Interplay between charge-lattice interaction and strong electron correlations in cuprates: phonon anomaly and spectral kinks
G. De Filippis, V. Cataudella, R. Citro, C.A. Perroni, A. S. Mishchenko, N. Nagaosa
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Interplay between charge-lattice interaction and strong electron correlations incuprates: phonon anomaly and spectral kinks
G. De Filippis , V. Cataudella , R. Citro , C.A. Perroni , A. S. Mishchenko , and N. Nagaosa , SPIN-CNR and Dip. di Scienze Fisiche - Universit`a di Napoli Federico II - I-80126 Napoli, Italy CNISM and Dip. di Fisica “E. R. Caianiello”- Universit`a di Salerno, I-84100 Salerno, Italy Cross-Correlated Materials Research Group (CMRG), ASI, RIKEN, Wako 351-0198, Japan RRC “Kurchatov Institute” - 123182 - Moscow - Russia Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan
We investigate the interplay between strong electron correlations and charge-lattice interactionin cuprates. The coupling between half breathing bond stretching phonons and doped holes in the t - t ′ - J model is studied by limited phonon basis exact diagonalization method. Nonadiabatic electron-phonon interaction leads to the splitting of the phonon spectral function at half-way to the zoneboundary at ~q s = { ( ± π/ , , (0 , ± π/ } and to low energy kink feature in the electron dispersion, inagreement with experimental observations. Another kink due to strong electron correlation effectsis observed at higher energy, depending on the strength of the charge-lattice coupling. PACS numbers: 71.10.Fd, 63.20.kk, 63.20.-e
There is a growing confidence that strong electron-phonon interaction (EPI) manifests itself both in vibra-tional [1] and electronic [2] spectra of cuprates. The mostpuzzling feature of the cuprate phonon spectra is theanomaly of the half breathing bond stretching (HBBS)phonon occurring at half-way to the Brillouin zone (BZ)boundary in the [100]-direction, while the most debatingfeature of the electronic spectra are the kinks observedin Angle Resolved Photoemission Spectra (ARPES).It was recently realized in the experimental commu-nity that it is highly important to measure both phononsspectra and ARPES on the same sample just to verifypossible link between HBBS phonons and ARPES [3].The results of the above studies support close connec-tion between HBBS phonon anomaly and lowest energykink in ARPES [3] and, thus, a model describing bothanomalies within the same approach is strongly needed.In the present Letter we study the low density limit(one hole on 4 × t - t ′ - J modelwhere holes are coupled to HBBS phonons [4]. In or-der to calculate the phonon spectral function (PSF) andhole spectral function (HSF), we generalize a recentlyintroduced approach [5], based on limited phonon basisexact diagonalization (LPBED), without adopting self-consistent Born (SCBA) and spin wave approximations[6, 7]. This method treats the non-adiabatic effects ofthe quantum phonon very effectively without approxi-mations for the magnetic degrees of freedom. Because ofexponential growth of the basis with size of the system,the 4 × √ × √
10 system [8], so that it is pos-sible to resolve fine structure of the PSF and HSF. Forthe first time we are able to reveal the shape of the PSFwhile previous studies [9–11] were restricted to at mostthe second moment of the response.We show that EPI can lead to the splitting of the PSFat half-way to the BZ boundary in the [100] direction. We demonstrate that the splitting can be easily smeared outby a very small broadening and, thus, the mysterious eva-sive behavior of double-peak structure can be attributedto tiny variations of chemical composition, crystal qualityand/or experimental setup. We argue that the splittingis a rather general phenomenon arising when the phononbranch interacts with a rather soft electronic excitationand we show that the HBBS phonon anomaly is linked tothe lowest energy kink observed in the ARPES. Finallywe emphasize that the same model supports the spec-tral kink at higher energy referred to colloquially as thewaterfall [12].McQueeney et al. [13] observed anomalous lineshape ofHBBS phonons around q = ( π/ , ,
0) and reported itsstrong temperature dependence. Subsequent studies con-firmed that the HBBS phonon anomaly is due to the cou-pling of the HBBS phonons to the doped carriers. Indeed,the anomaly is absent in undoped compounds [1] and thehardening of the phonon spectra with heating [13, 14] ex-cludes such sources of anomaly as anharmonicity or struc-tural inhomogeneity. It is also agreed that HBBS phononanomaly is located around q = ( π/ , ,
0) at any doping.On the other hand, there is a controversy on the rela-tion between the HBBS phonon anomaly and the lowestenergy kink in the ARPES. This relation is often deniedalthough the measurements of the phonon spectra andARPES on the same sample of Bi Sr . La . Cu O x suggest that the softening of the HBBS phonon modematches the energy and momentum of this kink [3]. An-other much debated question is the interpretation of thePSF structure at HBBS phonon anomaly. Actually thephonon peaks at ~q s are poorly defined and two scenar-ios are possible. According to the single-branch inter-pretation (SBI) there is softening and a very large in-trinsic linewidth of a single peak near q = ( π/ , , . Ba . CuO led authors toconclusion about relevance of DBI [15]. Likewise, mea-surement of HBBS in YBa Cu O x was interpreted inthe framework of DBI [18]. Recently a ”contaminationhypothesis” [19] has been suggested, claiming that theSBI is correct because the inelastic neutron scatteringmeasurement picks up the intensity from the k-vicinity( π/ , k,
0) of q = ( π/ , ,
0) point and, thus, the ”nor-mal” component comes from transverse phonons. How-ever, the above reasoning contradicts the results obtainedon La − x Ba x CuO δ at x = 0 . ± .
