Faculty of Mathematics and Physics, University of Ljubljana,

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Finite-temperature dynamics of small correlated
systems anomalous properties for cuprates
P. Prelovek, M. Zemljic, I. Sega and J. Bonca
Faculty of Mathematics and Physics, University of
Ljubljana, J. Stefan Institute, Ljubljana,
Slovenia
Sherbrooke, July 2005
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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, ?/T scaling, NFL-FL
  crossover

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Cuprates phase diagram
quantum critical point, static stripes, crossover
?
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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.Jaklic PP
T gt Tfs
finite size temperature
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Exact diagonalization of correlated electron
systems Tgt0

• Basis states system with N sites
• Heisenberg model states
• t J model states
• Hubbard model states

• different symmetry sectors

A) Full diagonalization T gt 0 statics and
dynamics
operations
me
memory and
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Finite temperature Lanczos method
FTLM Lanczos basis random sampling P.P., J.
Jaklic (1994)
Lanczos basis
Matrix elements
exactly with Mmax (k,l)
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Static quantities at T gt 0
High temperature expansion full sampling
calculated using Lanczos exactly for k lt M,
approx. for k gt M
Ground state T 0
FTLM gives correct T0 result
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Dynamical quantities at T gt 0
Short-t (high-?), high-T expansion full sampling
exact
and
M steps started with normalized
Random sampling
random
gtgt 1
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Finite size temperature
many body levels
2D Heisenberg model 2D t-J model
2D t-J model J0.3 t
optimum doping
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FTLM advantages and limitations

• Interpolation between the HT expansion and T0
  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 gt 0 dynamical
  quantities in strong
• correlation regime - competitors QMC has
  minus sign maximum
• entropy problems, 1D DMRG so far T0
  dynamics
• T gt 0 calculation controlled extrapolation
  to g.s. T0 result
• Easy to implement on the top of usual LM and
  very pedagogical
• Limitations very similar to usual T0 LM
  (needs storage of Lanczos
• wf. and calculation of matrix elements)
  small systems N lt 30

many static and dynamical properties within t-J
and other models calculated, reasonable
agreement with experimental results for cuprates
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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
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Example anomalous diffusion in the integrable 1D
t-V model insulating T0 regime
(anisotropic Heisenberg model)
T gtgt 0 huge finite-size effect (1/L)
! convergence to normal diffusion ?
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Resistivity and optical conductivity of cuprates
Takagi et al (1992)
Uchida et al (1991)
resistivity saturation
? aT
mid-IR peak at low doping
pseudogap scale T
universal marginal FL-type conductivity
normal FL ? cT2 , s(?) Drude form
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Low doping recent results
Ando et al (01, 04)
1/mobility vs. doping
Takenaka et al (02) Drude contribution at lower
TltT mid IR peak at TgtT
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FTLM boundary condition averaging
Zemljic and Prelovsek, PRB (05)
t-J model N 16 26 1 hole
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Intermediate - optimum doping
t-J model ch 3/20
van der Marel et al (03)
BSCCO
reproduces linear law
? aT
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deviation from the universal law
Origin of universality
assuming spectral function of the MFL form
increasing function of ? !
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Low doping
mid- IR peak for T lt 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 lt T
N 26, Nh 1
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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) ?

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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 ?(q,?)
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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
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large damping
collective AFM mode overdamped
Argument decay into fermionic electron-hole
excitations Fermi liquid
FTLM results for t-J model N20 sites
J0.3 t, T0.15 t gt Tfs 0.1 t Nh2, ch0.1
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Normal state ?/T scaling TgtTFL
PRL (04)
parameter
cuprates low doping
Fermi scale ?FL
normalization function
scaling function ?/T scaling for ? gt ?FL
Zn-substituted YBCO6.5 Kakurai different
energies
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Crossover FL NFL characteristic FL scale
PRB(04)
t-J model - FTLM N18,20
ch lt ch 0.15
non-Fermi liquid
ch gt ch
Fermi liquid
T0 Lanczos
FTLM
NFL-FL crossover
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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
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Summary

• FTLM Tgt0 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 TltT 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 !

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AFM inverse correlation length ? Balatsky,
Bourges (99) ? weakly T dependent and not small
even at low doping ? not critical
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Inelastic neutron scattering normal resonant
peak
Doping dependence
Bourges 99 YBCO
q - integrated
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Energy scale of spin fluctuations FL scale
characteristic energy scale of SF
T lt TFL ?FL FL behavior
T gt TFL ?FL
scaling
phenomenological theory
Kondo temperature ?
simulates varying doping
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Local spin dynamics
J.Jaklic, PP., PRL (1995)
marginal spin dynamics
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Hubbard model constrained path QMC