workshop “anderson localization in topological insulators”...
TRANSCRIPT
Exponential Orthogonality Catastrophe at Anderson Metal-Insulator Transitions
Stefan Kettemann Jacobs University Bremen, GermanyAMS, Postech, Pohang South Korea
1
Workshop “Anderson Localization in Topological Insulators” Center for Theoretical Physics of Complex Systems (PCS) IBS
Daejeon, Korea, September 5 to September 9, 2016.
Bremen
Jacobs University Bremen
1st Private Research University in EuropeEnglish Speaking Campus with Students from 110 Nations 80% international students
Metal-Insulator Transitions
in 3D
I. Transport Measurements
3
Metal-Insulator Transitions in 3D
4
Res
istiv
ity
P doped Silicon
Temperature
Insulator
Metal
Critical
T. F: Rosenbaum et al., Phys. Rev. B 27, 7509 (1983)
Quantum Phase TransitionTunable by - Doping - Pressure - Gate Voltage on Thin Films
Metal-Insulator Transition
5
Metal
Insulator
by changing Donor concentration ND in Si:P Metal for ND>NC: Conduction in Impurity Band
EF
EF
Increase Donor Concentration
CB
VB
Impurity Band
LocalisedDonor States
ND>NC
ND<NC
by Doping a Semiconductor like Si:P
Bohr Radius a0
6 co: Bustarret
Insulator to Metal Transition
by Doping Insulator
Localised Donor States Anderson MIT
in Impurity Band?
Mott MIT
in Impurity Band
??Rich Physics!
Anderson MIT in Merged
Bands?
compensated uncompensated
Questions• What drives the transition
– Interactions, disorder, impurity band formation?– Anderson or Mott or both?
• Where does it happen?– In the impurity band?– Or after merging with the conduction band?
• Quantum Critical phenomena– Can we understand the experimentally measured
critical exponents?
� ⇥ (n� nc)��
� = 1.58± .02
Disorder Driven Anderson-Metal-Insulator Transition: 2nd order quantum phase transition
T. F: Rosenbaum et al., Phys. Rev. B 27, 7509 (1983)
In metallic phase in 3D close to MIT:resistivity scales
with diverging correlation length
1-particle-numerical Theory yields
K. Slevin and T. Ohtsuki, Phys. Rev. Lett. 78,
4083 (1997)
Metal
Insulator⇥(T = 0) � �
h
e2
Res
istiv
ity
8
(Wegner 1979)
3D Anderson MIT1-particle theory with random potential V(r)
Metal
Critical multifractal state
Insulator
extended state Localised State
Quantum Critical Point
| �(r) |2Wavefunction Intensity
9
�p2
i
2m+ V (r)
⇥�n(r) = En�n(r)
E > EM E = EM Mobility Edge E < EM
⇥ � e2/h�
� = 1.58± .02
The Metal-
Stupp et al., PRL 72, 2122 (1994)
Theory: Wegner Scaling in 3D:
diverging correlation length
1-particle-Theory:
K. Slevin and T. Ohtsuki, Phys. Rev. Lett. 78, 4083 (1997)
Experiment: Finite T Scaling Ansatz
� = .48± .07
for uncompensated
doped SiSi:P
Strong Experimental Evidence for 2nd order Anderson MIT.Remaining Discrepancy in ν due to Many Body Corrections to 1-particle Anderson Localization Theory??
Theory vs. Experiment
10
�(N,T ) =
✓N �Nc
Nc
◆⌫
F
T
✓N �Nc
Nc
◆�z⌫!
