Power-law banded random matrices:
a testing ground for the Anderson transition
Imre Varga
Elméleti Fizika
Department
of Tanszék
Theoretical Physics
Budapesti University
Budapest
Műszaki ésofGazdaságtudományi
Technology and
Egyetem, Magyarország
Economics,
Hungary
collaborators: Daniel Braun (Toulouse)
Tsampikos Kottos (Middletown, CT)
José Antonio Méndez Bermúdez (Puebla)
Stefan Kettemann (Bremen, Pohang),
Eduardo Mucciolo (Orlando, FL)
thanks to: V.Kravtsov, B.Shapiro, A.Ossipov, A.Garcia-Garcia, Th.Seligman,
A.Mirlin, I.Lerner, Y.Fyodorov, F.Evers, B.Eckhardt, U.Smilansky, etc.
also to AvH, OTKA, CiC, Conacyt, DFG, etc.
Outline
 Introduction
 Anderson transition
 Intermediate statistics
 PBRM and the MIT
 Spectral statistics, multifractal states
 New results with PBRM at criticality
 Scattering
 Wave packet dynamics
 Entanglement
 Magnetic impurities
 Summary
Anderson model (1958)
 Hamiltonian:
 Energies en are uncorrelated, random numbers from uniform
(bimodal, Gaussian, Cauchy, etc.) distribution 
W
 Nearest-neighbor hopping  V (symmetry:
,

Bloch states for
V  W
,
)
W V, localized states for W  V
V W
 W V ??
V  W
One-parameter scaling (1979)
Two energy (time) scales: ETh and D (tD and tH)
tH/tD
g = ETh/D =
Gell-Mann – Low function
Metal – insulator
transition (MIT) for d>2.
(d=3)
Conductivity
Density of states
Mobility edge
Localized wave functions
A non-interacting electron moving in random potential
Quantum interference of scattering waves
Anderson localization of electrons
extended
localized
localized
localized
Ec
E
extended
critical
Spectral statistics
(d=3)
MIT
Zharekeshev ‘96
Spectral statistics
(d=3)

W < Wc
• extended states
• RMT-like

W > Wc
• localized states
• Poisson-like

W = Wc
• multifractal states
• intermediate
‘mermaid’
Anderson - MIT
 Dependence on symmetry parameter
superscaling relation

thru parameter g
with
and
are the RMT limit
IV, Hofstetter, Pipek ’99
Eigenstates for weak and strong W
extended state
weak disorder, band center
localized state
strong disorder, band edge
(L=240) R.A.Römer
Multifractality at the MIT (3d)
 Inverse participation numbers
• higher accuracy
• scaling with L
 Box counting technique
• fixed L
• state-to-state fluctuations
http://en.wikipedia.org/wiki/Metal-insulator_transition (L=240) R.A.Römer
Multifraktál állapotok a
valóságban
LDOS fluktuációk a fém-szigetelő
átalakulás közelében Ga1-xMnxAs-ban
Multifractality:
scaling behavior of moments of (critical) wave functions
Critical wave function at a metal-insulator transition point
multifractal exponents
In a metal
fractal dimension
Continuous set of independent and universal critical exponents
: anomalous scaling dimensions
singularity spectrum
: measure of r where
Unusual features of the MIT (3d)
 Interplay of eigenvector and spectral
statistics
 Chalker et al. ‘95
 Anomalous diffusion at the MIT
 Huckestein et al. ‘97
 Correlation dimension
 strong probability overlap (Chalker ’88)
 LDOS vs wave function fluctuations
 Huckestein et al. ‘97
D2
Unusual features of the MIT (3d)
Detect the MIT using a stopwatch!
Kottos and Weiss ‘02; Weiss, et al. ‘06
PBRM:
Power-law Band Random Matrix
 Model:
matrix with
 asymptotically
 parameters:
and
and
PBRM
 for


for 1/2 < a < 1  similar to metal with
d=1/(a-1/2)
 for

 for
RMT, as if
 BRM 
for a
a
Poisson, as if
> 3/2  power law localization with exponent
(cf. Yeung-Oono ‘87)
 criticality (cf. Levitov ‘90)

no mobility edge!

