in the presence of a magnetic fieldas the ground state ~in the half-filled band! remains a singletof the h-pairing SU~2!. ~3! All charged ~nonsinglet! excita-tions are gapped.We note that these conditions are fulfilled for the half-filled Hubbard model on a bipartite lattice in any dimension,provided that U is larger than the critical Mott-Hubbardvalue.8Analogous conclusions can be reached for spin-spin cor-relations in the attractive Hubbard model by employing dis-crete symmetries of the Hamiltonian ~2!. Under the particle-hole transformation for spin upc j ,↑→~21 ! jc j ,↑† , c j ,↓→c j ,↓ ~9!the Hamiltonian ~2! transforms according to H(U)→H(2U), whereas h-pairing and spin SU~2! symmetriesare interchanged Sa↔ha, a51 ,2 ,z . Furthermore, theground state of the attractive Hubbard model is a spin sin-glet. This implies that spin-spin correlation functions in theattractive Hubbard model decay exponentially at large dis-tances^Sz~ t ,x !Sz~0,0!&→exp~2bx !, x→‘ , ~10!where b.0 for any U,0.We are grateful to the Max-Planck-Institut fu¨r Physik ko-mplexer Systeme at Dresden for hospitality and to A. O.Gogolin and A. Schadschneider for important discussions.F.H.L.E. was supported by the EPSRC. H.F. was supportedin part by the Deutsche Forschungsgemeinschaft under GrantNo. Fr 737/2–3.1 V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev. B 59, 15641 ~1999!.2 P. Azaria, A. O. Gogolin, P. Le´cheminant, and A. A. Nersesyan,Phys. Rev. Lett. 83, 624 ~1999!; R. Assaraf, P. Azaria, M.Caffarel, and P. Le´cheminant, Phys. Rev. B. 60, 2299~1999!.3 V. J. Emery, in Highly Conducting One-Dimensional Solids, ed-ited by J. T. Devreese, R. P. Evrard, and V. E. van Doren ~Ple-num, New York, 1979!.4 J. Voit, J. Phys. C 21, L1141 ~1988!.5 H. Frahm and V. E. Korepin, Phys. Rev. B 42, 10 553~1990!.8542 PRB 60FABIAN H. L. EßLER AND HOLGER FRAHM6 O. J. Heilmann and E. H. Lieb, Ann. ~N.Y.! Acad. Sci. 172, 583~1971!.7 C. N. Yang, Phys. Rev. Lett. 63, 2144 ~1989!.8 E. H. Lieb, Phys. Rev. Lett. 62, 1201 ~1989!.9 F. H. L. Eßler and V. E. Korepin, Phys. Rev. Lett. 72, 908 ~1994!;Nucl. Phys. B 426, 505 ~1994!.10 E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445~1968!.11 A. A. Ovchinnikov, Zh. E´ ksp. Teor. Fiz. 57, 2137 ~1969! @Sov.Phys. JETP 30, 1160 ~1970!#.12 C. A. Stafford and A. F. Millis, Phys. Rev. B 48, 1409 ~1993!.13 F. Woynarovich, J. Phys. C 16, 5293 ~1983!.

A note on density correlations in the half-filled Hubbard model

Abstract

Similar works

Full text

Available Versions

Institutionelles Repositorium der Leibniz Universität Hannover

Oxford University Research Archive (ORA)

Oxford University Research Archive

Crossref