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Thermoelectric properties of heavy fermion systems calculated within a DMFT/NRG-treatment of the periodic Anderson model

Gerd Czycholl

Institut für Theoretische Physik, Universität Bremen Coworkers: Claas Grenzebach, Frithjof Anders

Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.1/9

Contents

• 1. Typical experimental results
• 2. Periodic Anderson Model (PAM)
• 3. Dynamical Mean field theory (DMFT), mapping on single impurity

Anderson model (SIAM)

• 4. Impurity solvers: modified perturbation theory (MPT) and numerical

renormalization group (NRG)

• 5. Results
• 6. Open problems
• 7. Discussion and Conclusion

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Typical experimental results

Temperature dependence of resistivity Jaccard et al. 1997

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Typical experimental results

Temperature dependence of thermopower

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Typical experimental results

Dynamical conductivity Marabelli et al. 1988

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Periodic Anderson Model (PAM)

Hamiltonian H =

• kσ

ǫ

kc†

• kσc

kσ +

• Rσ

(Eff †

• Rσf

Rσ + U

2 f †

• Rσf

Rσf †

• R−σf

R−σ + V (c†

• Rσf

Rσ+c.c.

periodic arrangement of localized f-levels, conduction band, hybridization, Coulomb correlation only between localized (f-) electrons Simplifying assumptions:

• 1. only 2-fold degeneracy of f- and conduction electrons
• 2. no crystal fields
• 3. no hybridization dispersion
• 4. no interaction between conduction and f- electrons
• 5. simple cubic lattice in dimension d: ǫ

k = 2t d l=1 cos(kla)

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Periodic Anderson Model (PAM)

Current operator (consistent with PAM) jx =

• kσ

∂ǫ

k

∂kx c†

• kσc

kσ[+

∂Vk ∂kx (f †

• kσc

kσ + c.c.)]

• =0, if V k-independent

(Czycholl and Leder 1981) in site representation: jx = ita

• R∆xσ

(c†

• R+∆xσc

Rσ − c†

• Rσc

R+∆xσ)

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Periodic Anderson Model (PAM)

Quantities to be calculated

• ne-particle Green function

Gab

• kσ(z) =≪ a

kσ; b†

• kσ ≫z= −i

∞ dteizt < [a

kσ(t), b†

• kσ(0)]+ >

(a, bǫ{c, f}) determines f-electron density of states, for instance: ρfσ(E) = − 1 πN

• k

ImGff

• kσ(E + i0)

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Periodic Anderson Model (PAM)

Quantities to be calculated f-electron selfenergy defined by

• ≪ c

kσ; c†

• kσ ≫z

≪ c

kσ; f †

• kσ ≫z

≪ f

kσ; c†

• kσ ≫z

≪ f

kσ; f †

• kσ ≫z
• =
• z − ǫ

k

−V −V z − Ef − Σ

kf(z)

−1 Kubo formula for dynamic conductivity: (two-particle Green function) σxx(ω) = − 1 ω + i0Imχjx,jx(ω + i0) χjx,jx(z) = −i ∞ dteizt < [jx(t), jx(0)]− >

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Dynamical mean-field theory (DMFT)

Large-d limit Idea: For lattice models of correlated and disordered electron systems mathematically a non-trivial limit d → ∞, t → 0 with dt2 = const. can be defined (Metzner and Vollhardt 1989). Simplifications:

• 1. Unperturbed density of states Gaussian (for d-dimensional

hypercubic lattice) (U. Wolff 1983; Metzner and Vollhardt 1989)

• 2. Selfenergy k-independent (site-diagonal, local)

(Müller-Hartmann 1989)

• 3. Current operator vertex corrections vanish

(Khurana 1989; Schweitzer und Czycholl 1991)

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Dynamical mean-field theory (DMFT)

Mapping on effective single-impurity problem Because of site-diagonality of selfenergy mapping on effective single-impurity problem possible (Brandt and Mielsch 1989; Georges, Kotliar 1993) Single-impurity Anderson model (SIAM) H =

• kσ

˜ ǫ

kc†

• kσc

kσ +

• σ

(Eff †

σfσ + U

2 f †

σfσf † −σf−σ +

˜ V √ N

• k

(c†

• kσfσ+c.c.))

