2 fermionic basis for the xxz model - research.kek.jp...
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![Page 1: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/1.jpg)
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![Page 2: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/2.jpg)
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Fermionic Basis for the XXZ model
![Page 3: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/3.jpg)
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Fermionic Basis for the XXZ model
T. Miwa
![Page 4: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/4.jpg)
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Fermionic Basis for the XXZ model
T. Miwa
joint work with
![Page 5: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/5.jpg)
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Fermionic Basis for the XXZ model
T. Miwa
joint work withH. Boos
![Page 6: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/6.jpg)
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Fermionic Basis for the XXZ model
T. Miwa
joint work withH. Boos, M. Jimbo
![Page 7: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/7.jpg)
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Fermionic Basis for the XXZ model
T. Miwa
joint work withH. Boos, M. Jimbo, F. Smirnov
![Page 8: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/8.jpg)
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Fermionic Basis for the XXZ model
T. Miwa
joint work withH. Boos, M. Jimbo, F. Smirnov, Y. Takeyama
1. Quantum XXZ Hamiltonian2. Quantum symmetry and integral formula3. Algebraic formula4. Quasi-local operators5. Annihilation operators6. Particle structure
![Page 9: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/9.jpg)
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1. Quantum XXZ Hamiltonian
![Page 10: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/10.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
![Page 11: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/11.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
![Page 12: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/12.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2
![Page 13: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/13.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2HXXZ ‘acts’ on · · · ⊗ C2 ⊗ C2 ⊗ C2 ⊗ · · ·
![Page 14: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/14.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2HXXZ ‘acts’ on · · · ⊗ C2 ⊗ C2 ⊗ C2 ⊗ · · ·• correlation functions
![Page 15: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/15.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2HXXZ ‘acts’ on · · · ⊗ C2 ⊗ C2 ⊗ C2 ⊗ · · ·• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉
![Page 16: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/16.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
where ∆ =q + q−1
2HXXZ ‘acts’ on · · · ⊗ C2 ⊗ C2 ⊗ C2 ⊗ · · ·• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ
![Page 17: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/17.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ
![Page 18: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/18.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
![Page 19: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/19.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
![Page 20: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/20.jpg)
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1. Quantum XXZ Hamiltonian• quantum spin chain
HXXZ =1
2
∞∑
k=−∞
(σ1
kσ1k+1+σ2
kσ2k+1+∆σ3
kσ3k+1
)
• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
![Page 21: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/21.jpg)
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1. Quantum XXZ Hamiltonian• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
![Page 22: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/22.jpg)
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1. Quantum XXZ Hamiltonian• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
• general correlation functions
![Page 23: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/23.jpg)
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1. Quantum XXZ Hamiltonian• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
• general correlation functionsfor local operator O
![Page 24: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/24.jpg)
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1. Quantum XXZ Hamiltonian• correlation functions
〈σ31σ
3n〉 =
〈vac|σ31σ
3n|vac〉
〈vac|vac〉|vac〉 : the lowest eigenvector of HXXZ• results for small n for ∆ = 1
〈σ31σ
32〉 ∼ log 2 (Hulthen)
〈σ31σ
33〉 ∼ ζ(3) (Takahashi)
• general correlation functionsfor local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
![Page 25: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/25.jpg)
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
• inhomogeneous model
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
• inhomogeneous modelspectral parameters ξ1, . . . , ξn
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉 =〈vac|O|vac〉〈vac|vac〉
• inhomogeneous modelspectral parameters ξ1, . . . , ξn|vac〉 → |vac〉ξ1,...,ξn
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
