We show how the addition of a phonon field to the Hubbard model deforms the superconducting su(2) part of the global symmetry Lie algebra su(2)⊗su(2)/openZ2, holding at half filling for the customary model, into a quantum [su(2)]q symmetry, holding for a filling which depends on the electron-phonon interaction strength. Such symmetry originates in the feature that in the presence of phonons the hopping amplitude turns out to depend on the coupling strength. The states generated by resorting to this q symmetry exhibit both off-diagonal long-range order and pairing

A. MONTORSI

M. RASETTI

English

MONTORSI, Arianna

RASETTI, Mario

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEQuantum symmetries induced by phonons in the Hubbard model / A. MONTORSI; M. RASETTI. - In: PHYSICALREVIEW LETTERS. - ISSN 0031-9007. - 72:11(1994), pp. 1730-1733.OriginalQuantum symmetries induced by phonons in the Hubbard modelPublisher:PublishedDOI:10.1103/PhysRevLett.72.1730Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1402804 since:APSVOLUME 72, NUMBER 11 PHYSICAL REVIE%' LETTERS 14 MARCH 1994Quantum Symmetries Induced by Phonons in the Hubbard ModelArianna Montorsi ' and Mario Rasetti'Theory Oitision, Los Alamos IVational Laboratories, Los Alamos, New Mexico 87545(Received 3 I August l 993)We show how the addition of a phonon field to the Hubbard model deforms the superconducring su(2)part of the global symmetry Lie algebra su(2) Ssu(2)/Z2, holding at half filling for the customary model,into u quantum [su(2)]q symmetry, holding for a filling which depends on the electron-phonon interac-tion strength. Such symmetry originates in the feature that in the presence of phonons the hopping am-plitude turns out to depend on the coupling strength. The states generated by resorting to this ne~ qsymmetry exhibit both of-diagonal long-range order and pairing.PACS numbers: 74.20.—z, 05.30.FkThe interest in itinerant interacting electron systemshas been greatly revived since the discovery of high-T,materials, for which the interplay between itinerantmagnetism and insulating behavior is believed to play arelevant role. The Hubbard model [I] provides the sim-plest description of such systems, by assuming that theitinerant electrons interact only via an on-site Coulombrepulsion term, and it has therefore been intensively stud-ied. In particular, it is known that at half filling the mod-el has a su(2) @su(2) symmetry algebra [2], where su(2)refers to the magnetic part (spin conserving) and su(2) tothe superconducting part (number conserving) of the glo-bal symmetry. Korepin and co-workers [3] utilized thecorresponding SO(4) =SU(2) SU(2)/Z2 group to provecompleteness of the solution for the one-dimensionalmodel given by Lieb and Wu [4]. Also, in ID, such sym-metry extends to an -dimensional one, generated by de-formations of the corresponding Kac-Moody algebra [5].Yang [6] showed that, due to the algebraic structure un-derlying the symmetry, at any filling it is possible to con-struct in any dimension eigenstates of the Hubbard Ham-iltonian characterized by off-diagonal long-range order(ODLRO), and these can be labeled in terms of twoquantum numbers connected with the above symmetry,one of which is related to the existence of superconduc-tivity. These eigenstates can be shown to be of high ener-gy compared to the global ground state, but for U &0turn out to be metastable states. Because of the above re-sults, even though there is still no exact solution in morethan one dimension, it has by now become clear that, atleast as far as s-wave pairing is concerned, the repulsiveHubbard model itself is not able to describe a supercon-ducting phase. Many generalizations have then been pro-posed in order to make it capable of exhibiting the latterfeature, some of which, having in mind the role whichphonons play for conventional superconductivity, explicit-ly consider the electron-phonon coupling as a possiblemechanism for the formation of electron pairs [7].In this Letter, we show that the addition of a phononfield to the Hubbard model, although breaking the super-conductive part of the symmetry, restores it as a de-formed (quantum group) [su(2)l~ symmetry, where q iswhereH, ~" =g [ p(n; I+—n; I)+Un; ln; I],JH.']"'"' = — rg g—(ajf~ak, ~+af,,~;,e),(ho ) (jk) a (3)the first representing the contributions to the electron lo-cal energy, the second the nonlocal kinetic part of theenergy. In (2) and (3) a&,a are fermionic creationand annihilation