Charge-Density-Excitation Spectrum in the t-t'-J-V Model
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Charge-Density-Excitation Spectrum in the t - t ′ - J - V Model
Andr´es Greco † , Hiroyuki Yamase ‡ , and Mat´ıas Bejas †† Facultad de Ciencias Exactas, Ingenier´ıa y Agrimensuraand Instituto de F´ısica Rosario (UNR-CONICET),Av. Pellegrini 250, 2000 Rosario, Argentina ‡ National Institute for Materials Science, Tsukuba 305-0047, Japan (Dated: August 29, 2016)
Abstract
We study the density-density correlation function in a large- N scheme of the t - t ′ - J - V model.When the nearest-neighbor Coulomb interaction V is zero, our model exhibits phase separation ina wide doping region and we obtain large spectral weight near momentum q = (0 ,
0) at low energy,which originates from the proximity to phase separation. These features are much stronger forelectron doping than for hole doping. However, once phase separation is suppressed by includinga finite V , the low-energy spectral weight around q = (0 ,
0) is substantially suppressed. Insteada sharp zero-sound mode is stabilized above the particle-hole continuum. We discuss that thepresence of a moderate value of V , which is frequently neglected in the t - J model, is important tounderstand low-energy charge excitations especially close to q = (0 ,
0) for electron doping. Thisinsight should be taken into account in a future study of x-ray scattering measurements.
PACS numbers: 75.25.Dk, 78.70.Ck, 74.72.-h, 71.10.Fd . INTRODUCTION Charge order and charge excitation spectra attract a renewed interest in cuprate su-perconductors. While the presence of spin-charge stripe order is well known in La-basedcuprates , charge order discovered recently is not accompanied by spin order. Such chargeorder is observed not only in hole-doped cuprates ( h -cuprates) such as Y- , Bi- , andHg-based compounds, but also in electron-doped cuprates ( e -cuprates) .While charge order phenomena are now ubiquitous in cuprates, there is a clear particle-hole asymmetry between h - and e -cuprates. Charge order occurs inside the pseudogap phasein h -cuprates. In contrast, in e -cuprates the charge order is observed below 340K (Ref. 13)and seems to occur from the normal metallic phase, since the pseudogap effect is absentor very weak in this system. On the other hand, theoretical insights are controversial withregard to particle-hole asymmetry of charge excitations. A recent density-matrix renormal-ization group (DMRG) study for the Hubbard model with U = 8 t predicts an enhancementof the low-energy charge excitations in e -cuprates, whereas a similar enhancement does notoccur for h -cuprates . In contrast, a leading-order theory of the large- N expansion to the t - J model shows that a tendency to charge order is stronger in h -cuprates than e -cuprates ,implying that low-energy charge excitations tend to be enhanced more in h -cuprates.These different theoretical conclusions could be understood consistently by consideringthe well-known insight that e -cuprates are expected to be closer to phase separation (PS)than h -cuprates as shown by various theoretical studies . However, charge excitationsassociated with PS are hardly known even theoretically. The proximity to PS is expected tohave a strong influence on charge excitations especially near momentum q = (0 , and the modenear q = (0 , in e -cuprates in terms of a large- N expansion to the t - J model , wecompute the density-density correlation function in the same large- N scheme as previousones . To clarify charge excitations associated with PS, we study the impact of thenearest-neighbor Coulomb interaction V and show dramatic changes of the low-energy charge2xcitations around q = (0 ,
0) especially for electron doping. We compare our results withrecent theoretical work and Ref. 23, both of which predict the presence of low-energycharge excitations for a small q .In Sec. 2 we describe the model and summarize the formalism. Sec. 3 contains the resultswhich are discussed in Sec. 4. Conclusions are given in Sec. 5. II. MODEL AND FORMALISM
