vincent pasquier - thp.uni-koeln.deklevtsov/pasquier.pdf · vincent pasquier q-boson koln december...
TRANSCRIPT
![Page 1: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/1.jpg)
Toda chain
Vincent PASQUIER
IPhT Saclay
Collaboration with Olivier Babelon, Simon Ruisjenaars
Vincent Pasquier q-Boson Koln December 2015
![Page 2: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/2.jpg)
Former contributions
Former contributions:
• Gutzwiller• Gaudin• Sklyanin• Pasquier Gaudin• Karchev Lebedev• Shatashvili, Nekrasov• Kozlowski Techner•· · ·
Vincent Pasquier q-Boson Koln December 2015
![Page 3: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/3.jpg)
Schroedinger in periodic potential
Hamiltonian:
H = p2 + 2 cos(2πx)
Schroedinger equation:
ω2ψ(x) = −ψ′′(x) + 2 cos(2πx)
• electrons moving in a periodic potential.• Floquet theory ψ(x) = e iµx
∑n ane
2πinx .
Vincent Pasquier q-Boson Koln December 2015
![Page 4: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/4.jpg)
Schroedinger in periodic potential
Hamiltonian:
H = p2 + 2 cos(2πx)
Schroedinger equation:
ω2ψ(x) = −ψ′′(x) + 2 cos(2πx)
• electrons moving in a periodic potential.• Floquet theory ψ(x) = e iµx
∑n ane
2πinx .
Vincent Pasquier q-Boson Koln December 2015
![Page 5: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/5.jpg)
Schroedinger in periodic potential
Hamiltonian:
H = p2 + 2 cos(2πx)
Schroedinger equation:
ω2ψ(x) = −ψ′′(x) + 2 cos(2πx)
• electrons moving in a periodic potential.• Floquet theory ψ(x) = e iµx
∑n ane
2πinx .
Vincent Pasquier q-Boson Koln December 2015
![Page 6: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/6.jpg)
Hill determinant
Recursion:
an+1 + an−1 + Vnan = 0
with:
Vn = (n + µ)2 − ω2
• Two decaying solutions at infinity need to be matched.• Wronskian equal to zero is the infinite determinant of a tridiagonalmatrix:
W = |1/Vn, 1, 1/Vn|
Vincent Pasquier q-Boson Koln December 2015
![Page 7: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/7.jpg)
Hill determinant
Recursion:
an+1 + an−1 + Vnan = 0
with:
Vn = (n + µ)2 − ω2
• Two decaying solutions at infinity need to be matched.• Wronskian equal to zero is the infinite determinant of a tridiagonalmatrix:
W = |1/Vn, 1, 1/Vn|
Vincent Pasquier q-Boson Koln December 2015
![Page 8: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/8.jpg)
Hill determinant
Recursion:
an+1 + an−1 + Vnan = 0
with:
Vn = (n + µ)2 − ω2
• Two decaying solutions at infinity need to be matched.• Wronskian equal to zero is the infinite determinant of a tridiagonalmatrix:
W = |1/Vn, 1, 1/Vn|
Vincent Pasquier q-Boson Koln December 2015
![Page 9: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/9.jpg)
Hill determinant-2
Periodicity and asymptotic:
W (µ) = sin(µ−δ) sin(µ+δ)sin(µ−ω) sin(µ+ω)
Determines the spectrum as the zeros of W :
µ = δ(ω)
Vincent Pasquier q-Boson Koln December 2015
![Page 10: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/10.jpg)
Hofstadter
Hamiltonian:
H = X + X † + Y + Y †
X = e iαn+θ, Yf (n) = f (n + 1)
Schroedinger equation:
Eψn = ψn+1 + ψn−1 + 2 cos(αn + θ)ψn
• Lattice electrons moving in a periodic potential α times the latticeperiod.• 2D lattice electrons in a strong magnetic field with α fluxes per unitcell.
Vincent Pasquier q-Boson Koln December 2015
![Page 11: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/11.jpg)
Hofstadter
Hamiltonian:
H = X + X † + Y + Y †
X = e iαn+θ, Yf (n) = f (n + 1)
Schroedinger equation:
Eψn = ψn+1 + ψn−1 + 2 cos(αn + θ)ψn
• Lattice electrons moving in a periodic potential α times the latticeperiod.• 2D lattice electrons in a strong magnetic field with α fluxes per unitcell.
Vincent Pasquier q-Boson Koln December 2015
![Page 12: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/12.jpg)
Hofstadter
Hamiltonian:
H = X + X † + Y + Y †
X = e iαn+θ, Yf (n) = f (n + 1)
Schroedinger equation:
Eψn = ψn+1 + ψn−1 + 2 cos(αn + θ)ψn
• Lattice electrons moving in a periodic potential α times the latticeperiod.• 2D lattice electrons in a strong magnetic field with α fluxes per unitcell.
