Dynamical properties of quantum impurity systems...
Transcript of 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
NRG Review R. Bulla, T. Costi and Th. Pruschkecond-mat/0701105to be published in RMP
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Kondo effect in nano-devices
Kondo effect in a single electron transistor (SET)
SET
D. Goldhaber-Gordon, Nature 98
weak coupling
M.Kastner RMP 1992
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Kondo effect in nano-devices
Kondo effect in a single electron transistor (SET)
SET
D. Goldhaber-Gordon, Nature 98
strong coupling
van der Wiel et al. Science 289
(2000)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Kondo effect in nano-devices
=⇒
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, · · ·
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Kondo effect in nano-devices
=⇒
dynamicalbath
~G(z)
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, · · ·
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Λ−n/2
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Discretization of the bath contiuum
Wilson’s NRG (1975)
Impurity
ξ3
ξ2
ξ 1
321 N
ξΝ ∼Λ −Ν/2
switching on iteratively the couplings ξm ∝ Λ−m/2
recursion relation (RG transformation)
HN+1 =√
ΛHN +∑
σ
ξN
(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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Discretization of the bath contiuum
Wilson’s NRG (1975)
Impurity
ξ3
ξ2
ξ 1
321 N
ξΝ ∼Λ −Ν/2
switching on iteratively the couplings ξm ∝ Λ−m/2
recursion relation (RG transformation)
HN+1 =√
ΛHN +∑
σ
ξN
(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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Discretization of the bath contiuum
Wilson’s NRG (1975)
Impurity
ξ3
ξ2
ξ 1
321 N
ξΝ ∼Λ −Ν/2
switching on iteratively the couplings ξm ∝ Λ−m/2
recursion relation (RG transformation)
HN+1 =√
ΛHN +∑
σ
ξN
(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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Discretization of the bath contiuum
Wilson’s NRG (1975)
Impurity
ξ3
ξ2
ξ 1
321 N
ξΝ ∼Λ −Ν/2
switching on iteratively the couplings ξm ∝ Λ−m/2
recursion relation (RG transformation)
HN+1 =√
ΛHN +∑
σ
ξN
(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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Discretization of the bath contiuum
Wilson’s NRG (1975)
Impurity
ξ3
ξ2
ξ 1
321 N
ξΝ ∼Λ −Ν/2
switching on iteratively the couplings ξm ∝ Λ−m/2
recursion relation (RG transformation)
HN+1 =√
ΛHN +∑
σ
ξN
(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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
T/Γ0
0.5
1
1.5
2
2.5
Entro
py S
/log(
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 ∗
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Fixed points
Numerical Renormalization Group
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?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Fixed points
Numerical Renormalization Group
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?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Fixed points
Numerical Renormalization Group
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?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Fixed points
Numerical Renormalization Group
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?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Fixed points
Numerical Renormalization Group
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?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
)Z
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 ?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
)Z
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 ?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
)Z
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 ?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
)Z
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 ?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
)Z
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 ?
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
321 N
|l,e,1>
|l,e,2>
|l,e,3>
|e>
complete basis: {|e〉} = {|αimp, α1, α2, α3, α4, · · · , αN〉}
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
eEnvironmentξ 1
321 N
|l,e,1>
|l,e,2>
|l,e,3>
|e>
|k,e,1>
|k’,e,1>
complete basis: {|e〉} = {|k, e; 1〉}
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
eEnvironmentξ2ξ 1
321 N
|l,e,2>
|l,e,2>
|l,e,3>
|e>
|k,e,2>
complete basis: {|e〉} = {|k, e; 2〉}+ {|l , e; 2〉}
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
eξ2ξ 1
321 N
ξ3
|e>
|l,e,3>
|l,e,2>
|k,e,3>
complete basis: {|e〉} = {|k, e; 3〉}+∑3
m=2{|l , e;m〉}
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
ξ2ξ 1
ξ3
321 N
ξΝ
|e>
|l,e,3>
|l,e,2>
|l,e,N>
complete basis: {|e〉} =∑N
m=2{|l , e;m〉}