Cuprate superconductors are doped Mott insulators and their minimal model is the so-called t - J model . Although the nearest-neighbor Coulomb interaction is frequently ne-glected in a study of the t - J model, its effect proves to be crucially important to understandcharge excitations close to q = (0 , t - t ′ - J - V model H = − X i,j,σ t ij ˜ c † iσ ˜ c jσ + J X h i,j i (cid:18) ~S i · ~S j − n i n j (cid:19) + V X h i,j i n i n j , (1)where the sites i and j run over a square lattice. The hopping t ij takes a value t ( t ′ ) betweenthe first (second) nearest-neighbors sites. h i, j i indicates a nearest-neighbor pair, and J and V are the spin exchange and Coulomb interaction, respectively. ˜ c † iσ (˜ c iσ ) is the creation(annihilation) operator of electrons with spin σ ( σ = ↓ , ↑ ) in the Fock space without doubleoccupancy. n i = P σ ˜ c † iσ ˜ c iσ is the electron density operator and ~S i is the spin operator.We analyze the model (1) in terms of the large- N expansion formulated in Ref. 26. Afull formalism is described in Ref. 27 where charge instabilities, not charge excitations, werestudied in the same framework as the present one. Hence leaving the details to Sec. II A inRef. 27, we here keep our presentation minimal. In the large- N approach, charge excitationswith momentum q and bosonic Matsubara frequency ω n are described by a 6 × D − ab ( q , i ω n ) = [ D (0) ab ( q , i ω n )] − − Π ab ( q , i ω n ) . (2)3ere a and b run from 1 to 6, D (0) ab ( q , i ω n ) is a bare bosonic propagator[ D (0) ab ( q , i ω n )] − = N F ( q ) δ δ J J J
00 0 0 0 0 J , (3)and Π ab ( q , i ω n ) are self-energy corrections at leading order; F ( q ) = ( δ / V ( q ) − J ( q )], V ( q ) = V (cos q x + cos q y ) and J ( q ) = J (cos q x + cos q y ); for a given doping rate δ , themean-field value of a bond-field ∆ is determined self-consistently.We compute the density-density correlation function in the present large- N scheme. Sum-ming all contributions up to O (1 /N ), we obtain χ c ( q , i ω n ) = N (cid:18) δ (cid:19) D ( q , i ω n ) . (4)Thus, the density-density correlation function is connected with the component (1 ,
1) of the D ab . The factor N in front of Eqs. (3) and (4) comes from the sum over the N fermionicchannels after the extension of the spin index σ from 2 to N .Although the physical value is N = 2, the large- N expansion has several advantagesover usual perturbation theories. First, charge degrees of freedom are generated by carrierdoping into a Mott insulator and thus the effect of the Coulomb interaction should vanishat half-filling. This feature is not reproduced in weak coupling theory but in the presentleading order theory; see the (1 ,
1) component in Eq. (3). Second, the large- N expansionyields results consistent with those obtained by exact diagonalization for charge excitations including plasmons . As we will discuss later, the present formalism also predicts chargeexcitations similar to those obtained in the dynamical DMRG method . Third, we actuallyshowed that the present large- N framework can capture short-range charge order recentlyobserved by resonant x-ray scattering and also a mysterious mode around q = (0 , e -cuprates . 4 IG. 1: (Color online) (a)-(c) q - ω maps of Im χ c ( q , ω ) at T = 0 and δ = 0 .
20 for several choices ofCoulomb repulsion V . q is scanned along the symmetry axes: ( π, π ) → (0 , → ( π, → ( π, π ).(d) q - ω map of Im χ c ( q , ω ) in the same condition of (a), except that a sign of t ′ (= − .
3) is changedto mimic the hole-doped case.
III. RESULTS
As it is well known , a tendency toward PS is stronger for e -cuprates than h -cuprates.We therefore choose parameters appropriate to e -cuprates such as J/t = 0 . t ′ /t = 0 . t . We computeIm χ c ( q , ω ) after analytical continuation in Eq. (4)i ω n → ω + iΓ , (5)where Γ( >
0) is infinitesimally small and we set Γ = 10 − for numerical convenience. Tem-perature T is fixed at T = 0. Since PS occurs below δ c = 0 .
