Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke,...
Transcript of Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke,...
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Numerical renormalization group andmulti-orbital Kondo physics
Theo Costi
Institute for Advanced Simulation (IAS-3) andInstitute for Theoretical Nanoelectronics (PGI-2)
Research Centre Jülich
25th September 2015Autumn School on Correlated Electrons:
“From Kondo to Hubbard”
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Outline
1 Introduction
2 Wilson’s NRG
3 Complete Basis Set
4 Recent Developments
5 Summary
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Collaborations
Dr Hoa Nghiem (Postdoc, Juelich), Lukas Merker (PhD,Juelich)RWTH Aachen colaborators Dr Dante Kennes, Prof. VolkerMeden (complementary methods, FRG/DMRG), Uni.Tuebingen, Prof. Sabine Andergassen (FRG)
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
References on NRG and codes
K.G. Wilson, Rev. Mod. Phys. 1975 [Kondo model]Krishna-murthy et al, PRB 1980 [Anderson model]R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 2008[further developments/models]A. C. Hewson, The Kondo Problem to Heavy Fermions,C.U.P. (1997)Recommended public domain NRG codes:
“flexible DM-NRG” code (G. Zarand, et al., Budapest)“NRG-Ljubljana” code (R. Zitko, Ljubljana)
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Quantum impurity systems
Quantum impurity systems
U Environment
Impurity: small system, discrete spectrumEnvironment: large system, quasi-continuous spectrumCoupling: hybridization, Kondo exchange,...General structure:
H = Himp + Hint + Hbath
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Quantum impurity systems
Anderson model
Single-level Anderson impurity model (nd� = d†�d�):
HAM =X
�
"dnd� + Und"nd#
| {z }Himp
+X
k�
Vkd(c†k�d� + d†
�ck�)
| {z }Hint
+X
k�
✏kc†k�ck�
| {z }Hbath
(1)
We take constant hybridization function:
�(!) = ⇡X
k
V 2kd�(! � ✏k ) ⇡ �(✏F = 0)
non-constant hybridization can also be dealt with in NRG(e.g., for applications to DMFT, pseudogap systems,...)
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Quantum impurity systems
Multi-orbital and multi-channel Anderson Model
Rotationally symmetric case, e.g., for transition metal ionsl = 2, and m = �2, �1, 0,+1,+2
H =X
m�
"dmnm� +12
UX
m�
nm�nm��+12
U 0X
m 6=m0�
nm�nm0��
+12(U 0 � J)
X
m 6=m0�
nm�nm0� � J2
X
m 6=m0�
d†m�dm��d†
m0��dm0�
�J 0
2
X
m 6=m0�
d†m�d†
m��dm0��dm0�
+X
km�
Vkm�(c†km�dm� + h.c.) +
X
km�
✏km�c†km�ckm�
Crystal field and spin-orbit interactions )Further lowering of symmetry, fewer channels
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Experiments
Magnetic impurities in non-magnetic metals
Fe in Au: de Haas et al 1934
Au - sample 1
Au -sample 2
T [K]
Fe in Ag: Mallet et al 2006
de Haas et al: resistivity minimum and increase as T ! 0 !Kondo 1964: explanation in terms of magnetic impuritiesKondo 1964: RK(T ) = �cimp ln(kBT/D) ! 1 as T ! 0 !!
