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Disorder and chaos in Disorder and chaos in quantum systems II.quantum systems II.
Lecture 2.
Boris Altshuler Physics Department, Columbia
University
Previous Lecture:Previous Lecture:1. Anderson Localization as Metal-Insulator Transition
Anderson model. Localized and extended states. Mobility edges.
2. Spectral Statistics and Localization. Poisson versus Wigner-Dyson. Anderson transition as a transition between different types of spectra. Thouless conductance
Conductance g s
P(s)
Lecture2.Lecture2.1. 1. Quantum Quantum Chaos, Chaos, Integrability Integrability and and
LocalizationLocalization
sP(s)
Particular nucleus
166Er
Spectra of several nuclei combined (after spacing)rescaling by the mean level
P(s)
N. Bohr, Nature 137 (1936) 344.
Q: ?Why the random matrix theory (RMT) works so well for nuclear spectra
Original answer:
These are systems with a large number of degrees of freedom, and therefore the “complexity” is high
Later itbecameclear that
there exist very “simple” systems with as many as
2 degrees of freedom (d=2), which
demonstrate RMT - like spectral statistics
Integrable Systems
Classical ( =0) Dynamical Systems withd degrees of freedom
The variables can be separated and the problem reduces to d one-dimensional problems
d integrals of motion
ExamplesExamples1. A ball inside rectangular billiard; d=2• Vertical motion can be
separated from the horizontal one
• Vertical and horizontal components of the
momentum, are both integrals of motion
2. Circular billiard; d=2• Radial motion can be separated from the angular one
• Angular momentum and energy are the integrals of motion
Integrable Systems
Classical Dynamical Systems with d degrees of freedom
Rectangular and circular billiard, Kepler problem, . . . , 1d Hubbard model and other exactly solvable models, . .
The variables can be separated d one-dimensional problems d integrals of motion
Chaotic Systems
The variables can not be separated there is only one integral of motion - energy
ExamplesExamples
Sinai billiard
Kepler problem in magnetic field
B
Stadium
Classical Chaos =0
•Nonlinearities•Exponential dependence on the original conditions (Lyapunov
exponents)
•Ergodicity
Q: What does it mean Quantum Chaos ?
Quantum description of any Quantum description of any System with a finite number of System with a finite number of the degrees of freedom is a the degrees of freedom is a linear problem – Shrodinger linear problem – Shrodinger equation equation
Bohigas – Giannoni – Schmit conjecture
Chaotic classical analog
Wigner- Dyson spectral statistics
0
No quantum numbers except
energy
Wigner-Dyson
?
Classical
Poisson
Quantum
?Chaotic
Integrable
Lecture1.Lecture1.
2. 2. Localization Localization beyond real beyond real spacespace
Andrey Kolmogorov
Vladimir Arnold
JurgenMoser
Kolmogorov – Arnold – Moser (KAM) theoryKolmogorov – Arnold – Moser (KAM) theory
Integrable classical Hamiltonian , d>1:
Separation of variables: d sets of action-angle variables
1 1 1 2 2 2, 2 ; ... , , 2 ;..I t I t
A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 1954. Proc. 1954 Int. Congress of Mathematics, North-Holland, 1957
0H
0
1D classical motion – action-angle variables
Andrey Kolmogorov
Vladimir Arnold
JurgenMoser
Kolmogorov – Arnold – Moser (KAM) theoryKolmogorov – Arnold – Moser (KAM) theory
Integrable classical Hamiltonian , d>1:
Separation of variables: d sets of action-angle variables
Quasiperiodic motion: set of the frequencies, which are in general incommensurate. Actions are integrals of motion
1 2, ,.., d
11I
22I …=>
1 1 1 2 2 2, 2 ; ... , , 2 ;..I t I t
iI0 tI i
A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 1954. Proc. 1954 Int. Congress of Mathematics, North-Holland, 1957
0H
0
tori
space Number of dimensions
real space d
phase space: (x,p) 2denergy shell 2d-1
tori d
For d>1 each torus has measure zero on the energy shell !