18 in the present parametersfor V = 0 [see the inset of Fig. 2 (a)], we choose the doping δ as δ = 0 .
20, which is in theparamagnetic phase but close to PS.Figure 1 shows intensity maps of Im χ c ( q , ω ) for several choices of V along the symmetry5xes: ( π, π ) → (0 , → ( π, → ( π, π ). The dotted line denotes the upper bound ofparticle-hole excitations, above which there is a sharp dispersive mode. This is a particle-holebound state, called as the zero-sound mode. The zero-sound mode was actually obtainedin different approximations to the t - J model . Since it is a sound wave, it featuresa gapless linear dispersion around q = (0 , V = 0 [Fig. 1(a)], there exists largelow-energy spectral weight inside the particle-hole continuum slightly below the zero-soundmode around q = (0 , δ c , leading to the divergence of the compressibility there. Therefore this low-energy spectral weight originates from the proximity to PS. Because of the mixture of thelarge low-energy spectral weight, the zero-sound mode is overdamped around q = (0 , V , PS is suppressed as expected [see the inset ofFig. 2 (a)]. Concomitantly low-energy spectral weight around q = (0 ,
0) is also suppressedas shown in Figs. 1 (b) and (c) for V = 0 . V = 1, respectively. We then obtain onlythe zero-sound mode as dominant charge excitations around q = (0 , FIG. 2: (Color online) (a) ω dependence of Im χ c ( q , ω ) at q = ( π/ , π/
7) and (b) q dependenceof the integrated spectral weight with respect to ω ( >
0) for V = 0, 0 . · · · , 1 .
0. The inset in(a) shows the V dependence of the critical doping, below which PS occurs at T = 0, whereas theinset in (b) the V dependence of the total spectral weight integrated over ω ( >
0) and q along( π, π ) → (0 , → ( π, → ( π, π ). V dependence of the spectral weight are presented in Fig. 2(a) as a repre-sentative, where we present Im χ c ( q , ω ) at low momentum q = ( π/ , π/
7) for various choicesof V . With increasing V , the spectral weight is transferred to a high energy region and thepeak associated with the zero-sound mode is enhanced. However, when the spectral weightis integrated for each q with respect to energy ( > q -resolved total spectral weight S ( q ), which corresponds to the equal-time correlation function at T = 0, is suppressedaround q = (0 ,
0) with increasing V as shown in Fig. 2(b). This suggests that the contri-bution from the proximity to PS is indeed sizable in the spectral weight around q = (0 , q = ( π, π ). This is because alarge V favors a checkerboard-type charge-density-wave . In fact, such charge order wouldoccur for V > V c ≈ .