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Experiments
Quantum dots, Kondo effect and nanoscale size SET’s
100nm lateral quantum dotsGoldhaber-Gordon et al1998
DotGate
source
drain
) Tunable Kondo effect
εd + U
εd
~∆
~TK
Vg
V I
TK =p
�U/2e⇡"d ("d+U)/2�U ) exponential sensitivitySingle-electron transistor based on Kondo effect
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Experiments
High/low spin molecules in nanogaps
C60: Roch et al 2009 Underscreened Kondo effect
Asymmetric lead couplings: partial screening of S = 1 )Underscreened Kondo effect (N. Roch et al PRL 2009)
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Experiments
Magnetic atoms on surfaces: Kondo + spin anisotropy
Co (S=3/2): Kondo+anisotropy
Otte et al 2008
Ti (S=1/2): usual Kondo effect
Otte et al 2008
Himp = �gµBSz + DS2z for Co adatom with S=3/2
D > 0 ) m = 1/2 $ �1/2 transitions ) Kondo effectm = ±1/2 ! ±3/2 transitions, side-bands to Kondo
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Linear chain form
NRG: linear chain form of the Anderson model
HAM =X
�
"dnd� + Und"nd# +
Vf0�=P
k Vkd ck�z }| {VX
�
(f †0�d� + d†�f0�)+
X
k�
✏kc†k�ck�
Tridiagonalize Hbath =P
k� ✏kc†k�ck� using Lanczos with starting
vector defined by Vf0� =P
k Vkdck� to obtain linear chain form:
HAM =X
�
"dnd� + Und"nd# + VX
�
(f †0�d� + d†�f0�)
+1X
n=0,�
✏nf+n�fn� +1X
n=0,�
tn(f+n�fn+1� + H.c.)
...suitable for numerical treatment by adding one orbital at atime, starting with d�, then f0, then f1, ...
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Linear chain form
Atomic and zero-bandwidth limits
Hm=�1AM (V = 0) =
X
�
"dnd� + Und"nd#
Hm=0AM (✏k = 0) =
X
�
"dnd� + Und"nd# + VX
�
(f †0�d� + d†�f0�)
A�(!) =1Z
X
p,q|hp|d†
�|qi|2(e��Eq + e��Ep)�(! � (Eq � Ep)
V = 0:atomic limit
A(!)
!
✏F
"d
"d + U
✏k = ✏F = 0:zero bandwidth limit
A(!)
!
✏F
"d
"d + U
O(V 2/U)
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Linear chain form
Anderson model: the Kondo resonance
HAM =X
�
"dnd� + Und"nd# + VX
k�
(f †0�d� + d†�f0�)
+1X
n=0,�
✏nf+n�fn� +1X
n=0,�
tn(f+n�fn+1� + H.c.)
Finite bandwidthAdd f1�, f2�, ...
) broadening O(�) of "d ,and "d + U excitations) Kondo resonanceTK ⇠ e�1/J⇢, J ⇠ V 2/U
Kondo resonance
A(!)
!
✏F
"d
"d + UTK ⇠ e�1/J�
2�
2�
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Linear chain form
Iterative diagonalization/recursion relationDefine truncated Hamiltonian (m conduction orbitals):
HAM ⇡ Hm = "dnd + Und"nd# + VX
�
(f+0�d� + d+� f0�)
+mX
�,n=0
✏nf+n�fn� +m�1X
�,n=0
tn(f+n�fn+1� + f+n+1�fn�)
satisfying recursion relation:
Hm+1 = Hm +X
�
✏m+1f †m+1�fm� + tmX
�
(f+m�fm+1� + f+m+1�fm�).
diagonalize Hm =P
p Emp |pimmhp|
use product basis |r , em+1i ⌘ |rim|em+1i with|em+1i = |0i, | "i, | #i, | "#i to set up matrix of Hm+1 viarecursion relation and diagonalize, ...
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Iterative diagonalization/truncation
Iterative diagonalization
...Himp
ε0 ε1 ε2 εm
t0 t1 t2 tm-1vm ...