01 [20] by in-elastic x-ray scattering technique which has higher mo-mentum resolution than that in neutron scattering mea-surements. ”Contamination hypothesis” implies that the”normal” component must disappear by improving mo-mentum resolution. Instead, the results of Ref. [20] sup-port DBI. One can guess that the SBI vs DBI dependson the chemical composition [19], Ba-doped [15, 20] vsSr-doped [14, 17] compound. Besides, since the softeningis very sensitive to the broadening caused by decay chan-nels and/or experimental resolution, SBI vs DBI mustbe strongly dependent on the compound, sample quality,and experimental setup.We will show that many of the questions discussedabove can be understood within the t - t ′ - J model includ-ing the coupling with HBBS. On the contrary densityfunctional theory calculations [21, 22] do not predict anyHBBS phonon anomaly [23]. Other approaches, espe-cially those associating the anomaly with stripes, havedifficulty with the position of HBBS phonon anomaly.Scenario suggested in Ref. [24] relates the HBBS soften-ing to the Kohn anomaly at double Fermi momentum2 k F along the Fermi surface of stripes [24]. Here, in con-trast to experiment, the softening must be θ -independent.In a different theoretical proposal, HBBS anomaly is as-sociated with stripe mediated collective charge excita-tions [25] or incommensurate low energy spin-fluctuations[26]. Within these scenarios, in contrast with experi-ment [14, 15, 20, 27, 28], wave vector of phonon anomalystrongly depends on doping level. Finally we note thatphonon softening [9, 10], broadening [9], and correct po-sition of the anomaly [11] have already been qualitativelyexplained by the coupling between phonons and densityresponse of the t - J model although none of those studiesconsidered the shape of the phonon spectral function.The Hamiltonian of the t - t ′ - J -Holstein model in 2D isthe sum of the electronic part and hole-phonon couplingHamiltonian H tt ′ J = − t X i,δ,σ c † i + δ,σ c i,σ − t ′ X i,δ ′ ,σ c † i + δ ′ ,σ c i,σ + J X i,δ S i + δ S i − J X i,δ n i + δ n i , (1) H ′ = ω X q,µ a † q,µ a q,µ + X i,q,µ (cid:16) M q,µ e i~q · ~R i (1 − n i ) a q,µ + H.c. (cid:17) . Here J is the exchange interaction constant of the spin-spin interaction, t and t ′ are hopping amplitudes to near-est and next nearest neighbors. At site i (Cu atoms), S i is the -spin operator, c i,σ is the fermionic operatorsin the space without double occupancy, and n i is thenumber operator. a q,µ is the phonon annihilation oper-ator with momentum q and µ = x or y indicates thelongitudinal polarization of the oxygen vibrations alongthe direction of the nearest neighbor Cu atoms (Oxy-gen atoms are located at ~R i + a/ µ , a being the lat-tice parameter). M q,µ is the matrix element of the EPI: M q,µ = gω / √ N i sin( q µ /
2) where N is the number oflattice sites and ω is the frequency of dispersionless op-tical phonon. The strength of EPI is characterized bydimensionless coupling constant λ = P ~q,µ | M q,µ | / ω t .We chose parameters which correspond to hole dopedcuprates: J = 0 . t , t ′ = − . t , ω = 0 . t , and set t = a = ~ = 1. The PSF is expressed as D ( ~q, ω + iη ) = − π ℑ D − ( ~q, ω + iη ) − Σ( ~q, ω + iη ) , (2)where the self-energy Σ( ~q, ω + iη ) is given byΣ( ~q, ω + iη ) = | M q,x | Π( ~q, ω )(1 + | M q,x | Π( ~q, ω ) /D − ( ~q, ω + iη )) . (3)Here D ( ~q, ω + iη ) is the bare phonon Green function andΠ( ~q, ω ) = P ( ~q, ω + iη ) + P ( ~q, − ω − iη ) is the polarizationinsertion with P ( ~q, ω + iη ) = h ψ | O † ω + iη − H + E O | ψ i . (4)We choose the ground state (GS) | ψ i as a linear super-position with equal weights of the 4 degenerate statescorresponding to ~k = ( ± π , ± π ) with energy E , η isa broadening factor that shifts the poles of D ( ~q, ω ) inthe complex plane, and O = P i e i~q · ~R i (1 − n i ). The in-crease of broadening factor η has the physical meaningof phonon damping or limited experimental resolution.The ground state | ψ GS i and the function P ( ~q, ω + iη )are obtained by modified [29] and standard Lanczosmethods, respectively, within the LPBED method [5] in-troduced for the t - t ′ - J -Holstein model.In Fig. 1 we plot the PSF D ( ~q, ω ) for different wavevec-tors along the (1 ,