⇠ ⇠✓N �Nc
Nc
◆�⌫
⌫ = 1.3, z = 2.4N(1018cm�3)
Effect of
Electron-Electron Interactions in Doped Semiconductors
• Long Range: Coulomb Interactions Metal: Althshuler-Aronov Corrections to DOS and σ(T) Insulator: Coulomb Gap in DOS, Efros-Shklovskii VRH
May reduce critical expnent (Harashima,Slevin 𝝂 =1.3(1); Amini, Kravtsov, Müller 1.2 (1))
• Short Range: on site Hubbard Interactions Hubbard Splitting of Impurity Band (Exp. Evidence Reflectance Measurments Gaymann et al 94)
Formation of Magnetic Moments (Exp. Evidence from Magnetic Susceptibility and Specific Heat fe. Schlager et al 97)
11
Orthogonality Catastrophe
12
h | 0i ! 0
PW Anderson PRL 1967
Fermi Sea
Fermi level EF
E
| i =Y
✏n<EF
c+n |0i
� = F 2= |h | 0i|2 < exp (�IA)
IA =1
2
X
✏n0,✏m0>0
|hn|m0i|2
Single Impurity mixes in continuum of unoccupied states to form new Eigenstates |m’>
| 0i =Y
✏m0<EF
c+m0 |0i
with
Orthogonality Catastrophe
in a Metal
PW Anderson PRL 1967
co Eugene Demler
Fermi Sea
Fermi level EF
E
� = F 2= |h | 0i|2 < exp (�IA)
IA =1
2
X
✏n0,✏m0>0
|hn|m0i|2with
Orthogonality Catastrophe
in a Metal
PW Anderson PRL 1967Y Gefen, R Berkovits, IV Lerner, BL Altshuler PR B (2002)
IA =(2⇡�)2
2⇢2
X
En<EF
X
Em>EF
| n(x)|2| m(x)|2
(En � Em)2
For short range impurity of strength λ adding up continuum of electron-hole excitations:
Fermi Sea
Fermi level EF
E
� = F 2= |h | 0i|2 < exp (�IA)
for clean metal
Orthogonality Catastrophe
in a Metal
PW Anderson PRL 1967
IA = 2(⇡�)2 lnN
� = |h | 0i|2 < N�2⇡2�2
|N!1 ! 0
where N is number of particles
Fermi Sea
Fermi level EF
E
� = F 2= |h | 0i|2 < exp (�IA)
for short range impurity in dirty metal
Orthogonality Catastrophe
in a dirty Metal
� = |h | 0i|2 < N�2⇡2�2
|N!1 ! 0
Y Gefen, R Berkovits, IV Lerner, BL Altshuler PR B (2002)
for long range impurity in dirty metal enhanced Orthogonality Catastrophe
hIAi = 2(⇡�)2 lnN
hIAi ⇠ �2 ln2 N
same as in clean metal because what matters is only continuum of states no matterwhich states, so
Orthogonality Catastrophe
18
h | 0i ! 0?
but this is a silicon crystal, a semiconductor!
Fermi Sea
Fermi level EF
E
� = F 2= |h | 0i|2 < exp (�IA)
impurity can mix states with unoccupied continuum only above band gap
No Orthogonality Catastrophe
in band insulator, undoped semicond.
Many body state is hardly changed by impurity
� = |h | 0i|2 remains finite
Fermi level EF
E
� = F 2= |h | 0i|2 < exp (�IA)
impurity can mix states only with discrete number of unoccupied states
No Orthogonality Catastrophe
in Anderson insulator, like weakly doped SC
Many body state is hardly changed by impurity
� = |h | 0i|2 remains finite
Orthogonality Catastrophe
21
h | 0i ! 0?
And what if Doping Concentration is critical n = nc?
Fermi level EF
E
� = F 2= |h | 0i|2 < exp (�IA)
impurity can mix states with continuum of unoccupied states
Orthogonality Catastrophe
at mobility edge
Many body state is mixed by impurity
= Mobility Edge EM
h | 0i ! 0But, how does it depend on N??
