Power-law banded random matrices
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Transcript Power-law banded random matrices
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)