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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 in the low-doping limit J. Bonča 1 Collaborators: S. Maekawa 2, T. Tohyama 3, and P.Prelovšek 1 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

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

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www-f1.ijs.si/~bonca LAW3M-05 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 Z k in the t-J model? The size of Z k in the t-J model? The influence of el. ph. interaction on correlated hole motion The 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 ARPES The origin of the ‘famous’ kink seen in ARPES 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. 59, 2110, (1990) 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). J. Bonča, S.A. Trugman and I. Batistić, Phys. Rev. B, 60, 1663 (1999).

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach Create Spin-flip fluctuations and phonon quanta in the vicinity of the hole: Start with one hole in a Neel state 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. Apply kinetic part of H as well as the off- diagonal phonon part to create new states. LFS Neel state LFS Neel state {f kl (Nh) } =(H t +H g M ) Nh | f k (0) > Total # of phonons : Nh*M

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >: = N h =1 N h =2

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 EDLFS approach (graphic representation of the LFS generator) Application of the kinetic part of H: H t Nh | f k (0) >:

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www-f1.ijs.si/~bonca LAW3M-05 E(k) and Z(k) for the 1-hole system, no phonons, t-J model E k =E k 1h - E 0h Polaron energy Quasiparticle weight Good agreement of E k with allGood agreement of E k with all known methods Best agreement of Z k with ED on 32-sites cluster for J/t~0.3Best agreement of Z k with ED on 32-sites cluster for J/t~0.3 J.B., S.M., and T.T., PRB 76, 035121 (2007)

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www-f1.ijs.si/~bonca LAW3M-05 E(k) and Z(k) for the 1-hole system, no phonons

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www-f1.ijs.si/~bonca LAW3M-05 Stability of E k and Z k against the choice of functional space J/t=0.3

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www-f1.ijs.si/~bonca LAW3M-05 Spectral function A(k, w ) J/t=0.3 J.B., S.M., and T.T., PRB 76, 035121 (2007)

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www-f1.ijs.si/~bonca LAW3M-05 Finite electron-phonon coupling l =g 2 /8t w J/t=0.4 TJH: t’=t’’=0, TJHH: t’/t=-0.34, t’’/t=0.23 TJHH TJHE: t -t Linear decrease of Z k at small lLinear decrease of Z k at small l Crossover to the strong coupling regime becomes bore abrupt as the quasi- particle becomes more coherentCrossover 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))Qualitative agreement with DMC method (Mishchenko & Nagaosa, PRL 93, (2004)) N h =8, M=7, N st =8.1 10 6

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www-f1.ijs.si/~bonca LAW3M-05 E k, Z k, N k t’=-0.34t, t’’=0.23t J/t=0.4 Ca 2-x Na x CuO 2 Cl 2 T. Tohyama et al., J. Phys. Soc. Jpn. 69 (200) 9 Increasing l leads to: flattening of E k flattening of E k decreasing of Z k decreasing of Z k increasing of N k increasing of N k Z k in the band minimum is much larger in the electron- than in the hole- doped case in part due to stronger antiferomagnetic correlations. Larger Z k 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.

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www-f1.ijs.si/~bonca LAW3M-05 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) 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 Z k ). Positions of quasiparticle peaks with increasing l shift below the low-energy peaks and loose their spectral weight (diminishing Z k ).

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www-f1.ijs.si/~bonca LAW3M-05 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 W TJH /t ~ 0.64 and W TJHH /t~ 0.75. Widths of low-energy peaks at M- point are comparable to respective bandwidths, G TJH /t ~ 0.82 and G TJHH /t~ 0.52. Widths of low-energy peaks at M- point are comparable to respective bandwidths, G TJH /t ~ 0.82 and G TJHH /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 ). 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) Results consistent with Shen et al. PRL 93 (2004)

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www-f1.ijs.si/~bonca LAW3M-05 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. 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. Ripples due to phonon excitations as well become visible. TJHH model, w 0 /t=0.2

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www-f1.ijs.si/~bonca LAW3M-05 Spectral function at half-filling and different EP interaction l 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 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. Even in the strong coupling regime, l >=0.4 the dispersion roughly follows the dispersion at l =0. TJHH model, w 0 /t=0.2, U/t=10, J/t=0.4

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www-f1.ijs.si/~bonca LAW3M-05 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. Low-energy incoherent peaks disperse along M G. Widths of low-energy peaks are comparable to respective bandwidths Widths 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 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. 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. 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.

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