Optical Conductivity of Strongly Correlated Electrons in High ...

Optical Conductivity of 

Strongly Correlated Electron Systems 

in High Dimensions 

N. Blümer 

Universität Mainz 

Outline 

Introduction 

Optical Conductivity near a MIT for d → ∞ 

General Dispersion Formalism 

Numerical QMC/MEM Results for Half-filled Hubbard Model 

Conclusion

Introduction 

Definition and Measurement of the Optical Conductivity σ(ω) 

linear response 

homogeneous system 

limit q → 0 

� 

specular reflectivity r(ω) = 

⇒ Jα(ω) = 

� 

� 

� 

� 1−√ɛ(ω) 1+ √ ɛ(ω) 

� 

� 

� 

� 

d� 

σαβ(ω) Eβ(ω) 

β=1 

2 

K-K 

4πi 

−→ dielectric function ɛ(ω) = 1 + ω σ(ω) 

Nils Blümer · Johannes-Gutenberg-Universität Mainz · February 27, 2003 ⊳ ←↪ △ ⊲ 1

Introduction 

Definition and Measurement of the Optical Conductivity σ(ω) 

linear response 

homogeneous system 

limit q → 0 

� 

specular reflectivity r(ω) = 

The f-sum rules 

bounded absorption spectrum: 

2 

π 

ɛ(ω) ω→∞ 

−→ 1 − ω2 p 

� ∞ 

0 

ω 2 

⇒ Jα(ω) = 

� 

� 

� 

dω ω Im ɛ(ω) = ω 2 p 

� 1−√ɛ(ω) 1+ √ ɛ(ω) 

Full universal f-sum rule: ω 2 p = 4πne2 

m 

� 

� 

� 

� 

d� 

σαβ(ω) Eβ(ω) 

β=1 

2 

K-K 

4πi 

−→ dielectric function ɛ(ω) = 1 + ω σ(ω) 

=⇒ 

� ∞ 

0 

dω Re σ(ω) = ω2 p 

8 

(independent of T , interactions etc.) 

Optical 

f-sum rule 


Kubo formalism 

For continuum systems: 

σαβ(ω) = 

V 

�(ω + i0 + ) 

� ∞ 

Lattice case: ˆ H0 = ˆ K + ˆ Hint ≡ � 

0 

n e 2 

dt e i(ω+i0+ )t † 

〈[ˆj α(t), ˆj β (0)]〉 + i 

m(ω + i0 + ) δαβ 

t 

ij,σ 

0 ij ĉ † 

iσĉjσ + ˆ � � 

Hint ˆniσ 

Peierls construction tij = t0 ij exp � −i e 

derive ˆj from ˆ H0 −→ ˆ H0 − V 

c ˆj · A + O(A2 ) 

Bravais lattice, one-band: ˆ K = 1 

V 

� 

k,σ ɛk ˆnk,σ, ˆj = e 

V � 

c� (Ri − Rj) · A � 

� 

k,σ vk ˆnk,σ (vk = 1 

� ∇ɛk) 




σαβ(ω) = 

V 

�(ω + i0 + ) 

� ∞ 


0 

n e 2 

dt e i(ω+i0+ )t † 

〈[ˆj α(t), ˆj β (0)]〉 + i 

m(ω + i0 + ) δαβ 

t 

ij,σ 

0 ij ĉ † 





c ˆj · A + O(A2 ) 


V 

Hypercubic lattice, NN hopping: 

ˆK = 

d� 

α=1 

� ∞ 

0 

ˆKα ; ˆ Kα = 2 

N 

dω σαα(ω) = − π 

2 

� 


V � 

� 

Re ĉ † 

Ri+eαĉRi ; ˆjα = e 

iσ 

e2a2 V �2 〈 ˆ Kα〉 = − π 

2 

e 2 a 2 

V � 2 

2 

� N 

c� (Ri − Rj) · A � 

� 


� ∇ɛk) 

Im � 

iσ 

ĉ † 

Ri+eα ĉ Ri 

1 

d 〈 ˆ K〉 ≡ − σ0 

4d 〈ɛ〉 




σαβ(ω) = 

V 

�(ω + i0 + ) 

