Effects of the electron-phonon coupling near and within the insulating Mott phase
a r X i v :c o n d -m a t /0412243v 1 [c o n d -m a t .s t r -e l ] 9 D e c 2004
E?ects of the electron-phonon coupling near and within the insulating Mott phase
C.A.Perroni,V.Cataudella,G.De Filippis,and V.Marigliano Ramaglia
Coherentia-INFM and Dipartimento di Scienze Fisiche,
Universit`a degli Studi di Napoli “Federico II”,Complesso Universitario Monte Sant’Angelo,
Via Cintia,I-80126Napoli,Italy
(Dated:February 2,2008)
The role of the electron-phonon interaction in the Holstein-Hubbard model is investigated in the metallic phase close to the Mott transition and in the insulating Mott phase.The model is studied by means of a variational slave boson technique.At half-?lling,mean-?eld static quantities are in good agreement with the results obtained by numerical techniques.By taking into account gaussian ?uctuations,an analytic expression of the spectral density is derived in the Mott insulating phase showing that an increase of the electron-phonon coupling leads to a sensitive reduction of the Mott gap through a reduced e?ective repulsion.The relation of the results with recent experimental observations in strongly correlated systems is discussed.
I.
INTRODUCTION
The understanding of the properties of Mott insula-tors such as CaT iO 3and V 2O 3represents a long-standing problem.As suggested by Mott,1the strong Coulomb re-pulsion among the electrons can cause a metal-insulator transition opening a gap in the density of states that is usually known as Mott gap.The correlation-driven metal-insulator transition is often investigated within the framework of the Hubbard lattice Hamiltonian which takes into account the electron interaction through the on-site repulsion term U .As developed by Hubbard,2two subbands generically called lower and upper Hub-bard bands separated by the Mott gap of the order of the energy U arise in the excitation spectrum for large enough repulsion.
The correlation e?ects are important not only on the insulating side but also on the metallic phase where a great deal of insight has been obtained by using the Gutzwiller wave function.3A progress toward the under-standing of the metal-insulator transition has been made by the reformulation of the problem in terms of slave bosons,4which reproduces the Gutzwiller approximation at the saddle-point level and allows to study the e?ects of gaussian ?uctuations in the metallic 5and in the in-sulating phase.6In this state the resulting single-particle spectral function consists of two broad incoherent con-tributions reminiscent of the lower and upper Hubbard bands.An overall description of the Mott transition in the Hubbard model can be obtained by means of the dy-namical mean ?eld theory (DMF T )that is exact in the limit of in?nite dimension.7The study of the in?nite di-mensional model has essentially con?rmed the validity of the description obtained when slave boson ?uctuations about the mean-?eld (MF )are considered.
Although the Hubbard Hamiltonian captures the fun-damental properties of systems near the Mott transition,it does not take into account the role of the lattice degrees of freedom.Actually,the presence of strong electron-phonon (el ?ph )interactions has been pointed out in several systems,such as cuprates,8,9manganites,10and
V 2O 3.11.In addition to the on-site repulsion,the most natural model incorporates a short-range interaction of the electrons with local phonon modes of constant fre-quency ω0.12This model is described by the Holstein-Hubbard Hamiltonian H
H =?t
c ?iσc jσ+U
i
n i ↑n i ↓+
ω0
i
a ?i a i
+gω0
i
n i
a i +
a ?i .