1, but we consider that the region of
V < V c is relevant to cupratesuperconductors. On the other hand, the total weight integrated over ω ( >
0) and q alongthe symmetry axes ( π, π ) → (0 , → ( π, → ( π, π ) does not depend on V [see the insetin Fig. 2(b)]. Therefore the spectral weight around q = (0 ,
0) is transferred to the regionaround q = ( π, π ) with increasing V . This spectral weight transfer occurs already for Vmuch smaller than V c and we do not observe any charge dynamics, which could be associatedwith other types of charge orders such as stripes from the mechanism of frustrated PS .For the hole-doped case, we may change the sign of t ′ . In this case, the tendencytoward PS becomes much weaker than that for a positive t ′ . As seen in Fig. 1 (d), there isno enhancement of low-energy spectral weight around q = (0 ,
0) even for V = 0. In fact,there occurs no PS for any doping. Hence the effect of V becomes much weaker around q = (0 , q = ( π, π ) is common to bothsign of t ′ for a large V . IV. DISCUSSIONS
We have found that the nearest-neighbor Coulomb repulsion V plays a crucial role tounderstand the low-energy charge excitations around q = (0 ,
0) especially in a model calcu-lation for e -cuprates, because e -cuprates are expected to be close to PS .Recently low-energy charge excitations were reported by a DMRG method for a 6 × U = 8 t (Ref. 14). In particular, the presence of thelow-energy peak was predicted in e -cuprates at energy lower than the currently available7IXS data .The Hubbard model with a large U should share similar properties to the t - J model andindeed exhibits phase separation . Hence the predicted low-energy charge excitationslikely originate from the proximity to PS and would be strongly suppressed once the Coulombrepulsion is included. To demonstrate the connection between the present work and Fig. 4in Ref. 14, we took the same momenta, doping, and a large damping Γ = 0 . t to mimic thebroadening of Ref. 14; V is set to zero. We then computed Im χ c ( q , ω ) within the presenttheory for both t ′ = 0 . t (electron-doped case) and t ′ = − . t (hole-doped case). Since alarge Γ was invoked, PS does not occur even for the electron-doped case in our framework,but charge fluctuations associated with the proximity to PS are expected. FIG. 3: (Color online) ω dependence of Im χ c ( q , ω ) at q = ( π/ , π/ π/ , π/ π/ , π ) forseveral choices of doping δ for the electron-doped case (a) and the hole-doped case (b). To makea direct comparison with Fig. 4 in Ref. 14, V is set to zero, a value of Γ is chosen to be Γ = 0 . t ,and the horizontal axis is taken to be 2 ω because t is scaled as t in the large- N theory. Figures 3 (a) and (b) capture major features of Figs. 4(c) and (d) in Ref. 14, respectively.The low-energy spectral weight is strongly suppressed at q = ( π/ , π/
3) and ( π/ , π ) inFig. 3(a), but there is strong enhancement in a low-energy region at q = ( π/ , π/ q = ( π/ , π/
3) and ( π/ , π ), similar to the corresponding results in the electron-doped case[Fig. 3(a)]. At small momentum q = ( π/ , π/ q = ( π/ , π/
3) and ( π/ , π ), because of the presence of zero-sound modeat relatively low energy [see Fig. 1(d)]. In a lower energy region (2 ω/t . . V is introduced, such low-energy spectral weightis substantially suppressed and both electron- and hole-doped cases show similar chargeexcitations at low energy for a small q , as we have already seen in Fig. 1. Low-energycharge excitations at small q are also reported in the t - J model with t ′ = 0 and V = 0 inRef. 23. Such excitations may also come from the proximity to PS.The zero-sound mode is stabilized in the present model, consistent with theliterature . We have found that the presence of the zero-sound mode is independent ofthe value of V . Recalling our parameters J = 0 . ≤ V ≤ .
0, the effective interaction V ( q ) − J ( q ) in the (1 ,
1) component of Eq. (3) changes a sign at V = 0 .