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Dynamical mean-field theory (DMFT)

Mapping on effective single-impurity problem

• ne-particle f-electron Green function of SIAM

Gff

SIAM(z)

= 1 z − Ef − Σf(z) − ∆(z) with ∆(z) = ˜ V 2 N

• k

1 z − ˜ ǫ

k

Correlated f-impurity coupled to effective ”bath” (conduction band) described by ∆(z)

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Dynamical mean-field theory (DMFT)

DMFT selfconsistency loop Effective bath Green function ∆(z) to be determined selfconsistently so that f-electron Green function of PAM and effective SIAM agree: Gff

SIAM(z)

= 1 z − Ef − Σf(z) − ∆(z) = Gff

P AM(z)

= 1 N

• k

1 z − Ef − Σf(z) −

V 2 z−ǫ

k

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Dynamical mean-field theory (DMFT)

DMFT selfconsistency loop Initial value: e.g. Σf(z) = 0 Gf(z) = Gff

P AM ; ∆(z) = z − Ef − Σf(z) − G−1 f (z)

Gff

SIAM(z) = Gf(z)?

solve SIAM (for Ef, U, ∆(z)) by suitable ”impurity solver” (approximation or numerical method) New value for Σf

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Dynamical mean-field theory (DMFT)

Transport quantities in DMFT Because of vanishing of vertex corrections in DMFT transport quantities can be expressed by single-particle Green functions: dynamical conductivity σxx(ω) ∼

• dE f(E) − f(E + ω)

ω L(E, E + ω) L(E, E + ω) = 1 N

• R

R′σ∆x

ImGcc

• R

R′σ(E + i0)ImGcc

• R′+∆x

R+∆x(E + ω + i0)

Gcc

• k (z)

= 1 z −

V 2 z−Ef −Σf (z) − ǫ k

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Dynamical mean-field theory (DMFT)

Transport quantities in DMFT static conductivity: σxx(ω = 0) ∼

• dE
• − d

f dE

• L(E, E)

Resistivity: R(T) = 1 σxx(0) Thermoelectric power (TEP): S(T) = 1 eT

• dE
• − d

f dE

• (E − µ)L(E, E)
• dE
• − d

f dE

• L(E, E)

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Impurity solver

Possible methods for SIAM

• 1. Non-crossing approximation (NCA) (Keiter and Kimball 1971)
• 2. Second order perturbation theory (SOPT) (termed IPT within

DMFT-scheme)

• 3. Exact diagonalization (ED)
• 4. Quantum Monte Carlo (QMC) (Hirsch and Fye 1983)
• 5. Numerical renormalization group (NRG) (Krishnamurty, Wilkens,

Wilson 1980)

• 6. modified perturbation theory (MPT) (Martin-Rodero et al. 1982)

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Impurity solver

Possible methods for SIAM Main properties of impurity solvers:

• 1. NCA has correct low-T scale (Kondo temperature, but leads to

unphysical singularity at Fermi level for low T, not appropriate for low-T transport quantities

• 2. SOPT fulfills Fermi liquid properties, valid in weak-coupling situation
• nly, violates atomic limit, does not lead to correct Kondo temperature
• 3. ED leads to discrete spectrum, no static conductivity, not suitable for

transport quantities

• 4. QMC applicable only for high T and not too large U, no low-T

transport quantities describable Therefore, here MPT and NRG as impurity solvers.

