Bipolaron in the t-J model coupled to longitudinal and transverse quantum lattice vibrations
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Bipolaron in the t − J model coupled to longitudinal and transverse quantum latticevibrations L. Vidmar, J. Bonˇca,
2, 1
S. Maekawa,
3, 4 and T. Tohyama J. Stefan Institute, 1000 Ljubljana, Slovenia Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan CREST, Japan Science and Technology Agency (JST), Tokyo 102-0075, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: November 14, 2018)We explore the influence of two different polarizations of quantum oxygen vibrations on the spacialsymmetry of the bound magnetic bipolaron in the context of the t − J model by using exact diag-onalization within a limited functional space. Linear as well as quadratic electron phonon couplingto transverse polarization stabilize d − wave symmetry. The existence of a magnetic background isessential for the formation of a d − wave bipolaron state. With increasing linear electron phononcoupling to longitudinal polarization the symmetry of a d -wave bipolaron state changes to a p -wave.Bipolaron develops a large anisotropic effective mass. PACS numbers: 71.27.+a,71.38.Mx, 71.38.-k,74.20.Rp
Soon after the discovery of high- T c superconductivitythe quest for the pairing mechanism focused on magneticfluctuations due to a broadly accepted conjecture thatphonon mechanism alone is not strong enough to pro-duce high transition temperatures as observed in high- T c compounds. Recently, a growing evidence is emerg-ing in favor of the significance of lattice degrees of free-dom in high- T c compounds [1, 2, 3]. The interplay be-tween strong correlations and lattice degrees of freedom[4] seems to be responsible for many unusual propertiesof cuprates in the low doping regime, such as kinks [1],stripes [5], and waterfall [6, 7] structures.A long-standning objection against phonon-basedmechanism for high- T c superconductivity is based on awidely accepted notion that coupling to phonon degreesof freedom is predominantly consistent with s − wave pair-ing, not characteristic for cuprates. Despite recent dis-covery that weak EP coupling to acoustic phonons in thepresence of large on-site Coulomb interaction leads to d − wave pairing [8], the role of short-wavelength oxygenoscillations on the symmetry of the paired state remainsa challenging open problem.Investigation of correlated models coupled to phononswere based on exact diagonalization (ED) calculationson small lattice systems [9, 10, 11, 12], slave-boson ap-proaches [13, 14, 15], dynamical mean-field calculations[16, 17, 18], coherent states Lanczos method [19], andquantum Monte Carlo methods (QMC) [12, 20]. InRef. [9] authors present a detailed study of the influ-ence of EP coupling to a single phonon mode on for-mation of inhomogeneous charge structures in the t − J model. They show that half-breathing mode stabilizesa stripe phase. In contrast, using slave-boson approachauthors of Ref. [13] suggest, that half-breathing mode en-hances d − wave pairing. They furthermore underlinethe importance of off-diagonal EP coupling modulating the hopping and the spin-exchange terms. In contrast,authors of Ref. [21] find that diagonal EP terms exceedoff-diagonal ones by nearly two orders of magnitude.ED calculations of the t − J model show that the d − wave symmetry of a bipolaron [22, 23, 24, 25, 26]is not robust against addition of longer range hoppingterms [27] while recent QMC calculations of the Hub-bard model yield T c far below those of cuprates [28].There seems to be a need to uncover additional mech-anism that would help stabilize the d − wave symmetry ofa bound bipolaron state. In this Letter we show that EPcoupling to a transverse polarization (TP) of oxygen vi-bration provides an important mechanism that stabilizesthe d − wave symmetry of a bound hole pair with a smalleffective mass.We solve a system of two holes in the t − J model de-fined on an infinite two-dimensional lattice by extendingthe method for a single hole based on exact diagonaliza-tion within a limited functional space [7, 29]. We intro-duce diagonal EP coupling to either TP of oxygen (O)vibration relevant for the description of buckling modesor longitudinal polarization (LP) of O vibration relevantfor description of bond-streching modes. We investigatethe following Hamiltonian H = − t X h i , j i ,s (˜ c † i ,s ˜ c j ,s + H . c . ) + J X h i , j i ( S i S j − n i n j )+ g X i , δ ( n h i − n h i + δ )( a † i + δ / + a i + δ / )+ q β X i , δ ( n h i + n