15. Moreover becauseof the form factor, cos q x + cos q y , the sign of V ( q ) − J ( q ) depends on q . Nevertheless, thezero-sound mode is stabilized for any V and q in Fig. 1. In fact, as shown in Refs. 32 and34, the zero-sound exists even for J = V = 0. This cannot be understood in terms of ausual weak coupling analysis such as a random phase approximation (RPA).Instead, the robustness of the zero-sound mode originates from strong correlation effectscontained in the t - J model in the sense that the present leading order theory does not predicta RPA-like susceptibility, but predicts χ c ( q , ω ) ∼ Π ( q , ω )[2 F ( q ) − Π ( q , ω )]Π ( q , ω ) + [ δ − Π ( q , ω )] . (6)Here we have neglected the components a, b = 3 , , , F ( q ) − ReΠ ( q , ω )] ReΠ ( q , ω ) + [ δ − ReΠ ( q , ω )] = 0 . (7)The point is that the term, 2 F ( q ) − ReΠ ( q , ω ), is always positive in the parameter spacewe have studied and ReΠ ( q , ω ) is negative at high energy. Equation (7) is then fulfilled.This is the reason why the zero-sound mode is robust in the present model. We mayinterpret high-energy peaks of the density-density correlation function obtained by exactdiagonalization as the zero-sound mode, as pointed out in Refs. 32 and 33.The robustness of the zero-sound mode, however, should be taken carefully. In factthe zero-sound mode changes to a plasmon mode with an excitation gap at q = (0 ,
0) by9ncluding the long-range Coulomb interaction, as recently shown in the present large- N scheme for a layered system .The strong asymmetry of PS between e -cuprates ( t ′ >
0) and h -cuprates ( t ′ <
0) is alsounderstood from Eq. (6). To simplify our analysis, we put V = J = 0, i.e., F ( q ) = 0. Wethen obtain at T = 0 χ c ( q → , ω = 0) ∼ N F δ − µN F , (8)where N F is the density of states at the Fermi energy and µ is the chemical potential .Hence PS occurs when the following condition is fulfilled: δ < µN F . (9)We can check that µ becomes positive close to half-filling only for t ′ > V = J = 0in Ref. 15. Therefore the asymmetry of PS between e - and h -cuprates is controlled byEq. (9) and originates from strong correlation effects in the sense that Eq. (9) is obtainedfor V = J = 0. For a finite J and V we can perform a similar analysis and obtain the sameconclusion. V. CONCLUSIONS
We have studied charge excitations in the t - t ′ - J model by including the nearest-neighborCoulomb repulsion V . While the effect of V is frequently neglected in research of chargeexcitations in cuprates, we have found that the V term is crucially important to understandlow-energy charge excitations around q = (0 ,
0) [Figs. 1(a)-(c) and 2(a)] especially in a modelcalculation for e -cuprates. In line with the prediction in Ref. 14, we have also obtained low-energy spectral weight around q = (0 , V is sufficiently small. Given the increasing interest in the studyof low-energy charge excitations around q = (0 ,
0) in RIXS, it is crucial to consider thepresence of the Coulomb repulsion, which is expected to be finite in real systems. We havealso found that the zero-sound mode is stabilized above the particle-hole continuum, whichis independent of a value of V . 10 cknowledgments The authors thank G. Khaliullin and T. Tohyama for very fruitful discussions. H.Y.acknowledges support by JSPS KAKENHI Grant Number 15K05189. S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, andC. Howald, Rev. Mod. Phys. , 1201 (2003). T. Wu, H. Mayaffre, S. Kr¨amer, M. Horvati´c, C. Berthier, W. N. Hardy, R. Liang, D. A. Bon,and M.