• inhomogeneous modelspectral parameters ξ1, . . . , ξn|vac〉 → |vac〉ξ1,...,ξn
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1. Quantum XXZ Hamiltonian• general correlation functions
for local operator O
〈O〉ξ1,...,ξn =ξ1,...,ξn
〈vac|O|vac〉ξ1,...,ξnξ1,...,ξn
〈vac|vac〉ξ1,...,ξn• inhomogeneous model
spectral parameters ξ1, . . . , ξn|vac〉 → |vac〉ξ1,...,ξn
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2. Quantum symmetry and inte-gral formula
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations
![Page 34: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/34.jpg)
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2e1, t1e0t
−11 = q−2e0
![Page 35: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/35.jpg)
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2e1, t1e0t
−11 = q−2e0
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
![Page 36: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/36.jpg)
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
![Page 37: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/37.jpg)
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2. Quantum symmetry and inte-gral formula
Uq(sl2) symmetry (∆>1↔0<q<1)
(DFJMN,JMMN,JM)• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
![Page 38: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/38.jpg)
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2. Quantum symmetry and inte-gral formula• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
e0 =
(0
ζ
),
![Page 39: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/39.jpg)
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2. Quantum symmetry and inte-gral formula• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
e0 =
(0
ζ
), e1 =
(ζ
0
)
![Page 40: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/40.jpg)
40
2. Quantum symmetry and inte-gral formula• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
e0 =
(0
ζ
), e1 =
(ζ
0
)
t1 = t−10 =
(q
q−1
)
![Page 41: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/41.jpg)
41
2. Quantum symmetry and inte-gral formula• algebra generated by ei, fi, ti (i = 0, 1)
with certain defining relations, e.g.,
t1e1t−11 = q2ei, t1e0t
−11 = q−2ei
e0e31 − [3]e1e0e
21 + [3]e2
1e0e1 − e31e0 = 0
[3] = q2 + 1 + q−2
• admits two dimensional representation(C2)ζ depending on a spectral parameter ζ
e0 =
(0
ζ
), e1 =
(ζ
0
)
t1 = t−10 =
(q
q−1
)
• level 1 HWRs and intertwiners
![Page 42: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/42.jpg)
42
2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
![Page 43: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/43.jpg)
43
2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2
![Page 44: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/44.jpg)
44
2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
![Page 45: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/45.jpg)
45
2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ
![Page 46: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/46.jpg)
46
2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ• representation of correlation functions
using trace of product of intertwiners
![Page 47: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/47.jpg)
47
2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ• representation of correlation functions
using trace of product of intertwiners
trH
(q2dΦε1(ζ1) · · ·Φε2n(ζ2n)
)
![Page 48: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/48.jpg)
48
2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ• representation of correlation functions
using trace of product of intertwiners
trH
(q2dΦε1(ζ1) · · ·Φε2n(ζ2n)
)
• integral formula is obtained by bosoniza-tion
![Page 49: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/49.jpg)
49
2. Quantum symmetry and inte-gral formula• level 1 HWRs and intertwiners
· · ·C2 ⊗ C2 ⊗ C2 ' H = L(Λ0) ⊕ L(Λ1)
Φ(ζ) : H → H ⊗ (C2)ζ• representation of correlation functions
using trace of product of intertwiners
trH
(q2dΦε1(ζ1) · · ·Φε2n(ζ2n)
)
• integral formula is obtained by bosoniza-tion• leads to the qKZ equation ζi → q2ζi
![Page 50: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/50.jpg)
50
3. Algebraic formula
![Page 51: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/51.jpg)
51
3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)
![Page 52: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/52.jpg)
52
3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)
![Page 53: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/53.jpg)
53
3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)∑ω(ξi1/ξj1) · · ·ω(ξik/ξjk)︸ ︷︷ ︸Fi1j1···ikjk
(ξ1,...,ξn)︸ ︷︷ ︸↑ ↑
transcendental rational
![Page 54: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/54.jpg)
54
3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)∑ω(ξi1/ξj1) · · ·ω(ξik/ξjk)︸ ︷︷ ︸Fi1j1···ikjk
(ξ1,...,ξn)︸ ︷︷ ︸↑ ↑
transcendental rational
• transcendental function
![Page 55: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/55.jpg)
55
3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)∑ω(ξi1/ξj1) · · ·ω(ξik/ξjk)︸ ︷︷ ︸Fi1j1···ikjk
(ξ1,...,ξn)︸ ︷︷ ︸↑ ↑
transcendental rational
• transcendental function
ω(ζ) =
∫ i∞−0
−i∞−0ζu sin
π(1−ν)u2
sin πu2 cos πνu
2du
![Page 56: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/56.jpg)
56
3. Algebraic formula
• integral formula∆=1−→ log 2, ζ(3), ζ(5), . . .
