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

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

quantum critical point, static stripes, crossover ?

Cuprates: phase diagram

t – J model

interplay : electron hopping + spin exchangesingle 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 > Tfs

finite size temperature

n.n.n. hopping

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

me

memory and operations

Finite temperature Lanczos method

FTLM = Lanczos basis + random sampling: P.P., J. Jaklič (1994)

Lanczos basis

Matrix elements: exactly with M=max (k,l)

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

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

Finite size temperature

many body levels:

2D Heisenberg model 2D t-J model 2D t-J model: J=0.3 t

optimum doping

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

Microcanonical Lanczos method

Long, Prelovsek, El Shawish, Karadamoglou, Zotos (2004)

thermodynamic sum can be replaced with a single microcanonical statein 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

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 ?

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

resistivitysaturation

normal FL: ρ ~ cT2 , σ(ω) Drude form

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*

FTLM + boundary condition averaging

t-J model: N = 16 – 261 hole

Zemljic and Prelovsek, PRB (05)

Intermediate - optimum doping

van der Marel et al (03)

BSCCO

t-J model: ch = 3/20

ρ ~ aT

reproduceslinear law

deviation from the universal law

Origin of universality:

assuming spectral function ofthe MFL form

increasing function of ω !

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

Comparison with experiments

underdoped LSCO intermediate 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) ?

Cuprates – normal state: anomalous spin dynamics

LSCO: Keimer et al. 91,92Zn-substituted YBCO: Kakurai et al. 1993

Low doping:

inconsistent with normal Fermi liquid

~

normal FL: T-independent χ’’(q,ω)

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

large damping:

FTLM results for t-J model: N=20 sites

J=0.3 t, T=0.15 t > Tfs ~ 0.1 tNh=2, ch=0.1

Argument: decay into fermionic electron-hole excitations ~ Fermi liquid

collective AFM mode overdamped

scaling function: ω/T scaling for ω > ωFL

Zn-substituted YBCO6.5 : Kakuraidifferent energies

Normal state: ω/T scaling – T>TFL

parameter

‘normalization’ function

cuprates: low doping

Fermi scale ωFL

PRL (04)

Crossover FL: NFL – characteristic FL scale

ch < ch* ~ 0.15:

ch > ch* :

non-Fermi liquid

Fermi liquid

t-J model - FTLM N=18,20

PRB(04)

FTLMT=0 Lanczos

NFL-FL crossover

Re-analysis of NMR relaxation

spin-spin relaxation

+ INSUD

OD

Berthier et al 1996

+ CQ

from t-Jmodel

UD

OD

Balatsky, Bourges (99)

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 !

AFM inverse correlationlength κ

Balatsky, Bourges (99)

κ weakly T dependent and not small even at lowdoping

κ not critical

Inelastic neutron scattering: normal + resonant peak

Bourges 99: YBCO

q - integrated

Doping dependence:

Energy scale of spin fluctuations = FL scale

characteristic energy scale of SF:

T < TFL ~ ωFL : FL behavior

T > TFL ~ ωFL: scaling

phenomenological theory:

simulates varying doping

Kondo temperature ?

Local spin dynamics

‘marginal’ spin dynamics

J.Jaklič, PP., PRL (1995)

Hubbard model: constrained path QMC