Download Disordered Quantum Systems

Disordered Quantum Systems
Boris Altshuler
Physics Department, Columbia University and NEC Laboratories America
Collaboration: Igor Aleiner , Columbia University
Part 1: Introduction
Part 2: BCS + disorder
Centre Emile Borel
Gaz quantiques
23 avril - 20 juillet 2007
Finite size quantum physical systems
Atoms
Nuclei
Molecules
.
.
.
Quantum
Dots
Cold gas in a trap ?
e
Quantum Dot
×
e
e
e
×
×
e
×
×
1. Disorder (× − impurities)
×
2. Complex geometry
3. e-e interactions
Realizations:
• Metallic clusters
• Gate determined confinement in 2D gases (e.g. GaAs/AlGaAs)
• Carbon nanotubes
•
•
Finite number N of electrons:
ĤΨα = EαΨα
No interactions between electrons
Shrodinger eqn in d dimensions
In the presence of the interactions
between electrons
Shrodinger equation in dN dimensions
e
Quantum Dot
×
e
e
e
×
×
e
×
×
1. Disorder (× − impurities)
×
2. Complex geometry
3. e-e interactions
for a while
Realizations:
• Metallic clusters
• Gate determined confinement in 2D gases (e.g. GaAs/AlGaAs)
• Carbon nanotubes
•
•
I. Without interactions
Random Matrices,
Anderson Localization
Quantum Chaos
e
×
e
e
e
×
×
e
×
×
1. Disorder (× − impurities)
×
2. Complex geometry
How to deal with disorder?
•Solve the Shrodinger equation exactly
•Start with plane waves, introduce the
mean free path, and . . .
How to take quantum
interference into account
?
Dynamics
Integrable
Chaotic
Classical (h =0) Dynamical Systems with d degrees of freedom
Integrable
Systems
The variables can be
separated and the problem
reduces to d onedimensional problems
Examples
1. 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
d integrals
of motion
Classical Dynamical Systems with d degrees of freedom
Integrable
The variables can be separated
d one-dimensional
Systems
problems d integrals of motion
Rectangular and circular billiard, Kepler problem, . . . ,
1d Hubbard model and other exactly solvable models, . .
Chaotic
Systems
The variables can not be separated
integral of motion - energy
Examples
Sinai billiard
there is only one
B
Kepler problem
in magnetic field
Stadium
Chaotic
Systems
The variables can not be separated
integral of motion - energy
Examples
Sinai billiard
Yakov Sinai
there is only one
B
Kepler problem
in magnetic field
Stadium
Leonid Bunimovich
Johnnes Kepler
Integrable
d integrals of motion, d quantum numbers
d-dimensional
I
k
=
1
,
2
,...,
d
k
systems
Chaotic
d-dimensional
systems
The only conserved quantity is the energy
Each eigenstate is characterized only by
the eigenvalue of the Hamiltonian
Connection with the inverse problem:
Q:
A:
Why original conditions can not be
used as the integrals of motion ?
Not stable
Classical Chaos
h =0
•Nonlinearities
•Lyapunov exponents
•Exponential dependence on
the original conditions
•Ergodicity
Quantum description of any System
with a finite number of the degrees
of freedom is a linear problem –
Shrodinger equation
Q: What does it mean Quantum Chaos ?
RANDOM MATRICES
N×N
ensemble of Hermitian matrices
with random matrix element
Eα
ν (ε ) ≡
......
N→∞
- spectrum (set of eigenvalues)
δ (ε − Eα )
∑
α
- density of states
- ensemble averaging
N→∞
Gaussian ensembles (matrix
elements are normally distributed)
ν (ε )
Wigner Semicircle
ε
Spectral
statistics
RANDOM MATRICES
ensemble of Hermitian matrices
with random matrix element
N×N
Eα
- spectrum (set of eigenvalues)
δ 1 ≡ Eα +1 − Eα =
......
sα ≡
P (s )
N→∞
1
- mean level spacing
ν
- ensemble averaging
Eα +1 − Eα
δ1
Noncrossing
- spacing between nearest
neighbors
rule
- distribution function of spacings
between the nearest neighbors
Spectral Rigidity
P (s = 0) = 0
Level repulsion
P (s << 1) ∝ s β
β =1,2,4
Noncrossing rule (theorem)
Suggested by Hund (Hund F. 1927 Phys. v.40, p.742)
Justified by von Neumann & Wigner (v. Neumann J. & Wigner E.