(Boos-Korepin)• qKZ equation → algebraic formula
(BKS, BJMST)∑ω(ξi1/ξj1) · · ·ω(ξik/ξjk)︸ ︷︷ ︸Fi1j1···ikjk
(ξ1,...,ξn)︸ ︷︷ ︸↑ ↑
transcendental rational
• transcendental function
ω(ζ) =
∫ i∞−0
−i∞−0ζu sin
π(1−ν)u2
sin πu2 cos πνu
2du
+ rational in q, ζ
![Page 57: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/57.jpg)
57
4. Quasi-local operators
![Page 58: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/58.jpg)
58
4. Quasi-local operators• disorder parameter α
![Page 59: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/59.jpg)
59
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
![Page 60: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/60.jpg)
60
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
⊗ (∗ ∗ ∗) ⊗
![Page 61: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/61.jpg)
61
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
⊗ (∗ ∗ ∗) ⊗(
11
)⊗ · · ·
![Page 62: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/62.jpg)
62
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
![Page 63: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/63.jpg)
63
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operators
![Page 64: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/64.jpg)
64
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
![Page 65: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/65.jpg)
65
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
![Page 66: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/66.jpg)
66
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
=
![Page 67: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/67.jpg)
67
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
![Page 68: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/68.jpg)
68
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
where
![Page 69: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/69.jpg)
69
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
where
trα(X) =
![Page 70: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/70.jpg)
70
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
where
trα(X)=
∏
k
1
qα/2 + q−α/2trVk
q−ασ3k/2
(X)
![Page 71: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/71.jpg)
71
4. Quasi-local operators• disorder parameter α
quasi local operator qα∑0
k=−∞ σ3kO
· · · ⊗(
qα
q−α
)⊗ (∗ ∗ ∗) ⊗
(1
1
)⊗ · · ·
Wα: space of quasi local operatorsdim (Wα)[1,n] = 4n
• algebraic formula
〈vac|q2αS(0)O|vac〉〈vac|q2αS(0)|vac〉
= trα[eΩ
(q2αS(0)O
)]
where
trα(X)=
∏
k
1
qα/2 + q−α/2trVk
q−ασ3k/2
(X)
Ω : nilpotent
![Page 72: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/72.jpg)
72
5. Annihilation operators
![Page 73: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/73.jpg)
73
5. Annihilation operators• decomposition of Ω
![Page 74: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/74.jpg)
74
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξj
![Page 75: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/75.jpg)
75
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
![Page 76: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/76.jpg)
76
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
• annihilation operators
![Page 77: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/77.jpg)
77
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
![Page 78: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/78.jpg)
78
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
![Page 79: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/79.jpg)
79
5. Annihilation operators• decomposition of Ω
Ω = −∑
1≤i,j≤nresζ1=ξi, ζ2=ξjω(ζ1/ζ2, α)c−(ζ1, α − 1)c+(ζ2, α)
dζ1
ζ1
dζ2
ζ2
• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
![Page 80: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/80.jpg)
80
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
![Page 81: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/81.jpg)
81
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1]) =
![Page 82: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/82.jpg)
82
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
![Page 83: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/83.jpg)
83
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1
![Page 84: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/84.jpg)
84
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
