Anderson Localization Looking Forward

Anderson Localization – Looking Forward

Boris Altshuler

Physics Department, Columbia University

Collaborations:

Igor Aleiner

Also

Denis Basko, Gora Shlyapnikov,

Vincent Michal, Vladimir Kravtsov, …

Lecture 4

extended localized Localized f = 7.33 GHz f = 3.04 GHz j i I_ij Anderson Model

,

W I

DoS

 

 

ˆ

H x



E

_

x

x

E

I

2

I

₁ Finite motion

0



0 1 1 1 ˆ N _ˆz _{ˆ ˆ}z z N _ˆx ˆ N _ˆx i i ij i j i i i i j i i H B J   I  H I            



x

t

_

t

_ Irreversibility = delocalization  

insulator non-ergodic metal ergodic metal transition transition _or crossover

?

or control parameter 0

H



H





V



Central problem of the proposed research:

Delocalized but not ergodic systems

Boltzmann:

Ergodicity means that time-average is equivalent to the space-average

Time-average is equivalent to the energy-shell-average Quantum: spectrum-average is equivalent to the space-average

Localization and Ergodicity – one particle,





i i

i

i

H

ˆ

₀



, . .

ˆ

i j n n

V

i

j







Anderson Model,

H

ˆ



H

ˆ

₀



V

ˆ



W 2



_i



W 2



   _{- random}

 

a

i



_{random wave functions}

N

sites c

N

W

W

 



For



a

 

i

are

localized if

extended

ergodic

if

c

W



W

c

W



W

Critical behavior:

Critical volume:

 

c

c W W

N W



_

1

N

N

_c



_a

 

i

are extended but

_{non-ergodic multifractal}



fixed

2

  

d

c

3D Anderson transition

Disorder, W

Ergodic

Non-ergodic

Anderson transition in terms of level statistics

3D

Multifractality

Moments of the

inverse participation

ratio:

 

 

2q q a i

I

N







i

Scaling with

 

 

0  q q

I

N



O N

N



N

 

 

1

1

I

N



normalization

1

0





Multifractality

Moments of the

inverse participation

ratio:

 

 

2q q a i

I

N







i

Scaling with

 

 

0  q q

I

N



O N

N



N

 

 

1

1

I

N



normalization

1

0





Ergodicity:



 

q

 

q

1



_a

 

i

2



O N

 

1

Exponentially localized states:



 

q



0



q

Multifractality

 

1

q

q

D

q







Fractal dimensions

differ from

0

and

1

Spectrum of

fractal

dimensions

Statistics of the

onsite values of

the eigenfunctions

 

2

a

i



 

2

ln

ln

a i

i

N





 

random

_variable

 

P



Distribution

function

Spectrum of

fractal

dimensions

Statistics of the

onsite values of the

eigenfunctions

 

2 a

i



 

2 ln ln a i i N





  random _variable

 

P



Distribution function

 

0   1

(

)

f N

P





_

O N



N

 

Multifractal ansatz

 

1 lim

ln

 

ln

N

P

f

N









_

_





_

_



 

_

_









Spectrum

of Fractal

Dimensions

Legendre

transform

of



_q

Spectrum of

fractal

dimensions

Statistics of the

onsite values of the

eigenfunctions

 

2 a

i



 

2 ln ln a i i N





  random _variable

 

P



Distribution function

 

0   1

(

)

f N

P





_

O N



N

 

Multifractal ansatz

 

1 lim

ln

 

ln

N

P

f

N









_

_





_

_



 

_

_









Spectrum

of Fractal

Dimensions

Legendre

transform

of



_q

Properties

of

f

 



•

.

• . max

1

f



 

2 1 a i  

f

 



is a convex function

 

1

P

 

d





  f 



  f   1  : q q f q  





_    _ _    q q ₁

 

q q f D q







  q

q

 





Straight lines are

tangential to at points

f

 





q

2



 

q q q f q











q

q

 



q q I  N 2





 

2q q i I 





i I0  N



0  N

Typical spectrum of fractal dimensions

max 1 f 

 

2 1 i



 



_q  q 0

 

f



is convex

1 1 I  _f

 







is tangential to at

f

 



 

 ₁

f







for a d-dimensional lattice

f  1 0

I

M

C

1

 

 

def

q q q q

f

q

q

f

















1 1 q q q      Metal: Insulator:



  

0 1 1 q q q q D q        Critical point:

 

 

2 q q q a i

I

N







i



N

 0 0 0 0 0; 0; q q q q q               

Bethe Lattice

Cayley tree

not good for numeric: most of the sites are on the boundary

Random Regular Graph

with a fixed connectivity K+1

1 2 3 4

N

sites

K

=3

randomly connected 2, 100 K N  

Q:

Can extended eigenstates of the

Anderson model on the Bethe-Lattice be

non-ergodic outside the critical region

?