....
1929 Phys. Zeit. v.30, p.467)
Usually textbooks present a simplified version of the justification
due to Teller (Teller E., 1937 J. Phys. Chem 41 109).
Arnold V. I., 1972 Funct. Anal. Appl.v. 6, p.94
Mathematical Methods of Classical Mechanics
(Springer-Verlag: New York), Appendix 10, 1989
RANDOM MATRICES
N×N
ensemble of Hermitian matrices
with random matrix element
N→∞
Dyson Ensembles
Matrix elements
Ensemble
β
real
orthogonal
1
complex
unitary
2
2 × 2 matrices
simplectic
4
P ( s ) → 0 when s → 0 :
H12 ⎞
2
2
E2 − E1 = ( H 22 − H11 ) + H12
⎟
⎟
H 22 ⎠
small
Reason for
⎛ H11
Hˆ = ⎜
⎜ ∗
⎝ H12
small
small
1. The assumption is that the matrix elements are statistically
independent. Therefore probability of two levels to be
degenerate vanishes.
2. If
s to be small
two statistically independent variables ((H22- H11) and H12)
should be small and thus P ( s ) ∝ s
β =1
H12
is real (orthogonal ensemble), then for
P ( s ) → 0 when s → 0 :
H12 ⎞
2
2
E2 − E1 = ( H 22 − H11 ) + H12
⎟
⎟
H 22 ⎠
small
Reason for
⎛ H11
Hˆ = ⎜
⎜ ∗
⎝ H12
small
small
1. The assumption is that the matrix elements are statistically
independent. Therefore probability of two levels to be
degenerate vanishes.
2. If
3.
s to be small
two statistically independent variables ((H22- H11) and H12)
should be small and thus P ( s ) ∝ s
β =1
Complex H12 (unitary ensemble)
both Re(H12) and
Im(H12) are statistically independent
three independent
random variables should be small
P( s) ∝ s 2
β =2
H12
is real (orthogonal ensemble), then for
Wigner-Dyson; GOE
Poisson
Gaussian
Orthogonal
Ensemble
Orthogonal
β=1
Unitary
β=2
Simplectic
β=4
Poisson – completely
uncorrelated
levels
RANDOM MATRICES
N×N
ensemble of Hermitian matrices
with random matrix element
N→∞
No conservation laws
no
quantum numbers except the energy
N × N matrices with random matrix elements. N → ∞
Spectral Rigidity
Level repulsion
P ( s << 1) ∝ s
Dyson Ensembles
Matrix elements Ensemble
β = 1, 2, 4
Realizations
β
real
orthogonal 1
complex
unitary
2 × 2 matrices simplectic
β
T-inv potential
2
broken T-invariance
(e.g., by magnetic
field)
4
T-inv, but with spinorbital coupling
Main goal is to classify the eigenstates
ATOMS in terms of the quantum numbers
For the nuclear excitations this
NUCLEI program does not work
N. Bohr, Nature
137 (1936) 344.