Vincent Pasquier q-Boson Koln December 2015
![Page 13: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/13.jpg)
Hofstadter
Hamiltonian:
H = X + X † + Y + Y †
X = e iαn+θ, Yf (n) = f (n + 1)
Schroedinger equation:
Eψn = ψn+1 + ψn−1 + 2 cos(αn + θ)ψn
• Lattice electrons moving in a periodic potential α times the latticeperiod.• 2D lattice electrons in a strong magnetic field with α fluxes per unitcell.
Vincent Pasquier q-Boson Koln December 2015
![Page 14: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/14.jpg)
How to obtain spectrum:
Construct recursion relation for ψ:(ψn+1
ψn
)=
(E − Vk −1
1 0
)(ψn
ψn−1
)Vk = 2 cos(2παk)
Spectrum for |ψn| bounded when |n ±∞.if α = p/q,
−2 < tr(Mq) < 2
tr(Mq) = g(E )± 2 cos(qθ)
|g(E )| is a polynomial of degree q.Typically q bands. Fractal spectrum.
Vincent Pasquier q-Boson Koln December 2015
![Page 15: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/15.jpg)
How to obtain spectrum:
Construct recursion relation for ψ:(ψn+1
ψn
)=
(E − Vk −1
1 0
)(ψn
ψn−1
)Vk = 2 cos(2παk)
Spectrum for |ψn| bounded when |n ±∞.if α = p/q,
−2 < tr(Mq) < 2
tr(Mq) = g(E )± 2 cos(qθ)
|g(E )| is a polynomial of degree q.Typically q bands. Fractal spectrum.
Vincent Pasquier q-Boson Koln December 2015
![Page 16: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/16.jpg)
Hofstadter
Vincent Pasquier q-Boson Koln December 2015
![Page 17: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/17.jpg)
Toda chain
• Dynamics
H =∑N
i=1 p2i + eqi−qi+1
Two particles in the center of mass frame:
− d2ψdq2 + 4 cosh(q)ψ = Eψ
• Hyperbolique Mathieu equation
Vincent Pasquier q-Boson Koln December 2015
![Page 18: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/18.jpg)
Toda chain
• Dynamics
H =∑N
i=1 p2i + eqi−qi+1
Two particles in the center of mass frame:
− d2ψdq2 + 4 cosh(q)ψ = Eψ
• Hyperbolique Mathieu equation
Vincent Pasquier q-Boson Koln December 2015
![Page 19: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/19.jpg)
Hill determinant
• Model is integrable in addition to the Hamiltonian N conservedquantities can be encoded in a transfer matrix T (u)• The complete solution holds in a single Mathieu equation:
T (u)q(u) = iNq(u + i~) + i−Nq(u − i~)
• q(u) is an entire function vanishing as e−Nπ|u|/2 in the real axis.• Two solutions Q+,Q− with the incorrect asymptotic behavior can
be constructed by solving the recursion.• We can recover the correct asymptotic by dividing them with aproduct of sinh(u − uk):
Vincent Pasquier q-Boson Koln December 2015
![Page 20: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/20.jpg)
Hill determinant
• Model is integrable in addition to the Hamiltonian N conservedquantities can be encoded in a transfer matrix T (u)• The complete solution holds in a single Mathieu equation:
T (u)q(u) = iNq(u + i~) + i−Nq(u − i~)
• q(u) is an entire function vanishing as e−Nπ|u|/2 in the real axis.
• Two solutions Q+,Q− with the incorrect asymptotic behavior canbe constructed by solving the recursion.• We can recover the correct asymptotic by dividing them with aproduct of sinh(u − uk):
Vincent Pasquier q-Boson Koln December 2015
![Page 21: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/21.jpg)
Hill determinant
• Model is integrable in addition to the Hamiltonian N conservedquantities can be encoded in a transfer matrix T (u)• The complete solution holds in a single Mathieu equation:
T (u)q(u) = iNq(u + i~) + i−Nq(u − i~)
• q(u) is an entire function vanishing as e−Nπ|u|/2 in the real axis.• Two solutions Q+,Q− with the incorrect asymptotic behavior can
be constructed by solving the recursion.• We can recover the correct asymptotic by dividing them with aproduct of sinh(u − uk):
Vincent Pasquier q-Boson Koln December 2015
![Page 22: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/22.jpg)
Bethe equations
Q+(u)−ζQ−(u)∏Nk=1 sinh(u−uk )
• Analyticity? how to eliminate the poles?• Require the numerator to vanish, Wronskian:
W = |1/T (u + in), 1, 1/T (u + in)|
W =∏N
k=1 sinh(u−δk )∏Nk=1 sinh(u−uk )
• Quantization conditions:
ζ = Q+(δk )Q−(δk )
Does not depend on k.
Vincent Pasquier q-Boson Koln December 2015
![Page 23: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/23.jpg)
Bethe equations
Q+(u)−ζQ−(u)∏Nk=1 sinh(u−uk )
• Analyticity? how to eliminate the poles?• Require the numerator to vanish, Wronskian:
W = |1/T (u + in), 1, 1/T (u + in)|
W =∏N
k=1 sinh(u−δk )∏Nk=1 sinh(u−uk )
• Quantization conditions:
ζ = Q+(δk )Q−(δk )
Does not depend on k.