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Impurity solver

MPT Ansatz for selfenergy of effective SIAM: Σf(z) = Unf−σ + αΣSOP T

f

(z) 1 − βΣSOP T

f

(z) where ΣSOP T

f

(z) is the SIAM selfenergy in SOPT relative to the Hartree-Fock solution. Here the parameters α, β can be determined by the condition that the atomic limit (of vanishing V) and an additional criterion (Fermi liquid sum rule, first four moments) are fulfilled. (Martin-Rodero et al. 1982, Kajueter and Kotliar 1996, Nolting, Meyer et

• al. 1997-2000)

MPT fulfills atomic limit, applicable for all U, T, Fermi liquid properties for low T , but not correct Kondo scale.

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Impurity solver

NRG

1 −1 Λ−1 −Λ−1 Λ−2 −Λ−2 Λ−3 −Λ−3 … … Λ−n −Λ−n

Core of the NRG: logarithmic discretisation of the energy axis around the chemical potential µ = 0 with a parameter Λ. Hamiltonian of SIAM: limit of discrete Hamiltonians fulfilling a recursion

• formula. Solving this semi-infinite chain of discrete models recursively

leads to Green functions with discrete poles in spectral representation. To get a continuous Green functions broadening necessary, here broadening by logarithmic Gauss functions with b = 0.5 δ(E − E0) → exp(−b2/4) bE√π exp(−(ln E − ln E0)2/b2),

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Impurity solver

NRG For DMFT-loop selfenergy required, NRG does not immediately provide for f-electron selfenergy, here calculated via Σf(z) = U ≪ f↑f †

↓f↓, f † ↑ ≫z

≪ f↑, f †

↑ ≫z

(Bulla, Hewson, Pruschke 1998)

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Results

Density of states Parameters: ǫc = 0, ǫf = −U/2, V ≈ 5.6, dt2 = 10 Λ = 1.6, N = 1500 varying U, ntotal ntotal = 1.6

−10 −5 5 10 0.1 0.2 0.3 0.4 0.5 E f−DOS

T = 0.03; δ = 0.01 U = 10; NRG U = 10; MPT U = 5; NRG U = 5; MPT U = 2; NRG U = 2; MPT

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Results

Density of states

−0.5 0.5 0.1 0.2 0.3 0.4 0.5 E f−DOS

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Results

Selfenergy

−10 −5 5 10 1 2 3 4 5 6 7 8 9 E −Im Σ

T = 0.03; δ = 0.01 U = 10; NRG U = 10; MPT U = 5; NRG U = 5; MPT U = 2; NRG U = 2; MPT

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Results

Selfenergy

−0.4 −0.2 0.2 0.4 0.5 1 1.5 2 2.5 E −Im Σ

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Results

Resistivity

0.1 0.2 0.3 0.4 0.5 10 20 30 40 50 Temperature T Resistivity R

U = 10; NRG U = 10; MPT U = 5; NRG U = 5; MPT U = 2; NRG U = 2; MPT

Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.7/9

Results

Resistivity

10

−3

10

−2

10

−1

10 20 30 40 50 Temperature T Resistivity R

U = 10; NRG U = 10; MPT U = 5; NRG U = 5; MPT U = 2; NRG U = 2; MPT

Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.7/9

Results

TEP

0.1 0.2 0.3 0.4 0.5 −50 50 100 150 200 Temperature T Thermopower S[µV/K]

U = 10; NRG U = 10; MPT U = 5; NRG U = 5; MPT U = 2; NRG U = 2; MPT

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Results

Resistivity NRG

10

−3

10

−2

10

−1

10 100 200 300 400 500 600 700 Temperature T Resistivity R ntotal = 1.4 ntotal = 1.6 ntotal = 1.8 ntotal = 1.9 ntotal = 1.95 ntotal = 2

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Results

TEP NRG

0.1 0.2 −50 50 100 150 200 Temperature T Thermopower S(T) ntotal = 1.4 ntotal = 1.6 ntotal = 1.8 ntotal = 1.9 ntotal = 1.95 ntotal = 2