Main goal is to classify the eigenstates
ATOMS in terms of the quantum numbers
For the nuclear excitations this
NUCLEI program does not work
E.P. Wigner
(Ann.Math, v.62, 1955)
Study spectral statistics of
a particular quantum system
– a given nucleus
P(s)
Particular
nucleus
166Er
P(s)
N. Bohr, Nature
137 (1936) 344.
s
Spectra of
several
nuclei
combined
(after
spacing)
rescaling
by the
mean level
Main goal is to classify the eigenstates
ATOMS in terms of the quantum numbers
For the nuclear excitations this
NUCLEI program does not work
E.P. Wigner
(Ann.Math, v.62, 1955)
Study spectral statistics of
a particular quantum system
– a given nucleus
Random Matrices
• Ensemble
Atomic Nuclei
• Particular quantum system
• Ensemble averaging
• Spectral averaging (over α)
of the nuclear spectra
Nevertheless Statistics
are almost exactly the same as the
Random Matrix Statistics
Q:
Why the random matrix
theory (RMT) works so well
for nuclear spectra
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 it
became
clear that
there exist very “simple” systems
with as many as 2 degrees of
freedom (d=2), which demonstrate
RMT - like spectral statistics
Chaotic
Systems
The variables can not be separated
integral of motion - energy
Examples
Sinai billiard
Yakov Sinai
there is only one
B
Kepler problem
in magnetic field
Stadium
Leonid Bunimovich
Johnnes Kepler
Integrable
d integrals of motion, d quantum numbers
d-dimensional
I
k
=
1
,
2
,...,
d
k
systems
Chaotic
d-dimensional
systems
The only conserved quantity is the energy
Each eigenstate is characterized only by
the eigenvalue of the Hamiltonian
Connection with the inverse problem:
Q:
A:
Why original conditions can not be
used as the integrals of motion ?
Not stable
h≠0
Bohigas – Giannoni – Schmit conjecture
h≠0
Bohigas – Giannoni – Schmit conjecture
h≠0
Bohigas – Giannoni – Schmit conjecture
Chaotic
classical analog
Wigner- Dyson
spectral statistics
No quantum
numbers except
energy
Q: What does it mean Quantum Chaos ?
Two possible definitions
Chaotic
classical
analog
Wigner Dyson-like
spectrum
Quantum
Classical
Integrable
Chaotic
?
?
Poisson
WignerDyson
Poisson to Wigner-Dyson crossover
Poisson to Wigner-Dyson crossover
Important example: quantum
particle subject to a random
potential – disordered conductor
Scattering centers, e.g., impurities
e
Poisson to Wigner-Dyson crossover
Important example: quantum
particle subject to a random
potential – disordered conductor
Scattering centers, e.g., impurities
e
•As well as in the case of Random
Matrices (RM) there is a luxury
of ensemble averaging.
•The problem is much richer than
RM theory
•There is still a lot of universality.
Anderson
localization (1956)
At strong enough
disorder all eigenstates
are localized in space
f = 3.04 GHz
Anderson Insulator
f = 7.33 GHz
Anderson Metal
Poisson to Wigner-Dyson crossover
Important example: quantum
particle subject to a random
potential – disordered conductor
e
Scattering centers, e.g., impurities
Models of disorder:
Randomly located impurities
r
r r
U ( r ) = ∑ u ( r − ri )
i
Poisson to Wigner-Dyson crossover
Important example: quantum
particle subject to a random
potential – disordered conductor
e
Scattering centers, e.g., impurities
Models of disorder:
r
r r
U ( r ) = ∑ u ( r − ri )
Randomly located impurities
i
r
r
u
(
r
)
→
λδ
(
r
) λ → 0 cim → ∞
White noise potential
Anderson model – tight-binding model with onsite disorder
Lifshits model –
.
.
.
tight-binding model with offdiagonal disorder
• Lattice - tight binding model
Anderson
Model
• Onsite energies
j
i
• Hopping matrix elements Iij
Iij
-W < εi <W
uniformly distributed
εi - random
Iij =
I i and j are nearest
neighbors
0
otherwise
Anderson Transition
I < Ic
Insulator
All eigenstates are localized
Localization length ξ
I > Ic
Metal
There appear states extended
all over the whole system
Localization of single-electron wave-functions:
extended
localized
Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=2; All states are localized
d>2; Anderson transition
localized
Anderson Transition
I < Ic
Insulator
I > Ic
Metal
All eigenstates are localized
Localization length ξ
There appear states extended
all over the whole system
The eigenstates, which are
localized at different places
will not repel each other
Any two extended
eigenstates repel each other
Poisson spectral statistics
Wigner – Dyson spectral statistics
Zharekeschev & Kramer.
