How to quantify correlation in correlated electron system
Transcript of How to quantify correlation in correlated electron system
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How to quantify correlation
in correlated electron system
Krzysztof Byczuk
Institute of Theoretical PhysicsDepartment of Physics, University of Warsaw
andCenter for Electronic Correlations and Magnetism
Augsburg University
May 31st, 2010
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Collaboration
Dieter Vollhardt - Augsburg University
Walter Hofstetter - Frankfurt University
Jan Kunes - Prague, Academy of Sciences
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Aim of this talk
CORRELATION
• What is it?
• How to quantify it?
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Correlation
• Correlation [lat.]: con+relatio (“with relation”)
• Mathematics, Statistics, Natural Science:
〈xy〉 6= 〈x〉〈y〉
The term correlation stems from mathematical statistics and means that twodistribution functions, f(x) and g(y), are not independent of each other.
• In many body physics: correlations are effects beyond factorizing approximations
〈ρ(r, t)ρ(r′, t′)〉 ≈ 〈ρ(r, t)〉〈ρ(r′, t′)〉,
as in Weiss or Hartree-Fock mean-field theories
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Spatial and temporal correlations everywhere
car traffic
air traffic
human traffic
electron traffic
more .....
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Spatial and temporal correlations neglected
time/space average insufficient
〈ρ(r, t)ρ(r′, t′)〉 ≈ 〈ρ(r, t)〉〈ρ(r′, t′)〉 = disaster!
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Spatial and temporal correlations neglected
Local density approximation (LDA) disaster in HTC
T
T
T
N
*
x
SCAFSC
0
ρ~ T 2
ρ ~ T
−
T
Mot
t ins
ulat
or
LaCuO4 Mott (correlated) insulator predicted to be a metal
Partially curred by (AF) long-range order ... but correlations are still missed
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Correlated electrons
Narrow d,f-orbitals/bands → strong electronic correlations
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Electronic bands in solids
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Band insulators, e.g. NaCl
Correlated metals, e.g. Ni, V2O3, Ce
Simple metals, e.g. Na, Al
Atomic levels, localized electrons |Riσ〉
Narrow bands, |Riσ〉 ↔ |kσ〉
Broad bands, extended Bloch waves |kσ〉
Wave function overlap ∼ tij = 〈i|T̂ |j〉 → |Ek| ∼ bandwidth W
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Electronic bands in solids
Mean time τ spent by the electron on an atom in a soliddepends on the band width W
group velocity vk ≈lattice spacing
mean time=
a
τ
Heisenberg principle Wτ ∼ ~
a
τ∼
aW
~=⇒ τ ∼
~
W
Small W means longer interaction with another electron on the same atomStrong electronic correlations
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Optical lattices filled with bosons or fermionsGreiner et al. 02, and other works
atomic trap and standing waves of light create optical lattices a ∼ 400− 500nm
alkali atoms with ns1 electronic state J = S = 1/2
F = J + I
87Rb, 23Na, 7Li - I = 3/2: effective bosons6Li - I = 1, 40K - I = 4: effective fermions
dipol interaction − hopping
atom scattering − Hubbard U
Esolidint ∼ 1− 4eV ∼ 104K, Esolid
kin ∼ 1− 10eV ∼ 105K
Eopticalkin ∼ Eoptical
int ∼ 10kHz ∼ 10−6K
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Quantifying correlations
How many correlation is therein correlated electron systems?
We need information theory tools to address this issue.
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Classical vs. Quantum Information Theory
Probability distribution vs. Density operator
pk ←→ ρ̂ =∑
k
pk|k〉〈k|
Shannon entropy vs. von Neumann entropy
I = −〈log2 pk〉 = −∑
k
pk log2 pk ←→ S = −〈ln ρ̂〉 = −Tr[ρ̂ ln ρ̂]
Two correlated (sub)systems have relative entropy
I = I1 + I2 −∆I ←→ S = S1 + S2 −E
∆I(pkl||pkpl) = −∑
kl
pkl[log2
pkl
pkpl
]←→ E(ρ̂||ρ̂1 ⊗ ρ̂2) = −Tr[ρ̂(ln ρ̂− ln ρ̂1 ⊗ ρ̂2)]
Relative entropy vanishes in the absence of correlations (product states)
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Asymptotic distiguishability
Quantum Sanov theorem:Probability Pn that a state σ̂ is not distinguishable from a state ρ̂ in n measurementson σ̂, when n≫ 1, is
Pn ≈ e−nE(ρ̂||σ̂).