-H. Julien, Nature , 191 (2011). G. Ghiringhelli, M. L. Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. D.Luca, A. Frano, D. G. Hawthorn, F. He, et al., Science , 821 (2012). J. Chang, E. Blackburn, A. T. Holmes, N. B. Christensen, J. Larsen, J. Mesot, R. Liang, D. A.Bonn, W. N. Hardy, A. Watenphul, et al., Nat. Phys. , 871 (2012). A. J. Achkar, R. Sutarto, X. Mao, F. He, A. Frano, S. Blanco-Canosa, M. Le Tacon, G. Ghir-inghelli, L. Braicovich, M. Minola, et al., Phys. Rev. Lett. , 167001 (2012). D. LeBoeuf, S. Kr¨amer, W. N. Hardy, R. Liang, D. A. Bonn, and C. Proust, Nat. Phys. , 79(2013). E. Blackburn, J. Chang, M. H¨ucker, A. T. Holmes, N. B. Christensen, R. Liang, D. A. Bonn,W. N. Hardy, U. R¨utt, O. Gutowski, et al., Phys. Rev. Lett. , 137004 (2013). S. Blanco-Canosa, A. Frano, E. Schierle, J. Porras, T. Loew, M. Minola, M. Bluschke,E. Weschke, B. Keimer, and M. L. Tacon, Phys. Rev. B , 054513 (2014). R. Comin, A. Frano, M. M. Yee, Y. Yoshida, H. Eisaki, E. Schierle, E. Weschke, R. Sutarto,F. He, A. Soumyanarayanan, et al., Science , 390 (2014). E. H. da Silva Neto, P. Aynajian, A. Frano, R. Comin, E. Schierle, E. Weschke, A. Gyenis,J. Wen, J. Schneeloch, Z. Xu, et al., Science , 393 (2014). M. Hashimoto, G. Ghiringhelli, W.-S. Lee, G. Dellea, A. Amorese, C. Mazzoli, K. Kummer,N. B. Brookes, B. Moritz, Y. Yoshida, et al., Phys. Rev. B , 220511 (2014). W. Tabis, Y. Li, M. L. Tacon, L. Braicovich, A. Kreyssig, M. Minola, G. Dellea, E. Weschke,M. J. Veit, M. Ramazanoglu, et al., Nat. Commun. , 5875 (2014). E. H. da Silva Neto, R. Comin, F. He, R. Sutarto, Y. Jiang, R. L. Greene, G. A. Sawatzky, and . Damascelli, Science , 282 (2015). T. Tohyama, K. Tsutsui, M. Mori, S. Sota, and S. Yunoki, Phys. Rev. B , 014515 (2015). M. Bejas, A. Greco, and H. Yamase, New J. Phys. , 123002 (2014). R. J. Gooding, K. J. E. Vos, and P. W. Leung, Phys. Rev. B , 12 866 (1994). G. B. Martins, J. C. Xavier, L. Arrachea, and E. Dagotto, Phys. Rev. B , R1805 (2001). A. Macridin, M. Jarrell, and T. Maier, Phys. Rev. B , 085104 (2006). K. Ishii, M. Fujita, T. Sasaki, M. Minola, G. Dellea, C. Mazzoli, K. Kummer, G. Ghiringhelli,L. Braicovich, T. Tohyama, et al., Nat. Commun. , 3714 (2014). W. S. Lee, J. J. Lee, E. A. Nowadnick, S. Gerber, W. Tabis, S. W. Huang, V. N. Strocov, E. M.Motoyama, G. Yu, B. Moritz, et al., Nat. Phys. , 883 (2014). H. Yamase, M. Bejas, and A. Greco, Europhys. Lett. , 57005 (2015). A. Greco, H. Yamase, and M. Bejas, Phys. Rev. B , 075139 (2016). G. Khaliullin and P. Horsch, Phys. Rev. B , R9600 (1996). P. W. Anderson, Science , 1196 (1987). P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. , 17 (2006). A. Foussats and A. Greco, Phys. Rev. B , 205123 (2004). M. Bejas, A. Greco, and H. Yamase, Phys. Rev. B , 224509 (2012). A. T. Hoang and P. Thalmeier, J. Phys.: Condens. Matter , 6639 (2002). J. Merino, A. Greco, R. H. McKenzie, and M. Calandra, Phys. Rev. B , 245121 (2003). P. Prelovˇsek and P. Horsch, Phys. Rev. B , R3735 (1999). Z. Wang, Y. Bang, and G. Kotliar, Phys. Rev. Lett. , 2733 (1991). L. Gehlhoff and R. Zeyher, Phys. Rev. B , 4635 (1995). R. Zeyher and M. Kuli´c, Phys. Rev. B , 8985 (1996). A. Foussats and A. Greco, Phys. Rev. B , 195107 (2002). V. J. Emery and S. A. Kivelson, Physica C , 597 (1993). T. Tohyama and S. Maekawa, Phys. Rev. B , 3596 (1994). E. Koch and R. Zeyher, Phys. Rev. B , 094510 (2004). T. Tohyama, P. Horsch, and S. Maekawa, Phys. Rev. Lett. , 980 (1995)., 980 (1995).