� ∞ 


0 

n e 2 

dt e i(ω+i0+ )t † 

〈[ˆj α(t), ˆj β (0)]〉 + i 

m(ω + i0 + ) δαβ 

t 

ij,σ 

0 ij ĉ † 





c ˆj · A + O(A2 ) 


V 

Hypercubic lattice, NN hopping: 

ˆK �= 

d� 

α=1 

� ∞ 

0 

ˆKα ; ˆ Kα = 2 

N 

dω σαα(ω) = − π 

2 

� 


V � 

� 

Re ĉ † 

Ri+eαĉRi ; ˆjα = e 

iσ 

e2a2 V �2 〈 ˆ Kα〉 �= − π 

2 

e 2 a 2 

V � 2 

2 

� N 

c� (Ri − Rj) · A � 

� 


� ∇ɛk) 

Im � 

iσ 

ĉ † 

Ri+eα ĉ Ri 

1 

d 〈 ˆ K〉 ≡ − σ0 

4d 〈ɛ〉 

stacked 


DMFT Treatment of the Optical Conductivity 

Conductivity on Bravais lattice: 

σ ∝ vk 

k 

k 

k 

vk+vk Γ 

k 

k ′ 

k ′ 

v k ′ + vk 

k 

k 

Γ 

k ′′ 

k ′′ 

Γ 

k ′ 

k ′ 

v k ′ + . . . 




σ ∝ vk 

k 

k 

k 

vk+vk Γ 

k 

k ′ 

k ′ 

v k ′ + vk 

DMFT: limit d → ∞ / Z → ∞: scaling t ∝ 1/ √ Z, Σ(k, ω) → Σ(ω) 

local properties depend on lattice only via ρ(ɛ) := 1 

N 

k 

k 

Γ 

k ′′ 

k ′′ 

Γ 

� 

k δ(ɛ − ɛk) 

k ′ 

k ′ 

v k ′ + . . . 




σ ∝ vk 

k 

k 

k 

vk+vk Γ 

k 

k ′ 

k ′ 

v k ′ + vk 

DMFT: limit d → ∞ / Z → ∞: scaling t ∝ 1/ √ Z, Σ(k, ω) → Σ(ω) 

local properties depend on lattice only via ρ(ɛ) := 1 

N 

k 

k 

Γ 

k ′′ 

k ′′ 

Γ 

� 

k δ(ɛ − ɛk) 

d → ∞: Vertex corrections vanish [Khurana, PRL 64, 1990 (1990)] 

k sum → ɛ integral [Pruschke, Cox, and Jarrell, PRB 47, 3553 (1993)] 

� ∞ � ∞ 

σxx(ω) = σ0 dɛ ˜ρxx(ɛ) 

−∞ 

σ0 := 2πe2 N 

�2 V , ˜ρxx(ɛ) := 1 

N 

dω 

−∞ 

′ Aɛ(ω ′ ) Aɛ(ω ′ + ω) nf(ω ′ ) − nf(ω + ω ′ ) 

ω 

� 

k 

(vk) 2 x δ(ɛ − ɛk), Aɛ(ω) := − 1 

π Im 

k ′ 

k ′ 

v k ′ + . . . 

, where 

1 

ω − ɛ − Σ(ω) 


Optical Conductivity Near an MIT for d → ∞ 

Lattice and Density of States 

Nearest-neighbor (NN) hopping 

on hypercubic (hc) lattice: 

d = 3: square-root band edges 

ρ hc 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

-3 -2 -1 0 1 2 3 

Nils Blümer · Johannes-Gutenberg-Universität Mainz · February 27, 2003 ⊳ ←↪ △ ⊲ 4 

ε 

d=3






ρ hc 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

-3 -2 -1 0 1 2 3 


ε 

d=3 

d=4






ρ hc 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

-3 -2 -1 0 1 2 3 


ε 

d=3 

d=5






d → ∞: no band edges 

ρ hc 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

-3 -2 -1 0 1 2 3 


ε 

d=3 

d=∞







Bethe-”lattice” (Z=4) 

