1d Anderson localization: devil’s staircase of statistical anomalies

V.E.Kravtsov, Abdus Salam ICTP,

Trieste & Landau Institute

Collaboration: V.I.Yudson

Discussion: A.Ossipov arXiv:0806.2118

Hofstadter butterfly: hierarchy of spectral gaps

H

EF, f

Harper, Thouless… Wigmann &Zabrodin

Magnetic field opens up spectral gaps in a

2D lattice tight-binding model: effect

of commensurability of magnetic flux

through a unit cell and the magnetic flux

quantum

Anderson localization: hierarchy of statistical anomalies

n

1V

)( nP

2/W2/W

2

1f

3

1f

3

2f

E

l

a

Effect of commensurability of

the lattice constant a and the

Fermi-wavelength This effect is not present in the

continuous model.Localization length sharply increases

in the vicinity of E=0 and sharply decreases in the vicinity of E=1

F.Wegner, 1980; Derrida & Gardner,

1986 Titov, 2002 Deytch et al 2003

10.1/ At E=0 (f=1/2):

At E=1 (f=1/3 or 2/3): 1~/ 2 W

The width of anomalous region 1~ 2 W

What about the entire wavefunction What about the entire wavefunction statistics?statistics?

)|(| 2P ?

Generalized Fokker-Planck equationGeneralized Fokker-Planck equation

)2,(),(cos)!2(

2|| 2

0

2

0

2 fzzzdzd

q nNnqqq

n

The function is found from the generalized Fokker-Planck equation:

)|,(]),(ˆ[/)|,( xuuuLxxu f

where ),2/()|,( 00 zxu xn

2

2

0

)(sin2

W

f Is the localization length without

anomalous contributions

)cos(~ z Result of the super-symmetric quasi-sigma model (Ossipov, Kravtsov 2006)

Where are the anomalous terms?Where are the anomalous terms?

)|,(]),(ˆ[/)|,( xuuuLxxu f

...),(ˆ),(ˆ),(ˆ),(ˆ )2(4)1(2)0( uLWuLWuLuL ffff

The term has a part that emerges only at f=m/(p+2)),(ˆ )( uL p

f

p

mf

pm

p

m

regpf

pf uLuLuL

2

,

)(1

1

),()( ),(ˆ),(ˆ),(ˆ

(Derivation…

)…derivation

Center of the band anomaly f=1/2Center of the band anomaly f=1/2

222),0(

4

3ˆ u

regf uL

This term survives in the continuous limit

)4sin(2

3))4cos(3(

4

1)4sin(

)4cos(2)4cos(2)4cos(ˆ

2

22)0(

u

uuf

u

uuL This term is present only

for f=1/2

)cos(~ u Dependence of the Fokker-Plank equation on the phase at f=1/2

)|,(),(ˆ)|,( )0( xuuuLxu fx

Is this a mess?

Exact integrability of the Fokker-Plank equation

N-> infinity: ),()|,( uxu obeys zero-mode Fokker-

Planck Equation 0),(),(ˆ )0( uuuL f

),(~

2),(

~222222 vuu

vuvuvuvu vvuu

),(~

),( vuuu )2cos( uv

This equation is exactly integrable

Separation of variables and the inverse-square Hamiltonian

)()(),(~

),( 4 22 vuvuvu

0)()()](ˆ)(ˆ[),(),(ˆ hhvuvuH

)()(4

1

16

3)()(ˆ

22

h

Celebrated inverse-square Hamiltonian

)(cos2 u )(sin 2 u

Problem of continuous degeneracy

W is the Whittaker function

)(cos2 u )(sin2 u

How to fix the function CQ:

A: Smoothness of (uwith all the derivatives

+ some miraculous properties of the Whittaker function

Final result for the generating function

)(cos2 u )(sin 2 u

WHY SO MUCH OF A MIRACLE?

WHAT IS THE HIDDEN SYMMETRY?

What is the new properties of the anomalous statistics?

ConclusionConclusion

Statistical anomalies in 1d Anderson Statistical anomalies in 1d Anderson model at any rational filling factor f.model at any rational filling factor f.

Integrability of the TM equation for GF Integrability of the TM equation for GF determining all local statistics at a principal determining all local statistics at a principal anomaly at f=1/2.anomaly at f=1/2.

Unique solution for the GF in the infinite 1d Unique solution for the GF in the infinite 1d Anderson model.Anderson model.

Hidden symmetry that makes TM equation Hidden symmetry that makes TM equation integrable?integrable?