Post on 30-Apr-2020
Isosbestic Points and Kinks: Fingerprints of Electronic Correlations
Dieter Vollhardt
Supported by Deutsche Forschungsgemeinschaft through SFB 484
Symposium celebrating Prof. Dr. Hilbert v. Löhneysen‘s 60th Birthday;
Karlsruhe, October 27, 2006
• Crossing ("isosbestic") points
• Kinks in the effective electronic dispersion
Outline:
Characteristic energies/temperatures (“fingerprints”) due to electronic correlations
x
f(x,y)
y=0
0.1
0.2
0.3
x*(y)
crossing region
, x*(y) = const sharp crossing point
"isosbestic" point
I. Crossing (="isosbestic") points
What do they tell us?
Isosbestic point: “Specific wavelength at which two (or more) chemical species have the same extinction coefficient.”
Scheibe (1937)1. Isosbestic points in chemistry
Example: Solution of components A, B with concentrations nA, nB
Exti
ncti
on c
oeff
.
A: 1,4-naphtaquinone diphenylhydrazoneB: 4-dimethylaminoazobenzene
Isosbestic: from Greek isos “equal” + sbestos, verbal adj. from sbennynai“to quench, extinguish” = “equal attenuance”
Cohen and Fischer (1962)
Origin of isosbestic points in the absorption coefficient of a solution
const. ( , ) ( ) ( ) ( )A B A A A A Bn n n n n n nα ω α ω α ω= + = ⇒ = + −
)
Applications, e.g.:• Chemical kinetics: Reference points for reaction rates• Clinical chemistry: Test of spectrophotometer
Isosbestic point at ω*for all concentrations nA
If ( *) ( *)A Bα ω α ω= ⇒*
( , ) 0A
A
nn ω
α ω∂=
∂
1*ω 2*ω
Isosbestic points Special (linear) functional dependence
⇔
Systems sizes L=6–18
Conductance distribution near the Anderson transitionSlevin et al. (2003)
2. "Isosbestic points" in critical phenomena
Isosbestic points Scaling of variables/scale-invariance⇔
Heavy fermion systems
Brodale, et al. (1986) Phillips et al. (1987)
Specific heat of fermionic systems
Heavy fermion systems
Schlager, Schröder, Welsch, v. Löhneysen (1993)
Non-Fermi liquid behaviorSteglich, Geibel, Gloos, Olesch,Schank, Wassilew, Loidl, Krimmel,Stewart (1994)
Specific heat of fermionic systems
X: thermodynamic variable
DV (1996)A. Why do curves cross?B. Width of crossing regime?
Specific heat of fermionic systems
Sum rule
3He: X=P ⎫⎪⎬⎪⎭
curves must cross, FT T>
, FT T<<
0
0
CPCP
∂<
∂∂
>∂
DV (1996)
Correlation effect
Excitation of low-energy(spin) degrees of freedom:
00
TC SP P
→∂ ∂=
∂>
∂
A. Why do curves cross?
0T
1 ( , ) ' ( ', )lim constln 'B Bk
S T X X dT C T XX k T X
∞
→∞
∂ ∂= =
∂ ∂∫
Specific heat of fermionic systems
X: thermodyn. variableξ: conjugate variable
2
2
( , )T XT X
χ
ξ∂ ∂⎛ ⎞⎜ ⎟∂ ∂⎝ ⎠
∼Width
Susceptibility
Isosbestic points low + T-insensitive compressibility
⇔
B. Width of crossing regime?
SX T
ξ∂ ∂=
∂ ∂XPBμ
VMN
ξ−
Vanishing width if:
3He( , )T Xχa) Weak T-dependence of
-, TV VP
X P V χξ κ∂= = ⇒ = − =
∂
Specific heat of fermionic systems
X: thermodyn. variableξ: conjugate variable
Vanishing width if:
Isosbestic points linear susceptibility + turning point in χ(T)