A:

YES

Localized states – triangular shape of Extended states – gradually approach the ergodic limit, but reach it only at

 

f



0

Extended – non-ergodic regime,

W

<W

_c

=17,5:

The spectrum of the fractal dimensions is

gradually

evolving with the strength of disorder

W

,

but does not collapse to the ergodic limit, which is

( )

f



(

1)

f



  

(1) 1 f 

It is unlikely that this is a finite size effect:

1) Two fixed points

2) This is not a critical behavior:

depends

on both

N

and

W

.

( ,

,

)

Q:

Can extended eigenstates of the

Anderson model on the Bethe-Lattice be

non-ergodic outside the critical region

?

A:

YES

Localized states – triangular shape of Extended states – gradually approach the ergodic limit. When do they reach it reach it Only at

 

f



0

W



?

Localization at the Edge

of 2D Topological Insulator

by Kondo Impurities

2D Topological Insulator

Kane and Mele (2005);

Bernevig, T. L. Hughes, and S. C. Zhang (2006) CdTe-HgTe-CdTe

In the balk

(inside the plane) – gap in the

spectrum of charge excitations insulator

At the edge

excitations are gapless 1D metal

Insensitive

to any static disorder – topological

protection.

2D Topological Insulator

Kane and Mele (2005);

Bernevig, T. L. Hughes, and S. C. Zhang (2006) CdTe-HgTe-CdTe

 

ˆ

x F x z

H k



v k



Hamiltonian of the chiral states at

the helical edge

momentum Z-component

of the spin Fermi

velocity

Rashba term:

[ ] S O F H _   U



k  v k U 

k

) (k E Electric field

2D

Topological

Insulator

Kane and Mele (2005);

Bernevig, T. L. Hughes, and S. C. Zhang (2006) CdTe-HgTe-CdTe

 

ˆ

x F x z

H k



v k



Hamiltonian of the chiral edge states

momentum Z-component of the spin Fermi velocity

E

x

k

Chiral edge states:

Localization length ~ mean free path

Mott and Twose (1961):

Consequence:

a zero DC conductivity

Why all this is not directly applicable to 1D helical

edge electrons?

M. E. Gertsenstein and V. B. Vasiliev, “Wave guides with random inhomogeneities and Brownian motion in the Lobachevsky plane”, Theory of Probability & Its

Applications, 1959, Vol. 4, No. 4 : pp. 391-398

Exact

Solution!

Even a “weak” disorder

is not weak in d=1 !

Backscattering would mix the chiral states and

thus destroy chirality. One needs spin-flip for the backscattering.

Kramers degeneracy: one needs to violate the

time-reversal symmetry

to mix left and right movers

E

x

k

Chiral edge states:

Left and Right movers

E

x

Basic properties of a generic 2D Topological Insulator:

(strong spin-orbit coupling)

2D bulk = insulator: electron spectrum is gapped,

levels of impurities are

localized

Edge Modes are Helical

Statement:

Time Reversal Symmetry

protects Helical Edge

Modes from Backscattering and thus from

Quantum Hall Effect

Quantum Hall Effect:

Topological Insulator:

the back moving state is nearby

but

the back-scattering can not happen without a spin flip

spatially separated edge

Localization at the edge

Time Reversal Symmetry protects Helical Edge Modes from Backscattering and Anderson Localization

h

e

G

_ideal 2

2



Experimental Observations

 Large samples show large resistance at the gap.