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Results

Resistivity NRG

10

−3

10

−2

10

−1

10 10 20 30 40 50 Temperature T Resistivity R

U = 11 U = 10 U = 9 U = 8 U = 7

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Results

TEP NRG

0.2 0.4 0.6 0.8 1 −60 −40 −20 20 40 60 Temperature T Thermopower S(T)

U = 11 U = 10 U = 9 U = 8 U = 7

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Results

Dynamical conductivity

10

−4

10

−2

10 0.05 0.1 0.15 Frequency ω Conductivity σ(ω, T) U = 5; ntotal = 1.6; NRG T = 0.4000 T = 0.1200 T = 0.0300 T = 0.0120 T = 0.0045 T = 0.0005

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Results

specific heat γ-coefficient

10 10

1

10

2

50 100 150 Temperature gamma

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Open problems

Broadening in NRG For results presented an additional Lorentzian broadening, i.e. a finite energy imaginary part δ was introduced. Possible physical interpretation: finite lifetime due to couplings and interactions not included in the model, e.g. small selfenergy imaginary part due to disorder scattering (δ ∼ c impurity concentration), good results (also for T → 0) with δ = 0.01. Influence of this additional broadening:

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Open problems

Broadening in NRG

−10 −5 5 10 0.2 0.4 0.6 E f−DOS

T = 0.0005; U = 10 δ = 10−10; NRG δ = 10−6; NRG δ = 10−2; NRG δ = 10−5; MPT δ = 10−2; MPT

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Open problems

Broadening in NRG

−10 −5 5 10 −2 2 4 6 8 10 E −Im Σ

T = 0.0005; U = 10 δ = 10−10; NRG δ = 10−6; NRG δ = 10−2; NRG δ = 10−5; MPT δ = 10−2; MPT

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Open problems

Broadening in NRG

−0.02 −0.01 0.01 0.02 0.05 0.1 0.15 0.2 0.25 E −Im Σ

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Open problems

Fermi liquid behavior

0.5 1 1.5 2 x 10

−4

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Temperature T

2

Im Σ(µ) MPT

U = 10 U = 5 U = 2

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Open problems

Fermi liquid behavior

0.5 1 1.5 2 x 10

−4

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Temperature T

2

Im Σ(µ) NRG

U = 10 U = 5

Hvar2005 – Workshop on Correlated Thermoelectric Materials – p.8/9

Open problems

Fermi liquid behavior

0.5 1 1.5 2 x 10

−4

1.5 2 2.5 3 3.5 4 4.5 5 Temperature T

2

Resistivity R MPT

U = 10 U = 5 U = 2

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Open problems

Fermi liquid behavior

0.5 1 1.5 2 x 10

−4

5 10 15 20 25 30 35 40 45 Temperature T

2

Resistivity R NRG

U = 10 U = 5 U = 2

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Open problems

Open problems Broadening in NRG not yet fully satisfactory, broadening with logarithmic Gaussians not yet satisfactory for very low T and close to the Fermi energy. Relatively large additional Lorentzian broadening δ necessary for smooth behavior at Fermi level Appearant NFL-behavior in DMFT/NRG for sufficiently large U. Gradual transition from FL to NFL behavior with increasing U. FL-behavior in MPT not surprising as based on U-perturbation theory.

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Conclusion

• 1. DMFT-treatment of PAM qualitatively well reproduces transport

properties, in particular thermoelectric properties (even the absolute magnitude), typical for HFS.

• 2. DMFT/NRG and DMFT/MPT yield comparable results; for small U

perfect agreement, for larger U at least qualitative agreement, only characteristic low temperature scale (Kondo temperature) too large in DMFT/NRG.

• 3. Broadening in NRG still problematic at very low energies and

temperatures, additional Lorentzian broadening necessary to obtain meaningful low-T results.

• 4. Even with this Lorentzian broadening NFL-behavior in

DMFT/NRG-treatment of PAM for large enough U. Is this realistic or still an artifact of the NRG?

• 5. In the future application to disordered HFS and to extensions of PAM

with realistic f-degeneracy, crystal fields, etc.

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