Exact, if all states retained.In practice, no. states of Hm is O(4m+1) (= 4096 at m = 5)Symmetries help Hm =
P�Hm
Ne,S,Sz
([Ne, Hm] = [~S2, Hm] = [Sz , Hm] = 0)For m > 5, truncate to O(1000) lowest eigenstatesTruncation works, provided tm decay sufficiently fast with m
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Iterative diagonalization/truncation
Logarithmic discretization
−Λ−1 Λ−3 Λ−2 Λ−1−3−Λ−Λ−2 ... ω
N(ω)
−1 1
Logarithmic discretization of band✏kn/D = ±⇤�n, n = 0, 1, . . . with ⇤ > 1, e.g. ⇤ = 2) tm ⇠ ⇤�m/2, ✏m ⇠ ⇤�m
For ⇤ � 1, average over discretizations (Oliveira),✏kn/D = ±⇤�n�z+1, n = 1, . . . (±1, n = 0)
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Iterative diagonalization/truncation
NRG for multiple-channels
Transition metal ions l = 2, and m = �2, �1, 0,+1,+2
H = Himp({"dm} , U, U 0, J) +X
km�
Vmf0m�=P
k Vkmckm�z }| {Vkm(c
†km�dm� + h.c.)
+X
km�
✏km�c†km�ckm�
Convert to linear chain form using Lanczos:
H = Himp({"dm} , U, U 0, J) +X
m�
Vm(f0m�dm� + h.c.)
++1X
n=0
X
m�
✏nmf †nm�fnm� ++1X
n=0
X
m�
tnm(f †nm�fn+1m� + h.c.)
NRG: Hilbert space increases by 42l+1 = 1024 per shelladded !
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Iterative diagonalization/truncation
NRG for multiple-channels: exploiting symmetries
For 3-channel S = 3/2 Kondo model, without symmetries,fraction of kept states = 1/43 = 1/64.implementing SU(3) symmetries (Hanl, Weichselbaum etal, PRB 2013) increased this fraction to ⇡ 1/42 = 1/16,making the 3-channel Kondo calculation equivalent to a2-channel calculationallowed comparing to experimental resistivity (R(T )) anddephasing (⌧'(T )) data for Fe/Au for
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Iterative diagonalization/truncation
Multi-channel fully screened/single-channelunderscreened resistivities
Resistivity for 3 channelS = 3/2 Kondo model, Hanl etal PRB 2013
10−1 100 1010
0.5
1
1.5
2
T (K)
∆ρ(
T,B=
0)/∆ρ(
0,0)
TK(1)(K)
TK(2)(K)
TK(3)(K)
AuFe3 AgFe2 AgFe3
0.6±0.1
1.0±0.1
1.7±0.1
2.5±0.2
4.3±0.3
7.4±0.5
2.8±0.2
4.7±0.3
8.2±0.5
n=1
n=2
n=3
Resistivity for 1-channel spinS � 1/2 Kondo model, Parks etal Science 2010
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Physical properties
Output from NRG procedureKept/discarded states
Kept states Discarded states
EnergyLevels
m0-1 m0 m0+1 ...
(Toth et al)
Many-body eigenvalues Emp and eigenvectors |pim on all
relevant energy scales tm ⇠ ⇤�m/2, m = 0, 1, ..., N.From eigenvalues calculate Z =
Pp e��Ep and
thermodynamicsFrom eigenstates calculate matrix elements, hence Greenfunctions and transportDiscarded states not used in iterative diagonalization, butuseful for calculating physical properties (see later)
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Physical properties
Thermodynamics
Impurity entropy/specific heat
10-8
10-6
10-4
10-2
100
kBT /D
-0.4
-0.2
0
0.2
Cimp[kB]
0
0.5
1
1.5
2
Simp[kB]/ln(2)
z-averagedz=1
(a)
(b)
Merker et al2012
Entropy
S(T ) = (E � F )/T ,
E = hHAMi,
F = �1�
ln Z
Specific heat
C(T ) = kB�2hH2AM � hHAMi2i
Discretization oscillations at ⇤ � 1 eliminated byz-averaging (here ⇤ = 4, nz = 2).