Anderson MIT
Metal
Critical multifractal state
Insulator
extended state Localised State
Quantum Critical Point
| �(r) |2Wavefunction Intensity
23
�p2
i
2m+ V (r)
⇥�n(r) = En�n(r)
E > EM E = EM Mobility Edge E < EM
Fermi level EF
E
� = F 2= |h | 0i|2 < exp (�IA)
impurity can mix states with continuum of unoccupied states
Orthogonality Catastrophe
at mobility edge
= Mobility Edge EM
For short range impurity of strength λ adding up continuum of electron-hole excitations:
IA =(2⇡�)2
2⇢2
X
En<EF
X
Em>EF
| n(x)|2| m(x)|2
(En � Em)2
n(x)|2We need to know electron intensity at postion x, at all energies En
Multifractal states: sparse but extended
⇥q = d(q � 1) + (�0 � d)q(1� q)
All moments of wave function scale with length L with different powers
�0 � 43D MIT:
P (| l(r)|2) =1
| l(r)|2L�
(↵�↵0)2
4(↵0�d)
↵ = � ln | l(r)|2/ lnL
Wegner 1980, Aoki 1983
25with
wide distribution of wave function amplitudes, in good approximation lognormal:
↵0 ⇡ 4
Ldh | l(r)|2qi ⇠ L�dq(q�1)
dq = d� (↵0 � d)q
Intensities close to MIT are power-law correlated in energy: Stratification
Cnm = Ld
⇤d3r
�|�n(r)|2|�m(r)|2
⇥
=
⇧��⌥
��⌃
�Ec�
⇥�, |En � Em| < �,⇤
Ec|En�Em|
⌅�, � < |En � Em| < Ec,
⇤|En�Em|
Ec
⌅2, Ec < |En � Em|
� = 1� d2/3 d2 = 1.3± 0.05
critical exponent
⇥ =2(�0 � d)
d
(Chalker1990, Kravtsov, Muttalib 1997)
26
-
Fermi level EF
E
� = F 2= |h | 0i|2 < exp (�IA)
impurity can mix states with continuum of unoccupied states
Exponential Orthogonality Catastrophe
at mobility edge
= Mobility Edge EM
For short range impurity of strength λ adding up continuum of electron-hole excitations:
IA =(2⇡�)2
2⇢2
X
En<EF
X
Em>EF
| n(x)|2| m(x)|2
(En � Em)2
Ensemble Average yields
hIAi|E=EM =(2⇡�)2
2
ZZ
✏<��/2,✏0>�/2
d✏d✏0C✏,✏0
(✏� ✏0)2=
2(⇡�)2
�(1 + �)
✓Ec
�
◆�
⇠ N�d
�typ < exp(�const N
�d)
S. K., Phys. Rev. Lett, to appear Sep 23rd (2016)
Orthogonality Catastrophe
28
at Anderson MIT
InsulatornoOC
MetalPower LawOC
Critical:ExponentialOC
hIAi|EF<EM =2d(⇡�)2
⌘(1 + ⌘/d)
✓Ec
�⇠
◆⌘/d
hIAi|EF=EM =2d(⇡�)2
⌘(1 + ⌘/d)
✓Ec
�
◆⌘/d
hIAi|EF>EM = 2(⇡�)2✓
Ec
�⇠c
◆� ✓ 1
�(1 + �)+ ln
N
N⇠c
◆
� = D/N
�⇠ = D/N⇠
�⇠c = D/N⇠c
L=10
L=20
L=50
S. K., Phys. Rev. Lett, to appear Sep 23rd (2016)
Near “Orthogonality” Catastrophe
29
in 2D disordered Electron System
Insulator with localization length no OC
hIAi|EF<EM =2d(⇡�)2
⌘(1 + ⌘/d)
✓Ec
�⇠
◆⌘/d�⇠ = D/N⇠
⇠2D = g exp(⇡g) g = EF ⌧with
⌘2D = 1/(2⇡g)
Maximal “Orthogonality” when L = ξ S. K., Phys. Rev. Lett, to appear Sep 23rd (2016)
Fermi level EF
E
Orthogonality Catastrophe
at mobility edge
= Mobility Edge EM
So, the orthogonality catastrophe is typically exponentially enhanced at the MIT:
�typ < exp(�const N
�d)
This is the typical result. But the actual result may depend strongly where the impurity is placed, since the critical state is multifractal:
Depending where you place impurity there may be local pseudo gaps or divergencies
Conditional Intensity for Given intensity at Fermi energy:
SSK, I. Varga, E. R. Mucciolo, Phys. Rev. Lett. 103, 126401 (2009)
(�� �0)/dLocal pseudo gap or
Local divergency with power
|El � EM | < Ec
for ↵ = 2,
33.5,
4,
6
↵ = � ln | l(r)|2/ lnL
31
Energy
I↵ = Ldh| l(r)|2i| M (r)|2=L�↵
=����El � EM
Ec
����
↵�↵0d +
d�↵0d glM
,
Fermi level EF
E
Wide distribution of Fidelity
at mobility edge
= Mobility Edge EM
So, IA = - 2 log F has wide distribution itself:
↵ = � ln | l(r)|2/ lnLP (| l(r)|2) =1
| l(r)|2L�
(↵�↵0)2
4(↵0�d)
Intensity has close to lognormal distribution:
P (IA) =
Z Y
l
d↵lP ({↵l}) � (IA � I[{↵l}])
P (IA) =1p
8⇡⌘ lnL
1
IAe�
✓ln
IAhIAi+2⌘ lnL
◆2
8⌘ lnLyielding in pair approximation
Average Fidelity at MIT
33S. K., Phys. Rev. Lett, to appear Sep 23rd (2016)
hF i = 1p8⇡⌘ lnL
Z 1
�1dxe
�hIA
iL�⌘
exp(x)
e
� x
2
8⌘ lnL
.
η=2(𝛂0-3)=2.096 3D AMIT orth (Vasquez 2008)
η=2(𝛂0-2)=0.524 2D IQHT (Slevin;Evers,Mirlin 2008)
η=2(𝛂0-2)=.344 2D SOAMIT (Slevin;Evers,Mirlin,2008)
Average Fidelity is finite, even though typical fidelity is exponentially small!!Due to Multifractality
Exponential Orthogonality Catastrophe at MIT
34
S. K., Phys. Rev. Lett, to appear Sep 23rd (2016)
“Just a niche Effect, of purely academic interest” (anonymous referee for PRL (2016)) or measurable as- enhanced edge singularities in X-Ray absorption and emission?
(compare with Chen, Kroha 1992)- reduced relaxation at MIT (see Ovadyahu 2015)- direct measurement of fidelity in engineered many-body systems in
ensembles of ultracold atoms by parameter change
To be done: OC at AMIT with extended impurity, with global parametric change OC at AMIT of interacting disordered system (see Amini, Kravtsov,Müller (2014); Burmistrov, Gornyi, Mirlin (2013))
Metal-Insulator Transitions
in 3D
II. Magnetic Properties
35
Evidence for Magnetic Moments in
Si:P
36
Nc=3.5(as obtained from MIT)
Magnetic Susceptibility diverging for T➛0K: Evidence for paramagnetic Moments with Curie behavior:
⇥(T ) � µ0µ2B
kBTNFree(T )
with temperature dependent density of paramagnetic Moments
NFree(T )
finite even deep into the metallic regime N ⨠ Nc= 3.5 1018 cm-3
nega
tive
conn
stan
t dia
mag
netic
susc
eptib
ility
from
Si h
ost m
ater
ial
�(T ) ⇠ T�↵(N)
Schlager,Löhneysen EPL 97
N. F. Mott (Collège de France 1974)
Magnetic properties:
.... Formation of Local Magnetic Moments from strongly localised states in tail of impurity band.... Kondo frequency for the various sites will spread over a large range of values, perhaps from zero upwards. .... It seems likely that a quantitative explanation of the susceptibility could be given along these lines.