Integrable dynamics:Integrable dynamics:Each classical trajectory is quasiperiodic and confined to a particular torus, which is determined by a set of the integrals of motion
Andrey Kolmogorov
Vladimir Arnold
JurgenMoser
Kolmogorov – Arnold – Moser (KAM) theoryKolmogorov – Arnold – Moser (KAM) theory Integrable classical Hamiltonian , d>1: Separation of variables: d sets of action-angle variables Quasiperiodic motion: set of the frequencies,
which are in general incommensurate Actions are integrals of motion
1 2, ,.., d
11I
22I …=>
;..2,;..,2, 222111 tItI
iI 0 tI i
Q:Will an arbitrary weak perturbation of the integrable Hamiltonian destroy the tori and make the motion ergodic (when each point at the energy shell will be reached sooner or later)
?A:
Most of the tori survive weak and smooth enough perturbations
V 0H
KAM theorem
A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 1954. Proc. 1954 Int. Congress of Mathematics, North-Holland, 1957
0H
1I
2I
Each point in the space of the integrals of motion corresponds to a torus and vice versa
0ˆ V
1I
2I
Finite motion.Localization in the space of the integrals of motion
?
Most of the tori survive weak and smooth enough perturbations
KAM theorem:
1I
2I
0ˆ V
1I
2I
Most of the tori survive weak and smooth enough perturbations
KAM theorem:
1I
2I 0
Energy shell
Consider an integrable system. Each state is characterized by a set of quantum numbers.
It can be viewed as a point in the space of quantum numbers. The whole set of the states forms a lattice in this space.
A perturbation that violates the integrability provides matrix elements of the hopping between different sites (Anderson model !?)
Weak enough hopping: Localization - Poisson
Strong hopping: transition to Wigner-Dyson
Squarebilliard
Localized momentum
spaceextended
Localized real space
Disordered extended
Disordered localized
Sinaibilliard
Glossary
Classical Quantum
Integrable Integrable
KAM Localized
Ergodic – distributed all over the energy shell Chaotic
Extended ?
IHH
00 IEH
,ˆ0
Extended states:
Level repulsion, anticrossings, Wigner-Dyson spectral statistics
Localized states: Poisson spectral statistics
Invariant (basis independent)
definition
Squarebilliard
Sinaibilliard
Disordered localized
Disordered extended
Integrable ChaoticAll chaotic systems resemble each other.
All integrable systems are integrable in their own way
Q:
Consider a finite system of quantum particles, e.g., fermions. Let the one-particle spectra be chaotic (Wigner-Dyson).
What is the statistics of the many-body spectra? ?a.The particles do not interact with each other.Poisson: individual energies are conserving quantum numbers. b. The particles do interact. ????
Lecture Lecture 2. 2. 3. 3. Many-Body Many-Body excitation in excitation in finite systemsfinite systems
Fermi Sea
Decay of a quasiquasiparticleparticle with an
energy in Landau Fermi liquid
QuasiparticleQuasiparticle decay rate at T = 0 in a clean clean Fermi Liquid.
Fermi Sea
322
constant
coupling
dFee
Reasons:Reasons:• At small the energy transfer, , is small and the integration
over and gives the factor . …………………………………………………………………•The momentum transfer, q , is large and thus the scattering
probability at given and does not depend on , or
I. I. d=3d=3
II. II. Low dimensionsLow dimensions
Small moments transfer, q , become important at low dimensions because the scattering probability is proportional to the squared time of the interaction,
(qvF. )-2
evF
1/q
1
2log
3
2
2
d
d
d
FF
F
ee
QuasiparticleQuasiparticle decay rate at T = 0 in a clean clean Fermi Liquid.
Quasiparticle decay rate at T = 0 in a cleanclean Fermi Liquid.
1
2log
32
2
d
d
d
FF
F
ee
Conclusions: Conclusions:
1. For d=3,2 from it follows that , i.e., that the qusiparticles are well determined and the Fermi-liquid approach is applicable.
2. For d=1 is of the order of , i.e., that the Fermi-liquid approach is not valid for 1d systems of interacting fermions.
Luttinger liquids
Fermi Sea
QuasiparticleQuasiparticle decay rate at T = 0 in a clean clean Fermi Liquid.
F ee
ee
Fermi Sea
Decay of a quasiparticle with an
energy in Landau Fermi liquid
Quantum dot – zero-dimensional case ?
Fermi Sea
Decay of a quasiparticle with an
energy in Landau Fermi liquid
Quantum dot – zero-dimensional case ?