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Physical properties
Dynamics
d-spectral function
10-5
10-3
10-1
101
ω/∆0
0
0.5
1
1.5
2
π∆0A
(ω,T
=0)
Merker et al 2013
NRG spectra are discrete(�Eqq0 = Eq0 � Eq):
Ad�(!, T ) =X
q,q0
wqq0(T )��! � �Eqq0
�
broaden with logarithmicGaussians:
�(! � �Eq0q)
! e�b2/4
b|�Eq0q|p
⇡e�(ln(!/�Eq0q)/b)2
Discretization oscillations at ⇤ � 1 eliminated byz-averaging (here ⇤ = 10, nz = 8).
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Linear transport
Linear transportThermopower for U < 0
10-2 10-1 100 101 102 103
T/TK
0
10
20
30
40
50
60
70
80
S (µ
V/K
)
Vg/TK=0.1Vg/TK=1Vg/TK=10Vg/TK=100
Andergassen et al, 2011
Conductance: Merker et al
10-3
10-1
101
103
T/TK
0
0.2
0.4
0.6
0.8
1
G[2
e2/h
]
εd = -U/2
εd = 0
εd = U/2
Linear response transport from spectral functions A(!, T ):
G(T ) =e2
h
Zd!
✓� @f
@!
◆ X
�
/A(!,T )z }| {T�(!, T ),
S(T ) = � 1|e|T
Rd! ! (�@f/@!)
P� T�(!, T )R
d! (�@f/@!)P
� T�(!, T )
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Kept states in NRG
Kept/discarded statesKept states Discarded states
EnergyLevels
m0-1 m0 m0+1 ...
(Toth et al)
Z =P
p e��Ep
Z 6=PN
m=0
=Zm(T )z }| {X
pe��Em
p
Double countingConventional approach:Z (T ) ⇡ Zm(T ) forT ⌘ Tm ⇠ tm
Use discarded states to build a complete set of eigenstatesfor summing up unambiguously contributions from differentHm (Anders & Schiller 2005)All states must reside in same Hilbert space: that of HN
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Construction of the complete basis setEnvironment states
... ...m NHimp ε
0
e
ε1
ε2
εm
t0 t1 t2 tm-1v
Hm|qmi = Emq |qmi, q = (k , l) = (kept,discarded)
Product states |lemi = |lmi ⌦ |ei withe = (em+1, ..., eNi, m = m0, ..., N form a complete set(Anders & Schiller 2005)
NX
m=m0
X
le
|lemihlem| = 1
Allows unambiguous summation of contributions fromdifferent Hm
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Full density matrix
Full density matrix (FDM)Full density matrix and partition function (Weichselbaum &von Delft 2007):
⇢ =1
Z (T )
NX
m=m0
X
le
e��Eml |lemihlem|, Tr⇢ = 1 )
Z (T ) =NX
m=m0
4N�mX
l
e��Eml ⌘
NX
m=m0
4N�mZm(T )
Rewrite ⇢ as weighted sum of shell density matrices ⇢m:
⇢ =NX
m=m0
wm⇢m, ⇢m =X
le
|lemie��Eml
Zmhlem|.
Tr [⇢m] = 1, wm = 4N�m Zm
Z,
NX
m=m0
wm = 1
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Full density matrix
Thermodynamics
Local observables O = nd , nd"nd#, d†�f0�, Sz , ...