![Page 85: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/85.jpg)
85
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support property
![Page 86: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/86.jpg)
86
5. Annihilation operators• annihilation operators
c±[1,n](ζ, α) =1
2
n∑
j=1
c±j,[1,n]
1 − ζ/ξj
c±[1,n](ζ, α) : Wα → Wα∓1
• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
![Page 87: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/87.jpg)
87
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
![Page 88: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/88.jpg)
88
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
![Page 89: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/89.jpg)
89
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
![Page 90: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/90.jpg)
90
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
![Page 91: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/91.jpg)
91
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
sic±j = c±
σi,i+1(j)si
![Page 92: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/92.jpg)
92
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
sic±j = c±
σi,i+1(j)si
• overdetermined Grassmann relations
![Page 93: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/93.jpg)
93
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
sic±j = c±
σi,i+1(j)si
• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j ,
![Page 94: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/94.jpg)
94
5. Annihilation operators• reduction property
c±j,[1,n](X[1,n−1]) = c±j,[1,n−1](X[1,n−1])
c±j,[1,n](qασ3
1X[2,n−1])=q(α∓1)σ31c±j,[2,n−1](X[2,n−1])
c±j : Wα,s → Wα∓1,s±1 (s : total spin)
• support propertyc±j (X[1,n]) = 0 if j 6∈ [1, n]
• equivariance
action of the symmetric group Sn
si : (Wα)[1,n] → (Wα)[1,n]
sic±j = c±
σi,i+1(j)si
• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
![Page 95: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/95.jpg)
95
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
![Page 96: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/96.jpg)
96
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
![Page 97: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/97.jpg)
97
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR?
![Page 98: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/98.jpg)
98
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO
![Page 99: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/99.jpg)
99
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
![Page 100: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/100.jpg)
100
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
![Page 101: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/101.jpg)
101
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
![Page 102: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/102.jpg)
102
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
: 2n dimensional
![Page 103: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/103.jpg)
103
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
: 2n dimensional• strong annihilation property
![Page 104: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/104.jpg)
104
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
: 2n dimensional• strong annihilation property
c±n,[1,n]
(Wα)[1,n] ⊂ (Wα∓1)[1,n−1]
![Page 105: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/105.jpg)
105
5. Annihilation operators• overdetermined Grassmann relations
cε1j c
ε2k = −c
ε2k c
ε1j , cε
jc−εj = 0
• Grassmann variables c±j (j ∈ Z)
Grassmann relation ⊂ CAR? → NO• vacuum states
X ∈ (Wα)[1,n] ; c±j,[1,n]
(X) = 0
= ⊕nk=0C[Sn]q2αS(k)
: 2n dimensional• strong annihilation property
c±n,[1,n]
(Wα)[1,n] ⊂ (Wα∓1)[1,n−1]
IMPLIES →
![Page 106: 2 Fermionic Basis for the XXZ model - research.kek.jp ...research.kek.jp/group/riron/workshop/theory2007/slides/miwa.pdf · Fermionic Basis for the XXZ model T. Miwa joint work with](https://reader031.vdocuments.us/reader031/viewer/2022022711/5bff568709d3f2641b8c1941/html5/thumbnails/106.jpg)
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6. Particle structure
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6. Particle structure• n = 1 case
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= 1
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0)
