Power-law banded random matrices: a testing ground for the Anderson transition
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Imre VargaElméleti Fizika TanszékBudapesti Műszaki és Gazdaságtudományi Egyetem, Magyarország
Imre VargaDepartment of Theoretical PhysicsBudapest University of Technology and Economics, Hungarycollaborators: 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.
Power-law banded random matrices: a testing ground for the Anderson transition
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
Hamiltonian:
Energies en are uncorrelated, random numbers from uniform (bimodal, Gaussian, Cauchy, etc.) distribution W
Nearest-neighbor hopping V (symmetry: , , ) Bloch states for W V, localized states for W V
W V ??
Anderson model (1958)
WV WV WV
Two energy (time) scales: ETh and D (tD and tH) g = ETh/D = tH/tD
One-parameter scaling (1979)
Metal – insulator transition (MIT) for d>2.
Gell-Mann – Low function
Mobility edge (d=3)De
nsity
of s
tate
s
Cond
uctiv
ity
Localized wave functions
A non-interacting electron moving in random potential
Quantum interference of scattering waves
Anderson localization of electrons
E
extended
localizedlocalized
extended
localized
critical
Ec
Spectral statistics (d=3)
MITZharekeshev ‘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 relationthru parameter g
with and are the RMT limit
IV, Hofstetter, Pipek ’99
Eigenstates for weak and strong W
extended stateweak disorder, band center
localized statestrong disorder, band edge
(L=240) R.A.Römer
Multifractality at the MIT (3d) Inverse participation numbers
Box counting technique• fixed L• state-to-state fluctuations
• higher accuracy• scaling with L
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
Continuous set of independent and universal critical exponents
multifractal exponents
: anomalous scaling dimensions
singularity spectrum
fractal dimension
: measure of r where
In a metal
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
2D
Unusual features of the MIT (3d)
Kottos and Weiss ‘02; Weiss, et al. ‘06
Detect the MIT using a stopwatch!
PBRM: Power-law Band Random Matrix Model: matrix with and
asymptotically
parameters: and
PBRM for RMT, as if
for 1/2 < a < 1 similar to metal with d=1/(a-1/2)
for BRM Poisson, as if for a > 3/2 power law localization with exponent a
(cf. Yeung-Oono ‘87)
for criticality (cf. Levitov ‘90) no mobility edge! continuous line of transitions: b
PBRM transitionCuevas 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
β = 1
β = 2
joint distribution state-to-state fluctuation
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:
Measure multifractality using a stopwatch!
Scattering: PBRM + 1 lead JA Méndez-Bermúdez – Kottos ‘05 Ossipov –
Fyodorov ‘05:
JA Méndez-Bermúdez – IV 06:
Wave function and LDOSWave 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 dynamicsasymptotic wave packet profile survival probability
J.A. Méndez-Bermúdez and IV ‘08 (in prep.)
Wave packet dynamicseffective dimensionality changes
J. A. Méndez-Bermúdez and IV ‘08 (in prep.)
AB
A
1 qubit in a tight-binding lattice site i with or without an electron: A
2 qubits in a tight-binding lattice site i and j with or without an electron: A
Entanglement at criticality
concurrence [Wootters (1997)] (bipartite systems)
tangle [Meyer and Wallach (2002)]
(multipartite)
AB
ji
i
4321 0,max)( AC
)()( yyAyyAAR
)Tr1(2)( 2AAQ
BA Tr
IV and JA Méndez-Bermúdez ‘08
Entanglement at criticalityAverage concurrence in an eigenstate
||2 jiijC
1||112
ii
jiij M
CM
C
Average tangle
114 P
NQ 12 12 P
MC
where M=N(N-1)/2 and41
iiP
IPR of state
b=0.3
by 1
IV and JA Méndez-Bermúdez ‘08
?)1( 2DLC
Entanglement at criticality
)(1 bNfNQ )(1 bNfNC
IV and JA Méndez-Bermúdez ‘08(cf. Kopp et al. ’07; Jia et al. ’08)
T < TK alatt spin-flip szórás,szinglet alapállapot,Kondo-árnyékolás
Kondo effektus fémben (1964)
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
lognormálishullámfüggvény eloszlás
hullámfüggvény intenzitások együttes eloszlása
hullámfüggvényekenergiakorrelációja
Kondo effektus a kritikus pont körülA mágneses momentumokközül 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 pontszimmetria 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
Scattering Wave packet dynamics Entanglement Interplay with magnetic impurities
Outlook 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 interactionso S. Kettemann (Hamburg)o E. Mucciolo (Orlando)
interplay of multifractality and interaction
decoherence of qubits in critical environmento Th. Seligman (Cuernavaca)o J.A. Méndez-Bermúdez (Puebla)
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