Exact diagonalization of the Anderson model
⎡ ∇2
⎤
r
r
+ WU (r ) − ε α ⎥ψ α (r ) = 0
⎢−
⎣ 2m
⎦
Disorder
W
Q: What does it mean Quantum Chaos ?
Two possible definitions
Chaotic
classical
analog
?
Wigner Dyson-like
spectrum
Are the two definitions equivalent?
Maybe not because of the localization!
Quantum particle in a random potential (Thouless, 1972)
Energy scales
Mean level spacing
L
2.
Thouless energy
δ1 = 1/ν× L
energy
1.
ET = hD/L
d
δ1 L is the system size;
d
2
is the number of
dimensions
D is the diffusion const
ET has a meaning of the inverse diffusion time of the traveling
through the system or the escape rate. (for open systems)
g = ET / δ1
dimensionless
Thouless
conductance
g = Gh/e
2
Thouless Conductance and
One-particle Spectral Statistics
0
g
1
Localized states
Insulator
Extended states
Metal
Poisson spectral
statistics
Wigner-Dyson
spectral statistics
Quantum Dots
Ν ×Ν
with Thouless
Random Matrices
conductance g
The same statistics of the
random spectra and oneΝ→ ∞
g→ ∞
particle wave functions
(eigenvectors)
Scaling theory of Localization
(Abrahams, Anderson, Licciardello and Ramakrishnan 1979)
g = ET / δ1
Dimensionless Thouless
conductance
g = Gh/e
L = 2L = 4L = 8L ....
without quantum corrections
ET ∝ L-2 δ1 ∝ L-d
ET
δ1
ET ET ET
δ1 δ1 δ1
g
g
g
g
d (log g)
= β ( g)
d (log L)
2
β(g)
1
-1
β - function
d log g
= β (g )
d log L
unstable
fixed point
3D
gc ≈ 1
2D
g
1D
Metal – insulator transition in 3D
All states are localized for d=1,2
Conductance g
Anderson transition in terms of
pure level statistics
P(s)
Chaotic
Integrable
Square
billiard
Disordered
localized
All chaotic
systems
resemble
each other.
All integrable
systems are
integrable in
their own way
Sinai
billiard
Disordered
extended
Disordered
Systems:
Q:
ET > δ 1;
ET < δ 1;
g >1
Anderson metal;
Wigner-Dyson spectral
statistics
g <1
Anderson insulator;
Poisson spectral statistics
Is it a generic scenario for the
Wigner-Dyson to Poisson crossover
?
Speculations
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 !?)
Q:
Does Anderson localization provide
a generic scenario for the WignerDyson to Poisson crossover
?
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
The very definition of the localization is
not invariant – one should specify in which
space the eigenstates are localized.