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Complete basis set of the Wilson chain
Sum-rule conserving NRG Green functions
GA,B(z) =N∑
m=mmin
∑l
∑k,k ′
Al ,k ′(m)ρredk ′,k(m)Bk,l(m)
z + El − Ek
+N∑
m=mmin
∑l
∑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) =
∑e
〈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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Complete basis set of the Wilson chain
Sum-rule conserving NRG Green functions
GA,B(z) =N∑
m=mmin
∑l
∑k,k ′
Al ,k ′(m)ρredk ′,k(m)Bk,l(m)
z + El − Ek
+N∑
m=mmin
∑l
∑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) =
∑e
〈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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Complete basis set of the Wilson chain
Spectral function in the presents of CEF splitting
-10 -8 -6 -4 -2 0 2ω/Γ
0
0.05
0.1
0.15
0.2
0.25
Γ ρ(
ω)
E1=E
2
E1=E
g
E2=E
g+0.1
-0.4 -0.2 0 0.2 0.4ω/Γ
0
0.05
0.1
0.15
0.2
0.25
Γ ρ(
ω)
E1=E
2
E1=E
g
E2=E
g+0.1
Σα(z) causal
G−1α (z) = z − Eα − Γα(z)− Σα(z)
NCA: Σα(z) violates causality already for T � TK
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Complete basis set of the Wilson chain
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
0
1
2
3
4
5
6
7
8
Im Σ
(ω)
E1=E
2
E1=E
g
E2=E
g+0.1
Σα(z) causal
G−1α (z) = z − Eα − Γα(z)− Σα(z)
NCA: Σα(z) violates causality already for T � TK
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Time-dependent numerical renormalization group
New method: time-dependent NRG
〈O〉(t) =∑n,m
〈En|O|Em〉〈Em|ρ0|En〉e−i(Em−En)t
O: local operator, diagonal in e
reduced density matrix
ρredll ′ (m) =
∑e
〈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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Time-dependent numerical renormalization group
New method: time-dependent NRG
〈O〉(t) =∑m
l or l ′ discarded∑l ,l
〈l |O|l ′〉e i(El−El′ )tρredl ′l (m)
O: local operator, diagonal in e
reduced density matrix
ρredll ′ (m) =
∑e
〈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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Time-dependent numerical renormalization group
New method: time-dependent NRG
〈O〉(t) =∑m
l or l ′ discarded∑l ,l
〈l |O|l ′〉e i(El−El′ )tρredl ′l (m)
O: local operator, diagonal in e
reduced density matrix
ρredll ′ (m) =
∑e
〈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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Time-dependent numerical renormalization group
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Spin decay in the anisotropic Kondo model
Benchmark: decoherence of a pure state|s〉 = (| ↑〉+ | ↓〉)/
√2
TD-NRG
0,001 0,01 0,1 1 10t*T
0
0,2
0,4
0,6
0,8
1
ρ 01(t
)/ρ 01
(0)
s=1.5s=1.0s=0.8s=0.6s=0.4s=0.2
analytical exact solution and TD-NRG: excellent agreement
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Spin decay in the anisotropic Kondo model
Benchmark: decoherence of a pure state|s〉 = (| ↑〉+ | ↓〉)/
√2
TD-NRG plus analytic solution PRB 74,245113 (2006)
0.001 0.01 0.1 1 10t*T
0
0.2
0.4
0.6
0.8
1
ρ 01(t
)/ρ 01
(0)
s=1.5s=1.0s=0.8s=0.6s=0.4s=0.2
analytical exact solution and TD-NRG: excellent agreement
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Spin decay in the anisotropic Kondo model
0 1 2 3 4 5 6 7 8 9 10
0.44
0.46
0.48
0.5
Sz(t
)
Jz=-0.1
J⊥ =0.15 (analytic)
J⊥ =0.15(ana) O(t2)
100
102
104
106
108
1010
t*D
0
0,1
0,2
0,3
0,4
0,5
Sz(t
)
Jz=0.15
Jz=0.1
Jz=0.05
Jz=0.0
(a)
(b)
short-time dynamics: perturbative in J⊥
AFM regime: infrared divergence+ exponentially long time-scale 1/TK
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Spin decay in the anisotropic Kondo model
0 1 2 3 4 5 6 7 8 9 10
0.44
0.46
0.48
0.5
Sz(t
)
Jz=-0.1
J⊥ =0.15 (analytic)
J⊥ =0.15(ana) O(t2)
100
102
104
106
108
1010
t*D
0
0,1
0,2
0,3
0,4
0,5
Sz(t
)
Jz=0.15
Jz=0.1
Jz=0.05
Jz=0.0
(a)
(b)
short-time dynamics: perturbative in J⊥
AFM regime: infrared divergence+ exponentially long time-scale 1/TK
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Spin decay in the anisotropic Kondo model
10-4
10-3
10-2
10-1
100
101
t*TK
0
0.1
0.2
0.3
0.4
0.5S z(t
)2ρJ
z=0.15
2ρJz=0.1
2ρJz=0.05
2ρJz=0.0
2ρJz=-0.1
Flow Equation
(c)
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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
Spin decay in the anisotropic Kondo model
10-4
10-3
10-2
10-1
100
101
t*TK
0
0.1
0.2
0.3
0.4
0.5S z(t
)2ρJ
z=0.15
2ρJz=0.1
2ρJz=0.05
2ρJz=0.0
2ρJz=-0.1
Flow Equation
(c)
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)
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium Conclusion
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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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
Introduction The Numerical Renormalization Group Spectral functions at finite temperatures Real-time dynamics out of equilibrium 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