hOi⇢ = Trh⇢O
i=
X
l 0e0m0
hl 0e0m0|⇢O|l 0e0m0i
=X
l 0e0m0
X
lem
wm
�ll0ee0mm0z }| {hl 0e0m0|lemi e��Em
l
Zm
Omllz }| {
hlem|O|l 0e0m0i
=X
lem
wme��Em
l
ZmOm
ll =X
lm
4N�mwmOmll
e��Eml
4N�mZm
=NX
m=m0,l
wmOmll
e��Eml
Zm
![Page 29: Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 2008 [further developments/models] A. C. Hewson, The Kondo Problem](https://reader033.fdocuments.in/reader033/viewer/2022042920/5f64a4d7ac84ad06757f53aa/html5/thumbnails/29.jpg)
Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Full density matrix
Double occupancy
Double occupancy: hnd"nd#i ! hnd"ihnd#i, U ! 0Double occupancy: hnd"nd#i ⌧ 1, U � �
FDM (lines) and conventional (symbols) in excellentagreement: FDM simpler, obviates need to chose bestshell for given T
0
0.05
0.1
0.15
0.2
0.25
0.3
<n
d↑n
d↓>
0.01124681012
10-3
10-2
10-1
100
101
102
103
104
T/TK
10-3
10-2
10-1
<n
d↑n
d↓> -5
-3-10135
U/∆0=
εd/∆
0=
(a)
(b)
![Page 30: Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 2008 [further developments/models] A. C. Hewson, The Kondo Problem](https://reader033.fdocuments.in/reader033/viewer/2022042920/5f64a4d7ac84ad06757f53aa/html5/thumbnails/30.jpg)
Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Application to Dynamics
Dynamics
Evaluation of equilibrium retarded Green function(Weichselbaum & von Delft 2007; Peters et al 2006):
GAB(t) = �i✓(t)h[A(t), B]+i ⌘ �i✓(t)Tr [⇢(A(t)B + BA(t))]= �i✓(t)
⇥CA(t)B + CBA(t)
⇤
Consider correlation function CA(t)B:
CA(t)B = Tr [⇢A(t)B] = Tr [⇢A(t)B] = Trh⇢A(t)1B
i
=X
lem
hlem|e�iHtAeiHtX
l 0e0m0
|l 0e0m0ihl 0e0m0|B⇢|lemi
=X
lem
X
l 0e0m0
hlem|e�iHtAeiHt |l 0e0m0i| {z }ei(Em0
l0 �Eml )t hl 0e0m0|A|lemi
hl 0e0m0|B⇢|lemi
![Page 31: Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 2008 [further developments/models] A. C. Hewson, The Kondo Problem](https://reader033.fdocuments.in/reader033/viewer/2022042920/5f64a4d7ac84ad06757f53aa/html5/thumbnails/31.jpg)
Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Application to Dynamics
Dynamics
... Peters et al 2006
Cm0>mA(t)B =
X
lem
hlem|e�iHtAeiHt
=|kemihkem|z }| {X
l 0e0m0>m
|l 0e0m0ihl 0e0m0| B⇢|lemi
Avoid off-diagonal matrix elements Am 6=m0
ll 0 by using
1 = 1�m + 1+
m
1�m =
mX
m0=m0
X
l 0e0
|l 0e0m0ihl 0e0m0|
1+m =
NX
m0=m+1
X
l 0e0
|l 0e0m0ihl 0e0m0| =X
ke
|kemih kem|.
![Page 32: Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 2008 [further developments/models] A. C. Hewson, The Kondo Problem](https://reader033.fdocuments.in/reader033/viewer/2022042920/5f64a4d7ac84ad06757f53aa/html5/thumbnails/32.jpg)
Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Application to Dynamics
Dynamics
... only shell diagonal matrix elements appear: Amll 0
... and reduced density matrices Hofstetter 2000:Rm
red(k0, k) =
Pehk 0em|⇢|kemi
GAB(! + i�) =X
m
wm
Zm
X
ll 0Am
ll 0Bml 0l
e��Eml + e��Em
l0
! + i� � (Eml 0 � Em
l )
+X
m
wm
Zm
X
lk
Amlk Bm
kle��Em
l
! + i� � (Emk � Em
l )+ ...
+X
m
X
lkk 0
Amkl B
mlk 0
Rmred(k
0, k)! + i� � (Em
l � Emk )
+ ...