h i + δ )( a † i + δ / + a i + δ / ) β + ω X i + δ a † i + δ / a i + δ / , (1)where ˜ c i ,s = c i ,s (1 − n i , − s ) is a projected fermion opera-tor, t represents nearest neighbor overlap integral, thesum h i , j i runs over pairs of nearest neighbors, a i arephonon annihilation operators and n i = P s n i ,s . Thethird term represents linear EP coupling to LP of O vi-bration with respect to Cu-O-Cu bond, see also Fig. 1(a).Fourth term is chosen either linear ( β = 1) or quadratic( β = 2) in O displacement, describing TP of O vibration,Fig. 1(a). β = 2 is chosen to describe the CuO plane withno pre-buckling of O positions. Sums over δ in the lattertwo terms run over two orthogonal nearest neighbor Cupositions. Lattice vibrations on O sites are independent- we do not predispose any particular phonon mode withthe exception of limiting our calculation to either TP orLP of O oscillation. In treating quantum phonons wefollow well established approach of Ref. [30].The construction of the functional space starts from aN´eel state with two holes located on neighboring Cu sitesand with zero phonon quanta. Such a state representsa parent state of a translationally invariant state witha given momentum k . In the case of a high symmetrypoint k = (0 ,
0) the parent state can be chosen to have d − , s − , or p − wave symmetry as for the case of d − and s − shown in Fig. 1(b). The starting state is writtenas | φ (0) i a = P γ ( − M a ( γ ) c c γ | Neel; 0 i , where sum runsover four nearest neighbors in the case of d − and s − wavesymmetry and over two in the case of p x ( y ) -wave while M a ( γ ), a ∈ { d, s, p } sets the appropriate sign.We generate new parent states by apply-ing the generator of states n | φ ( n h ) l i a o = (cid:16) H kin + ˜ H J + H ph (cid:17) n h | φ (0) i a ; n h = 1 , . . . , N h where H kin represent the first term in Eq. 1, ˜ H J denotes apart of the second term in Eq. 1 which is only applied toerase spin flips that were generated through succeedingapplication of H kin , as for a particular case depicted inFig. 1(c). H ph represents either third or fourth term inEq. 1. This procedure generates exponentially growingbasis of states, consisting of different shapes of strings inthe vicinity of the hole with maximum lengths given by N h as well as phonon quanta created along paths of bothholes. Identical basis functions, generated by differentprocesses, are chosen only once. We have used N h = 8that lead to N st = 13 × states. Full Hamiltonian inEq. 1 is diagonalized within this limited functional spacetaking explicitly into account translational symmetry.In Fig. 2 we present the energy difference E p − E d between the lowest p − and the d − wave state for twodifferent values of J/t as a function of q β /t and g/t forthe case of TP and LP respectively. At J/t = 0 . q , = g = 0 [22, 23] and de-generate p − wave ground state is found, E p − E d < q β /t and g/t leadsto rather surprisingly distinct results. In both cases in-creasing EP coupling leads to a formation of a bipolaron,as also evident from Figs. 3(e) and (f) and the discus-sion later in the text. While coupling to TP leads to a H (c) (b) HH H (a) kin q kin ~ J ++− −− Figure 1: (Color online) (a) Schematic representation of LPand TP vibrations of O atom (middle) with respect to Cu-O-Cu bond, (b) schematic representation of a d x − y − wave( s − wave) top (bottom) signs two-hole starting wavefunction.Fermion sign convention places the first hole depicted with thefull circle to the left-most, if the pair is vertical, then bottom-most position. Only Cu sites are presented with small dots in(b); (c) schematic representation of succeeding applicationsof different off-diagonal parts of Hamiltonian in Eq. 1, for n h = 4, starting from a single hole-pair in the N´eel state withzero phonon quanta. Dots represent Cu and O atoms, holesare denoted by open circles, crosses represent spin-flips (over-turned spins with respect to the original N´eel configurationof spins, localized on Cu sites), vertical arrows indicate andpoint to the numbers of excited TP phonon quanta. formation of a bound state with the d − wave symmetry,coupling to LP in contrast favors a bound state with the p − wave symmetry. This effect is even more pronouncedat larger value of J/t = 0 . q , = g = 0 a boundmagnetic bipolaron is already formed [22, 23, 24] with a d − wave symmetry. By increasing q /t , d − wave symme-try is stabilized. Increasing linear EP coupling q /t leadsto an initial increase of E p − E d followed by a decrease, E p − E d → q /t & .