operators ( a&, ak ) =0, [aj,ak=&;,gB, ll, n. =a; ~; ) over a d-dimensional lattice A(j,k 6 A, cr c l, l]), and (j,k) stands for nearest neigh-bors (nn) in A. p is the chemical potential and U &0 isthe on-site Coulomb repulsion between electrons [1],whereas t denotes the hopping integral of the band elec-trons, given by the overlap integral [8]r =„dr&*(r—R;)where y(r —R;) denotes the atomic wave function for anelectron of mass m on the atom centered at site Rj of A,and V(r, [Rr]) the crystal ion potential seen by a singleelectron.As for the phonons, since only the local modes arerelevant [7], they are identified —for simplicity —with anensemble of independent Einstein oscillators, all with thesame frequency co, and may be thought of as describingnothing but the ion oscillations around the lattice posi-related to the strength of electron-phonon coupling. Also,the filling at which symmetry is restored turns out to de-pend on the same interaction strength. Moreover, at anyfilling we can still build eigenstates of the Hamiltonianwith nonvanishing pairing, which have energy lower thanthe corresponding (rf-paired) states of Yang, exhibitODLRO, and could possibly be metastable for someU & 0.The grand-canonical Hubbard Harniltonian readsyg (loc) + H (hop)1730 0031-9007/94/72 (1 1)/17 30(4)$06.001994 The American Physical SocietyVOLUME 72, NUMBER 11 PHYSICAL REVIEW LETTERS 14 MARCH 1994tions. When one switches on the phonon field, therefore,the Hamiltonian (I) is simply changed intoH (loc) +H +H (ho() (5)where2H q=g +—Ma) x~PJ 12M 2x y(r —R), —x),) . (8)The hopping amplitude t j), should then be thought of asan operator, commuting with all fermionic operators, de-pending on the phonon bosonic dynamical variables.There follows that the hopping Hamiltonian (7) turns outto be the sum of two terms: Hc()".g =HcI&)+H, )"P'The former, due to the direct contribution of V(r, [Rtj) to(8), is readily seen, via the usual Holstein mechanism[10] of linear variation of V(r, [Rtj) in (8) to consistessentially in a purely local term,H,(j,h = )g (n; I + n—; I )xi,J(9)which describes coupling of the phonons with strength kto the charge density of the band electrons. X (indepen-dent on site I) is given by1(, = J~ dr p*(r —R—;)[Vtt, V(r, [Rtj )]p(r —Rt ) .Clearly, A. &0 dynamically breaks the su(2), not the su(2)symmetry.On the other hand, Ho()"~h' ) contains the contributionof the ion displacement to the hopping amplitude t J g in-duced in (8) by the Laplacian, depending on the electronorbital gradients overlap integral:2dr V& (r —R.—x ) Vp(r —R), —x), ) .2m~ J JIn view of the exponential character of (S(r —R),) farfrom the kth ion, and of the fact that ~xq~((~Rq~, thecorresponding variation of the overlap integral t;g withrespect to the original amplitude t can be seen to consist[I I] in the factor exp[((x; —x),)], accounting for themodulus and exp[ttr(pi —p), )] for the phase. Here g andHere M is the ion mass and xJ,pJ the local ion displace-ment and momentum operators, respectively ([x&,p&]=i hb; ),), commuting with the Fermi operators, andH,'(" I),) = g g(t;),a)', ~;.+H.c.),&Jt) ~with t&), given by the obvious generalization [9] of (4)taking into account the fact that, due to the phonon field,the orbitals should be centered at the displaced ion sites:Q2dr&*(r —R& —x&) — 4+ V(r, [Rt+xtj)K are real constants, the former depending ultimately,once more, on the form of the potential V(r, [Rtj) thateach single ion exerts on the electron, which determinesthe exponential decay of the wave function (()(r) far fromthe origin; the latter depends on the lattice structure, inparticular on the interatomic distance S=~R& —R), ~, inview of the Bloch theorem and the fact that one can thinkof (p) —p), ) as the relative variation of the electronmomentum eigenvalues induced by the ion displacernents.We write thereforeH (ooo)oc) t g g ( [&( )]xexp[i)r(p& —pg)]ay ~) + H.c.) .