1 q-Q.H.E. and Topology Vincent Pasquier Service de Physique Théorique C.E.A. Saclay France
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q-Q.H.E. and Topology
Vincent PasquierService de Physique ThéoriqueC.E.A. SaclayFrance
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From the Hall Effect to integrability
1. Hall effect.2. Transfer Matrix.3. Annular algebras.4. deformed Hall effect wave function as a
toy model for TQFT.5. Conclusion.
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Hall effect
• Lowest Landau Levelwave functions
2
!)( l
zzn
n enzz −=ψ
nlr =
n=1,2, …, 22 lAπ
4
022n
lA
=π
Number of available cells alsothe maximal degree in each variable
nnzz λλ ...1
1 Is a basis of states for the system
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Interactions
mji zz −
m measures the strength of the interactions.
Competition between interactions which spread electrons apart and highCompression which minimizes the degree n.
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With adiabatic time QHE=TQFT
• Bulk and edge.
Compute Feynman path integrals
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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:mε
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Projection onto the singlet state
RVB-BASIS
- =
1 2 3 4 5 6
Crossings forbidden to avoid double counting
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RVB basis:Projection onto the singlet state
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RVB Basise e e e
1354
1 5
4
1 2 3 4 5 6
1 23 45 6
3
11
Also with fluxes
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Exemples of wave functions• Haldane-Rezayi: singlet state for 2 layer system.
))()((1 Perm jijijiji
yxyyxxyx
−−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
− ∏
When 3 electrons are put together, the wave function vanishes as:2ε
x x
y
1 2
1
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Moore Read
∏ −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−)(1 Pfaff ji
ji
zzzz
When 3 electrons are put together, the wave function vanishes as:2ε
No spin
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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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1 2 3 4
e2
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e =1
1 2
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
12112
2 1 0 0 12 3 2 2 0
0 1 2 0 10 1 0 2 1
2 0 2 2 3
H=
1’ 2’
Stochastic matrixIf d=1
Not hermitian
1
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Transfer MatrixConsider inhomogeneous transfer matrix:
1−− zz
))()....((),( 1
zzL
zzLtrzzT n
i =
L= +
11 −−− zqzq
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ΨΛ=Ψ )()(),( 11 nni zzzzzzT ……
Transfer Matrix
[ ] 0)(),( =wTzT
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I.K. Partition function
4
6
5
z 31
Total partition function can be expressed as a Gaudin Determinant
When d=q+1/q=1 symmetrical and given by a Schur function.
Stroganov
2
)( 1 nzzZ
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Hecke and Yang generators• Consider Hecke algebra generators:
• And Yang operators:
11
2
11211
1 zqqztztzY −
−
−−
=
0))((
1
11
212121
=+−
=−qtqt
tttttt Braid group relations
111
212121
==
YYYYYYYY Permutation relations
Or Yang-Baxter algebra
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T.L. (Jones)
22
Y4
1 2 3 4
0],[ =ji YY
2321
1 TyTy =−
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Content of Representation
Read eigenvalues of y i
Initial height
Final height
ut
u
ut2
u-1
u-1
1 2 3 4 …….
t
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Bosonic Ground State
,
,
1
ii
ii
yy
tt
Ψ=Ψ
Ψ=Ψ
⊗==Ψ∧
∑ ππ
Look for dual representation of AHA on polynomials
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AHA• Spin representation • Polynomial
representation
2223
1112
1
tytytyty
y
==
ini
ii
n
szzandzzwith
tty
==
=
+
+
: : 1
11
σ
σ
Triangular matrices
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Two q-layer system.
• Spin singlet projected system of 2 layers
mji
mji yqqyxqqx ∏∏ −− −− )()( 11
.
If i<j<k cyclically ordered, then 0),,( 42 ====Ψ zqzzqzzz kji
))(( 11jiji yqyqxqxq −− −−∏
Imposes for no new condition to occur 6qs =
(P)
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T.L. and Measure
21
121
1221 )),(),(()(zz
qzqzzzzze−−
Ψ−Ψ=Ψ+−
τ
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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yi
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Flux seen by particles
i
Initial height
Final height
ut
u
ut2
u-1
u-1
1 2 3 4 …….
t
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k q-layer system
• Spin singlet projected system of k layers
mji
mji yqqyxqqx ∏∏ −− −− )()( 11
.
If i<j<k cyclically ordered, then
0),,( 22 ====Ψ zqzzqzzz kkji
∏=
−−1
1 )(a
aj
ai xqxq
Imposes for no new condition to occur)1(2 += kqs
(P)
1
2
k
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Other generalizations
• q-Haldane-Rezayi
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−− − ))((1Det 1
jiji 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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b
1
2
3
4
5
b
b
b
12
23
34
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Kasatani wheel conditions
11 ++ = aa
a
a b
i
i tqzz
111 =−+ rk qt
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1
11
−≤
>⇒=
∑ +
++
rbiib
aa
aaaa
k and r
With r , Flux=1/r particle.
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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.
⎥⎥⎦
⎤
⎢⎢⎣
⎡
− −ji xqqx 1
1 Pfaff
1)1(2 =+ksq
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The case q root of unity
• When q+1/q=1, Hecke representation is no more semisimple and degenerates into a trivial representation.
• Stroganov Partition function (Schurfunction) can be recovered as the unique symmetrical polynomial of the minimal degree obeying (P).
• Other roots of unity?
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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. Proof of conjecture still missing.
• Relations with works of Feigin, Jimbo,Miwa, Mukhin and Kasatani on polynomials obeying wheel condition.
• Excited states of the Hall effect.
cond-mat/0506075
math.QA/0507364