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