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= qασ3
1
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= σ±1
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
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120
6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
• 1 particle states = orbit of q2αS(0)σ±1
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
• 1 particle states = orbit of q2αS(0)σ±1• n = 2 case
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
• 1 particle states = orbit of q2αS(0)σ±1• n = 2 case : (Wα)[1,2] is 42 dimensional
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6. Particle structure• n = 1 case : (Wα)[1,1] is 4 dimensional
vacuum states are 2 dimensional
v(0)def= q2αS(0), v(0)
def= q2αS(1)
spin ±1 state : v(±)def= q2αS(0)σ±1
• annihilation of particles
c+1 v(+) = 0, c+
1 v(−) = v(0)
c−1 v(+) = v(0), c−1 v(−) = 0
• 1 particle states = orbit of q2αS(0)σ±1• n = 2 case : (Wα)[1,2] is 42 dimensional
vacuum states are 22 dimensional
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6. Particle structure• n = 2 case : (Wα)[1,2] is 42 dimensional
vacuum states are 22 dimensional
· · · ⊗ qασ3⊗ ∗ ⊗ ∗ ⊗ 1 ⊗ · · ·
· · · 0 1 2 3 · · ·(0, 0) ↔ 1 ⊗ 1, (0, 0) ↔ qασ3
⊗ 1
(0, 0) ↔ qασ3⊗ qασ3
, (0, 0) ↔ s1(qασ3
⊗ 1)
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6. Particle structure• n = 2 case : (Wα)[1,2] is 42 dimensional
vacuum states are 22 dimensional
· · · ⊗ qασ3⊗ ∗ ⊗ ∗ ⊗ 1 ⊗ · · ·
· · · 0 1 2 3 · · ·(0, 0) ↔ 1 ⊗ 1, (0, 0) ↔ qασ3
⊗ 1
(0, 0) ↔ qασ3⊗ qασ3
, (0, 0) ↔ s1(qασ3
⊗ 1)
• other states
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6. Particle structure• n = 2 case : (Wα)[1,2] is 42 dimensional
vacuum states are 22 dimensional
· · · ⊗ qασ3⊗ ∗ ⊗ ∗ ⊗ 1 ⊗ · · ·
· · · 0 1 2 3 · · ·(0, 0) ↔ 1 ⊗ 1, (0, 0) ↔ qασ3
⊗ 1
(0, 0) ↔ qασ3⊗ qασ3
, (0, 0) ↔ s1(qασ3
⊗ 1)
• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
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6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
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6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
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6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
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6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
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6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
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6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
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6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
∼ means ‘up to sign’
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6. Particle structure• other states
(0,±), (0,±), (±, 0), (±, 0), (±,±), (±,∓)
• action of the symmetric group
s1(p1, p2) ∼ (p2, p1)
• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
∼ means ‘up to sign’
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
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6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
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6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
• fermionic basis
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6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
• fermionic basisvp (p = (pj)j∈Z)
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6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
• fermionic basisvp (p = (pj)j∈Z)
pj ∈ ±, 0, 0, pj =
0 if j << 0
0 if j >> 0
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6. Particle structure• fermionic action of c±
p1, p2 ∈ 0, 0, +,−
c±2 (p1, p2) ∼
(p1, 0) if p2 = ∓0 otherwise
c±1 (p1, p2) ∼
(0, p2) if p1 = ∓0 otherwise
• fermionic basisvp (p = (pj)j∈Z)
pj ∈ ±, 0, 0, pj =
0 if j << 0
0 if j >> 0
exists, though not unique
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1. Universal R matrix
main ingredient L operators= images of universal R matrix
L1,2 = (π1⊗π2)(R) where R ∈ Uq(b+)⊗Uq(b
−)
πi : Uq(b+) → ai Uq(b
+) = 〈e0, e1, t0, t1〉a1 : auxiliary space
a2 : quantum space = End(V1 ⊗ · · · ⊗ Vn)
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2. Transfer matrix inEnd(V1 ⊗ · · · ⊗ Vn)
R matrix auxiliary space = Va ' C2
R(ζ) = (qζ − q−1ζ−1)
1β(ζ) γ(ζ)γ(ζ) β(ζ)
1
β(ζ) =(1 − ζ2)q
1 − q2ζ2, γ(ζ) =
(1 − q2)ζ
1 − q2ζ2.
