Dynamical properties of quantum impurity systems...

Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion

Dynamical properties of quantum impuritysystems in and out of equilibrium: a numerical

renormalization group approach

Frithjof B. Anders

Institut fur Theoretische Physik · Universitat Bremen

Dresden, August 15, 2007

Collaborators R. Bulla, G. Czycholl, C. Grenzebach, R.Peters, Th. Pruschke, A. Schiller, S. Tautz,R. Temirov, S. Tornow, M. Vojta

NRG Review R. Bulla, T. Costi and Th. Pruschkecond-mat/0701105to be published in RMP

Contents

1 IntroductionKondo effect in bulk materialsKondo effect in nano-devices

2 The Numerical Renormalization GroupDiscretization of the bath contiuumFixed points

3 Spectral functions at finite temperaturesComplete basis set of the Wilson chain

4 Real-time dynamics out of equilibriumTime-dependent numerical renormalization groupSpin decay in the anisotropic Kondo model

5 Conclusion

Contents

5 Conclusion

Kondo effect in bulk materials

Resistivity in bulk materials

scattering increases for T → 0!de Haas, de Boer, van denBerg, Physica 1,1115 (1934)

but: saturation T < TK

Onuki et al 1987

Kondo effect in bulk materials

Resistivity in bulk materials

scattering increases for T → 0!de Haas, de Boer, van denBerg, Physica 1,1115 (1934)

but: saturation T < TK

Onuki et al 1987

Kondo effect in nano-devices

Zero bias anomaly

zero bias anomaly

G (0) ∝ ln(T ) for T → 0!Wyatt, PRL 13,401 (1964)

G (V ) in Ta-I-Al

Wyatt, PRL 13,401 (1964)

Kondo 1964

single spin + metal

AF coupling: HK = J~S~sband

Zero bias anomaly

zero bias anomaly

G (0) ∝ ln(T ) for T → 0!Wyatt, PRL 13,401 (1964)

G (V ) in Ta-I-Al

Wyatt, PRL 13,401 (1964)

Kondo 1964

single spin + metal

AF coupling: HK = J~S~sband

Kondo effect in a single electron transistor (SET)

D. Goldhaber-Gordon, Nature 98

weak coupling

M.Kastner RMP 1992

Kondo effect in a single electron transistor (SET)

D. Goldhaber-Gordon, Nature 98

strong coupling

van der Wiel et al. Science 289

(2000)

lattice problem

Mapping the lattice problem onto an effective site problem(quantum impurity problem) plus dynamical bath (DMFT)Kuramoto 85; Grewe 87; Metzner, Volhardt; Muller-Hartmann, Brand, Mielsch 89;

Jarrell, Kotliar, Georges 92, · · ·

dynamicalbath

lattice problem

Mapping the lattice problem onto an effective site problem(quantum impurity problem) plus dynamical bath (DMFT)Kuramoto 85; Grewe 87; Metzner, Volhardt; Muller-Hartmann, Brand, Mielsch 89;

Jarrell, Kotliar, Georges 92, · · ·

Contents

5 Conclusion

Quantum Impurity Problems

Quantum Impurity

finite number of localized DOF

interacting with a bathcontiuumbosonic bath: see Ingersent

problem:

infrared divergence inperturbation theory

+ indicator for a change of groundstateKondo singlet vs free moment

|α>|γ>

quantum impurity

bosonic bath

metallic host

Quantum Impurity

problem:

|α>|γ>

quantum impurity

bosonic bath

metallic host

Quantum Impurity

problem:

|α>|γ>

quantum impurity

bosonic bath

metallic host

Quantum Impurity

problem:

|α>|γ>

quantum impurity

bosonic bath

metallic host

Discretization of the bath contiuum

Numerical Renormalization Group

Numerical Renormalization GroupWilson 1975, Krishnamurthy et al. 1980

discretization of the bathcontiuum on a logarithmic grid:I+n = D[Λ−n−1,Λ−n]