Relative entropy E(ρ̂||σ̂) as a ’distance’ between quantum states.
We calculate
• von Neumann entropies and
• relative entropies
for and between different correlated and uncorrelated(product) states of the Hubbard model.
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Correlated fermions on crystal and optical lattices
H = −∑
ijσ
tijc†iσcjσ + U
∑
i
ni↑ni↓
t
U
t
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InIn
Out
TIME|i, 0〉 → |i, ↑〉 → |i, 2〉 → |i, ↓〉
fermionic Hubbard model
P.W. Anderson, J. Hubbard, M. Gutzwiller, J. Kanamori, 1960-63
Local Hubbard physics
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Origin of genuine many-body correlation
H = Hhopping + H interactionloc
[
Hhopping, H interactionloc
]
6= 0
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DMFT for lattice fermions
Replace (map) full many-body lattice problem by a single-site coupled to dynamicalreservoir and solve such problem self-consistently
All local dynamical correlations included exactly
Space correlations neglected - mean-field approximation
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Local Entropy and Local Relative Entropy
Local density operator:ρ̂i = Trj 6=iρ̂
Local entropy:
S[ρ̂i] = −
4∑
k=1
pk ln pk,
where
p1 = 〈(1−ni↑)(1−ni↓)〉, p2 = 〈ni↑(1−ni↓)〉, p3 = 〈(1−ni↑)ni↓〉, p4 = 〈ni↑ni↓〉.
A.Rycerz, Eur. Phys. J B 52, 291 (2006);
D. Larsson and H. Johannesson, Phys. Rev. A 73, 042320 (2006)
Generalized equations for local relative entropy.KB, D. Vollhardt, ’09
Expectation values for correlated states are determined from DMFT solutionand for uncorrelated states from Hartree-Fock solutions.
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Local Entropy and Local Relative Entropy
Local entropy
S(ρ̂) = −TrAρ̂A log ρ̂A = −∑
α
pα log pα.
Local relative entropy
E(ρ̂||σ̂) = −TrAρ̂A(log ρ̂A − log σ̂A) = −∑
α
pα(log pα − log pσα).
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Correlation and Mott Transition
0 0.5 1 1.5 2 2.5 3Interaction, U
0.001
0.01
0.1
1
Dou
ble
occu
panc
y, d
0 0.5 1 1.5 2 2.5Interaction, U
0
0.5
1
1.5
2
Loca
l ent
ropi
es, S
SS
1S
22 ln 2
ln 2
A(0)
0 0.5 1 1.5 2 2.5Interaction, U
0
0.5
1
1.5
2
Rel
ativ
e lo
cal e
ntro
pies
, E
E1
E2
E3
A(0)
S(ρ̂) = −Tr[ρ̂ ln ρ̂]
E(ρ̂||σ̂) = −Tr[ρ̂ ln ρ̂− ρ̂ ln σ̂]
Product (HF) states:
|0〉 =∏kF
kσ a†kσ|v〉 - U = 0 limit
|a〉 =∏NL
i a†iσi|v〉 - atomic limit
S = S(ρ̂DMFT )
S1 = S(ρ̂0)
S2 = S(ρ̂a)
E1 = E(ρ̂DMFT ||ρ̂0)
E2 = E(ρ̂0||ρ̂DMFT )
E3 = E(ρ̂a||ρ̂DMFT )
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Correlation and Antiferromagnetic OrderS(ρ̂) = −Tr[ρ̂ ln ρ̂]
E(ρ̂||σ̂) = −Tr[ρ̂ ln ρ̂− ρ̂ ln σ̂]