THE METAL-INSULATOR TRANSITION IN AN IMPURITY BAND
37
Goal: Find Density of magnetic moments and Distribution of Kondo Temperature
Magnetic Moments
Formation of MMs in Strongly Localised states due to onsite Hubbard U → Concentration of MMs nM (N) In deep insulator N≪Nc :
AFM MM-Interaction J due to direct exchange, Singlet Formation. Distribution P(J) yields free moments for J < temperature T (Strong Disorder RG)
38
with 𝛂 ≾ 1
In metal N>Nc : MMs AFM coupled by J0~t2/U to itinerant electrons screened by Kondo Effect for T > TK Kondo Temperature Divergence of χ due to distribution of TK?
t
Bhatt, Lee, PRL 81Singlets only for J > T
�(T ) ⇠ nFM (T )
T⇠ 1
T
Z T
0dJP (J) ⇠ T�↵
M. Milovanovic, S. Sachdev, and R. N. Bhatt, PRL 1989
Find actual distribution of Kondo Temperature at MIT: Kondo impurity in multifractal state
Kondo Temperature TK depends on position R
| �(r) |2
39
Local Intensity at mobility edge EM
R
1 =J
2N
X
l
Ld| l(R)|2
El � EFtanh
✓El � EF
2TK
◆
AND Depends on all local intensities at all energies El
not only at Fermi energy EF ! | l(R)|2
Distribution Function of Kondo Temperature:
Integration over all intensities
P (TK) =
Z Y
l
d↵lP ({↵l})�(1� F ({↵l}, TK))| dFdTK
|,
j�
2
X
l
Ld| l(r)|2
✏ltanh
✓✏l
2TK
◆⌘ F ({↵l}, TK),
with
40
↵ = � ln | l(r)|2/ lnL
| l(R)|2
where
Analytical Result for Distribution of Kondo Temperature at MIT and in Metal
analytical result, S.K., Varga, Mucciolo, Slevin PRB 2012
MITLow-TK-Tail at MIT!
Metal
power law divergence at MIT
in quantitative agreement with numerical results Cornaglia et al. PRL 2006, HY Lee, SK, 2012
41
P (TK) ⇡✓Max(TK ,�⇠)
Ec
◆ ⌘2d
1
TK
✓1 +
⌘
2d
ln
D
2TK
◆⇥
exp
(� 1
2c1
✓Max(TK ,�⇠)
Ec
◆⌘/d
ln
2
"TK
T
(0)K
#).
P (TK) ⇠ T�↵K
↵ = 1� ⌘
2d= 2� ↵0
d
�⇠ ⇠ 1/⇠dwith ξ corelation length
Multifractality and Power Law divergence of Magnetic susceptibility at MIT and in insulator
where
At MIT
S.K., Varga, Mucciolo, Slevin PRB 2012
with universal exponent
With Multifractality exponent for d=3, we find power 𝛂≈2/3, Exp at MIT of Si:P yields 𝛂≈.5-.6
for
nFM(T = 0K) = nM⇠�12⌘ (dj)2
⌘ = 2(↵0 � d)
↵0 ⇡ 4
�(T ) =nFM (T )
T⇠ nFM (0)
1
T+ nM
1
Ec
2d
⌘
✓T
Ec
◆ ⌘2d�1
T > �⇠
⇠finite on insulator side of MIT with localization length
⇠ ! 1 with
we find
↵ = 1� ⌘
2d= 2� ↵0
d
�(T ) ⇠ T�↵
Multifractality and Power Law divergence of Magnetic susceptibility at MIT and in insulator
At MIT
S.K., Varga, Mucciolo, Slevin PRB 2012
with universal exponent
With Multifractality exponent for d=3, we find
power 𝛂≈2/3=.66, Exp at MIT of Si:P yields 𝛂≈.5-.6
↵0 ⇡ 4
⇠
we find
↵ = 1� ⌘
2d= 2� ↵0
d
�(T ) ⇠ T�↵
Bhatt, Paalanen, Sachdev find α≈.6 (1987)
Schlager, Löhneysen EPL 97
44
감사합니다.!