ρ hc 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

-3 -2 -1 0 1 2 3 

ε 

d=3 

d=∞ 

"Bethe" 

transport properties of Bethe 

lattice a priori undefined 








Bethe-”lattice” (Z=4) 

ρ hc 

� ∞ 

0 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

-3 -2 -1 0 1 2 3 

ε 

d=3 

d=∞ 

"Bethe" 

transport properties of Bethe 

lattice a priori undefined 

previously coherent definition 

of σ(ω) only for periodically 

continued lattice: anisotropic 

dω σxx(ω) = − σ0 � ɛ 

4d 4 − ɛ2 � 


Optical conductivity for the Bethe lattice 

Tree level picture [Chung and Freericks, PRB 57, 11955 (1998)] 

Ansatz for |ɛ〉 ⇒ K ˆ |ɛ〉 = ɛ |ɛ〉, ˆj |ɛ〉 = 〈vk〉(ɛ) e |ɛ〉 

? 

=⇒ ˜ρxx(ɛ) = � 4 − ɛ 2� ρ(ɛ), 

� ∞ 

dω σxx(ω) = 3 σ0 

4d 〈−ɛ〉 

But: incomplete set of states � ρ(ɛ) = 

0 

√1 π 4−ɛ2 : 1-dimensional 


Optical conductivity for the Bethe lattice 

Tree level picture [Chung and Freericks, PRB 57, 11955 (1998)] 

Ansatz for |ɛ〉 ⇒ K ˆ |ɛ〉 = ɛ |ɛ〉, ˆj |ɛ〉 = 〈vk〉(ɛ) e |ɛ〉 

? 

=⇒ ˜ρxx(ɛ) = � 4 − ɛ 2� ρ(ɛ), 

� ∞ 

dω σxx(ω) = 3 σ0 

4d 〈−ɛ〉 

But: incomplete set of states � ρ(ɛ) = 

0 

√1 π 4−ɛ2 New: general dispersion approach 

: 1-dimensional 

Defines microscopic model with (e.g.) semi-elliptic DOS 

• regular Bravais lattice 

• derivation of all transport properties straightforward 

• conductivity coherent in the noninteracting limit 

• hc symmetry, i.e., isotropic transport (for q → 0) 



For translation-invariant hopping: ˆ K = � � 

i,σ 

τ 

tτ ĉ † 

Ri+τ ,σ ĉ Ri σ 

= � 

k,σ 

ɛ(k)ˆnkσ 

classify contributions to dispersion by taxi-cab hopping distance ||τ || = �d α=1 |τα|: 

∞� 

ɛ(k) = ɛD(k), ɛD(k) = � 

D=1 

||τ ||=D 

tτ e iτ ·k . 



For translation-invariant hopping: ˆ K = � � 

i,σ 

τ 

tτ ĉ † 

Ri+τ ,σ ĉ Ri σ 

= � 

k,σ 

ɛ(k)ˆnkσ 

classify contributions to dispersion by taxi-cab hopping distance ||τ || = �d α=1 |τα|: 

∞� 

ɛ(k) = ɛD(k), ɛD(k) = � 

D=1 

||τ ||=D 

tτ e iτ ·k . 

For d → ∞, almost all vectors with ||τ || = D have form τ = D� 

isotropy 

=⇒ 

ɛD(k) = 

2 

tD 

D � �D/2 d 

BD(k) 

D! 2 

BD(k) = 

� �D/2 2 

d 

� 

α D�=α D−1�=···�=α1 

Functions BD(k) fulfill a recursion relation . . . 

i=1 

eαi with αi �= αj. 

cos(kα D ) cos(kα D−1 ) . . . cos(kα1 ) 


BD+1(k) = B1(k) BD(k) − D BD−1(k) + O(1/ √ d) 

of the Hermite polynomial type. Consequently: BD(k) = HeD(B1(k)). 

With initial condition B1(k) = 

ɛ(k) = 

∞� 

D=1 

� 2 

d 

� 

α 

cos(kα) ≡ ɛ hc 

k 

t ∗ D 

√ D! HeD(ɛ hc 

k ) =: F(ɛ hc 

k ) t ∗ D = 

and orthogonality relation: 

1 

√ 2πD! 