⇔
B. Width of crossing regime?
2
2an( , ) ( ,0) ( , 0)d 0dT T TdT
X Xξ χ χ= =
, =MX B ξ=
Schlager, Schröder, Welsch,v. Löhneysen (1993)
( , )T Xχb) Linear susceptibility
SX T
ξ∂ ∂=
∂ ∂XPBμ
VMN
ξ−
2
2
( , )T XT X
χ
ξ∂ ∂⎛ ⎞⎜ ⎟∂ ∂⎝ ⎠
∼Width
Susceptibility
Specific heat of fermionic systems
Spectral function
Hubbard model,n=1, T=0,Bethe-DOS,DMFT(NRG)
Eckstein, Kollar, DV (cond-mat 0609464)Width of crossing region?
Sum rule: curves must cross:⇒
1( , ) ''( 0, ),A U G i Uω ωπ
= − +
U=0
Bulla, Hewson, Pruschke (1998)
Spectral function
Eckstein, Kollar, DV (cond-mat 0609464)
Analytic solution of in weak coupling theory
Sharp isosbestic points: only if A(ω,U=0) has van-Hove singularity at band-edge
⇒
Isosbestic point “free” band-width
⇔
U=0
Electrical conductivity σ(ω)
Uchida et al. (1991)
T= room temperature
Spinless Falicov-Kimball model T=0.005
Freericks, Zlatić (2003)
f-sum rule:2
Re ( , ) ed n nmπω σ ω
∞
−∞=∫
Isosbestic points “free” band-width?⇔
“Kinks” at 40-70 meV due to coupling of electrons to phonons or spin fluctuations ?
High-Tc cuprates
Zhou et al. (2006)
Metal (e.g., Tungsten) surface
160 meV:Surface phonon
Rotenberg et al. (2000)
Kink due to electron-phonon coupling
Electron-phonon (boson) correction of electronic dispersionAshcroft, Mermin; Solid State Physics (1976)
Kinks due to electron-boson coupling
Kinks due to electron-electron hybridization
f-electrons
d-electrons
PES of quasi-1D electronic structures on (3x1)-Br/Pt(110) surface
Menzel et al. (2005)
300meV: too high for phonons
High-energy kinks
Kinks due to coupling of electrons to what?
Kinks in strongly correlated electron systems
Yoshida et al. (2005)Ekaterinburg – Augsburg – Stuttgart collaboration,
Nekrasov et al. (2004, 2006)
Renormalization of LDA-bands by self-energy
* 0.2 eVω ≈Kinks at Origin of kinks in a purely electronic theorywith one type of electron ?
Byczuk, Kollar, Held, Yang, Nekrasov, Pruschke, DV; cond-mat 0609594
Strongly correlated paramagnetic metal
Byczuk, Kollar, Held, Yang, Nekrasov, Pruschke, DV; cond-mat 0609594
⇓KK
New energy scale
•Meaning of ω* ? •Range of FL region ? •Consequences of ω* ?
*linear flinear for
or
1
( )( ) ( [ ( ]) ( ))DMFT
GG G
ωω ω
ωω ω ω ωμ
≤Ω≤
⎯⎯⎯→Σ = − − Δ+ hybridization fct.
FL regime
FL regime
Byczuk, Kollar, Held, Yang, Nekrasov, Pruschke, DV; cond-mat 0609594 0* FLZ Dω =
Charact. Scaleof non-interact. system
analyt. given by Z + non-interact. quantities
0.2 eV , / * 0.35; ' , ' 0.65; 1.5 eV 0.2 e V
LDA
LDAFL FLZ E Z m m
EZ E c Z
ωω
⎧ = =⎪= ⎨ ± =⎪⎩
≤≥ ≥
kk
k
FL regime
outside FL regime
SrVO3 (U=5.55 eV, J=1.0 eV)
SrVO3
Kinks in effective dispersion: • Generic features of strongly correlated electrons• Provide quantitative microscopic information
HTS: * 0.2 0.4 eVω ≈ −Byczuk, Kollar, Held, Yang, Nekrasov, Pruschke, DV; cond-mat 0609594
Graf et al., cond-mat 0607319
Bi2212