1. Faculty of Mathematics and Physics, University of Ljubljana, J. Stefan Institute, Ljubljana, Slovenia P. Prelovšek, M. Zemljič, I. Sega and J. Bonča Finite-temperature dynamics of small correlated systems: anomalous properties for cuprates Sherbrooke, July 2005

2. Outline Numerical method: Finite temperature Lanczos method (FTLM) and microcanonical Lanczos method for small systems: static and dynamical quantities: advantages and limitations Examples of anomalous dynamical quantities (non-Fermi liquid –like) in cuprates: calculations within the t-J model : Optical conductivity and resistivity: intermediate doping – linear law, low doping – MIR peak, resistivity saturation and kink at T* Spin fluctuation spectra: (over)damping of the collective mode in the normal state, ω/T scaling, NFL-FL crossover

3. quantum critical point, static stripes, crossover ? Cuprates: phase diagram

4. t – J model interplay : electron hopping + spin exchange single band model for strongly correlated electrons projected fermionic operators: no double occupation of sites n.n. hopping finite-T Lanczos method (FTLM): J.Jaklič + PP T > T fs finite size temperature n.n.n. hopping

5. Exact diagonalization of correlated electron systems: T>0 Basis states: system with N sites Heisenberg model: states t – J model: states Hubbard model: states different symmetry sectors: A) Full diagonalization: T > 0 statics and dynamics meme memory and operations

6. Finite temperature Lanczos method FTLM = Lanczos basis + random sampling: P.P., J. Jaklič (1994) Lanczos basis Matrix elements:exactly with M=max (k,l)

7. Static quantities at T > 0 High – temperature expansion – full sampling: calculated using Lanczos: exactly for k M Ground state T = 0: FTLM gives correct T=0 result

8. Dynamical quantities at T > 0 Short-t (high-ω), high-T expansion: full sampling M steps started with normalized and exact Random sampling: random >> 1

9. Finite size temperature many body levels: 2D Heisenberg model 2D t-J model 2D t-J model: J=0.3 t optimum doping

10. FTLM: advantages and limitations Interpolation between the HT expansion and T=0 Lanczos calculation No minus sign problem: can work for arbitrary electron filling and dimension works best for frustrated correlated systems: optimum doping So far the leading method for T > 0 dynamical quantities in strong correlation regime - competitors: QMC has minus sign + maximum entropy problems, 1D DMRG: so far T=0 dynamics T > 0 calculation controlled extrapolation to g.s. T=0 result Easy to implement on the top of usual LM and very pedagogical Limitations very similar to usual T=0 LM (needs storage of Lanczos wf. and calculation of matrix elements): small systems N < 30 many static and dynamical properties within t-J and other models calculated, reasonable agreement with experimental results for cuprates

11. Microcanonical Lanczos method Long, Prelovsek, El Shawish, Karadamoglou, Zotos (2004) thermodynamic sum can be replaced with a single microcanonical state in a large system MC state is generated with a modified Lanczos procedure Advantage: no Lanczos wavefunction need to be stored, requirement as for T = 0

12. Example: anomalous diffusion in the integrable 1D t-V model insulating T=0 regime (anisotropic Heisenberg model) T >> 0: huge finite-size effect (~1/L) ! convergence to normal diffusion ?

13. Resistivity and optical conductivity of cuprates Takagi et al (1992)Uchida et al (1991) ρ ~ aT pseudogap scale T * mid-IR peak at low doping universal marginal FL-type conductivity resistivity saturation normal FL: ρ ~ cT 2, σ(ω) Drude form

14. Low doping: recent results Ando et al (01, 04) 1/mobility vs. doping Takenaka et al (02) Drude contribution at lower T<T * mid – IR peak at T>T *

15. FTLM + boundary condition averaging t-J model: N = 16 – 26 1 hole Zemljic and Prelovsek, PRB (05)

16. Intermediate - optimum doping van der Marel et al (03) BSCCO t-J model: c h = 3/20 ρ ~ aT reproduces linear law

17. deviation from the universal law Origin of universality: assuming spectral function of the MFL form increasing function of ω !

18. Low doping mid- IR peak for T < J: related to the onset of short-range AFM correlations position and origin of the peak given by hole bound by a spin-string resistivity saturation onset of coherent ‘nodal’ transport for T < T * N = 26, N h = 1

19. Comparison with experiments underdoped LSCOintermediate doping LSCO Ando et al. Takagi et al. normalized resistivity: inverse mobility agreement with experiments satisfactory both at low and intermediate doping no other degrees of freedom important for transport (coupling to phonons) ?

20. Cuprates – normal state: anomalous spin dynamics LSCO: Keimer et al. 91,92 Zn-substituted YBCO: Kakurai et al. 1993 Low doping: inconsistent with normal Fermi liquid ~ normal FL: T-independent χ’’(q,ω)

21. Spin fluctuations - memory function approach goal: overdamped spin fluctuations in normal state + resonance (collective) mode in SC state Spin susceptibility: memory function representation - Mori ‘mode frequency’‘spin stiffness’ – smoothly T, q-dependent at q ~ Q fluctuation-dissipation relation Less T dependent,saturates at low T damping function

22. large damping: FTLM results for t-J model: N=20 sites J=0.3 t, T=0.15 t > T fs ~ 0.1 t N h =2, c h =0.1 Argument: decay into fermionic electron-hole excitations ~ Fermi liquid collective AFM mode overdamped

23. scaling function: ω/T scaling for ω > ω FL Zn-substituted YBCO 6.5 : Kakurai different energies Normal state: ω/T scaling – T>T FL parameter ‘normalization’ function cuprates: low doping Fermi scale ω FL PRL (04)

24. Crossover FL: NFL – characteristic FL scale c h < c h * ~ 0.15: c h > c h * : non-Fermi liquid Fermi liquid t-J model - FTLM N=18,20 PRB(04) FTLM T=0 Lanczos NFL-FL crossover

25. Re-analysis of NMR relaxation spin-spin relaxation + INS UD OD Berthier et al 1996 + C Q from t-J model UD OD Balatsky, Bourges (99)

26. Summary FTLM: T>0 static and dynamical quantities in strongly correlated systems advantages for dynamical quantities and anomalous behaviour t – J model good model for cuprates (in the normal state) optical conductivity and resistivity: universal law at intermediate doping, mid-IR peak, resisitivity saturation and coherent transport for T<T * at low doping, quantitative agreement with experiments spin dynamics: anomalous MFL-like at low doping, crossover to normal FL dynamics at optimum doping small systems enough to describe dynamics in correlated systems !

29. AFM inverse correlation length κ Balatsky, Bourges (99) κ weakly T dependent and not small even at low doping κ not critical

30. Inelastic neutron scattering: normal + resonant peak Bourges 99: YBCO q - integrated Doping dependence:

31. Energy scale of spin fluctuations = FL scale characteristic energy scale of SF: T < T FL ~ ω FL : FL behavior T > T FL ~ ω FL : scaling phenomenological theory: simulates varying doping Kondo temperature ?

32. Local spin dynamics ‘marginal’ spin dynamics J.Jaklič, PP., PRL (1995)

33. Hubbard model: constrained path QMC