 _{Small samples (~1X1}_{m) show}

quantized conductance at the gap, indicating transport by edge states.  _{g: 20-50} Molenkamp’s group d (nm) L×W (μm2₎ I 5.5 20.0×13.3 II 7.3 20.0×13.3 III 7.3 1.0×1.0 IV 7.3 1.0×0.5

Problems with the interpretation:

•

Why the “quantization” of the conductance

takes place only in short samples?

d (nm) L×W (μm2₎ I 5.5 20.0×13.3 II 7.3 20.0×13.3 III 7.3 1.0×1.0 IV 7.3 1.0×0.5 NO NO YES Molenkamp et al. YES ?

Time Reversal Symmetry protects Helical Edge Modes from Backscattering and Anderson Localization

h

e

G

_ideal 2

2



In reality, conductance of only

extremely small

fraction

of

short

samples is somewhat close to

, while for most of the short samples and all

of the long samples

ideal

G

ideal

G

G



Problems with the interpretation:

•

Why the “quantization” of the conductance

takes place only in short samples?

d (nm) L×W (μm2₎ I 5.5 20.0×13.3 II 7.3 20.0×13.3 III 7.3 1.0×1.0 IV 7.3 1.0×0.5 NO NO YES YES

Problems with the interpretation:

•

Why the “quantization” of the conductance

takes place only in short samples?

•

Why the accuracy of the quantization is

so poor? Impressive only in the log scale

reproducible

Yacoby group Harvard

The same sample !

Questions:

•

How universal is the protection?

•

Can the “topological protection”

be softened?

•

Can helical edge electrons be

localized?

Time Reversal

Symmetry can be

trivially

broken

by an

External

Magnetic Field

k

) (k E

What about intrinsic sources of the

Time Reversal Symmetry Violation

Spatially homogeneous field – no effect!

Modulated field –> energy gap

Homogeneous field + potential disorder = backscattering!

~ cos(2kx)

B

Origin:

1.

Chemistry: dangling bonds, etc.

2. Localized energy levels close to the edge

Formation of the Kondo Spins

In the presence of disorder the “edge” is not single connected.

or

Hubbard repulsion

Formation of the Kondo Spins

No e-e interactions: Only empty or double occupied localized states Single occupied states spins

Localized Spins in the presence of itinerant electrons = Kondo Spins Chemical potential E_c E0 -E 1 E2 -E 1

Origin:

1.

Chemistry: dangling bonds, etc.

2. Localized energy levels close to the edge



E=0

E_c– energy of the repulsion E_c >m E0 > E1

E₂ > E₁

{

or

Anderson Model Kondo Model

Onsite Hubbard repulsion – Anderson Model

Interaction between the itinerant electrons and the Kondo Spins or

 

ˆ

ˆ







ˆ

ˆ

ˆ







ˆ

_ˆ

_ˆ

_ˆ

eS e S z z e S

H



J



S



r



r



J



S







S









S





r



r

Magnetic (spin) impurity near the helical edge

Electron-spin

interaction:

 || ||

(

)

:

)

2

1)

(

(

_z z z x x y y z z z

J

S

J

S

S

J

J

U

S

S

S











 



 













electron spin

No influence on the

T = 0

dc charge transport

Reasons:

single electron impurity

S

impurity spin

U(1)-symmetry: symmetry under rotation around z-axis in the spin space = conservation of the z-component of the total spin

U(1)

-

symmetric (xy-isotropic) electron-spin

interaction has no influence on T=0 dc transport

Spin down impurity can back-scatter a right-moving electron.

However, subsequent backscattering of right-moving electrons is impossible until some left-moving electron reverse the impurity spin!

The impurities can effect ac conductivity but not dc one! Reason: conservation of

1. Kinematic reason

(Tanaka, Furusaki, Matveev (2011))

Second reason:

Kondo effect -

screening of the impurity spin

Recovery of the Time Reversal Symmetry

2

2

i i

e



_{ }

e

 

_

Kondo exchange

U(1)

-

symmetric (xy-isotropic) electron-spin

interaction has no influence on zero-temperature

dc charge transport

Finite density of spins – competition between the Kondo effect and Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction of the spins

2 _{cos[2 ]} j k F jk S S d j k F jk S S k r J H v r   



d – number of dimensions rjk  rj  rk

Single spin:

T-invariance always survives due to the

Kondo effect

Finite density of spins:

T-invariance can be violated

spontaneously (Kondo RKKY)

but

the

backscattering would not appear

as long as the

z-component of the total spin is conserved = the system

remains

U(1)

-symmetric, i.e.

invariant under rotations

in spin space around z-axis.