continuous line of transitions: b
PBRM transition
Cuevas et al. ‘01
• asymmetric transition
• Kosterlitz-Thouless
Kottos and IV ‘01 (unpub.)
PBRM at criticality (
)
 for b  1 non-linear s-model RG, SUSY (Mirlin ‘00)
• large conductance: g*=4bb
 for b 
1 real-space RG, virial expansion, SUSY
(Levitov ‘99, Yevtushenko-Kravtsov ‘03,
Yevtushenko-Ossipov ‘07)
Mirlin ‘00
PBRM at criticality – DOS (
b=0.1
b=1.0
)
b=10.0
L=1024
PBRM at criticality (b=1)
semi-Poisson statistics is qualitatively valid only
IV and Braun ‘00
joint distribution
state-to-state fluctuation
β=1
β=2
IV ‘02
How does multifractality show up?
 Scattering (1 lead)
 LDOS vs wave function fluctuations
 Anomalous diffusion at the MIT
 Nature of entanglement
 Screening of magnetic impurities
Open system: PBRM + 1 lead
 scattering matrix
 Wigner delay time
 resonance width,
eigenvalues of
poles of
Perfect coupling
 distribution of phases for
b > 1:
with
 perfect coupling achieved:
Scattering: PBRM + 1 lead
 JA Méndez-Bermúdez – Kottos ‘05
Ossipov – Fyodorov ‘05:
Measure multifractality using a stopwatch!
 JA Méndez-Bermúdez – IV 06:
Wave function and LDOS
Wave functions
LDOS
J.A. Méndez-Bermúdez and IV ‘08 (in prep.)
Wave function and LDOS
J.A. Méndez-Bermúdez and IV ‘08 (in prep.)
Wave packet dynamics
asymptotic wave packet profile
survival probability
J.A. Méndez-Bermúdez and IV ‘08 (in prep.)
Wave packet dynamics
effective dimensionality changes
J. A. Méndez-Bermúdez and IV ‘08 (in prep.)
Entanglement at criticality
 1 qubit in a tight-binding lattice
site i with or without an electron: A
 A  Tr B  

i
B
concurrence [Wootters (1997)]
(bipartite systems)
A

C (  A )  max 0,
 2 qubits in a tight-binding lattice
site i and j with or without an electron: A
B
i
j
A
A
1   2   3   4 

R A   A ( y   y )  A ( y   y )
tangle [Meyer and Wallach (2002)]
(multipartite)
Q (  A )  2 (1  Tr  A )
2
IV and JA Méndez-Bermúdez ‘08
Entanglement at criticality
Average concurrence in an eigenstate 


C ij  2 |  i 
C


1
M
C
b=0.3

|
j

ij
i j
2


1 


   |  i |   1
M  i



y 1 b
Average tangle
Q


4
N
C
1  P 
1

where M=N(N-1)/2 and
C 
2
1
P 
2
M

 L
 (1 D 2 )
1  P 
1

i

4
IPR of state 
i
IV and JA Méndez-Bermúdez ‘08
?
Entanglement at criticality
C  N
1

f ( bN )
(cf. Kopp et al. ’07; Jia et al. ’08)
Q  N
1

f ( bN )
IV and JA Méndez-Bermúdez ‘08
Kondo effektus fémben (1964)
T < TK alatt spin-flip szórás,
szinglet alapállapot,
Kondo-árnyékolás
Kondo effektus rendezetlen fémben
TK helyfüggő
P(TK) széles, bimodális
1-hurok (Nagaoka – Suhl):
Árnyékolatlan (szabad) mágneses momentumok,
ha
Kissé rendezetlen vezető:
Szigetelő:
Kondo effektus a kritikus pontban
lognormális
hullámfüggvény
eloszlás
hullámfüggvény
intenzitások
együttes eloszlása
hullámfüggvények
energiakorrelációja
Kondo effektus a kritikus pont körül
A mágneses momentumok
közül pontosan egy szabad:
A kritikus pontban nincsenek szabad momentumok
A szigetelő oldalon:
A kritikus ponttól távolodva léteznek szabad momentumok
A fém-szigetelő átalakulás
szimmetriája
Kritikus pont
szimmetria függő:
esetén
Magnetic impurity
S Kettemann, E Mucciolo, IV ‘09



Summary

PBRM: a good testing ground for the Anderson transition
 d=1 → scaling with L
 no mobility edge (!)
 features similar to Anderson MIT → deviations found
 tunable transition → b serve as 1/d or g
 multifractal states induce unusual behavior





Outlook
Scattering
Wave packet dynamics
Entanglement
Interplay with magnetic impurities
 Effect of interactions on the HF level
 Dynamical stability versus chaotic environment
Thank you for your attention
Outlook:
Current and future problems
 free magnetic moments + e-e interactions
o
o
S. Kettemann (Hamburg)
E. Mucciolo (Orlando)
 interplay of multifractality and interaction
 decoherence of qubits in critical environment
o
o
Th. Seligman (Cuernavaca)
J.A. Méndez-Bermúdez (Puebla)