Www-f1.ijs.si/~bonca SNS2007 SENDAI Spectral properties of the t-J- Holstein model in the low-doping limit Spectral properties of the t-J- Holstein model

www-f1.ijs.si/~bonca SNS2007 SENDAI

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

limitlimit 　　J. BonJ. Bonččaa11

Collaborators:Collaborators:S. MaekawaS. Maekawa22, T. Tohyama, T. Tohyama33, and P.Prelov, and P.Prelovššekek11

11 Faculty of Mathematics and Physics, University of Ljubljana, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, and J. Stefan Institute,Ljubljana, and J. Stefan Institute,

Ljubljana, SloveniaLjubljana, Slovenia22 Institute for Materials Research, Tohoku University, Sendai 980- Institute for Materials Research, Tohoku University, Sendai 980-8577, and CREST, Japan Science and Technology Agency (JST), 8577, and CREST, Japan Science and Technology Agency (JST),

Kawaguchi, Saitama 332-0012, JapanKawaguchi, Saitama 332-0012, Japan3 3 Institute for Theoretical Physics, Kyoto University, Kyoto 606-Institute for Theoretical Physics, Kyoto University, Kyoto 606-

8502, Japan8502, Japan

www-f1.ijs.si/~bonca LAW3M-05

The modelThe model


EDLFS approachEDLFS approach Problem of one hole in the t-J model remains unsolved Problem of one hole in the t-J model remains unsolved

except in the limit when Jexcept in the limit when J0.0. Many open questions:Many open questions:

The size of The size of ZZkk in the t-J model? in the t-J model? The influence of el. ph. interaction on correlated hole motionThe influence of el. ph. interaction on correlated hole motion Unusually wide QP peak at low doping Unusually wide QP peak at low doping The origin of the ‘famous’ kink seen in ARPESThe origin of the ‘famous’ kink seen in ARPES

Method is based on Method is based on S.A. Trugman, Phys. Rev. B 37, 1597 (1988).S.A. Trugman, Phys. Rev. B 37, 1597 (1988). J. Inoue and S. Maekawa, J. Phys. Soc. Jpn. J. Inoue and S. Maekawa, J. Phys. Soc. Jpn. 5959, 2110, (1990), 2110, (1990) J. BonJ. Bončča, S.A. Trugman and I. Batistia, S.A. Trugman and I. Batistićć, Phys. Rev. B, , Phys. Rev. B, 6060, ,

1663 (1999).1663 (1999).


EDLFS approachEDLFS approach Create Create Spin-flip fluctuationsSpin-flip fluctuations and and phonon phonon

quantaquanta in the vicinity of the hole: in the vicinity of the hole: Start with one hole in a Neel stateStart with one hole in a Neel state Apply Apply kinetic partkinetic part of H as well as the of H as well as the off-off-

diagonal phonondiagonal phonon part to create new states. part to create new states.

LFSLFS Neel stateNeel state

kkll((NhNh) ) =(H=(Htt+H+Hgg

MM))NhNh | |kk(0)(0) >>

Total # of phonons : Total # of phonons : NhNh*M*M


EDLFS approach EDLFS approach (graphic representation of the LFS generator)(graphic representation of the LFS generator)

Application of the kinetic partApplication of the kinetic part of H: of H:

HHttNhNh | |kk

(0)(0) >:>:

=

Nh=1

Nh=2


E(k) and Z(k) for the 1-hole system, no E(k) and Z(k) for the 1-hole system, no phonons, phonons, t-J modelt-J model

Ek=Ek1h - E0h

Polaron energyPolaron energy

Quasiparticle weightQuasiparticle weight

• Good agreement of Good agreement of EEkk

with all with all known methodsknown methods

• Best agreement of Best agreement of ZZkk with with

ED ED on 32-sites cluster on 32-sites cluster for Jfor J/t~0.3/t~0.3 J.B., S.M., and T.T., PRB J.B., S.M., and T.T., PRB 76, 035121 (2007)76, 035121 (2007)


E(k) and Z(k) for the 1-hole system, no E(k) and Z(k) for the 1-hole system, no phononsphonons


Spectral function A(Spectral function A(kk,,))

J/t=0.3

J.B., S.M., and T.T., PRB 76, 035121 (2007)


Finite electron-phonon couplingFinite electron-phonon coupling

=g2/8t J/t=0.4

TJH: t’=t’’=0, TJHH: t’/t=-0.34, t’’/t=0.23TJHHTJHE: t -t

• Linear decrease of Linear decrease of ZZkk at small at small • Crossover to the strong coupling Crossover to the strong coupling

regime becomes bore abrupt as the regime becomes bore abrupt as the quasi-particle becomes more coherent quasi-particle becomes more coherent

• Qualitative agreement with DMC Qualitative agreement with DMC method method (Mishchenko & Nagaosa, PRL (Mishchenko & Nagaosa, PRL 93, (2004))93, (2004))

Nh=8, M=7, Nst=8.1 106


EEkk, Z, Zkk, N, Nkk

t’=-0.34t, t’’=0.23t

J/t=0.4

Ca2-xNaxCuO2Cl2T. Tohyama et al.,

J. Phys. Soc. Jpn. 69 (200) 9

Increasing Increasing leads to: leads to:• flattening of flattening of EEkk

• decreasing of decreasing of ZZkk

• increasing of increasing of NNkk

ZZkk in the band minimum is in the band minimum is much largermuch larger in in

the electron- than in the hole- doped case in the electron- than in the hole- doped case in part due to stronger antiferomagnetic part due to stronger antiferomagnetic correlations. correlations.