75 due to a crossover to astrong EP coupling regime, Fig. 2(a). In contrast, linearEP coupling to LP drives even a bound d − wave bipo-laron state at J/t = 0 . g = 0 to a bound state witha p − wave symmetry at g/t ∼ .
61, see Fig. 2(b).Effective bipolaron mass m αα = t (cid:2) ∂ E ( k ) /∂ k ∂ k (cid:3) − αα ,computed in its eigen-directions, presented in Figs. 2(c)and (d), is isotropic in the case of d − wave symmetry andanisotropic, with the anisotropy ratio m yy /m xx ∼ − p − wave state. At J/t = 0 . m αα further-more shows only a weak increase with q /t , see Fig. 2(c).Even more surprising is the decrease of the effective massat J/t = 0 . d − wave bipolaron, i.e. for q /t & .
5. Note, that the nonanalytic behaviorof m xx is a consequence of the symmetry change from p − to d − state at q /t ∼ . m xx at J/t = 0 . p − wave state, around g/t ∼ .
57, see Fig. 2(d). Asthe system enters p − wave state the mass again becomesanisotropic.In Fig. 3 we present the probability of find-ing a hole-pair at a distance of r: P ( r ) = Transverse -0.100.10.20.3 E p - E d J/t=0.1J/t=0.4
Longitudinal -0.0500.050.10.15
J/t=0.1J/t=0.4 q β / t m αα g/t ω /t= (a) (b)(c) (d) J/t αα β=2β=2β=1 Figure 2: (Color online) E p − E d at ω /t = 0 .
2, and twodifferent strengths of
J/t vs. q β /t in the case of EP couplingto TP (a) and vs. g/t in the case of linear EP coupling to LP(b). Effective masses m αα vs. q /t in (c) and vs. g/t in (d).The ground state wave-vector is k = 0 except in the regime g/t & .
57 where k = ( π, h P h i = j i n h i n h j δ [ | i − j | − r ] i / h P h i = j i n h i n h j i , and averagehole distance h d i = P r rP ( r ) . We first focus on the effectof EP coupling to TP, see Figs. 3(a,c,e). At
J/t = 0 . q /t . .
5, and q /t . . h d i remains finite due to alimited Hilbert space where the maximal inter-hole dis-tance is given by l max = N h +1 = 9. Increasing N h wouldlead to further increase of h d i in this regime as well asto further spread of P ( r ) towards larger r , see Fig. 3(c)for J/t = 0 . q , = 0. In this range of parameterswe observe no exponential decay of P ( r ), see Fig. 3(e).In contrast, in the regime of a bound bipolaron, i.e. for J/t = 0 . q /t & . q /t & .
22) as well as at
J/t = 0 . h d i and P ( r ) do not change much with furtherincreasing N h and exponential decay is clearly observedin Fig. 3(e). Our tests performed on smaller systems( N h = 4 and 6) reaffirm that results in the regime of abound bipolaron have indeed converged close to a ther-modynamic limit. Good agreement of P ( r ) at J/t = 0 . q , = g = 0 is found with ED calculation on 32-sitescluster, Ref. [22, 24]. Structure of a bound bipolaron, re-vealed by P ( r ) at J/t = 0 . q , = 0, is remarkablysimilar to that computed at J/t = 0 . q /t = 1 . β = 2, Fig. 3(c). Both, quadratic and linear EPcoupling to TP lead to a formation of a bound bipolaronwith the d − wave symmetry even in the case of smallexchange interaction J/t = 0 . d − wave state in the absence of a magneticbackground, we have solved a problem with two spinlessparticles quadratically coupled to TP or linearly to LPusing topology of a Cu-O plane. By increasing q /t or g/t we obtain in both cases a bipolaron with a p − wavesymmetry, Figs. 3(a) and (b). We thus emphasize an im-portant conclusion: EP coupling to TP stabilizes d − wavesymmetry of a hole-pair, however, the existence of a mag- netic background as found in the t − J model seems tobe essential precondition for the formation of a d − wavestate. q β / t < d > J/t=0.1J/t=0.4SL P ( r ) g/t J/t=0.1J/t=0.4SL
SL 1 0.1 0 0.4 0 0.1 0.7 0.4 0.7 r -5 -4 -3 -2 -1 r -5 -4 -3 -2 -1 ω /t= (a) (b)(c) (d) Transverse Longitudinal
J/t q β / t J/t g/t p- wave d- wave d- wave p- wave (e) (f) β=1β=2β=2β=1 Figure 3: (Color online) Average hole distance h d i computedat ω /t = 0 . J/t vs. q β /t ; β = 1 , g/t in (b). Results for the system of spinlessparticles (SL) are shown with dotted lines in (a) and (b); P ( r )at chosen values of q , /t in (c) and g/t in (d). P ( r ) is nor-malized to P r P ( r ) = 1; corresponding exponential scalingsof P ( r ) for r & Turning to linear EP coupling to LP phonons we startfrom
J/t = 0 .
