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
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. Bonca, S.A. Trugman and I. Batistic, 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.
• 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

J.B., S.M., and T.T., PRB 76, 035121 (2007)
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
lg2/8tw
J/t0.4
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-0.34t, t0.23t
Ca2-xNaxCuO2Cl2

• 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 et al., J. Phys. Soc. Jpn. 69 (200) 9
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
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
28
Spectral function at half-filling and different
EP interaction l
TJHH model, w0/t0.2, U/t10, J/t0.4

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

29
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.