(1)
Here t is the electron transfer integral between nearest
neighbor sites
σn iσ= σc ?iσc iσis the local density operator.In Eq.(1)a ?i (a i )is the creation (anni-hilation)phonon operator at the site i,and the parame-ter g represents the coupling constant between electrons and local displacements.The dimensionless parameter λ=g 2ω0/W ,with W =2dt bare electron half band-width,is typically used to measure the strength of the el ?ph interaction in the adiabatic regime characterized by small values of the ratio γ=ω0/t .Within this regime the Coulomb repulsion is found to dominate the prop-erties of the metallic phase also with a sizable el ?ph coupling.13Furthermore,there is a very little softening of the phonon frequency on the approach to the Mott transition since the Hubbard term U suppresses charge ?uctuations.13Actually within the Mott phase the spec-tral density shows phonon side bands whose peaks are separated by multiples of the bare frequency ω0.14Fi-nally,close to the Mott transition at ?nite density,an intermediate el ?ph coupling leads to the phase separa-tion between a metal and an insulator.15
In this paper the study of the Holstein-Hubbard model focuses on the role of the el ?ph interaction in mod-ifying the physical properties of the electrons close to the metal-insulator transition and in the insulating Mott phase.The starting point of the approach is the gener-
2
alized Lang-Firsov transformation16U=e V,with
V=g i[
tortions.The next step is the functional-integral repre-sentation in terms of the four-slave bosons e i,p iσ,and d i
obtained by imposing the equivalence with the original model through the Lagrange multipliersλ(1)i andλ(2)iσ.4
First we consider the MF solution at half-?lling,then slave boson gaussian?uctuations on the top of the MF ?nding that the resulting Mott gap is sensitively reduced
with increasingλsince it is determined by the el?el repulsion renormalized by the e?ects of the el?ph cou-
pling.Finally the relation of the results with recent ex-perimental observations in strongly correlated systems is
discussed.
The MF solution in the paramagnetic homogeneous
phase is obtained by replacing the Bose?elds with their mean values( e i =e0, d i =d0, p iσ =p0)and by assuming f i=f,λ(1)i=λ(1)0,andλ(2)iσ=λ(2)0.17The
resulting mean-?eld Hamiltonian is characteristic of free fermion quasi-particles and phonons and the mean double occupation is controlled by U eff=U+2g2ω0(f2?2f). At half-?lling the MF energy per site u is given by a simple functional depending on f and d0
u=qe?f2g2ˉε+U eff d20?g2ω0,(3) with q=8d20?16d40andˉεmean bare kinetic energy. The Mott phase is the insulating state for large U eff>0 characterized by q=0,d0=0,and the energy per site equal to?g2ω0(characteristic energy of the limit U/t=∞).When U eff becomes negative,a on-site bipolaron solution sets in corresponding to d20=0.5,f=1,and energy u=U/2?2g2ω0.
In Fig.1the phase diagram at half-?lling is reported forγ=0.2in the three-dimensional case.By analyzing the behavior of d0,it is found that the transition to MI is always found to be second order,that to the bipolaronic insulator(BI)is?rst order,?nally that between MI and BI is again discontinuous in excellent agreement with the results derived by the DMF T solution.13In particular we notice that,with increasingλ,the line separating the M and MI phases shifts to values of U larger than U c,the critical value in absence of the el?ph coupling.Actually the Mott transition is in?uenced by the el?ph interac-tion since it is U eff that governs the transition and it becomes smaller with increasingλ.Therefore,the con-dition U eff?U c,characteristic of a transition driven by the el?el interaction,implies that the transition occurs for larger values of the bare U.Within the MF approach the interplay between el?el and el?ph interactions in a?ecting the Mott phase is essentially linked to the value of the parameter f that,at?xed adiabaticity ratio and λ,is weakly decreasing with increasing U and it is of the order ofγ/4d in the adiabatic regime near the Mott tran-sition.The transition line between M and MI phase is given byλ?(U?U c)/(2fW)?(U?U c)/γt.Therefore, as shown in the inset of Fig.1,the dependence of the Mott transition line onλbecomes also more pronounced with increasing the adiabaticity ratioγ.In the atomic limit(γ=∞)we recovers the exact solution without metallic phase with f=1and U c=0(dotted line in Fig.1corresponding toλ=U/2W).