1. A p − state of two separate holes changesto a bound bipolaron state with increasing of g/t at about g/t ∼ .
5, see Fig. 3(b). Transition to a bound state isnot sharp as in Fig. 3(a) since there is no change of a sym-metry. Nevertheless, in the N h → ∞ limit, we anticipatea sharp transition from an unbound to a bound bipolaronstate. Stabilization of a p − wave state under the influenceof LP is even more evident when starting from a d − wavebound bipolaron state at J/t = 0 .
4. With increasing g/t , a change of symmetry occurs around g/t ∼ . d − to a p − wave state, see Figs. 3(b,d,f). A de-tailed inspection of a bound p − wave state in the regime g/t & .
61 reveals unusually simple structure where theprobability of finding holes at a distance r = 2 is morethan 0.6. This is in a sharp contrast with the structureof a d − wave bound state where P ( r = 2) < . P ( r ) is at r = √
2, compare alsoFigs. 4(a) and (b). Our calculations of hopping termmodulated by LP phonons, as suggested in Ref. [13], aswell leads to stabilization of a bipolaron with a p -wavesymmetry, nonetheless, with a distinct spacial structure.From Fig. 3 is as well evident that linear couplingto TP leads a stronger attraction between holes thancoupling to LP as seen from Figs. 3(e) and (f) thatshow steeper decay of P ( r ) for TP at comparable values q /t = g/t = 0 .
7. At small
J/t = 0 . λ q = q / ω t ∼ . λ g = g / ω t ∼ .
31 in theLP case. This result suggests that even a small pre-buckling within the CuO plane, that generates non-zerolinear EP coupling term to TP, may have a pronouncedeffect on the attraction between magnetic polarons. (b)(a)
Figure 4: (Color online) (a) C ( r ) and N ( r ) at J/t = 0 . q /t = 1 .
0, and β = 2 ( d − wave). Radii of circles, representing C ( r ), located on Cu-sites are, proportional to the probabilityof finding a hole-pair at a distance of r . Empty circle is locatedat r = 0. Sides of squares, representing N ( r ), located on O-sites, are proportional to average numbers of phonon quantaat a distance r from a hole; (b) C ( r ) and X ( r ) at J/t = 0 . g/t = 0 . p − wave). Lengths of arrows, representing X ( r ),are proportional to displacements of O atoms along Cu-Cubonds relative to the hole position. To investigate in more detail the nature ofthe magnetic-lattice bipolaron, we simultaneouslypresent two correlation functions: hole-hole density C ( r ) = h P i n h i n h i + r i , and hole-phonon number N ( r ) = h P i n h i a † i + r a i + r i , in Fig. 4(a), for the case of a bound d − wave bipolaron state. Largest phonon numbers arefound at the closest possible distance from the hole. Thestructure of C ( r ) is consistent with d x − y symmetry de-spite its largest value at a distance of r = √
2, as alreadypointed out in Refs. [23, 24]. In Fig. 4(b) we show C ( r )and X ( r ) = h P i n h i (cid:16) a † i + r + a i + r (cid:17) i , measuring displace-ments along Cu-Cu bonds relative to the position of thehole, for the case of a p − wave ground state. Both cor-relations display the unidirectional spacial distribution.Correlation functions, presented in Figs. 4(a) and (b),show detectable values only up to r .
