dynamical properties of quantum impurity systems...
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![Page 1: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/1.jpg)
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
![Page 2: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/2.jpg)
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
![Page 3: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/3.jpg)
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
![Page 4: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/4.jpg)
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
![Page 5: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/5.jpg)
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
![Page 6: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/6.jpg)
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
![Page 7: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/7.jpg)
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
![Page 8: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/8.jpg)
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
![Page 9: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/9.jpg)
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
![Page 10: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/10.jpg)
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)
![Page 11: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/11.jpg)
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, · · ·
![Page 12: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/12.jpg)
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, · · ·
![Page 13: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/13.jpg)
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
![Page 14: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/14.jpg)
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
![Page 15: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/15.jpg)
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
![Page 16: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/16.jpg)
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
![Page 17: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/17.jpg)
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
![Page 18: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/18.jpg)
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
![Page 19: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/19.jpg)
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
![Page 20: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/20.jpg)
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
![Page 21: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/21.jpg)
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
![Page 22: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/22.jpg)
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
![Page 23: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/23.jpg)
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
![Page 24: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/24.jpg)
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
![Page 25: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/25.jpg)
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
![Page 26: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/26.jpg)
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 ∗
![Page 27: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/27.jpg)
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?
![Page 28: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/28.jpg)
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?
![Page 29: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/29.jpg)
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?
![Page 30: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/30.jpg)
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?
![Page 31: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/31.jpg)
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?
![Page 32: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/32.jpg)
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
![Page 33: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/33.jpg)
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 ?
![Page 34: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/34.jpg)
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 ?
![Page 35: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/35.jpg)
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 ?
![Page 36: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/36.jpg)
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 ?
![Page 37: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/37.jpg)
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 ?
![Page 38: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/38.jpg)
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〉}
![Page 39: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/39.jpg)
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〉}
![Page 40: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/40.jpg)
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〉}
![Page 41: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/41.jpg)
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〉}
![Page 42: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/42.jpg)
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〉}
![Page 43: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/43.jpg)
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)
![Page 44: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/44.jpg)
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)
![Page 45: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/45.jpg)
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
![Page 46: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/46.jpg)
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
![Page 47: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/47.jpg)
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
![Page 48: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/48.jpg)
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
![Page 49: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/49.jpg)
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
![Page 50: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/50.jpg)
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
![Page 51: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/51.jpg)
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
![Page 52: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/52.jpg)
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)
![Page 53: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/53.jpg)
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)
![Page 54: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/54.jpg)
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)
![Page 55: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/55.jpg)
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)
![Page 56: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/56.jpg)
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)
![Page 57: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/57.jpg)
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)
![Page 58: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/58.jpg)
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
![Page 59: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/59.jpg)
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
![Page 60: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/60.jpg)
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
![Page 61: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/61.jpg)
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
![Page 62: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/62.jpg)
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
![Page 63: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/63.jpg)
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)
![Page 64: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/64.jpg)
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)
![Page 65: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/65.jpg)
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
![Page 66: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/66.jpg)
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
![Page 67: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/67.jpg)
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
![Page 68: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/68.jpg)
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
![Page 69: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/69.jpg)
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
![Page 70: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/70.jpg)
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
![Page 71: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/71.jpg)
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
![Page 72: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/72.jpg)
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
![Page 73: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/73.jpg)
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
![Page 74: Dynamical properties of quantum impurity systems …t2.physik.tu-dortmund.de/files/talks/anders/nrg-tutorial.pdfIntroduction The Numerical Renormalization Group Spectral functions](https://reader033.vdocuments.net/reader033/viewer/2022050110/5f479f2cb301aa3d9817c1ba/html5/thumbnails/74.jpg)
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