Workshop “Anderson Localization in Topological Insulators”...

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


in a Metal


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


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


in a Metal


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


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


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


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


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)



at mobility edge

= Mobility Edge EM


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)


Exponential Orthogonality Catastrophe

at mobility edge

= Mobility Edge EM


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)


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


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


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


“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.!

Workshop “Anderson Localization in Topological Insulators”...

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