Level statistics is invariant:
Poissonian
statistics
∃
basis where the
eigenfunctions are localized
Wigner -Dyson
statistics
∀
basis the eigenfunctions
are extended
Example 1
Doped semiconductor
Low concentration
of donors
Electrons are localized on
donors
Poisson
Higher donor
concentration
Electronic states are
extended Wigner-Dyson
e
Lx
Example 2
Rectangular billiard
Lattice in the
momentum space
py
Two
πn
;
integrals px =
Lx
of motion
Line (surface)
of constant
energy
px
πm L
py =
y
Lx
Ideal billiard – localization in the
momentum space
Poisson
Deformation or – delocalization in the
momentum space
smooth random
Wigner-Dyson
potential
Localization
and diffusion
in the angular
momentum
space
ε > 0 Chaotic
stadium
a
ε≡
R
ε → 0 Integrable circular billiard
Poisson
ε=0.01
g=0.012
Angular momentum is
the integral of motion
h = 0; ε << 1
Diffusion in the
angular momentum
space
52
D ∝ε
Wigner-Dyson
ε=0.1
g=4
D.Poilblanc, T.Ziman, J.Bellisard, F.Mila & G.Montambaux
Europhysics Letters, v.22, p.537, 1993
1D Hubbard Model on a periodic chain
H = t ∑ (ci+,σ ci +1,σ + ci++1,σ ci ,σ ) + U ∑ ni ,σ ni , −σ + V
i ,σ
V =0
V ≠0
Hubbard
model
extended
Hubbard
model
i ,σ
integrable
nonintegrable
12 sites
3 particles
Zero total spin
Total momentum π/6
Onsite
interaction
U=4 V=0
n σn
∑
σσ
i, , ′
i,
i +1,σ ′
n. neighbors
interaction
U=4 V=4
Finite number N of electrons:
ĤΨα = EαΨα
No interactions between electrons
Shrodinger eqn in d dimensions
Integrable system – each energy is conserved
Poissonian many-body spectrum
In the presence of the interactions
between electrons
Shrodinger eqn in dN dimensions
Finite number N of electrons:
ĤΨα = EαΨα
No interactions between electrons
Shrodinger eqn in d dimensions
Integrable system – each energy is conserved
Poissonian many-body spectrum
In the presence of the interactions
between electrons
Shrodinger eqn in dN dimensions
Can interaction between the particles drive
Q: this system into chaos and make it ergodic ?
Random Matrics statistics of nuclear spectra
II. With interactions
Fermi Liquid and Disorder
Zero Dimensional Fermi Liquid
Fermi Liquid
ƒ Fermi statistics
ƒ Low temperatures
ƒ Not too strong interactions
ƒ Translation invariance
What does it mean?
Fermi
Liquid
ƒ
ƒ
ƒ
ƒ
Fermi statistics
Low temperatures
Not too strong interactions
Translation invariance
Fermi
Liquid
It means that
1. Excitations are similar to the excitations in a Fermi-gas:
a) the same quantum numbers – momentum, spin ½ , charge e
b) decay rate is small as compared with the excitation energy
2. Substantial renormalizations. For example, in a Fermi gas
∂n ∂μ , γ = c T , χ gμ B
are all equal to the one-particle density of states.
These quantities are different in a Fermi liquid
Signatures of the Fermi - Liquid state ?!
1. Resistivity is proportional to T2 :
L.D. Landau & I.Ya. Pomeranchuk “To the properties of metals at very
low temperatures”; Zh.Exp.Teor.Fiz., 1936, v.10, p.649
…The increase of the resistance caused by the interaction between
the electrons is proportional to T2 and at low temperatures exceeds
the usual resistance, which is proportional to T5.
… the sum of the moments of the interaction electrons can change
by an integer number of the periods of the reciprocal lattice.
Therefore the momentum increase caused by the electric field can
be destroyed by the interaction between the electrons, not only by
the thermal oscillations of the lattice.
Signatures of the Fermi - Liquid state ?!
1. Resistivity is proportional to T2 :
L.D. Landau & I.Ya. Pomeranchuk “To the properties of metals at very
low temperatures”; Zh.Exp.Teor.Fiz., 1936, v.10, p.649
Umklapp electron – electron scattering dominates the
r
charge transport (?!)
n( p )
2.
Jump in the momentum distribution
function at T=0.
2a.
Pole in the one-particle Green function
r
G (ε , p ) =
p
pF
Z
r
iε n − ξ ( p )
Fermi liquid = 0<Z<1 (?!)
Landau Fermi - Liquid theory
Q:
Momentum
r
p
Momentum distribution
r
n( p )
Total energy
r
E{n ( p )}
Quasiparticle energy
r
r
ξ ( p ) ≡ δE δ n ( p )
Landau f-function
r r
r
r
f ( p, p′) ≡ δξ ( p ) δn ( p′)
Can Fermi – liquid survive without the momenta
?
Does it make sense to speak about the Fermi –
liquid state in the presence of a quenched disorder
Q:
Does it make sense to speak about the Fermi –
liquid state in the presence of a quenched disorder
1. Momentum is not a good quantum number – the
momentum uncertainty is inverse proportional to the
elastic mean free path, l. The step in the momentum
distribution function is broadened by this uncertainty
?
