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```QHE, ASM
Vincent Pasquier
Service de Physique Théorique
C.E.A. Saclay
France
1
From the Hall Effect to integrability
1. Hall effect.
2. Annular algebras.
3. deformed Hall effect and ASM.
2
Most striking occurrence of a
quantum macroscopic effect in the
real word
• In presence of of a magnetic field, the
conductivity is quantized to be a simple
fraction up to 10 10
• Important experimental fractions for
p
electrons are:
2 p 1
• Theorists believe for bosons:
p
p 1
• Fraction called filling factor.
3
Hall effect
• Lowest Landau Level
wave functions
n
z
 n ( z)  e
n!
zz
 2
l
r l n
n=1,2, …,
A
2l 2
4
A

n
0
2
2l
1
n
Number of available cells also
the maximal degree in each variable
S ( z1 ... zn )  m
Is a basis of states for the system
Labeled by partitions
5
N-k particles in
the orbital k are
the occupation
numbers
N0 N1N 2 .... N k .....
Can be represented by a partition with N_0 particles in orbital 0,
N_1 particles in orbital 1…N_k particles in orbital k…..
There exists a partial order on partitions, the squeezing order
6
Interactions translate into repulsion
between particles.
z  z 
m
i
j
m universal measures the strength of the interactions.
Competition between interactions which spread electrons apart and high
compression which minimizes the degree n. Ground state is the minimal degree
symmetric polynomial compatible with the repulsive interaction.
7
Laughlin wave function
occupation numbers:
Keeping only the dominant weight of the expansion
100100100
1 particle at most into m orbitals (m=3 here).
8
Important quantum number
with a topological interpretation
• Filling factor equal to number of particles
per unit cell:
•
 
1
3
Number of variables
Degree of polynomial
In the preceding case.
9
Jack Polynomials:
n

i 1
( zi  z ) 
2
i

i j

zi  z j
zi  z j
Jack polynomials are eigenstates of the CalogeroSutherland Hamiltonian on a circle with 1/r^2
potential interaction.
10
Jack polynomials at
r 1
 
k 1
Feigin-Jimbo-Miwa-Mukhin
Generate ideal of polynomials“vanishing as the r power
of the Distance between particles ( difference
Between coordinates) as k+1 particles come together.
11
Exclusion statistics:
• No more than k particles into r consecutive
orbitals.   
i
i k  r
For example when
k=r=2, the possible ground states (most dense
packings) are given by:
20202020….
11111111…..
02020202…..
r
Filling factor is
k
12
When 3 electrons are put together, the wave function vanishes as:

2
 1 
Pfaff 
  ( zi  z j )
 zi  z j 
13
.
When 3 electrons are put together, the wave function vanishes as:
x 1

2
2 x
y
1
14
One must consider the space P(x1,x2,x3|zi)
of polynomials vanishing when zi-xj goes
to 0.
x1
x2
x3
Compute Feynman path integrals
15
1
2
3
4
Y3
[Yi , Y j ]  0
16
1
y1T1  T1 y2
17
Non symmetric polynomials
• When additional degrees of freedom are
present like spin, it is necessary to
consider nonsymmetric polynomials.
• A theory of nonsymmetric Jack
polynomials exists with similar vanishing
conditions.
18
Two layer system.
• Spin singlet projected
system of 2 layers
m
m
(
x

x
)
(
y

y
)
 i j i j
.
When 3 electrons are put together, the wave function vanishes as:

m
19
Exemples of wave functions
• Haldane-Rezayi: singlet state for 2 layer system.


1
Det 

2
 ( xi  y j ) 

( xi  y j ) 2
When 3 electrons are put together, the wave function vanishes as:
x 1

2
2 x
y
1
20
RVB-BASIS
Projection onto the singlet state
-
=
Crossings forbidden to avoid double counting
Planar diagrams.
1 2
3
4
5
6
21
RVB basis:
Projection onto the singlet state
22
q-deformation
• RVB basis has a natural q-deformation
known as the Kazhdan Lusztig basis.
• Jack polynomials have a natural
deformation Macdonald polynomials.
• Evaluation at z=1 of Macdonald
polynomials in the KL basis have
mysterious positivity properties.
23
e=
2
1
e=
  (q  q )
2
ei
 ei
ei ei 1ei  ei
24
Razumov Stroganov Conjectures
I.K. Partition function:
6
1
5
2
4
3
6
Also eigenvector of:
Stochastic matrix
H   ei
i 1
25
1’
2’
1
e=
1
1
H=
Stochastic matrix
If d=1
Not hermitian
2
 3 2 2 0 2  2 

 
1 2 0 1 0 1 
1 0 2 1 0 1 

 
 0 2 2 3 2  2 
1 0 0 1 2 1 

 
26
Transfer Matrix
Consider inhomogeneous transfer matrix:
zn
z1
T ( z , zi )  tr ( L(
)....L(
))
z
z
L=
+
1 1
q z q z
z z
1
27
Transfer Matrix
T ( z, zi )( z1  zn )  
T ( z), T (w)  0
Di Francesco Zinn-justin.
28
Vector indices are patterns, entries are
polynomials in z , …, z
1
n

    
ei  ei 
yi  yi 
Bosonic condition
29
T.L.( Lascoux Schutzenberger)
z1 q  z2 q 1
(e   )  ( ( z1 , z2 )   ( z2 , z1 ))
z1  z2
e+τ projects onto polynomials divisible by:
z1q  z2 q
1
e Measures the Amplitude for 2
electrons to be In the same layer
30
Two q-layer system.
• Spin singlet projected
system of 2 layers
 (qx  q
i
1
x j ) m  (qyi  q 1 y j )
m
.
 (q x  q
i
1
1
x j )(q yi  q y j )
(P) If i<j<k cyclically ordered, then
Imposes
s  q6
( zi  z, z j  q 2 z, z k  q 4 z)  0
for no new condition to occur
31
Conjectures generalizing R.S.
• Evaluation of these
polynomials at z=1
have positive integer
coefficients in d:
F5  1,
F4  2d ,
F3  F2  d ,
2
F1  d
3
 d.
32
Other generalizations
• q-Haldane-Rezayi


1
Det 

1
 ( xi  y j )( qxi  q y j ) 
Generalized Wheel condition, Gaudin Determinant
Related in some way to the Izergin-Korepin partition function?
Fractional hall effect Flux ½ electron
33
• Property (P) with s arbitrary.
• Affine Hecke replaced by Birman-WenzlMurakami,.
• R.S replaced by Nienhuis De Gier in the
symmetric case. sq 2( k 1)  1


1
Pfaff 

AS

1
 qxi  q x j

34
Conclusions
• T.Q.F.T. realized on q-deformed wave functions of
the Hall effect .
• All connected to Razumov-Stroganov type
conjectures.
• Relations with works of Feigin, Jimbo,Miwa,
Mukhin and Kasatani on polynomials obeying
wheel condition.
• Understand excited states (higher degree
polynomials) of the Hall effect.
35
```
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