Mapping onto a semi-finitechain for an arbitrary bathcoupling function ∆(ω), J(ω)

|α>|γ>

quantum impurity

bosonic bath

metallic host

|α>|γ>

quantum impurity

|α>|γ>

quantum impurity

Λ−n/2

Wilson’s NRG (1975)

Impurity

ξΝ ∼Λ −Ν/2

switching on iteratively the couplings ξm ∝ Λ−m/2

recursion relation (RG transformation)

HN+1 =√

ΛHN +∑

(f †NσfN+1σ + f †N+1σfNσ

)iteratively diagonalize the series of Hamiltonians Hm

RG: elimination of the high energy states, rescaling by√

Λtemperature: Tm ∝ Λ−m/2

stop at chain length N, when desired TN ∝ Λ−N/2 is reached

Impurity

ξΝ ∼Λ −Ν/2

HN+1 =√

ΛHN +∑

Impurity

ξΝ ∼Λ −Ν/2

HN+1 =√

ΛHN +∑

Impurity

ξΝ ∼Λ −Ν/2

HN+1 =√

ΛHN +∑

Impurity

ξΝ ∼Λ −Ν/2

HN+1 =√

ΛHN +∑

Fixed points

10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102

2.) ∆=0

∆=0.01∆=0.1∆=0.5

CEF Splitting in the SU(4) SIAM

local moment fixed point J=3/2

local moment fixed point J=1/2

strong coupling FP

free orbital FP

NRG not only a numerical tool! Wilson 1975, Krishnamurty et al. 1980

analysis of the fixed points H∗ = T 2RG [H∗]:

deep insight into the physics of a model, crossover scales T ∗

Fixed points

NRG Review: R. Bulla, T. Costi and Th. Pruschke, cond-mat/0701105

Extensions of Wilson’s method in recent years

bosonic baths: Tong, Bulla, Vojta 2003

bosonic and fermionic baths : Glossop, Ingersent 2005

non-equilibrium: Costi, 1997, Anders, Schiller 2005

Calculation of spectral functions

Frota, Olivera 1986

Sakai et al 1989

Costi, Hewson 1992, 1994

Bulla et al., 1998

Hofstetter 2000

Problem:

dynamical properties unsystematic:how are different energy scale connected?

Fixed points

Frota, Olivera 1986

Sakai et al 1989

Bulla et al., 1998

Hofstetter 2000

Problem:

Fixed points

Frota, Olivera 1986

Sakai et al 1989

Bulla et al., 1998

Hofstetter 2000

Problem:

Fixed points

Frota, Olivera 1986

Sakai et al 1989

Bulla et al., 1998

Hofstetter 2000

Problem:

Fixed points

Frota, Olivera 1986

Sakai et al 1989

Bulla et al., 1998

Hofstetter 2000

Problem:

Contents

5 Conclusion

Spectral functions at finite temperatures

Assumption: solve the Wilson chain exactly, i.e HN |n〉 = En|n〉Then: Lehmann representation of ρ(ω) (text book)

ρA,B(ω) =∑n,m

(e−βEn + e−βEm

AnmBmnδ(ω + En − Em)

The challenge

1 discrete spectrum =⇒ continous ρ(ω), broading of δ(ω)

2 how do we gather the information from different iterations?

3 how do we guarantee the sum-rule∫ ∞

−∞dω ρσ(ω) = 1 ?

ρA,B(ω) =∑n,m

The challenge

−∞dω ρσ(ω) = 1 ?

ρA,B(ω) =∑n,m

The challenge

−∞dω ρσ(ω) = 1 ?

ρA,B(ω) =∑n,m

The challenge

−∞dω ρσ(ω) = 1 ?

ρA,B(ω) =∑n,m

The challenge

−∞dω ρσ(ω) = 1 ?