Product (HF) states:
|0〉 =∏kF
k∈(A,B) a†kA↑a
†kB↓|v〉 - Slater limit
|a〉 =∏NL
i∈(A,B) a†iA↑a
†iB↓|v〉 - Heisenberg limit
S = S(ρ̂DMFT )
S0 = S(ρ̂0)
Sa = S(ρ̂a)
D1 = E(ρ̂DMFT ||ρ̂0)
D2 = E(ρ̂0||ρ̂DMFT )
D4 = E(ρ̂a||ρ̂DMFT )
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Correlation in Transition Metal-Oxides
MnO FeO CoO NiO
[Ar]3d5s2 [Ar]3d6s2; [Ar]3d7s2; [Ar]3d8s2
0 0.2 0.4 0.6 0.8 1 1.2Doping
0
1
2
3
4
5
Ent
ropi
es
S(dmft)
S(lda)
E(dmft|lda)
E(lda|dmft)
NiO with doping
LDA entropy represents number of local states - maximum at d5
Interaction reduces this number and it becomes almost the same
Non-interacting system chemistry decides how much it is correlated
Doping reduces number of local states in noninteracting system
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Summary
• We used entropy and relative entropies to quantify in numbers correlation incorrelated electron systems.
• Examples for Hubbard model.
• Different correlations in paramagnetic and in antiferromagnetic cases.
• Different amount of correlation in transition metal oxides: MnO 3 times morecorrelated then NiO.
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Calculation details
Consider a pure state (maximal information)
|Ψ〉 =∑
αβ
Ψαβ|α〉|β〉
of a system which is composed of two subsystems A = {|α〉} and B = {|β〉}.
Density operator (Schmidt decomposition)
ρ̂ =∑
k
pk|k〉〈k| = |Ψ〉〈Ψ|.
Entropy
S(ρ̂) = −〈log ρ̂〉 = −Trρ̂ log ρ̂ = −∑
k
pk log pk = 0,
becausepk = δk,Ψ.
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Calculation details
Trace out the B subsystem, reduced density operator
ρ̂A = TrB|Ψ〉〈Ψ| =∑
β
〈β|Ψ〉〈Ψ|β〉 =∑
α1,α2
|α1〉∑
β
Ψα1,βΨ†β,α2〈α2| =
∑
α1,α2
|α1〉ρα1,α2〈α2|.
Subsystem A is in a mixed state (reduced information).
Introduce projector and transition operators
P̂i = |i〉〈i|, T̂ij = |i〉〈j|,
thenρα1α2 =
∑
β
Ψα1,βΨ†β,α2
= 〈Ψ|P̂α1T̂α1,α2P̂α2|Ψ〉†.
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Calculation details
Consider a single lattice site (DMFT) as the A subsystem
|α〉 = {|0〉, | ↑〉, | ↓〉, | ↑↓〉},
then
P̂α =
(1− n̂↑)(1− n̂↓)n̂↑(1− n̂↓)(1− n̂↑)n̂↓
n̂↑n̂↓,
and
T̂α1,α2 =
1 c↑ c↓ c↓c↑c†↑ 1 c†↑c↓ −c↓c†↓ c†↓c↑ 1 c↑
c†↑c†↓ −c†↓ c†↑ 1
.
Assuming absence of any off-diagonal order 〈Ψ|cσ|Ψ〉 = 〈Ψ|cσc−σ|Ψ〉 the reduceddensity operator is diagonal
ρα1α2 = p1|0〉〈0|+ p2| ↑〉〈↑ |+ p3| ↓〉〈↓ |+ p4| ↑↓〉〈↑↓ |,
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Calculation details
with matrix elementspα = 〈Ψ|P̂α|Ψ〉
determined with an arbitrary pure state |Ψ〉 (exact, DMFT, HF, etc.) of the fullsystem.
It is straightforward to derive for an arbitrary mixed state ρ̂ of the full system.