Vincent Pasquier q-Boson Koln December 2015
![Page 24: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/24.jpg)
Bethe equations
Q+(u)−ζQ−(u)∏Nk=1 sinh(u−uk )
• Analyticity? how to eliminate the poles?• Require the numerator to vanish, Wronskian:
W = |1/T (u + in), 1, 1/T (u + in)|
W =∏N
k=1 sinh(u−δk )∏Nk=1 sinh(u−uk )
• Quantization conditions:
ζ = Q+(δk )Q−(δk )
Does not depend on k.
Vincent Pasquier q-Boson Koln December 2015
![Page 25: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/25.jpg)
deformed Toda Lattice
• Is there some analogous of lattice Schroedinger for Toda?• Ruisjenaars Hamiltonian:
H =∑N
1 Yk(1− Xk−1/Xk)
X = ex , Yf (x) = f (x + iα)
• α is the analogous of the flux per unit cell.• i has migrated from X to Y :X † = X , Y † = Y .• Can easily be truncated on Harmonic oscillator basis.
Vincent Pasquier q-Boson Koln December 2015
![Page 26: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/26.jpg)
deformed Toda Lattice
• Is there some analogous of lattice Schroedinger for Toda?• Ruisjenaars Hamiltonian:
H =∑N
1 Yk(1− Xk−1/Xk)
X = ex , Yf (x) = f (x + iα)
• α is the analogous of the flux per unit cell.• i has migrated from X to Y :X † = X , Y † = Y .• Can easily be truncated on Harmonic oscillator basis.
Vincent Pasquier q-Boson Koln December 2015
![Page 27: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/27.jpg)
Hill determinant
• Model is integrable in addition to the Hamiltonian N conservedquantities can be encoded in a transfer matrix T (u)• The complete solution holds in a single Mathieu equation:
T (u)q(u) = iNq(u + iα) + i−Nq(u − iα)
• T (u) is now a trigonometric polynomial: T =∏
sinh(u − uk)• q(u) is an entire function.• Two solutions Q+,Q− with the incorrect asymptotic behavior
cannot be constructed by solving the recursion.
Vincent Pasquier q-Boson Koln December 2015
![Page 28: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/28.jpg)
Hill determinant
• Model is integrable in addition to the Hamiltonian N conservedquantities can be encoded in a transfer matrix T (u)• The complete solution holds in a single Mathieu equation:
T (u)q(u) = iNq(u + iα) + i−Nq(u − iα)
• T (u) is now a trigonometric polynomial: T =∏
sinh(u − uk)• q(u) is an entire function.• Two solutions Q+,Q− with the incorrect asymptotic behavior
cannot be constructed by solving the recursion.
Vincent Pasquier q-Boson Koln December 2015
![Page 29: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/29.jpg)
The Hill determinant ??
W = |1/T (u + iαn), 1, 1/T (u + iαn)|
• T (u) is now a trigonometric polynomial, so, the situation is similar toHofstadter, in particular periodicity for α rational.
• Natural guess for the Hill determinant:
W =∏N
k=1 θ1(u− δk )∏Nk=1 θ1(u−uk )
• Period τ should be iα Absurd!
Vincent Pasquier q-Boson Koln December 2015
![Page 30: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/30.jpg)
The Hill determinant ??
W = |1/T (u + iαn), 1, 1/T (u + iαn)|
• T (u) is now a trigonometric polynomial, so, the situation is similar toHofstadter, in particular periodicity for α rational.
• Natural guess for the Hill determinant:
W =∏N
k=1 θ1(u− δk )∏Nk=1 θ1(u−uk )
• Period τ should be iα Absurd!
Vincent Pasquier q-Boson Koln December 2015
![Page 31: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/31.jpg)
The Hill determinant ??
W = |1/T (u + iαn), 1, 1/T (u + iαn)|
• T (u) is now a trigonometric polynomial, so, the situation is similar toHofstadter, in particular periodicity for α rational.
• Natural guess for the Hill determinant:
W =∏N
k=1 θ1(u− δk )∏Nk=1 θ1(u−uk )
• Period τ should be iα Absurd!
Vincent Pasquier q-Boson Koln December 2015
![Page 32: Vincent PASQUIER - thp.uni-koeln.deklevtsov/Pasquier.pdf · Vincent Pasquier q-Boson Koln December 2015. Hofstadter Hamiltonian: H = X + Xy+ Y + Yy X = ei n+ ; Yf(n) = f(n + 1) Schroedinger](https://reader033.vdocuments.site/reader033/viewer/2022060908/60a304688f9a740c957c71c5/html5/thumbnails/32.jpg)
Conclusion
• Thank You Semyon for organizing such a nice conference!
Vincent Pasquier q-Boson Koln December 2015