Spectral properties of

1
Spectral properties of  the t-J-Holstein model in
the low-doping limit

• J. Bonca1
• Collaborators
• S. Maekawa2, T. Tohyama3, and P.Prelovšek1
• 1 Faculty 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
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. Bonca, S.A. Trugman and I. Batistic, Phys.
  Rev. B, 60, 1663 (1999).

3
The model
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.
• LFS Neel state
• fkl(Nh) (HtHgM)Nh fk(0) gt
• Total of phonons NhM

5
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

Nh2
Nh1
6
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

7
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

8
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

9
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

10
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

11
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

12
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

13
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

14
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

15
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

16
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

17
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

18
EDLFS approach (graphic representation of the
LFS generator)

• Application of the kinetic part of H
• HtNh fk(0) gt

19
E(k) and Z(k) for the 1-hole system, no phonons,
t-J model
Polaron energy
EkEk1h - E0h
Quasiparticle weight

• Good agreement of Ek with all
• known methods
• Best agreement of Zk with ED on 32-sites cluster
  for J/t0.3

EDLFS J.B., S.M., and T.T., PRB 76, 035121
(2007), ED Leung Gooding, PRB 51, R15711
(1995), WMC Mishchenko et al., PRB 64, 033101
(2001), QMC Brunner et al., PRB 62, 15480
(2000), CE P.Prelovšek et al., PRB 42,
10706 (1990).
20
E(k) and Z(k) for the 1-hole system, no phonons
21
Stability of Ek and Zk against the choice of
functional space
J/t0.3
22
Spectral function A(k,w)
J/t0.3
J.B., S.M., and T.T., PRB 76, 035121 (2007)
23
Finite electron-phonon coupling
J/t0.4
lg2/8tw
TJH tt0, TJHH t/t-0.34,
t/t0.23 TJHH??TJHE t ??-t

• Linear decrease of Zk at small l
• 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))

Nh8, M7, Nst8.1 106
24
Ek, Zk, Nk
J/t0.4
t -0.34 t, t 0.23 t

• Increasing l leads to
• flattening of Ek
• decreasing of Zk
• increasing of Nk
• Zk in 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.

T. Tohyama, PRB 70, 174517 (2004)
Ca2-xNaxCuO2Cl2
25
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).

26
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)

27
Spectral function A(k,w)

• Shen et al. PRL 93 (2004)

Ca2-xNaxCuO2Cl2
28
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, l0.4 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/t0.2
29
Can electron-phonon coupling lead to anomalous
spectral features seen in ARPES?
TJHH model, w0/t0.2
F.Ronning et al, PRB, 71 094518 (2005)
30
Spectral function at half-filling and different
EP interaction l
TJHH model, w0/t0.2, U/t10, J/t0.4, T.
Tohyama, PRB 70, 174517 (2004)

• 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, lgt0.4 the
  dispersion roughly follows the dispersion at l0.

31
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, l0.4 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.