Faculty of Mathematics and Physics, University of Ljubljana,

1. Finite-temperature dynamics of small correlated systems: anomalous properties for cuprates P. Prelovšek, M. Zemljič, I. Sega and J. Bonča Faculty of Mathematics and Physics, UniversityofLjubljana, J. Stefan Institute, Ljubljana, Slovenia 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. Cuprates: phase diagram quantum critical point, static stripes, crossover ?

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

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 operations memory and me

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, approx. 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 exact and M steps started with normalized 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 quantitiesin 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) resistivity saturation ρ ~ aT mid-IR peak at low doping pseudogap scale T* universal marginal FL-type conductivity normal FL:ρ ~ cT2 , 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 Zemljic and Prelovsek, PRB (05) t-J model: N = 16 – 26 1 hole

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

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, Nh = 1

19. Comparison with experiments normalized resistivity: inverse mobility underdoped LSCO intermediate doping LSCO Ando et al. Takagi et al. • 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 Low doping: Zn-substituted YBCO: Kakurai et al. 1993 LSCO: Keimer et al. 91,92 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 damping function ‘mode frequency’ ‘spin stiffness’ – smoothly T, q-dependent at q ~ Q fluctuation-dissipation relation Less T dependent,saturates at low T

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

23. Normal state: ω/T scaling – T>TFL PRL (04) parameter cuprates: low doping Fermi scale ωFL ‘normalization’ function scaling function: ω/Tscaling for ω > ωFL Zn-substituted YBCO6.5 : Kakurai different energies

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

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

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 Doping dependence: Bourges 99: YBCO q - integrated

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

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

33. Hubbard model:constrained path QMC