(i 2)In a recent paper [12] it was pointed out how —in thelimit in which Hg~h is retained, but H, ("~g' is approxi-mated by H,~"' —the Hamiltonian H can be rewritten asa purely Hubbard-like anyonic oscillator Hamiltonian(whose building operators are no longer fermionic elec-tron operators, but satisfy a q-deformed Gel'fand-Fairliealgebra [13])plus H~h.In the following we shall show how a global symmetryof the model described by the complete H in (5) might infact be restored at the quantum group level [14]. In oth-er words we look for a global symmetry algebra of theform su(2)8[su(2)]z. We shall do this following thesame scheme used by Yang and Zhang in [2], construct-ing first a local [su(2)j)q, j E A, and requiring that itcommute with the jth art of the local Hamiltonian,=Hph+ H ) +H ) ph and extending it successive-ly to a global [su(2)]v generating a symmetry of thewhole Hamiltonian H.Dealing first with the local symmetry, we setg (+) itltPj y y g ( —) g (+)t=e aJ laJ I, J (i3)K)') =—,' (n& I + n& I —I ) .The three operators KJ", y=+, z, close both under aquantum [su(2)]~ algebra [15] with q=e ', a E C arbi-trary:[K)',K) — ] = ~ K) —,[K.+,K) ] = [2K)' ]q (14)(where the notation, standard in quantum group the-ory, [x]z =(q"—q ")/(q —q ') =sinh(ax)/sinha wasused), as well as under a customary su(2) Lie alge-bra ([K.+,K. ] =2K.' ), since [2K-' ] "+' =2K.'(n ~ 0) in the representation adopted.It is straightforward to check that H ' commuteswith all the above K;" 's, provided the conditions p —2(and tu = z (U —4)(() are satisfied, with (=A/Ma)~ [ob-serve that for (=0 the former leads to the customary fer-mionic realization of su(2) and the latter reduces to thecondition of half filling]. Even when the two above condi-tions do not hold, the local Hamiltonian commutes with1731VoLUME 72, NUMBER 11 PH YSICAL REVIE% LETTERS 14 MARCH 1994K)', in which case the local symmetry reduces to u(l).%'e intend to show first that the global extension to thewhole lattice A of the above local algebras gives rise to aphysically nontrivial symmetry of the Hamiltonian (5)only in the quantum algebra case.In view of the adoption of the quantum group scheme,the correct extension must naturally be performed resort-ing to the notion of coproduct. For [su(2)]q the Cartanelement K ' is primitive, and its coproduct h(K )=le|3K ' +K ' Slf, ~hereasNotice that in the limiting case a=0 such a coproductturns into the usual tensor product. For the sake of sim-plicity, and without any loss of generality in the thermo-dynamic limit, we assume the lattice number of sites 4of the form %=2' hence the extension to the wholeA of the local [su(2)]~ is given by K " =6 "(K " ).y =,z, with the iterated coproduct defined through& "+' (K " ) =A(A" (K " )). Explicitly,(i 6)Here the multiple tensor products consist of A factors,the single one of which not equal to the identity I (or tothe exponential of K ' ) is at position j. Also, following[2], we redefined the local generator K)+ (13) with amultiplicative phase factor exp(iG j) (G is a vector withall its components equal to )r). This of course does notaffect the structure of the algebra and will allow us todeal with the nn hopping term in the correct way. More-over, we introduced the notation k& j or k ( j withrespect to the arbitrary order of the lattice sites adoptedin the direct product.The global [su(2)]~ defined above manifestly com-mutes with H ' ). The requirement that H has su(2)&[su(2)]q as global symmetry thus reduces to checkingthat [H,)"&g', [su(2)]q] =0. On the other hand, as thehopping term in H conserves the total number of elec-trons, in view of its definition K ' always commutes withH. The symmetry condition implies therefore simply that[H,)"~~', K ] =0. A lengthy but straightforward cal-culation shows that this happens if the parameter a, sofar undefined, has real part Rea=2()r/h, and )r is con-strained by the condition )r=g. Naturally the formercondition suggests that a can be chosen to be real. Theabove discussion shows that only (=0 implies a =0; thusthe quantum symmetry can be ignored only if the eAectof ion displacements due to the phonon field on the hop-ping amplitude is neglected. It should be remarked, onthe other hand, that setting (=0, but still keeping )r=g,generates a Hamiltonian which, by a unitary transforma-tion R essentially equal to the inverse of that of Lang-Firsov [l6, i7] [R =expi(gp)(n; I+n) I)], can be recon-ducted to one describing a system of interacting electronsplus a field of phonons (uncorrelated with electrons). Itshould moreover be pointed out that the conditions thatguarantee quantum symmetry impose two constraints,K =g, p = —,' (U —4X /Mro ), among the four physical pa-rameters k, U, (o, and the filling no (through the chemicalpotential p), in terms of the atomic parameters M, (, andAs discussed above, among the latter, ( has only therole of fixing the quantum group deformation parameterwhere, however, the g-pairing operator is phonon depen-dent, defined, in terms of the set of parameters w, [0)], byand JV„'=n!JV!