Complete basis set of the Wilson chain

All discarded states: a complete basis set for Wilson chainAnders, Schiller PRL 95, 196801 (2005), PRB 74,245113 (2006)

Impurity

eEnvironment

|l,e,1>

|l,e,2>

|l,e,3>

complete basis: {|e〉} = {|αimp, α1, α2, α3, α4, · · · , αN〉}

Impurity

eEnvironmentξ 1

|l,e,1>

|l,e,2>

|l,e,3>

|k,e,1>

|k’,e,1>

complete basis: {|e〉} = {|k, e; 1〉}

Impurity

eEnvironmentξ2ξ 1

|l,e,2>

|l,e,3>

|k,e,2>

complete basis: {|e〉} = {|k, e; 2〉}+ {|l , e; 2〉}

Impurity

eξ2ξ 1

|l,e,3>

|l,e,2>

|k,e,3>

complete basis: {|e〉} = {|k, e; 3〉}+∑3

m=2{|l , e;m〉}

Impurity

ξ2ξ 1

|l,e,3>

|l,e,2>

|l,e,N>

complete basis: {|e〉} =∑N

m=2{|l , e;m〉}

Sum-rule conserving NRG Green functions

GA,B(z) =N∑

m=mmin

∑k,k ′

Al ,k ′(m)ρredk ′,k(m)Bk,l(m)

z + El − Ek

m=mmin

∑k,k ′

Bl ,k ′(m)ρredk ′,k(m)Ak,l(m)

z + Ek − El

reduced density matrix (Feynman 72, White 92, Hofstetter 2000)

ρredk,k ′(m) =

〈k, e;m|ρ|k ′, e;m〉 ,

Peters, Pruschke, FBA, Phys. Rev. B 74, 245114 (2006)

Weichelbaum, von Delft: cond-mat/0607497

extension to NEQ GF G (t, t‘) possible (Anders 2007)

Sum-rule conserving NRG Green functions

GA,B(z) =N∑

m=mmin

∑k,k ′

Al ,k ′(m)ρredk ′,k(m)Bk,l(m)

z + El − Ek

m=mmin

∑k,k ′

Bl ,k ′(m)ρredk ′,k(m)Ak,l(m)

z + Ek − El

reduced density matrix (Feynman 72, White 92, Hofstetter 2000)

ρredk,k ′(m) =

〈k, e;m|ρ|k ′, e;m〉 ,

Peters, Pruschke, FBA, Phys. Rev. B 74, 245114 (2006)

Weichelbaum, von Delft: cond-mat/0607497

extension to NEQ GF G (t, t‘) possible (Anders 2007)

Spectral function in the presents of CEF splitting

-10 -8 -6 -4 -2 0 2ω/Γ

Γ ρ(

-0.4 -0.2 0 0.2 0.4ω/Γ

Γ ρ(

Σα(z) causal

G−1α (z) = z − Eα − Γα(z)− Σα(z)

NCA: Σα(z) violates causality already for T � TK

Spectral function in the presents of CEF splitting

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1ω/Γ

Σα(z) causal

G−1α (z) = z − Eα − Γα(z)− Σα(z)

NCA: Σα(z) violates causality already for T � TK

Contents

5 Conclusion

Real-time dynamics of an observable

〈O〉(t) = Tr[Oρ(t)

]Equilibrium: single condition ρ(t) = ρ0 = exp(−βHf )/Z

Non-equilibrium: two conditions: ρ0 and Hf

ρ(t) = e−iHf t ρ0eiHf t

Calculation of the trace using an energy eigenbasis of Hf

〈O〉(t) =∑n,m

〈En|O|Em〉〈Em|ρ0|En〉e−i(Em−En)t

〈O〉(t) =∑n,m

The challenge

Non-equilibrium dynamics in quantum impurity systems

The problem

evaluation of all energy scales

avoid overcounting

NEQ-NRG spectral functions: Costi, PRB 55, 3003 (1997)

relaxation into the new thermodynamic ground state?