![Page 33: Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 2008 [further developments/models] A. C. Hewson, The Kondo Problem](https://reader033.fdocuments.in/reader033/viewer/2022042920/5f64a4d7ac84ad06757f53aa/html5/thumbnails/33.jpg)
Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Time-dependent NRG (TDNRG)
Motivation: pump-probe spectroscopies
Loth et al 2010 Fe on CuN/Cu Spin relaxation
![Page 34: Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 2008 [further developments/models] A. C. Hewson, The Kondo Problem](https://reader033.fdocuments.in/reader033/viewer/2022042920/5f64a4d7ac84ad06757f53aa/html5/thumbnails/34.jpg)
Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Time-dependent NRG (TDNRG)
Transient response following a sudden quench
H(t) = ✓(�t)Hi + ✓(t)Hf
O(t) = Trhe�iHf t⇢eiHf t O
i
=NX
m=m0
X
rs/2KK 0
⇢i!fsr (m)e�i(Em
s �Emr )tOm
rs
(Anders & Schiller 2005FDM gen.: Nghiem & Costi 2014)
"d(t) = �✓(t)U/2:Nghiem et al 2014
0.4
0.5
0.6
0.7
0.8
0.9
1
10-2 10-1 100 101 102
n d(t)
tΓ
N z=32
O(t ! 0+) exact in TDNRG (Nghiem et al PRB 2014)O(t ! 1) not exact in TDNRG (Anders & Schiller 2005,Nghiem et al PRB 2014)
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Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Time-dependent NRG (TDNRG)
Multiple quenches and general pulses
n-Quantum quenches
⇢(t) = e�iHQp+1 (t�⌧p)e�iHQp ⌧p ...e�iHQ1⌧1⇢
⇥ eiHQ1⌧1 ...eiHQp ⌧peiHQp+1 (t�⌧p)
O(t) = Trhe�iHf t⇢eiHf t O
i
=/2KK 0X
mrs⇢
i!Qp+1rs (m, ⌧p)e�i(Em
r �Ems )(t�⌧p)Om
sr
Nghiem et al 2014
......... -0 ⌧1 ⌧2 ⌧n t
Hi = HQ0
HQ1
HQ2
HQn H f = HQn+1
⌧1z}|{
⌧2z}|{
⌧nz}|{
.........
Formalism rests ONLY on NRG (truncation) approximation!Other approaches use hybrid (DMRG-TDNRG) andadditional approximations (Eidelstein et al 2013)
![Page 36: Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 2008 [further developments/models] A. C. Hewson, The Kondo Problem](https://reader033.fdocuments.in/reader033/viewer/2022042920/5f64a4d7ac84ad06757f53aa/html5/thumbnails/36.jpg)
Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Time-dependent NRG (TDNRG)
General pulses and periodic switching
linear ramp: Nghiem & Costi 2014
0.4
0.6
0.8
1
0 0.5 1 1.5 2
nd(t
)
tΓ
0
2
4
6
0 0.8 1.6
|εd(t
)|/Γ
Quench
Ramp
periodic switching
0.4
0.5
0.6
0.7
0 1 2 3 4 5 6 7
nd(t
)
tΓ
-1
0
1
εd(t
)/Γ
Linear ramp: time-delay in occupation nd(t)Periodic switching (U = 0): coherent control ofnanodevices (qubits,...)
![Page 37: Numerical renormalization group and multi-orbital Kondo ... · R. Bulla, T.A. Costi, T. Pruschke, Rev. Mod. Phys. 2008 [further developments/models] A. C. Hewson, The Kondo Problem](https://reader033.fdocuments.in/reader033/viewer/2022042920/5f64a4d7ac84ad06757f53aa/html5/thumbnails/37.jpg)
Introduction Wilson’s NRG Complete Basis Set Recent Developments Summary
Summary
NRG: powerful method for strongly correlated systemsNRG: still in development for:
time-dependent & non-equilibrium phenomenacomplex multi-channel models (see Mitchell et al PRB2013)
Plenty of work still to do for smart PhD students !