Q:
Does it make sense to speak about the Fermi –
liquid state in the presence of a quenched disorder
1. Momentum is not a good quantum number – the
r
n( p )
momentum uncertainty is inverse proportional to the
elastic mean free path, l. The step in the momentum
distribution function is broadened by this uncertainty
~
?
h
l
pF
p
2. Neither resistivity nor its temperature dependence is determined by the umklapp
processes and thus does not behave as T2
3. Sometimes (e.g., for random quenched magnetic field) the disorder averaged oneparticle Green function even without interactions does not have a pole as a
function of the energy, ε. The residue , Z, makes no sense.
Nevertheless even in the presence of the disorder
I. Excitations are similar to the excitations in a disordered Fermi-gas.
II. Small decay rate
III. Substantial renormalizations
e
Quantum Dot
×
e
e
e
×
×
e
×
×
1. Disorder (×impurities)
×
2. Complex geometry
chaotic
one-particle
motion
}
3. e-e interactions
Realizations:
• Metallic clusters
• Gate determined confinement in 2D gases (e.g. GaAs/AlGaAs)
• Carbon nanotubes
•
•
Finite
System
ε << ET
Thouless
energy
def
ET
0D
At the same time, we want the typical energies, ε , to
exceed the mean level spacing, δ1 :
δ 1 << ε << ET
g≡
ET
δ1
>> 1
Two-Body
Interactions
|α,σ>
Hˆ 0 = ∑ ε α aα+,σ aα ,σ
Hˆ int =
α
∑
α βγ δ
M αβγδ aα+,σ aβ+ ,σ ′aγ ,σ aδ ,σ ′
, , ,
σ ,σ ′
εα -one-particle orbital energies
Nuclear
Physics
Set of one particle states. σ
and α label correspondingly
spin and orbit.
εα
Mαβγδ
-interaction matrix elements
are taken from the shell model
M αβγδ are assumed to be random
εα
RANDOM; Wigner-Dyson statistics
Quantum
Dots M
????????
αβγδ
Thouless Conductance and
One-particle Quantum Mechanics
0
g
1
Localized states
Insulator
Extended states
Metal
Poisson spectral
statistics
Wigner-Dyson
spectral statistics
Ν ×Ν
Quantum Dots with
dimensionless
Random Matrices
conductance g
The same statistics of the
random spectra and oneΝ→ ∞
g→ ∞
particle wave functions
(eigenvectors)
Matrix Elements
Hˆ int =
∑
α βγ δ
M αβγδ aα+,σ aβ+ ,σ ′aγ ,σ aδ ,σ ′
, , ,
σ ,σ ′
Matrix
M αβγδ
Elements
It turns
out that
in the limit g → ∞
Diagonal - α,β,γ,δ are equal pairwise
α=γ and β=δ or α=δ and β=γ or α=β and γ=δ
Offdiagonal - otherwise
• Diagonal matrix elements are much bigger
than the offdiagonal ones
M diagonal >> M offdiagonal
• Diagonal matrix elements in a particular
sample do not fluctuate - selfaveraging
Toy model:
Short range e-e interactions
r λ r λ is dimensionless coupling constant
U (r ) = δ (r )
ν is the electron density of states
ν
r
ψ α (r )
r
r
r
r
λ r
M αβγδ =
ψα
dr ψ ∗α (r )ψ ∗β (r )ψ γ (r )ψ δ (r )
∫
ν
electron
wavelength
x
one-particle
eigenfunctions
Ψα (x) is a random
function that
rapidly oscillates
|ψα (x)|2 ≥ 0
ψα (x)2 ≥ 0 as long as