0) direction in the BZ. For the cho-sen values of the model parameters the system under-goes, in agreement with [30], a crossover towards strongEPI regime for λ c ≃ .
5. The anomalous softening ofthe phonon mode at ~q s = ( π/ ,
0) is already observedat moderate values λ < λ c of the hole-phonon coupling( λ = 0 . η = 0 .
08 the phonon peak softens and broadens at ~q s (Fig. 1a), supporting, thus, SBI. On the other hand, H a L
0. 0.1 0.20.1234 Ω D H b L
0. 0.1 0.20.4812 Ω D FIG. 1: (Color online) (a) The PSF for λ = 0 . ~q ( ~q = (0 ,
0) (dashed green line), ~q = ( π/ , ~q = ( π,
0) (dotted blue line)) with η = 0 .
08; (b) the PSF with a smaller broadening factor η , η = 0 .
03. The vertical lines indicate the bare phonon fre-quency ω . H L L H H L Ω
0. 0.1 0.2 0.3 0.40.0.40.81.21.6 01 Ω A D (cid:144) k GS H -Π , 0 L H
0, 0 LH Π LH -Π , Π L q s k f FIG. 2: (Color online) HSF A ( ~k, ω ), for λ = 0 . ~k ( ~k = ( π/ , π/
2) (dashed green line), ~k = (0 , π/
2) (solid blue line)) with η = 0 .
03. The energy ismeasured with respect to the GS energy. The phonon Greenfunction at ~q = ( π/ ,
0) (with the same value of η ) is plottedtoo for comparison (red dotted line). In the inset sketch ofone of transitions involved in the PSF calculation. reducing the broadening factor to η = 0 .
03 two-peakstructure turns out(Fig. 1b), that is in agreement withthe experimental observations reported in Ref.[15, 20].To clarify the physical nature of the splitting, we studythe hole spectral weight function A ( ~k, ω ) at the wavevec-tors ~k f that are reached starting from GS wavenumber at( ± π/ , ± π/
2) through phonon momentum ~q s = ( π/ , ~k =( − π/ , π/
2) we get ~k f = (0 , π/
2) after absorbing (emit-ting) a phonon with momentum − ~q s ( ~q s ). The lowest en-ergy peak in D ( ~q s , ω ) (dotted red curve in Fig. 2) is closeto the lowest energy peak of A ( ~k f , ω ) (solid blue curvein Fig. 2). Measuring energy from the GS, denoted byvertical line in Fig. 2, one can see that the lowest peaksin phonon and hole spectral functions are softer than thephonon energy ω =0.15t. We note that the significantrenormalization of the hole spectral weight, with respectto GS one, indicates strong coupling between phonon at ~q s and hole at ~k f . The high energy peak in D ( ~q s , ω ) is located at energy close to ω , and no peak at the sameenergy is found in A ( ~k, ω ). It is shown below that it isdue to the phononic nature of the high energy resonanceof the PSF.To give a simple explanation of the above scenarioone has to realize that the excited electronic state | ψ i ,π/ | i with the momentum (0 , π/
2) and withoutphonons is linked by the matrix elements of EPI with agroup of 8 degenerate states | ψ i ± π/ , ± π/ a † q,µ | i wherethe electronic subsystem is in the GS and one phononis excited. The momentum conservation ( ± π/ , ± π/
2) + ~q = (0 , π/
2) determines the phonon momenta ~q . Theenergies of the electronic subsystem at correspondingmomenta are ǫ (0 , π/
2) and ǫ ( π/ , π/ ǫ (0 , π/ ≈ ǫ ( π/ , π/
2) + ω is satisfied. Hence, even small matrixelements induce strong effects both in electronic and vi-brational subsystems. Analytic diagonalization of thisdegenerate 9 × L has energy below ω and has components on both one-phonon and zero-phonon states. This state correspondsto the peaks of electronic A ( ~k = (0 , π/ , ω ) and bosonic D ( ~q = ( π/ , , ω ) spectral functions at ω < . L in Fig. 2). In the highest energyrange, (see the area around letter H in in Fig. 2), bothspectral functions collect contributions at ω ≥ .