While the Mott transition is driven by the growth of the spin susceptibility,the transition from M to BI is characterized by the enhancement of the charge?uctu-ations inducing a decrease of the e?ective repulsion.In Fig.2we report the MF results for the spectral weight Z at the Fermi energy(equal to m/m?)and the local magnetic moment M=<(M z i)2>stressing the e?ects of the el?ph interaction.Within the MF approach we simply have Z=qe?f2g2and M=1?2d0.Far from the Mott transition(U/U c small)Z decreases with increasing the el?ph coupling as expected for any localizing inter-action.However,near the Mott phase,the e?ects due to the reduction of U eff become more marked and are able to induce the enhancement of Z with increasingλthat,again,is in good agreement with DMF T results.13 In Fig.2we also show that the M phase is reduced in comparison with its value atλ=0for any ratio U/U c implying that the double occupation d0increases by ap-proaching the BI phase.Therefore,while M increases as function of U,it decreases as function ofλ.
The MF solution can be readily generalized at densi-ties di?erent from half-?lling.Within the MF approach some of us have shown that the interplay of strong local el?el and el?ph interactions can push the system toward a phase separation between states characterized by dif-ferent lattice distortions.17The phase coexistence occurs for intermediate values of the el?ph coupling and its relevance within the Hubbard-Holstein model has been con?rmed also by DMF T works.15
The task of including charge?uctuations described by the e and d?elds is simpli?ed in the Mott phase by the fact that at MF level the Bose?elds e and d are not condensed(e0=d0=0),while p0=1/
√
3 analytic expression of the spectral function A(ω)yielding
at zero temperature
A(ω)=e?g2f2?A(ω)+(4)
e?g2f2
∞
n=1
(g2f2)n
n!
θ(ω?nω0)?A(ω?nω0),(5)
withθ(x)Heaviside function and the function?A(ω)akin to that obtained for the Hubbard model with U=U eff.6 The?rst term of Eq.(5)is the product of two quanti-ties,with e?f2g2renormalization factor due to el?ph coupling.Through?A(ω)this term is able to recon-struct lower and upper Hubbard bands.The second and the third term in Eq.(5)represent the contribu-tion due to the phonon replicas of hole-and particle-type, respectively.18In Fig.3(a)we report the spectral density N(ω)=A(ω)/2π(solid line)together with the resulting ?rst term of Eq.(5)(dotted line)and the contribution from the phonon replicas(dashed line).We notice that this last term provides a non negligible spectral weight to the total spectral density at energies out of the gap of?A(ω).Therefore,the Mott gap of the system is deter-mined by the gap of the function?A(ω)and it is simply given by?=U effξ.As seen in Fig.3(b),the reduc-tion of the gap with increasingλcan be also of the order of bare half bandwidth W.19Finally we stress that the calculated gap is traceable to the di?erence of the chem-ical potentials at n=1and n=1?,in analogy with the results of the Hubbard model.6For n=1we have μ(1)=U/2?2g2ω0,while for n=1?the chemical po-tential isμ(1?)=μ(1)?U effξ/2.Since the system has particle-hole symmetry,the gap?can be obtained as ?=2[μ(1)?μ(1?)]=U effξ.
In Fig.4we report the di?erence between the gap at ?niteλand that atλ=0as a function of the adiabatic-ity ratio.Since the attraction between the electrons gets larger with increasingγ,the resulting Mott gap is more reduced.However in the strong el?ph coupling regime(λlarger than1)there is no dependence on the adiabaticity ratio since the particles localized by the strong correla-tion are small polarons.In fact in this regime the MF solution at?nite density is minimized for f=1yield-ing the e?ective interaction U eff=U?2g2ω0for any ?nite value of adiabatic ratio.In the inset of Fig.4we show the variation of the e?ective interaction U eff as a function of the adiabaticity ratio making the comparison with the behavior of the Mott gap.In the regime where λ<1,the quantityξa?ects the magnitude of the gap and its dependence on the adiabaticity ratio,while in the strong coupling regimeξ?1implying that the gap in units of W is just U eff/W=U/W?2λ.