3, despite maximaldistance l m ax = N h + 1 = 9, allowed in our calculations. In conclusion significantly different bipolaron statesare found when EP coupling to either TP or LP isswitched on. Linear as well as quadratic EP coupling toTP stabilizes a d − wave bipolaron state. The magneticbackground is essential for the formation of a d − wavebipolaron. The effective bipolaron mass remains small inthe case of quadratic EP coupling despite lattice drivenbinding of the bipolaron.In contrast, increasing linear EP coupling to LPphonons changes the symmetry of a bound bipolaronfrom a d − wave state at zero EP coupling to a p − wavestate followed by a substantial change of the density-density correlation function. Since this state also hasa large and anisotropic effective mass and unidirectionalspacial distribution we may speculate, that in a system with finite doping linear EP coupling to LP of O vibra-tion would lead to formation of stripe states. This findingis consistent with inelastic neutron experiments show-ing strong coupling to the bond-streching mode in andaround the vicinity of the stripe phase in copper oxidesuperconductors [5].J.B. acknowledges financial support of the SRA undergrant P1-0044. S.M. and T.T. acknowledge the finan-cial support of the Next Generation Super ComputingProject of Nanoscience Program, CREST, and Grant-in-Aid for Scientific Research from MEXT. This work wasalso supported by JPSJ and MHEST under the Japan-Slovenia Research Cooperative Program. [1] A. Lanzara et al. , Nature , 510 (2001).[2] A. Alexandrov and N. F. Mott, Rep. Prog. Phys. ,1197 (1994).[3] D. M. Newns and C. C. Tsuei, Nature Physics , 184(2007).[4] O. Gunnarsson and O. Rosch, Journal of Physics: Con-densed Matter , 043201 (22pp) (2008).[5] D. Reznik et al. , Nature , 1170 (2006).[6] F. Ronning et al. , Phys. Rev. B , 094518 (2005).[7] J. Bonˇca et al. , Phys. Rev. B , 054519 (2008).[8] A. S. Alexandrov, Phys. Rev. B , 094502 (2008).[9] J. Riera and A. Moreo, Phys. Rev. B , 014518 (2006).[10] T. Sakai, D. Poilblanc, and D. J. Scalapino, Phys. Rev.B , 8445 (1997).[11] P. Prelovˇsek, R. Zeyher, and P. Horsch, Phys. Rev. Lett. , 086402 (2006).[12] M. Hohenadler et al. , Phys. Rev. B , 245111 (2005).[13] S. Ishihara and N. Nagaosa, Physi. Rev. B , 144520(2004).[14] B. Kyung et al. , Phys. Rev. B , 13167 (1996).[15] A. Ramˇsak, P. Horsch, and P. Fulde, Phys. Rev. B ,14305 (1992).[16] G. Sangiovanni et al. , Phys. Rev. Lett. , 046404 (2006).[17] A. Macridin et al. , Phys. Rev. Lett. , 056402 (2006).[18] E. Cappelluti, S. Ciuchi, and S. Fratini, Phys. Rev. B ,125111 (2007).[19] G. D. Filippis et al. , Phys. Rev. Lett. , 146405 (2007).[20] A. S. Mishchenko and N. Nagaosa, Phys. Rev. Lett. ,036402 (2004).[21] O. Rosch and O. Gunnarsson, Phys. Rev. Lett. ,146403 (2004).[22] A. L. Chernyshev, P. W. Leung, and R. J. Gooding, Phys.Rev. B , 13594 (1998).[23] P. Wr´obel and R. Eder, Phys. Rev. B , 15160 (1998).[24] J. Riera and E. Dagotto, Phys. Rev. B , 8609 (1998).[25] H. Barentzen and V. Oudovenko, Europhys. Lett. , 227(1999).[26] d -wave symmetry of a bipolaron is a many-body effect. Abound state of two particles with attractive interactionis always nodeless ( s -wave in 2D), see also Ref. [8].[27] P. W. Leung, Phys. Rev. B , 205101 (2002).[28] T. Aimi and M. Imada, J. Phys. Soc. Jpn. , 113708(2007).[29] J. Bonˇca, S. Maekawa, and T. Tohyama, Phys. Rev. B , 035121 (2007).[30] J. Bonˇca, S. A. Trugman, and I. Batisti´c, Phys. Rev. B60