Q:

What if there is a small but

of localized and

anisotropic

spins

finite density

?

(1) :

J

_z z

S

z

(

J

_x x

S

x

J

_y y

S

y

);

J

_x

J

_y

Q:

What if there is a small but

of localized and

anisotropic

spins

finite density

?

Kondo

RKKY

Spin glass

density of spins

electron-spin coupling constant

T-invariant _T-invarianceBroken

Spontaneous breaking of the time-reversal symmetry

Indirect exchange of the localized spins -“RKKY” interaction ) (x_j S _{( )} k x S

No magnetic anisotropy U(1) is conserved Magnetic anisotropy is not conserved U(1) No disorder (regular spin chain)

Perfect 1D metal Band Insulator:

Charged excitations are gapped even at the

edge Disorder Goldstone mode perfect 1D metal Anderson Insulator:

Edge states are localized tot z

S

tot z

S

2

2e h





2

2 e h





~𝑎

Helical edge Helical edge Localized spins Insulating bulk x y z Left mover Spin down Right mover Spin up

Localized spin impurities located at S_j,

Linear spin density ( ) 



(  );

j j S x  x x  a x S S 1 ) (    

Averaged spin density:

1

,

j j j

Free

helical

electrons:

Electron-spin interaction:

 || ||

(

)

(

)

2

z x y z z x y z z z

J

S

J

S

S

J

J

S

S

S















  













|| 1 1 ( ) , ( ) 2 x y 2 x y J  J  J J  J  J - electron spin ( ) || | J | J

:

)

1

(

U

( ) 2 J S S        

Hamiltonian:

S e e

H

H

H





_

Electron operator:

1 ˆ ( ) 2 F F ik x R ik x L e x e         _{ }    † ˆ ˆ ˆ    

Note:

the Hamiltonian is

T-invariant

† † † † ( ) ( ) ( ) ( ) 0 ˆ _{( )} ˆ _{( )} 0 ( ) ( ) x e F x F R x R L x L H iv dx x x iv dx _ x _{ } x _       _{ }         _       _





add

:

)

1

(

U

:

)

1

(

U

Effective spin-spin

(“RKKY”) interaction











                    _              j L x ik R j R x ik L j L L R R z z S e j F j F S S J S J H _{( ) ( ) ( ) ( )} || _{( )} e2 _{( ) ( )} e 2 _{( )} 2

Electron-spin interaction

) (x_j S _{( )} k x S



     _    k j j k x x ik k j F S S x x c h S S v J H k j F | | . . e 8 ) ( 2 2 || 

Helical (chiral) electrons



  _   k j j k k j F k j F S S x x x x k S S v J H | | )] ( 2 cos[ 2  

:

)

1

(

U

Effective spin-spin

(“RKKY”) interaction











                    _              j L x ik R j R x ik L j L L R R z z S e j F j F S S J S J H _{( ) ( ) ( ) ( )} || _{( )} e2 _{( ) ( )} e 2 _{( )} 2

Electron-spin interaction

) (x_j S _{( )} k x S



     _    k j j k x x ik k j F S S x x c h S S v J H k j F | | . . e 8 ) ( 2 2 ||  Features of chirality: 1. No interaction z k z jS S 2.Factorization 2 ( ) ₂ 2 instead of

e

ikF xixj



e

ik x_{F i}

e

 ik xF j )] ( 2 cos[ k_F x_j  x_j Can be removed by 2 e ik xF j j j j S  S  S

Helical (chiral) electrons



  _   k j j k k j F k j F S S x x x x k S S v J H | | )] ( 2 cos[ 2  

Helical edge Helical edge Localized spins Insulating bulk

x

y

z

j Fx ik j j

S

S







e

2

black

red

~𝑎

Helical edge Helical edge Localized spins Insulating bulk x y z Left mover Spin down Right mover Spin up 2

e

F j j ik x j

S

S





 Ferromagnet (_{Spin Glass) without mean} rather than

magnetization Effective spins with

ferromagnetic exchange interaction

2

k x

_{F j} 2 || ( )

. .