2. How-ever, the H -state is of predominantly electronic originwith a large peak in the hole spectral function while itgenerates a weak structure in the PSF, hardly observ-able in the numerical data. The last 7 degenerate levelswith energy ǫ ( π/ , π/
2) + ω have no projection on thevacuum boson state: this explains why the hole spectralweight function does not show any peak at this energy, ata distance ω from GS energy. The energies of all levelsof the above simple analytic solution are in qualitativeagreement with that provided by the numeric LPBEDmethod in Fig. 2. So, we conclude that the doubling ofthe phonon peak is due to coupling between holes and lat-tice, that lifts a degeneracy and produces one additionalstate with energy less than ω .Generically, the EPI involving electronic states withenergies considerably larger than the phonon frequencydoes not lead to any splitting since the adiabatic ap-proximation is valid and electron produces only a renor-malization of the adiabatic potential [31]. On the otherhand, when the electronic excitation is soft and its energyis comparable with phonon frequency, the nonadiabaticcorrections play a crucial role [32] and one can observeexotic spectral functions as it is seen in Fig. 1b and inexperiments on cuprates.The clear connection, established by the above consid-erations, between phonon at ~q s and hole at ~k f encouragedus to search for an interplay between the HBBS phononanomaly and the lowest energy kink in the dispersion ofquasiparticles in cuprates [33]. In Fig.3a we plot hole dis- H a L Λ= Λ= - k F E H k L - - - - - - k F E H k L - - - - H b L O H L FullO H L Dashed Λ= - - - - - - k F E H k L FIG. 3: (Color online) (a) Energy vs ( ~k − ~k F ) /π dispersionrelations along nodal direction from ~k F to Γ point at λ = 0 . λ = 0 (dashed). Arrows mark kink positions. (b)Dispersion relation as a function of oxygen isotope exchange( O (solid) and O (dashed)) at λ = 0 . persion derived from momentum distribution curves, i.e.we find the ~k for which the hole Green function A ( ~k, ω )has maximum at fixed ω [34]. The energy, measuredfrom GS, is plotted versus the wavenumber (cid:16) ~k − ~k F (cid:17) ,where ~k F is the momentum corresponding to the mini-mum of the hole dispersion relation. We find that, when E k − E k F is about ω = 0 .