/(JV —n)!. A direct calculation showsthat the state among the ~y„)'s, which minimizes the en-ergy expectation 8„=(y„~H~y„) with respect to w hasw =2( (—=p) and gives 8„= i ))'iro/V, which are the samevalues as for the q-symmetric eigenstates ~p„) =(I/JJV')x (K + ) "~vac) of H. Here the normalization isn —I I'=H ZI 0 p 0 q,[n],![)V],![/V n]q!—Both Yang's rl-paired state [6] and the BCS ground stateturn out to have —for the filling corresponding to fullsymmetry —energy 6'„+2k, /Mro & 6'„.If the chemical potential does not fulfill the symmetrycondition, it is still possible to work out ei~enstates of H,due to the fact that [H, K ] =b( K —,with bD=U—2p —4i(. /M(o, and starting from any eigenstate ~( ) ofH one can construct a sequence of other eigenstates in theform [K ]"~8), n ~ JV, with decreasing energy eigen-values 6„' =('+nbÃ, provided p+2)(, /Mro & 2 U. phere is to be intended as a parameter, and a detailedanalysis of its dependence on the filling would require theknowledge of the ground state of the system, or some an-satz about it, which is beyond the purposes of the presentFollowing an argument quite similar to that of Yang[6], we now consider the family of states~y„), character-ized by nonvanishing pairing (referred to as rl pairing), ofthe generic form1 (gt) "~vac),1732VOLUME 72, NUMBER 11 P H YSICAL R EV I E% LETTERSi6 (r —I) a(N+ l —(r+s() [N ][N] (19)whereN [N] t[n] ![N—n] !is the quanrum binomial coeII]cient. The states ~tl„) man-ifestly have pairing as well. It is worth noticing that (III„)cannot be the ground state because, provided that w =2(,~tir„) gives the same 8„ for whatever value of the phase (}Iin (17), whereas only for (}j=n~lir„) is an eigenstate of H.Nevertheless one could argue that, similarly to what hap-pens in Yang's case, for b@ (0, ~4I„) turns into a meta-stable state.In conclusion, one can thus observe that the supercon-ducting eigenstates (III„) of a Hamiltonian of the form (5)for systems whose ionic structure implies that the con-straint tc=( is satisfied by the parameters could be favor-able also for some range of positive U values, immediatelyupon doping away from the filling at which the q symme-try holds. Moreover, switching oA' of the phonon fieldraises the paired states energy.The authors gratefully acknowledge stimulating discus-sions with D. Baeriswyl, C. Castellani, V. E. Korepin, G.Kotliar, J. Ranninger, and C. M. Varma.paper. Moreover, as in the case analyzed by Yang, thestate (p„) exhibits ODLRO [18], in that for large(6(N)) ~r —s~, the expectation value& 0n I a s. I a s, I a r. I a rlI ,4n &equals[I] J. Hubbard, Proc. R. Soc. London A 276, 238 (1963);277, 237 (1964); M. C. Gutzwiller, Phys. Rev. Lett. 10,159 (1963).[2] C. N. Yang and S. C. Zhang, Mod. Phys. Lett. B 4, 759(1990};P. W. Anderson, Phys. Rev. 112, 1900 (1958};O.J. Heilmann and E. H. Lieb, Trans. N. Y. Acad. Sci. 33,116 (1971).[3] F. H. L. Essler, V. E. Korepin, and K. Shoutens, Phys.Rev. Lett. 67, 3848 (1991).[4] E. H. Lich and F. Y. Wu, Phys. Rev. Lett. 20, 1445(1968).[5] D. B. Uglov and V. E. Korepin, Stony Brook Universityreport (to be published).[6] C. N. Yang, Phys. Rev. Lett. 63, 2144 (1989).[7] A. S. Alexandrov, J. Ranninger, and S. Robaszkiewicz,Phys. Rev. B 33, 4526 (1986).[8] For an extended review of this point see, e.g. , D. Baer-iswyl, D. K. Campbell, and S. Mazumdar, in ConjugatedConducting Polymers, Springer-Verlag Series in SolidState Science Vol. 102, edited by H. Kiess (Springer-Verlag, Berlin, 1992).[9] A. L. Kuzemsky, A. Holas, and N. M. Plakida, Physica(Amsterdam} 122B, 168 (1983).[10] T. Holstein, Ann. Phys. (N.Y.) 8, 325 (1959).[I I] E. Hiickel, Z. Phys. 70, 204 (1931);76, 628 (1932).[12] A. Montorsi and M. Rasetti, in Superconductivity andStrongly Correfated Electronic Systems, edited by A.Romano and G. Scarpetta (World Scientific, Singapore,1994).[13] A. Lerda and S. Sciuto, Nucl. Phys. B401, 613 (1992).[14] V. G. Drinfeld, Sov. Math. Dokl. 32, 254 (1985).[15] M. Jimbo, Lett. Math. Phys. 10, 63 (1985); ll, 247(1986).[16] S. Robaszkiewicz, R. Micnas, and J. Ranninger, Phys.Rev. B 36, 180 (1987).[17] I. G. Lang and Yu.A. Firsov, Zh. Eksp. Teor. Fiz. 43,1843 (1962) [Sov. Phys. JETP 16, 1301 (1963)].[18] C. N. Yang, Rev. Mod. Phys. 34, 694 (1962).1733

Quantum symmetries induced by phonons in the Hubbard model

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Quantum symmetries induced by phonons in the Hubbard model

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