8

|

|

j k eff S S j k F j k

S S

h c

J

H

v

x

x



   



 





Coherent representation

of the spins:

2 , ,

1

1

;

1

e

2

2

j i z j z j

S

j

n

z j

S



n







 

1

; | | 1

2

j j j

S



n

n



n

S

z

n

2

1



n

_z

sin





2

1



n

_z

cos



j

S

is parametrized by and j



n

_{j z,}

x

y

z

Wess, Zumino. (1971). Witten. (1983)

Need to

1. Integrate over

2. Integrate over

3. Take into account the chiral anomaly

ˆ ( )

x



z

n

 

 





2





2 0 1 1 2 S x dx d u x K u            _    _  





S || ln 2 F B u K v J _E u      _{ }   

n

S

z

n

2

1



n

_z

sin





2

1



n

_z

cos



x

y

z

Free chiral edge electrons

El-spin exchange, U(1) -symmetry

Fermions acquire a gap;

A bosonic mode remains gapless

Ideal Metallic Conductance in the presence of a potential disorder

Mapping for isotropic exchange

1

F

u

K

v



Effective velocity Luttinger parameter || ||

ln

2

B F S

J

_E

u

v

S

J











_

_

_

_





1

2

el x













Bosonic field Goldstone mode Luttinger Liquid with small velocity

Electron-spin interaction: ₂|| ( j j j j ) ₂ j ( j j j j ) J J S S



S S



  _



  _



  _



  k j

d

jk

 



_



4 ||

e

ik xF j j j

J

J







Anisotropy of

j

-th spin If is independent of then the spectrum is gapfullj,

More likely is random!

j



j



Measure of the U(1) breaking disorder

 

   





j

x

x

x

S

d

x

x



_









_

_

 

_

_

Continuous limit

Matsubara Action (broken U(1)-symmetry)

Modified boson action:

2 2 2 2 ( , ) 2

[ ]

[(

)

(

) ]

8

Re

( ) e

F i x x

x

v

d dx

u

u

  



























_









_







S

Electron-spin interaction:

|| ( ) ( ) 2 2 j j j j j j j j j J J S S



S S



  _



  _



  _



  4 ||

e

ik xF j j j

J

J







2 2 2 2 ( , ) 2

[ ]

[(

)

(

) ]

8

Re

( ) e

F i x x

x

v

d dx

u

u

  



























_









_







S

* ( ) ( ')

x

x

_S

d

(

x

x

')







 



(Giamarchi & Schulz, 1988 )

Mapping on the problem of the pinning of

one-dimensional charge-density wave by

potential disorder

.

1 2 || ||

ln

2

B F

J

_aE

u

v

J









_

_

_

_







1 2 2 2 3 3 2 || ||

ln

K K K F B loc

v

aE

L

a

d

J

J

  















_

_

_

_

_

_

_

_

_

_









_





_

Localization length

1

F

u

K

v



Mapping for anisotropic exchange



||

 

exp 4



j

J

j

J

ik x

F j

1 2 || || ln 2 B F J _aE u v J     _{ }    1 2 2 2 3 3 2 || ||

ln

K K K F B loc

v

aE

L

a

d

J

J

  















_

_

_

_

_

_

_

_

_

_









_





_

Localization length can be of the order or even larger than the system size something like a quantization

F u K v  Mesoscopic fluctuations are probably caused by the rearrangement of the spins with the change of the gate voltage

Conclusions

Interaction of helical edge electrons with closely

located spin (Kondo) impurities lead to Anderson

localization of electrons

if the total z-component

of spin is not conserved (broken

U(1)

symmetry)

1. In the presence of

U(1)

symmetry spins rotate

in the xy-plane. This restores effectively the

time reversal symmetry and prevents localization;

2. If

U(1)

symmetry is randomly broken, the spins

are pinned, which means a spontaneous breaking of

T

-invariance, i.e. there remains no protection

against Anderson localization

Q:

Is the topological insulator a

state of matter

_{Anderson insulator with the localization}

or

it is a conventional

distinct

new

length at the edge can substantially

exceed the one in the bulk.

?

Note

that an arbitrary small concentration of

Kondo impurities with arbitrary weak anisotropy

eventually destroys the quantization.

Albeit electrons at 1d helical edge of a 2D topological

insulator are more protected from an influence of random

imperfections, they are still subjected to Anderson