2
1TE
( U.Sivan, Y.Imry & A.Aronov,1994 )Fermi Golden rule:
Mean level spacing
Thouless energy
Decay rate of a quasiparticle with energy
2
1TE
( U.Sivan, Y.Imry & A.Aronov,1994 )Fermi Golden rule:
Mean level spacing
Thouless energy
Decay rate of a quasiparticle with energy in 0d.
Recall: 1
TE g
Zero dimensional system
Thouless conductanc
e
Def:
One particle states are extended all over the system
11 gET
Fermi Sea
zero-dimensional case
one-particle spectrum is discrete
equation
’’can not be satisfied exactly
Recall: in the Anderson model the site-to-site hopping does not conserve the energy
Decay rate of a quasiparticle with energy in 0d.Problem:
’
’+
Offdiagonal matrix element
11,, g
M
Decay rate of a quasiparticle with energy in 0d.
Fermi Sea
generations
1 2 3 4 5 6
. . . .
Delocalization in Fock space
’
’
Can be mapped (approximately) to the problem of localization on Cayley tree
Chaos in Nuclei – Delocalization?
1 2 3 4 5
Conventional Anderson Model
Basis: ,i i
i
i iiH 0ˆ
..,
ˆnnji
jiIV
Hamiltonian: 0ˆ ˆH H V
•one particle,•one level per site, •onsite disorder•nearest neighbor hoping
labels sites
many (N ) particles no interaction:Individual energies and thus occupation numbers are conserved
N conservation laws“integrable system”
EH
nE
n
n
0d system; no interactions
integrable system
0H E
BasisBasis: 0,1n
HamiltonianHamiltonian:0
ˆ ˆH H V
n occupatio
n numbers
labels levels
,
V I
.., 1,.., 1,.., 1,.., 1,..n n n n
0d system with interactions
Conventional Conventional Anderson Anderson
ModelModel
Many body Anderson-Many body Anderson-like Modellike Model
Basis:Basis: ilabels sites
, . .
ˆi
i
i j n n
H i i
I i j
,
H E
I
.., 1,.., 1,.., 1,.., 1,..n n n n
BasisBasis: ,0,1n
n occupatio
n numbers
labels levels
i
“nearest neighbors”:
Isolated quantum dot – 0d system of fermions
Exact many-body states:Ground state, excited states
Exact means that the imaginary part of the energy is zero!
Quasiparticle excitations:Finite decay rate
Q:What is the connection?
source drain
gate
QDQD
current
S
D
No e-e interactions – resonance tunneling
source drain
gate
QDQD
current
S
D
VSD
gNo e-e interactions – resonance tunneling
Mean level spacing
source drain
gate
QDQD
current
S
D
VSD
gNo e-e interactions – resonance tunneling
The interaction leads to additional peaks – many body excitations
D
S S
D
Resonance tunnelingPeaks
Inelastic cotunnelingAdditional peak
source drain
gate
QDQD
current
S
S
VSD
g The interaction leads to additional peaks – many body excitations
source drain
gate
QDQD
current
S
D
VSD
g
loc
Ergodic - WDNE
Landau quasiparticle with
the widthSIA
VSDloc
Ergodic - WDNE
Landau quasiparticle with
the widthSIA
Localized - finite # of the satelites
Extended - infinite # of the satelites
(for finite the number of the satelites is always finite)
Ergodic – nonergodic crossover!
extended
Anderson Model on a Cayley tree
Anderson Model on a Cayley tree
I, W K – branching number
1
lnc
WI
K K
WI
K Resonance at
every generation
ln
W WI
K K K Sparse
resonances
Definition: We will call a quantum state ergodic if it occupies the number of sites on the Anderson lattice, which is proportional to the total number of sites :
N
N
0 NN
N
nonergodic
0 constN
NN
ergodic
Localized states are obviously not ergodic: constN
N
Is each of the extended state ergodicQ:Q: ??A:A: In 3D probably yes
For d>4 most likely no
Example of nonergodicity: Anderson ModelAnderson Model Cayley treeCayley tree:
nonergodic states
Such a state occupies infinitely many sites of the Anderson
model but still negligible fraction of the total number of sites
Nlnn
– branching number
KK
WIc ln
K
ergodicity
WIerg ~
transition
crossover
KWIW Typically there is a resonance at every step
WI Typically each pair of nearest neighbors is at resonance
Nn ln~
Nn ~
nonergodic
ergodic
lnW K I W K K Typically there is no resonance at the next step
Nn ln~ nonergodic
lnI W K KResonance is typically far n const localized