In this work we have seen that in the metallic phase close to the Mott transition the spectral weight Z is en-hanced and the gap in the insulating Mott phase is re-duced asλis raised.These behaviors can be related to some recent experimental and theoretical studies in V2O3and Cr-doped V2O311,20,21where the Mott gap is unexpectedly small and in the metallic phase the quasi-particle peak in the photoemission spectrum has a signif-icantly large weight in comparison with that theoretically predicted.We suggest that the inclusion of the el?ph interaction could be able to partially?ll the discrepancies between the experimental observations and the theoreti-cal studies.Clearly the single orbital model is not su?-cient to fully explain the electronic and magnetic struc-ture of such systems,22so a proper multi-orbital theory has to be considered in order to obtain a better agree-ment of the theory with experiments.21The results due to the reduction of the e?ective repulsion caused by the el?ph coupling are valid when the lattice distortions are coupled to charge?uctuations like in the model of Eq.(1).However for general models the issue is delicate since there are interactions such as the Jahn-Teller cou-pling for which the phonon-mediated attraction could be even diminished by strong correlations.23
Finally we can argue the modi?cations of the spectral properties at densities near the MI phase and at half-?lling just under the edge of localization.Clearly there is a?nite spectral weight at the Fermi energy where quasi-particle states begin to appear.Therefore,in addition to the incoherent contribution at high energy,the spec-tral density presents also the coherent term.Close to the metal-insulator transition the coherent term could not be strongly a?ected by the el?ph coupling.In fact for the combined e?ect of the strong correlation and el?ph cou-pling the quasi-particle band can be narrower than the phonon energyω0implying the impossibility of the single phonon scattering between the quasi-particles.There-fore near the Mott transition the el?ph coupling af-fects mainly the incoherent term of the spectral density. This result is in agreement with recent experiments on Bi2Sr2CaCu2O8+δmade using angle-resolved photoe-mission spectroscopy.9In fact it has been found that the oxygen isotope substitution mainly in?uences the broad high energy humps.
In conclusion,we have discussed the role of the el?ph interaction in modifying the physical properties of the electrons in the metallic phase close to the Mott transi-tion and in the MI phase.The approach to study the Holstein-Hubbard model has been based on a variational slave boson technique that provides results in agreement with DMF T.An analytic expression of the spectral den-sity is derived in the Mott phase showing that due to the reduced e?ective repulsion the Mott gap decreases as the el?ph coupling constant increases.In this paper we have mainly discussed the phases without long-range or-der.The study of broken symmetry phases is possible within the slave-boson formalism4and it is left for future work.
4
Figure captions
F1The phase diagram U/W versusλat half-?lling forγ=0.2in the three-dimensional case.The transition lines separate the metallic state M from the Mott insulator MI and the bipolaronic in-sulator BI.The dotted line is the locus where U?2g2ω0=0.In the inset,the transition line between the metallic and Mott insulating phase is shown for di?erent adiabaticity ratiosγ.
F2The di?erence between the spectral weight Z at the Fermi energy(a)and the local magnetic moment M at?niteλ(b)and their respective values at λ=0for some values of the ratio U/U c in the three-dimensional simple cubic lattice.
F3(a)The renormalized density of states N(solid line)
together with the dominant contribution e?g2f2
?A(ω)(dotted line)and the term due to the phonon replicas(dashed line)as function of the frequency ω.
(b)The renormalized density of states N for di?er-
ent values of the el?ph coupling constantλas a function of the frequencyω.
F4The di?erence between the gap atλ=0and that at ?niteλin units of the bare half bandwidth W as function of the adiabaticity ratio for di?erent values of the el?ph coupling.In the inset the e?ective repulsion U eff and the Mott gap?as a function of the adiabaticity ratioγ.
1N.F.Mott,Philos.Mag.6287(1961).
2J.Hubbard,Proc.R.Soc.London Ser.A276,238(1963). 3M.C.Gutzwiller,Phys.Rev.137,A1726(1965);W.F.
