Multichain Approach for the Single-Impurity Anderson Modelmn119/Vortrag-Mainz.pdf · 2015-12-16 ·...

Multichain Approach for theSingle-Impurity Anderson Model

Marlene Nahrgang Florian Gebhard

Fachbereich Physik, Philipps-Universitat Marburg,Marburg-Mainz Workshop on the Physics of Strongly Correlated Electron Systems

February 8, 2008

Outline

1 Introduction

2 Dynamical Mean-Field TheorySingle Impurity Anderson Model

3 Single-Chain MappingNon-interacting LimitExpansion in Chebyshev Polynomials

4 Double-Chain Mapping: Mott InsulatorHarris-Lange TransformationSolution to O(1/U)

5 Conclusions and Outlook

IntroductionHubbard Model

Hubbard Hamiltonian

H = −t ∑σ,〈i,j〉

c†iσcjσ + U ∑

i

(ni↑ −

12

) (ni↓ −

12

)

fermionic operators [ciσ, c†jσ′]+ = δij δσσ′ and [c(†)

iσ , c(†)jσ′ ]+ = 0

t: nearest-neighbour hoppingU: onsite Coulomb interactionhalf filling is guaranteed by choosingµ = 0 for the chemical potential

IntroductionMetal-Insulator Transition

• Mott-Hubbard transition between paramagnetic phases• not understandable in a single-particle picture −→ induced by

correlation effects

• competition between:kinetic energy t ←→ local interaction U

charge fluctuations←→ collective localization

Dynamical Mean-Field Theory

• each lattice site has infinitely many neighbours Z = 2d• finite kinetic energy =⇒ rescaling: t = t/

√Z

• for translationally invariant systems the self-energy is purely local(momentum independent)

Σσ(~k , ω) = Σσ (ω) = Σskeletonσ (ω, {G})

• local Green function

G (ω) =∫ ∞

−∞dε

D0 (ε)ω− ε− Σσ (ω) + iηsgn (ω)

• non-interacting density of states: D0 (ε)

Dynamical Mean-Field Theory

In the skeleton expansion the self-energy depends only on the localGreen function.Therefore, there exists an effective single-site problem with Hubbardinteraction with

ΣHubσ (ω) = Σss

σ (ω)

GHubσ (ω) = Gss

σ (ω)

⇒ self-consistent mapping is possible

Dynamical Mean-Field TheorySingle Impurity Anderson Model

SIAM-Hamiltonian in star geometry

H =U(

nd↑ −12

) (nd↓ −

12

)+

∞

∑σ,m=0

ξma†mσamσ

+ ∑σ

d†σ

∞

∑m=0

Vmamσ + ∑σ

∞

∑m=0

Vma†mσdσ

bath degrees of freedom: amσ (fermionic operators),bath energies: ξm, hybridization strength: Vm

GSIAMσ (ω) =

1ω− Ed − ∆(ω)− ΣSIAM

σ (ω)Green function

∆ (ω) = ∑m

V 2m

ω− ξmhybridization function

Single-Chain Mapping

SIAM in star geometry

SIAM in single-chain geometry

Single-Chain MappingMapping Procedure

Construction of a set of mutually orthogonal operators (Lanzcosalgorithm)In this basis

H = Himp + ∑σ

[d†

σ γ0σ + γ†0σdσ

]+ ∑

σ,n

[εnγ†

nσγnσ +(

tnγ†nσγn+1σ + t∗n γ†

n+1σγnσ

)]

Single-Chain MappingNon-interacting Limit

Density of states of a Bethe lat-tice for U = 0:

D0 (ω) =4

πW

√1−

(2ω

W

)2

where W = 4t bandwidth.−→ the self-consistency equa-tions reduce to

∆ (ω) = G (ω)

Single-Chain MappingNon-interacting Limit

Hamiltonian in the Lanzcos basis

H =

0 1 0 · · · 0

1 0 1...

0 1. . . . . . 0

.... . . 1

0 · · · 0 1 0

= T

D0 (ω < 0) = − 1π=〈γ†

0σ

1ω + T

γ0σ〉

=4

πW

√1−

(2ω

W

)2

⇒ Self-consistency achieved!

Single-Chain MappingChebyshev Polynomials

Chebyshev polynomials are orthogonal under the scalar product

〈f |g〉 =∫ 1

−1π

√1− x2f (x) g (x) dx

〈Un|Um〉 =π2

2δnm

any function on the intervall [−1, 1] can be expanded as

f (x) =2π

√1− x2

∞

∑n=0

µ(f )n Un (x)

µ(f )n =

∫ 1

−1Un (x) f (x) dx

Single-Chain MappingChebyshev Polynomials

The expansion of D0 (ω) on the intervall[−W

2 , W2

]reads

D0 (ω) =4

πW

√1−

(2ω

W

)2 ∞

∑n=0

µ0nUn

(2ω

W

)

µ0n =

∫ W2

−W2

Un

(2ω

W

)D0 (ω) dω

We can immediately read off thatµ0

0 = 1 , µ0n = 0 for all n ≥ 1.