15, the curve exhibits a slopechange related to the coupling between the bare phononenergy and the electronic band in the bare t - t ′ - J model.This kink does not appear in absence of the hole-phononcoupling. On the other hand, we obtain another kink athigher energy (see Fig.3a) [12]. This distinctive featureis related to the strong electron correlations since it isobserved also in the bare t - t ′ - J model. At λ = 0 the kinkis located around the exchange interaction energy J , inagreement with results by Chakraborty et al. [35] withinthe Hubbard model. Furthermore, the kink is shiftedat higher energy by increasing the hole-phonon coupling.We also investigated the effect of the oxygen isotope sub-stitution [5] on ARPES. The results (Fig.3b) show a neg-ligible (below 0 . t ) shift upon oxygen isotope exchange,that is in agreement with recent experimental observa-tions by Douglas et al. [36]. All these data point outthat the scenario based on the interplay between strongelectron correlations and hole-phonon interaction is ableto capture many physical distinctive features of high tem-perature superconductors.In conclusion, we showed that the EPI, in the presenceof strong correlations, can lead to the splitting of thephonon spectral function at half-way to the BZ bound-ary in the [100] direction. We demonstrated that thesplitting can be easily smeared out by very small broad-ening of the eigenstates. The same physical mechanismcan explain both HBBS phonon anomaly and lowest en-ergy kink in the ARPES. Finally we found that the iso-tope effect on ARPES is negligible in accordance withexperiment. These results support the claim that strong electron correlations and charge lattice interaction arecrucial in understanding cuprate experimental features.Work supported by RFBR 07-02-0067a (A.S.M.), andGrant- in-Aids No. 15104006, No. 16076205, No.17105002, No. 19048015, and NAREGI Japan (N.N.);G.D.F., V.C. and C.A.P. received financial support fromMIUR-PRIN 2007 under Prot. No. 2207FW3MJX003. [1] L. Pintschovius, Phys. Status Solidi b , 30 (2005)[2] O. Gunnarsson and R¨osch, J. Phys.: Condens. Matter , 043201 (2008)[3] J. Graf et al., Phys. Rev. Lett. , 227002 (2008)[4] N. Bulut and D. J. Scalapino, Phys. Rev. B , 14971(1996); S. Ishihara and N. Nagaosa, Phys. Rev. B ,144520 (2004)[5] G. De Filippis et al., Phys. Rev. B , 195104 (2009)[6] C. L. Kane et al., Phys. Rev. B , 6880 (1989)[7] Z. Liu and E. Manousakis, Phys. Rev. B , 2425 (1992)[8] B. Bauml et al., Phys. Rev. B , 3663 (1998)[9] G. Khaliullin and P. Horsch, Phys. Rev. B , R9600(1996); Physica C , 1751 (1997)[10] O. R¨osch and O. Gunnarsson, Phys. Rev. Lett. ,146403 (2004); ibid , 237001 (2004)[11] P. Horsch et al., Physica C , 117 (2000)[12] J. Graf et al., Phys. Rev. Lett. , 067004 (2007)[13] R. J. McQueeney et al., Phys. Rev. Lett. , 628 (1999)[14] D. Reznik et al., J. Low. Temp. Phys. , 353 (2007)[15] D. Reznik et al., Nature , 1170 (2006)[16] L. Pintschovius et al., Phys. Rev. B , 174514 (2006)[17] L. Pintschovius and M. Braden, Phys. Rev. B , R15039(1999)[18] F. Stercel et al., Phys. Rev. B , 014502 (2008)[19] D. Reznik, Adv. Cond. Matt. Phys. , 523549 (2010)[20] M. d’Astuto et al., Phys. Rev. B , 140511(R) (2008)[21] F. Giustino et al., Nature , 975 (2008)[22] K. P. Bohnen, R. Heid and M. Krauss, Europhys. Lett. , 104 (2003)[23] D. Reznik et al., Nature , E6 (2008)[24] S.I. Mukhin et al., Phys. Rev. B , 174521 (2007); R.Citro et al., Eur. Phys. J. B , 179–185 (2008)[25] E. Kaneshita et al., Phys. Rev. Lett. , 115501 (2002)[26] S. A. Kivelson et al., Rev. Mod. Phys. , 1201 (2003)[27] J. Graf et al., Phys. Rev. B , 172507 (2007)[28] H. Uchiyama et al., Phys. Rev. Lett. , 197005 (2004)[29] E. Dagotto and A. Moreo, Phys. Rev. D , 865 (1985)[30] A. S. Mishchenko and N. Nagaosa, Phys. Rev. Lett. ,036402 (2004); Phys. Rev. B , 092502 (2006); A. S.Mishchenko et al, Phys. Rev. Lett. , 166401 (2008)[31] M. Born and R. Oppenheimer, Ann. d. Phys. , 457(1927)[32] K. A. Kikoin and A. S. Mishchenko, Zh. Eksp. Teor. Fiz. , 3810 (1993) [JETP , 828 (1993)][33] A. Lanzara et al., Nature , 510 (2001)[34] The full ~k dependence is facilitated by twisted boundaryconditions (see T. Xiang and J. M. Wheatley, Phys. Rev.B , R12653 (1996))[35] S. Chakraborty et al., Phys. Rev. B , 212504 (2008)[36] J. F. Douglas et al., Nature446