Spectral properties of the t-J-Holstein model in the low-doping limit

1. Spectral properties of  the t-J-Holstein model in the low-doping limit　 J. Bonča1 Collaborators: S. Maekawa2, T. Tohyama3, and P.Prelovšek1 1Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, and J. Stefan Institute, Ljubljana, Slovenia 2 Institute for Materials Research, Tohoku University, Sendai 980-8577, and CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan 3 Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

2. The model

3. EDLFS approach • Problem of one hole in the t-J model remains unsolved except in the limit when J0. • Many open questions: • The size of Zk in the t-J model? • The influence of el. ph. interaction on correlated hole motion • Unusually wide QP peak at low doping • The origin of the ‘famous’ kink seen in ARPES • Method is based on • S.A. Trugman, Phys. Rev. B 37, 1597 (1988). • J. Inoue and S. Maekawa, J. Phys. Soc. Jpn. 59, 2110, (1990) • J. Bonča, S.A. Trugman and I. Batistić, Phys. Rev. B, 60, 1663 (1999).

4. EDLFS approach • Create Spin-flip fluctuations and phonon quanta in the vicinity of the hole: • Start with one hole in a Neel state • Apply kinetic part of H as well as the off-diagonal phonon part to create new states. LFSNeel state {fkl(Nh) }=(Ht+HgM)Nh |fk(0)> Total # of phonons : Nh*M

5. EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: HtNh |fk(0)>: = Nh=2 Nh=1

6. E(k) and Z(k) for the 1-hole system, no phonons, t-J model Polaron energy Ek=Ek1h - E0h Quasiparticle weight • Good agreement of Ek with all • known methods • Best agreement of Zkwith ED on 32-sites cluster for J/t~0.3 J.B., S.M., and T.T., PRB 76, 035121 (2007)

7. E(k) and Z(k) for the 1-hole system, no phonons

8. Spectral function A(k,w) J/t=0.3 J.B., S.M., and T.T., PRB 76, 035121 (2007)

9. Finite electron-phonon coupling l=g2/8tw J/t=0.4 TJH: t’=t’’=0, TJHH: t’/t=-0.34, t’’/t=0.23 TJHHTJHE: t -t • Linear decrease of Zk at smalll • Crossover to the strong coupling regime becomes bore abrupt as the quasi-particle becomes more coherent • Qualitative agreement with DMC method (Mishchenko & Nagaosa, PRL 93, (2004)) Nh=8, M=7, Nst=8.1 106

10. Ek, Zk, Nk J/t=0.4 t’=-0.34t, t’’=0.23t • Increasing l leads to: • flattening of Ek • decreasing of Zk • increasing of Nk • Zkin the band minimum is much larger in • the electron- than in the hole- doped case in part due to stronger antiferomagnetic correlations. • Larger Zk indicates that the quasiparticle is much more coherent and has smaller effective mass in the electron-doped case which leads to less effective EP coupling and higher l is required to enter the small-polaron (localized) regime. Ca2-xNaxCuO2Cl2 T. Tohyama et al., J. Phys. Soc. Jpn. 69 (200) 9

11. Spectral function A(k,w) • Low-energy peaks roughly preserve their spectral weight with increasing l. At large values of l they appear as broadened quasiparticle peaks. • Low-energy peak in the strong coupling regime of the TJHH model remains narrower than the corresponding peak in the pure t-J-Holstein model (TJH) • Positions of quasiparticle peaks with increasing l shift below the low-energy peaks and loose their spectral weight (diminishing Zk).

12. Spectral function A(k,w) • Low-energy incoherent peaks disperse • along MG. Dispersion qualitatively tracks the dispersion of respective t-J and t-t'-t''-J models yielding effective bandwidths WTJH/t ~ 0.64 and • WTJHH/t~ 0.75. • Widths of low-energy peaks at M-point are comparable to respective bandwidths, GTJH/t ~ 0.82 and GTJHH/t~ 0.52. • Peak widths increase with increasing binding energy. This effect is even more evident in the TJHH case, see for example (M G). • Results consistent with Shen et al. PRL 93 (2004)

13. Can electron-phonon coupling lead to anomalous spectral features seen in ARPES? • At rather small value of l = 0.2 the signature of the QP in the vicinity of G point vanishes while the rest of the low energy excitation broadens and remains dispersive. On the other hand, the bottom band loses coherence. • In the strong coupling regime, l=0.4 and 0.6, the qualitative behaviour changes since the dispersion seems to transform in a single band with a waterfall-like feature at k ~ (p/4,p/4), connecting the low-energy with the high-energy parts of the spectra. • Ripples due to phonon excitations as well become visible. TJHH model, w0/t=0.2

14. Spectral function at half-filling and different EP interaction l TJHH model, w0/t=0.2, U/t=10, J/t=0.4 • Largest QP weight at the bottom of the upper Hubbard band. • QP weight decreases with increasing l, while the incoherent part of spectral weight increases • Even in the strong coupling regime, l>=0.4 the dispersion roughly follows the dispersion at l=0.

15. Conclusions • We developed an extremely efficient numerical method to solve generalized t-J-Holstein model in the low doping limit. • The method allows computation of static and dynamic quantities at any wavevector. • Spectral functions in the strong coupling regime are consistent with Shen et al., PRL 93 (2004) and Ronning et al., PRB 71 (2005). • Low-energy incoherent peaks disperse along MG. • Widths of low-energy peaks are comparable to respective bandwidths • Peak widths increase with increasing binding energy. • At rather small value of l = 0.2 the signature of the QP in the vicinity of G point vanishes while the rest of the low energy excitation broadens and remains dispersive. • In the strong coupling regime, l=0.4 and 0.6, the dispersion seems to transform in a single band with a waterfall-like feature at k ~ (p/4,p/4), connecting the low-energy with the high-energy parts of the spectra.