T-invariance
is preserved
In the limit
g→∞
• Diagonal matrix elements are much bigger than
the offdiagonal ones
M diagonal >> M offdiagonal
• Diagonal matrix elements in a particular sample
do not fluctuate - selfaveraging
M αβαβ
r 2
r 2
λ r
= ∫ dr ψ α (r ) ψ β (r )
ν
r 2
ψ α (r ) ⇒
1
volume
M αβαβ = λδ 1
r
More general: finite range interaction potential U (r )
Mαβαβ
λ
r 2
r 2 r r r r The same
= ∫ ψ α (r1 ) ψ β (r2 ) U(r1 − r2 )dr1 dr2
conclusion
ν
Universal (Random Matrix) limit - Random
Matrix symmetry of the correlation functions:
All correlation functions are invariant under
arbitrary orthogonal transformation:
r
r ν r r
r
~
ψ μ (r ) = ∑ ∫ dr1Oμ (r , r1 )ψ ν (r1 )
ν
r ν r r η r r
r r
′
′
(
)
(
)
(
)
d
r
O
r
,
r
O
r
,
r
=
δ
δ
r
−
r
1
μ
1
ν
1
μη
∫
There are only three operators, which are quadratic in
+
the fermion operators a , a , and invariant under RM
transformations:
total number of particles
n̂ = ∑ aα ,σ aα ,σ
+
α ,σ
Ŝ =
∑
ασ σ
,
r
aα ,σ1σ σ1 ,σ 2 aα ,σ 2
+
1, 2
K̂ + = ∑ aα+ ,↑ aα+ ,↓
α
total spin
????
Charge conservation
(gauge invariance)
+
+
ˆ
ˆ
ˆ
ˆ
-no K or K only K K
Invariance under
- no
rotations in spin space
ˆS
only
ˆS 2
Therefore, in a very general case
2
2
+ ˆ
ˆ
ˆ
ˆ
H int = eVnˆ + Ec nˆ + JS + λBCS K K .
Only three coupling constants describe all of
the effects of e-e interactions
In a very general case only three coupling constants
describe all effects of electron-electron interactions:
Hˆ = ∑ ε α nα + Hˆ int
α
2
2
+ ˆ
ˆ
ˆ
ˆ
H int = eVnˆ + Ec nˆ + JS + λBCS K K .
I.L. Kurland, I.L.Aleiner & B.A., 2000
See also
P.W.Brouwer, Y.Oreg & B.I.Halperin, 1999
H.Baranger & L.I.Glazman, 1999
H-Y Kee, I.L.Aleiner & B.A., 1998
In a very general case only three coupling constants
describe all effects of electron-electron interactions:
Hˆ = ∑ ε α nα + Hˆ int
α
2
2
+ ˆ
ˆ
ˆ
ˆ
H int = eVnˆ + Ec nˆ + JS + λBCS K K .
For a short range interaction with a coupling constant
Ec =
λδ1
δ
2
J = −2λδ1
λ
λ BCS = λδ1 (2 − β )
where 1 is the one-particle mean level spacing
Hˆ = Hˆ 0 + Hˆ int
Hˆ 0 = ∑ εα nα
α
2
2
+ ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
H int = eVn + Ec n + JS + λBCS K K .
Only one-particle part of
ˆ ,
the Hamiltonian, H
0
contains randomness
Hˆ = Hˆ 0 + Hˆ int
Hˆ 0 = ∑ εα nα
α
2
2
+ ˆ
ˆ
ˆ
ˆ
H int = eVnˆ + Ec nˆ + JS + λBCS K K .
Ec
determines the charging energy
(Coulomb blockade)
J
describes the spin exchange interaction
λ BCS
determines effect of superconducting-like
pairing
Hˆ = Hˆ 0 + Hˆ int
Hˆ 0 = ∑ εα nα
α
2
2
+ ˆ
ˆ
ˆ
ˆ
H int = eVnˆ + Ec nˆ + JS + λBCS K K .
I. Excitations are similar to the excitations in a disordered Fermi-gas.
II. Small decay rate
III. Substantial renormalizations
Isn’t it a Fermi liquid ?
Fermi liquid behavior follows from the fact that
different wave functions are almost uncorrelated