Brinkman and T.M.Rice,Phys.Rev.B2,4302(1970). 4G.Kotliar and A.E.Ruckenstein,Phys.Rev.Lett.57, 1362(1986).
5J.W.Rasul and T.Li,J.Phys.C21,5119(1988);M.
Lavagna,Phys.Rev.B41,142(1990).
6R.Raimondi and C.Castellani,Phys.Rev.B48,11453 (1993).
7A.Georges,G.Kotliar,W.Krauth,and M.J.Rozenberg, Rev.Mod.Phys.68,13(1996).
8R.J.McQueeney,J.L.Sarrao,P.G.Pagliuso,P.W.
Stephens,and R.Osborn,Phys.Rev.Lett.87,77001 (2001).
9G.-H.Gweon,T.Sasagawa,S.Y.Zhou,J.Graf,H.Takagi,
D.-H.Lee,and https://www.sodocs.net/doc/7213898752.html,nzara,Nature430,187(2004).
https://www.sodocs.net/doc/7213898752.html,lis,Nature392,147(1998);M.B.Salamon and M.Jaime,Rev.Mod.Phys.73,583(2001).
11G.Kotliar,Science302,67(2003);P.Limelette, A.
Georges,D.Jerome,P.Wzietek,P.Metcalf,and J.M.
Honig,Science302,89(2003).
12T.Holstein,Ann.Phys.(Leipzig)8,325(1959);ibid.8, 343(1959).
13W.Koller,D.Meyer,Y.Ono,and A.C.Hewson,Euro-phys.Lett.66,559(2004);W.Koller,D.Meyer,and A.
C.Hewson,Phys.Rev.B70,155103(2004);G.S.Jeon,
T.-H.Park,J.H.Han,H.C.Lee,and H.-Y.Choi,Phys.
Rev.B70,125114(2004).14H.Feshke,A.P.Kampf,M.Sekania,and G.Wellein,Eur.
Phys.J.B31,11(2003);H.Fehske,G.Wellein,G.Hager,
A.Weisse,and A.R.Bishop,Phys.Rev.B69,165115
(2004).
15A.Deppeler and https://www.sodocs.net/doc/7213898752.html,lis,Phys.Rev.B65,100301 (2002);A.Deppeler and https://www.sodocs.net/doc/7213898752.html,lis,cond-mat/0204617;
M.Capone,G.Sangiovanni,C.Castellani,C.Di Castro, and M.Grilli,Phys.Rev.Lett.92,106401(2004).
https://www.sodocs.net/doc/7213898752.html,ng and Yu.A.Firsov,Soviet Physics JETP16,1301 (1963);Yu.A.Firsov,Polarons(Moskow,Nauka,1975). 17V.Cataudella,G.De Filippis,G.Iadonisi,A.Bianconi, and N.L.Saini,Int.J.Mod.Phys.B14,3398(2000).
18C.A.Perroni,G.De Filippis,V.Cataudella,and G.Iadon-isi,Phys.Rev.B64,144302(2001)and references therein. 19The spectral density is not shown for values ofλclose to1since the system shows phase separation at densities di?erent from half-?lling.
20S.-K.Mo,J.D.Denlinger,H.-D.Kim,J.-H.Park,J.W.
Allen,A.Sekiyama,A.Yamasaki,K.Kadono,S.Suga,Y.
Saitoh,T.Muro,P.Metcalf,G.Keller,K.Held,V.Eyert, V.I.Anisimov,and D.Vollhardt,Phys.Rev.Lett.90, 186403(2003).
21G.Keller,K.Held,V.Eyert,D.Vollhardt,and V.Anisi-mov,Phys.Rev.B70,205116(2004).
22R.Shiina,https://www.sodocs.net/doc/7213898752.html,a,F.-C.Zhang,and T.M.Rice,Phys.
Rev.B63,144422(2001).
23J.E.Han and O.Gunnarsson,Physica B292,196(2000).