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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