The solution

complete NRG basis set of the Wilson chainFBA and A Schiller, PRL 95, 196801 (2005), Phys. Rev. B 74, 245113 (2006)

The challenge

The problem

avoid overcounting

The solution

The challenge

The problem

avoid overcounting

The solution

Time-dependent numerical renormalization group

New method: time-dependent NRG

〈O〉(t) =∑n,m

O: local operator, diagonal in e

reduced density matrix

ρredll ′ (m) =

〈l , e;m|ρ0|l ′, e;m〉

RG upside down: elimited states contain the information onthe time evolution

discretization averaging simulates continuum

FBA, A Schiller, PRL 95, 196801 (2005), Phys. Rev. B 74, 245113 (2006)

〈O〉(t) =∑m

l or l ′ discarded∑l ,l

〈l |O|l ′〉e i(El−El′ )tρredl ′l (m)

ρredll ′ (m) =

〈l , e;m|ρ0|l ′, e;m〉

〈O〉(t) =∑m

l or l ′ discarded∑l ,l

〈l |O|l ′〉e i(El−El′ )tρredl ′l (m)

ρredll ′ (m) =

〈l , e;m|ρ0|l ′, e;m〉

Discussion of the method

resolving the contradiction: RG and including all energy scale

no accumulated error in time in contrary to td-DMRG

exponentially long time scales accessable (up to t ∗ T ≈ 1)

calculation of time-dependent NEQ Green functions G (t, t ′)for steplike Hamiltionians possible

Spin decay in the anisotropic Kondo model

Benchmark: decoherence of a pure state|s〉 = (| ↑〉+ | ↓〉)/

TD-NRG

0,001 0,01 0,1 1 10t*T

ρ 01(t

)/ρ 01

s=1.5s=1.0s=0.8s=0.6s=0.4s=0.2

analytical exact solution and TD-NRG: excellent agreement

Benchmark: decoherence of a pure state|s〉 = (| ↑〉+ | ↓〉)/

TD-NRG plus analytic solution PRB 74,245113 (2006)

0.001 0.01 0.1 1 10t*T

ρ 01(t

)/ρ 01

s=1.5s=1.0s=0.8s=0.6s=0.4s=0.2

analytical exact solution and TD-NRG: excellent agreement

0 1 2 3 4 5 6 7 8 9 10

Jz=-0.1

J⊥ =0.15 (analytic)

J⊥ =0.15(ana) O(t2)

Jz=0.15

Jz=0.1

Jz=0.05

Jz=0.0

short-time dynamics: perturbative in J⊥

AFM regime: infrared divergence+ exponentially long time-scale 1/TK

0 1 2 3 4 5 6 7 8 9 10

Jz=-0.1

J⊥ =0.15 (analytic)

J⊥ =0.15(ana) O(t2)

Jz=0.15

Jz=0.1

Jz=0.05

Jz=0.0

short-time dynamics: perturbative in J⊥

AFM regime: infrared divergence+ exponentially long time-scale 1/TK

0.5S z(t

z=0.15

2ρJz=0.1

2ρJz=0.05

2ρJz=0.0

2ρJz=-0.1

Flow Equation

flow equation solution: Kehrein 2005

long time relaxation: tspin ∝ 1/TK

conformal field theory and flow equationexponential decay only for t � 1/TK

details in: FBA and Schiller, Phys. Rev. B 74, 245113 (2006)

0.5S z(t

z=0.15

2ρJz=0.1

2ρJz=0.05

2ρJz=0.0

2ρJz=-0.1

Flow Equation

flow equation solution: Kehrein 2005

long time relaxation: tspin ∝ 1/TK

conformal field theory and flow equationexponential decay only for t � 1/TK

details in: FBA and Schiller, Phys. Rev. B 74, 245113 (2006)

Contents

5 Conclusion

Conclusion

The numerical renormalization group

accurate, non-perturbative solution to any QIP

fixed points: insight into the physics of a model

thermodynamics and quantum phase transitions

equilibrium spectral function

extendable to non-equilibriumTD-NRG: no accumulated error in time

Applications

one, two-site, multi-channel impurity models

zoo of coupled quantum dot models

impurity solver for DMFT calculations

real-time dynamics

Conclusion

Applications

real-time dynamics

Conclusion

Applications

real-time dynamics

Conclusion

Applications

real-time dynamics

Conclusion

Applications

real-time dynamics

Conclusion

Applications

real-time dynamics

Conclusion

Applications

real-time dynamics

Conclusion

Applications

real-time dynamics

Conclusion

Applications

real-time dynamics

Dynamical properties of quantum impurity systems...

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