� ∞ 

dɛ F(ɛ) HeD(ɛ) e 

−∞ 

−ɛ2 /2 

so far: completely general (for equivalent dimensions and usual DMFT scaling) 


BD+1(k) = B1(k) BD(k) − D BD−1(k) + O(1/ √ d) 

of the Hermite polynomial type. Consequently: BD(k) = HeD(B1(k)). 

With initial condition B1(k) = 

ɛ(k) = 

∞� 

D=1 

� 2 

d 

� 

α 

cos(kα) ≡ ɛ hc 

k 

t ∗ D 

√ D! HeD(ɛ hc 

k ) =: F(ɛ hc 

k ) t ∗ D = 

and orthogonality relation: 

1 

√ 2πD! 

� ∞ 

dɛ F(ɛ) HeD(ɛ) e 

−∞ 

−ɛ2 /2 

so far: completely general (for equivalent dimensions and usual DMFT scaling) 

Choice of monotonic function F(x) implies ρ(ɛ) = 

F −1 (ɛ) = √ 2 erf −1 

� � ɛ 

2 dɛ 

−∞ 

′ ρ(ɛ ′ � 

) − 1 

1 

F ′ (F −1 (ɛ)) ρhc (F −1 (ɛ)) and 


Redefinition of the Bethe Lattice 

√ 4 − ɛ 2 : 

For semi-elliptic DOS, ρ(ɛ) = 1 

2π 

F −1 (ɛ) = √ 2 

π erf−1� 

� 

ɛ 1 − ( ɛ 

2 )2 + 2 arcsin( ɛ 

2 ) 

numerical inversion � F(˜ɛ) 

� 

ε 

2 

1.5 

1 

0.5 

0 

-0.5 

-1 

-1.5 

-2 

F(˜ɛ) 

-4 -2 0 

~ 

ε 

2 4 



√ 4 − ɛ 2 : 


2π 

F −1 (ɛ) = √ 2 

π erf−1� 

� 

ɛ 1 − ( ɛ 


2 ) 


numerical integration � t ∗ D 

� 

ε 

2 

1.5 

1 

0.5 

0 

-0.5 

-1 

-1.5 

-2 

F(˜ɛ) 

-4 -2 0 

~ 

ε 

2 4 

D t ∗ D 

� D 

n=1 t∗ n 2 

1 0.98731 0.974773 

3 -0.15353 0.998345 

5 0.03893 0.999861 

7 -0.01125 0.999987 

9 0.00343 0.999999 



√ 4 − ɛ 2 : 


2π 

F −1 (ɛ) = √ 2 

π erf−1� 

� 

ɛ 1 − ( ɛ 


2 ) 


numerical integration � t ∗ D 

fast convergence with hopping cutoff Dmax 

ρ 

0.3 

0.2 

0.1 

0 

D max = 5 

D max = 7 

D max = 9 

0.3 

0.2 

0.1 

exact 0 

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 

ε 

1.6 1.8 2 2.2 

� 

ε 

2 

1.5 

1 

0.5 

0 

-0.5 

-1 

-1.5 

-2 

F(˜ɛ) 

-4 -2 0 

~ 

ε 

2 4 

D t ∗ D 

� D 

n=1 t∗ n 2 

1 0.98731 0.974773 

3 -0.15353 0.998345 

5 0.03893 0.999861 

7 -0.01125 0.999987 

9 0.00343 0.999999 


Chain rule yields Fermi velocity: vk = F ′ (F −1 (ɛ))v hc 

k . 

Specifically, for Bethe semi-elliptic DOS: 

〈|vk| 2 〉(ɛ) := ˜ρ(ɛ) 

ρ(ɛ) = 

π 

2(1 − ɛ2 /4) exp 

� � 

− 2 

 

2 

1.5 

1 

0.5 

0 

hypercubic / stacked 

Millis / Freericks 

this work 

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 

ε 

erf −1� ɛ � 1 − ɛ 2 /4 + 2 arcsin(ɛ/2) 

π 

〈|vk| 2 〉 → 0 at band edges 

��2 � 

. 

band center: max. transport contribution 

other transport properties: analogous 


Chain rule yields Fermi velocity: vk = F ′ (F −1 (ɛ))v hc 

k . 

