1 qhe, asm vincent pasquier service de physique théorique c.e.a. saclay france
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QHE, ASM
Vincent PasquierService de Physique ThéoriqueC.E.A. SaclayFrance
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From the Hall Effect to integrability
1. Hall effect.
2. Annular algebras.
3. deformed Hall effect and ASM.
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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
• Important experimental fractions for electrons are:
• Theorists believe for bosons:
• Fraction called filling factor.
1010
12 pp
1pp
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Hall effect
• Lowest Landau Level wave functions
2
!)( l
zzn
n en
zz
nlr
n=1,2, …, 22 l
A
5
022n
l
A
Number of available cells alsothe maximal degree in each variable
mzzS nn )...( 1
1Is a basis of states for the systemLabeled by partitions
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N-k particles in the orbital k are
the occupation numbers
.........210 kNNNNCan 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
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Interactions translate into repulsionbetween particles.
mji zz
m universal measures the strength of the interactions.
Competition between interactions which spread electrons apart and highcompression which minimizes the degree n. Ground state is the minimal degreesymmetric polynomial compatible with the repulsive interaction.
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Laughlin wave functionoccupation numbers:
100100100
Keeping only the dominant weight of the expansion
1 particle at most into m orbitals (m=3 here).
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Important quantum numberwith a topological interpretation
• Filling factor equal to number of particles per unit cell:
• Number of variables
Degree of polynomial
3
1In the preceding case.
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ji
ji
ji
n
izi
zz
zzz
i
2
1
)(
Jack polynomials are eigenstates of the Calogero-Sutherland Hamiltonian on a circle with 1/r^2
potential interaction.
Jack Polynomials:
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Jack polynomials at
1
1
k
r
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.
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Exclusion statistics:
• No more than k particles into r consecutive
orbitals.
For example when
k=r=2, the possible ground states (most dense packings) are given by:
20202020….
11111111…..
02020202…..
Filling factor is
rkii
k
r
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Moore Read (k=r=2)
)(
1 Pfaff ji
ji
zzzz
When 3 electrons are put together, the wave function vanishes as:2
14
.
When 3 electrons are put together, the wave function vanishes as:
2x x
y
1 2
1
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With adiabatic time QHE=TQFT
One must consider the space P(x1,x2,x3|zi) of polynomials vanishing when zi-xj goes to 0.
Compute Feynman path integrals
x1 x2 x3
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Y3
1 23 4
0],[ ji YY
17
211
11 yTTy
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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.
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Two layer system.
• Spin singlet projected system of 2 layers
mji
mji yyxx )()(
.
When 3 electrons are put together, the wave function vanishes as:m
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Exemples of wave functions• Haldane-Rezayi: singlet state for 2 layer system.
22
)()(
1Det ji
ji
yxyx
When 3 electrons are put together, the wave function vanishes as:
2x x
y
1 2
1
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Projection onto the singlet state
RVB-BASIS
- =
1 2 3 4 5 6
Crossings forbidden to avoid double countingPlanar diagrams.
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RVB basis:
Projection onto the singlet state
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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.
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e=
e=2
)( 1 qq
iiii
ii
eeee
ee
1
2
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Razumov Stroganov Conjectures
1
2
3
4
5
6
6
1iieHAlso eigenvector of:
Stochastic matrix
I.K. Partition function:
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e =1
1 2
1
2
1
1
2
2 1 0 0 1
2 3 2 2 0
0 1 2 0 1
0 1 0 2 1
2 0 2 2 3
H=
1’ 2’
Stochastic matrix
If d=1
Not hermitian
1
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Transfer MatrixConsider inhomogeneous transfer matrix:
1 zz
))()....((),( 1
z
zL
z
zLtrzzT n
i
L= +
11 zqzq
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)(),( 1 ni zzzzT
Transfer Matrix
0)(),( wTzT
Di Francesco Zinn-justin.
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Vector indices are patterns, entries are polynomials in z , …, z
1 n
ii
ii
yy
eeBosonic condition
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T.L.( Lascoux Schutzenberger)
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121
1221 )),(),(()(zz
qzqzzzzze
121 qzqz
e+τ projects onto polynomials divisible by:
e Measures the Amplitude for 2 electrons to be In the same layer
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Two q-layer system.
• Spin singlet projected system of 2 layers
m
jim
ji yqqyxqqx )()( 11
.
If i<j<k cyclically ordered, then 0),,( 42 zqzzqzzz kji
))(( 11jiji yqyqxqxq
Imposes for no new condition to occur 6qs
(P)
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Conjectures generalizing R.S.
• Evaluation of these polynomials at z=1
have positive integer coefficients in d:
.
,
,2
,1
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223
4
5
ddF
dFF
dF
F
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Other generalizations
• q-Haldane-Rezayi
))((
1Det
1jiji yqqxyx
Generalized Wheel condition, Gaudin Determinant
Related in some way to the Izergin-Korepin partition function?
Fractional hall effect Flux ½ electron
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Moore-Read
• Property (P) with s arbitrary.• Affine Hecke replaced by Birman-Wenzl-
Murakami,.• R.S replaced by Nienhuis De Gier in the
symmetric case.
ASxqqx ji
1
1 Pfaff
1)1(2 ksq
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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.
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