Larger Larger ZZkk indicates that the quasiparticle is indicates that the quasiparticle is

much more coherent and has smaller effective much more coherent and has smaller effective mass in the electron-doped case which leads to mass in the electron-doped case which leads to less effective EP coupling and higher l is less effective EP coupling and higher l is required to enter the small-polaron (localized) required to enter the small-polaron (localized) regime.regime.



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



• Low-energy incoherent peaks Low-energy incoherent peaks dispersedispersealong Malong M. Dispersion qualitatively . Dispersion qualitatively tracks the dispersion of respective t-J tracks the dispersion of respective t-J and t-t'-t''-J models yielding effective and t-t'-t''-J models yielding effective bandwidths Wbandwidths WTJHTJH/t ~ 0.64 and/t ~ 0.64 and

WWTJHHTJHH/t~ 0.75./t~ 0.75.• Widths of low-energy peaks at M-Widths of low-energy peaks at M-point are comparable to respective point are comparable to respective bandwidths, bandwidths, TJHTJH/t ~ 0.82 and /t ~ 0.82 and TJHHTJHH/t~ /t~

0.52.0.52.• Peak widths increase with increasing Peak widths increase with increasing binding energy. This effect is even binding energy. This effect is even more evident in the TJHH case, see for more evident in the TJHH case, see for example (M example (M ).).• Results consistent with Results consistent with Shen et al. Shen et al. PRL 93 (2004)PRL 93 (2004)


Can electron-phonon coupling lead to anomalous Can electron-phonon coupling lead to anomalous spectral features seen in ARPES?spectral features seen in ARPES?

• At rather small value of At rather small value of = 0.2 the = 0.2 the signature of the QP in the vicinity of signature of the QP in the vicinity of point vanishespoint vanishes while the rest of the low while the rest of the low energy excitation broadens and remains energy excitation broadens and remains dispersive. On the other hand, the dispersive. On the other hand, the bottom band loses coherence. bottom band loses coherence.

• In the strong coupling regime, In the strong coupling regime, =0.4 =0.4 and 0.6, the qualitative behaviour and 0.6, the qualitative behaviour changes since the dispersion seems to changes since the dispersion seems to transform in a single band with transform in a single band with a a waterfall-like featurewaterfall-like feature at k ~ ( at k ~ (/4,/4,/4), /4), connecting the low-energy with the connecting the low-energy with the high-energy parts of the spectra. high-energy parts of the spectra.

• Ripples due to phonon excitationsRipples due to phonon excitations as as well become visible.well become visible.

TJHH model, /t=0.2


Spectral function at half-filling and different EP Spectral function at half-filling and different EP interaction interaction

• Largest QP weight at the bottom Largest QP weight at the bottom of the upper Hubbard band.of the upper Hubbard band.• QP weight decreases with QP weight decreases with increasing increasing while the incoherent while the incoherent part of spectral weight increasespart of spectral weight increases• Even in the strong coupling Even in the strong coupling regime, regime, >=0.4 the dispersion >=0.4 the dispersion roughly follows the dispersion at roughly follows the dispersion at =0.=0.

TJHH model, /t=0.2, U/t=10, J/t=0.4


ConclusionsConclusions We developed an extremely efficient numerical method to solve We developed an extremely efficient numerical method to solve

generalized t-J-Holstein model in the low doping limit. generalized t-J-Holstein model in the low doping limit. The method allows computation of static and dynamic quantities at The method allows computation of static and dynamic quantities at

any wavevector.any wavevector. Spectral functions in the strong coupling regime are consistent with Spectral functions in the strong coupling regime are consistent with

Shen et al., PRL 93 (2004) and Ronning et al., PRB 71 (2005).Shen et al., PRL 93 (2004) and Ronning et al., PRB 71 (2005). Low-energy incoherent peaks disperse along MLow-energy incoherent peaks disperse along MG.G. Widths of low-energy peaks are comparable to respective bandwidthsWidths of low-energy peaks are comparable to respective bandwidths Peak widths increase with increasing binding energy.Peak widths increase with increasing binding energy. At rather small value of At rather small value of = 0.2 the = 0.2 the signature of the QP in the vicinity of signature of the QP in the vicinity of

point vanishespoint vanishes while the rest of the low energy excitation broadens and while the rest of the low energy excitation broadens and remains dispersive.remains dispersive.

In the strong coupling regime, l=0.4 and 0.6, the dispersion seems to In the strong coupling regime, l=0.4 and 0.6, the dispersion seems to transform in a single band with transform in a single band with a waterfall-like featurea waterfall-like feature at k ~ (p/4,p/4), at k ~ (p/4,p/4), connecting the low-energy with the high-energy parts of the spectra. connecting the low-energy with the high-energy parts of the spectra.

Documents

Www-f1.ijs.si/~bonca SNS2007 SENDAI Spectral properties of the t-J- Holstein model in the low-doping limit Spectral properties of the t-J- Holstein model