Specifically, for Bethe semi-elliptic DOS: 

〈|vk| 2 〉(ɛ) := ˜ρ(ɛ) 

ρ(ɛ) = 

π 

2(1 − ɛ2 /4) exp 

� � 

− 2 

 

2 

1.5 

1 

0.5 

0 

f-sum: 

hypercubic / stacked 

Millis / Freericks 

this work 

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 

� ∞ 

dω σxx(ω) = σ0 

4d 

0 

ε 

� ˜ρ ′ (ɛ) � σ0 

= 

ρ(ɛ) 4d 

erf −1� ɛ � 1 − ɛ 2 /4 + 2 arcsin(ɛ/2) 

π 

〈|vk| 2 〉 → 0 at band edges 

��2 � 

. 

band center: max. transport contribution 

other transport properties: analogous 

�� F ′′ � F −1 (ɛ) � − F −1 (ɛ)F ′� F −1 (ɛ) �� 


Numerical QMC/MEM Results for Half-filled 

crossover region 

Hubbard Model 

For orientation: 

MIT phase diagram 

Bethe DOS, W = 4 

results for T = 0.05 

0.07 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

MIT at Uc ≈ 4.7 0 

T 

critical end point 

coexistence region 

metal insulator 

U c1 U c U c2 

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 

Quantum Monte-Carlo (QMC) with discretization ∆τ = 0.1 

Analytical continuation using Maximum Entropy method 


U

A(ω) 

σ(ω) 

0.3 

0.2 

0.1 

Local spectral function A(ω) 

0 

-5 -4 -3 -2 -1 0 1 2 3 4 5 

0.3 

0.2 

0.1 

Optical conductivity σ(ω) 

U=4.0 

U=4.6 

U=5.0 

U=5.5 

isotrop 

ω 

∫0 dω’ σ(ω’) 

0.4 

0.3 

0.2 

0.1 

ω 

U=4.0 

U=4.6 

U=5.0 

U=5.5 

0 

0 1 2 3 4 

ω 

5 6 7 8 

0 

0 1 2 3 4 5 6 7 8 

|ω| 

σ(ω=0) 

∞ 

∫0 dω σ(ω) 

2 

1.5 

1 

0.5 

0 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

dc conductivity / f-sum 

-π/2 E kin 

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 

U 

isotropic 

stacked 

disordered 


Conclusion 

hc+NN DMFT transport formulae strongly modified for general lattices / hopping 

e.g.: t − t ′ model 

construction method for lattice models (hc symmetry) with arbitrary DOS in large d 

first Bravais lattice tb model with finite band edges in d→∞ and nonsingular vk 

first definition of isotropic and coherent optical conductivity σ(ω) in high d 

consistent with Bethe semi-elliptic DOS 

small impact of truncation (Dmax) and of application in finite d 

works as heuristic scheme in finite d (also multiple bands) 

new DMFT f-sum rule 

numerical results for σ(ω) based on high-precision QMC/MEM spectra 

not discussed: vertex corrections, reduced umklapp scattering in finite d 

http://www.physik.uni-augsburg.de/theo3/diss.de.shtml 

http://komet337.physik.uni-mainz.de/Bluemer/talks.de.shtml 


(ε) 

General Dispersion Formalism as Heuristic Scheme 

3 

2.5 

2 

1.5 

1 

0.5 

0 

Test: t − t ′ model 

d=3 

d=4 

t’ * /t * = -0.33 

exact 

-1 0 1 2 3 4 

ε 

ρ(ε), ~ ρ(ε) 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

Application for t2g bands of 

La1-xSrxTiO3 based on LDA data 

3 

2 

1 

0 

−1 −0.5 0 0.5 

ε 

1 1.5 2 

0 

−1 −0.5 0 0.5 1 1.5 2 


2 

ε ε 

ρ(ε) 

~ 

ρ(ε)

Optical Conductivity of Strongly Correlated Electrons in High ...

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