Double-Chain Mapping: Mott Insulator

SIAM in star geometry

Density of states

SIAM in double-chain geometry

Double-Chain Mapping: Mott InsulatorHarris-Lange Transformation

Use a Harris-Lange transformation (canonical transformation) todecouple the lower and the upper Hubbard band perturbativelyEffective Hamiltonian to O(1/U)

Heff = H0 (α, β, d) + T 0 +1

2U[T +, T−

].

Double-Chain Mapping: Mott InsulatorHarris-Lange Transformation

with a potential-energy change by (+U)

T + = ∑σ

(β†

0,σdσ (1− nd ,−σ) + d†σ α0,σnd ,−σ

)and a potential-energy change by (−U)

T− = ∑σ

(α†

0,σdσnd ,−σ + d†σ β0,σ (1− nd ,−σ)

)No change in the potential energy

T 0 =1√2

∑σ

[(β†

0,σdσ + d†σ β0,σ

)nd ,−σ +

(α†

0,σdσ + d†σ α0,σ

)(1− nd ,−σ)

]+ ∑

σ,n

(t−n α†

n,σαn+1,σ + h.c.)

+ ∑σ,n

(t+n β†

n,σβn+1,σ + h.c.)


Hamiltonian matrix in the Lanzcos basis with starting vector|φ0〉 = d↑|ψ0〉

Heff − E0 =

U2 + 1

U 1

1 −ε0 − 12U t0 0

t0 −ε1 t1

t1. . .

0 tN−3

tN−3 −εN−2


Educated guess: tn = 1, ε0 = −U/2 and εn = −U/2 + 1/(2U)

Heff − E0 =

U2 −

12U 1

1 U2 −

12U 1 0

. . .

0 1

1 U2 −

12U

︸︷︷︸

T + a

+3

2U|φ0〉〈φ0|︸︷︷︸1U V0

with a = U2 −

12U .


We are interested in

=G0l0 (ω) = π〈φl |δ

(ω + T

)|φ0〉

=√

1− (ω/2)2∞

∑n=0Un(ω/2)〈φl | Un(−T /2)|φ0〉︸︷︷︸

(−1)n |φn〉

=√

1− (ω/2)2 (−1)l Ul (ω/2)

from the Kramers-Kronig relations

<G0l0 (ω) = − 1

πP

∫ ∞

∞dω′=G0

l0 (ω′)ω′ − ω

= Tl+1(ω/2) (−1)l


D (ω) =1π=G00 (ω)

=1π

√1−

(ω

2

)2 (1− 1

2Uω

)to O(1/U) this is the Chebyshev expansion of the known result(obtained by Florian and Eva):

D (ω) =1π

√1− 1

4

(U −

√U2 + 2Ux + 4

)2

with x = ω + U2 −

12U and |x | < 2, this implies a = −U

2 + 12U


Ready? No! Must prove self-consistency! We’ve started with this matrix.


Self-consistency: ∆ (ω) = G (ω)

Start Lanzcos-Algorithm with |φ0〉 =1√2

∞

∑σm=0

Vma†mσ|vac〉 ⇒

H∆ =

ε0 t0

t0 ε1 t1 0. . .

0. . .

semi-infinite chain

Calculate ∆ (ω) = 〈φ0| 1ω−H∆ |φ0〉 = G (ω)


The matrices for Heff − E0 and H∆ differ from the matrix T + a only inthe (1,1) component.ε0 is chosen so that

D (ω) =1π=G00 (ω)

=1π

√1−

(ω

2

)2 (1− 1

2Uω

)from Heff − E0 and H∆.⇒ self-consistency is achieved by fixing a single number ε0!

Conclusions and OutlookInsulator

Conclusion:• Double-chain approach works nicely for the insulator.• For strong coupling, only a few Chebyshev moments are

necessary.Outlook:

• higher-orders in Harris-Lange expansion (or Kato-Takahashitheory)⇒ third order in hops?

• expectation for the Chebyshev moments of the density of states:µl = O(1/U)l with no corrections

Conclusions and OutlookMetal

When there is weight in the Hubbard bands and in the quasi-particlepeak, we need three chains.

Hope:No numerical power is wasted for limiting cases=⇒ triple-chain geometry is best for the investigation of the metaland of the metal-insulator transition

Multichain Approach for the Single-Impurity Anderson Modelmn119/Vortrag-Mainz.pdf · 2015-12-16 ·...

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