t(N)(ζ) = tra (Ran(ζ) · · ·Ra1(ζ))
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3. Commuting family
Yang-Baxter equation
R1,2R1,3R2,3 = R2,3R1,3R1,2
implies [t(N)(ζ1), t(N)(ζ2)] = 0
quantum Hamiltonian
t(N)(1)−1t(N)(ζ) = 1 + const(ζ − 1)H(N)
XXZ + · · ·inhomogeneous model
t(N)(ζ) = tra (Ran(ζ/ξn) · · ·Ra1(ζ/ξ1))
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4. Adjoint action
transfer matrix acting on
X ∈ End(V1 ⊗ · · · ⊗ Vn)
Ta(ζ) = Ra,n(ζ/ξn) · · ·Ra,1(ζ/ξ1)
t(ζ, α)(X) = tra(q−ασ3aTa(ζ)−1XTa(ζ))
tautological reduction to the right
t[1,n](ζ, α)(X[1,n−1]) = t[1,n−1](ζ, α)(X[1,n−1])
analytic structure
tra(q−ασ3a∗) =
∑±q±α(rational in q, ζ, ξi)
simple poles at (ζ/ξi)2 = q±2
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5. Baxter’s TQ relation
(Bazhanov-Lukyanov-Zamolodchikov)second order difference equation
t(ζ, α)Q±(ζ, α) = Q±(q−1ζ, α) + Q±(qζ, α)
n = 0 case
(qα + q−α)ζ±α = (q−1ζ)±α + (qζ)±α
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6. q-Oscillator algebra
algebra Osc with generators and relations
qDaq−D = q−1a, qDa∗q−D = qa∗
aa∗ = 1 − q2D+2, a∗a = 1 − q2D
two morphisms
o±ζ : Uq(b+) → Osc
o±ζ (e0) =ζ
q − q−1
a∗
ao±ζ (e1) =
ζ
q − q−1
a
a∗
o±(t0) = q±2D, o±(t1) = q∓2D
two representations of Osc
W+ = ⊕k≥0C|k〉, W− = ⊕k≤−1C|k〉qD|k〉 = qk|k〉, a|k〉 = (1 − q2k)|k − 1〉,a∗|k〉 = (1 − δk,−1)|k + 1〉.
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7. Q operators
L operators
L±(ζ) = iζ−1/2q−1/4
×(1 − ζa∗σ± − ζaσ∓ − ζ2q2D+2τ∓
)q±σ3D
σ+=
(1
0
), σ−=
(0
1
), τ+=
(1
0
), τ−=
(0
1
)
T±A (ζ) = L±
A,n(ζ/ξn) · · ·L±A,1(ζ/ξ1)
Q operators
Q±(ζ, α)(X) = ±(1 − q±2(α−S))ζ±(α−S)
×Tr±A
(q±2αDAT±
A(ζ)−1(X))
where T±A(ζ)−1(X) = T±
A (ζ)−1(X)T±A (ζ)
and S(X) = [S[∞], X ]
analytic structure
Tr±A(q±2αDAqmDA) = ±(1 − q±2α+m)−1
simple poles at (ζ/ξj)2 = 1
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8. Triangularity
TQ relation follows from triangular de-composition
L+A,a,j(ζ) = (G+
A,a)−1L+A,j(ζ)Ra,j(ζ)G+
A,a
= (qζ − q−1ζ−1) ×L+
A,j(q−1ζ)q
−σ3j/2
0
∗ β(ζ)L+A,j(qζ)q
σ3j/2
a
G+A,a = q−σ3
aDA(1 + a∗Aσ+a )
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9. Reduction to the left
q difference operator
∆q(F (ζ)) = F (qζ) − F (q−1ζ)
off diagonal element
±(1 − q±2(α−S))−1k±(ζ, α)(X)def= ζ±(α−S)
×Tr±Atra
(q±2(α∓1)DAσ±a Ta(ζ)−1T±
A(ζ)−1(X))
= ζ±(α−S)Tr±Atra
(q±2αDAσ±a T±
A,a(ζ)−1(X))
σ±a picks up the off-diagonal elements
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10. Annihilation operators
Reduction modulo q exact form
k±[1,n](ζ,α)(qασ31X[2,n])=q(α∓1)σ3
1k±[2,n](ζ,α)(X[2,n])
+σ±1 ∆q
(q−q−1
ζ/ξ1−(ζ/ξ1)−1Q±[2,n](ζ, α ∓ 1)(X[2,n])
)
annihilation operators
c±(ζ, α)def= (normalization)×
n∑
j=1
singζ=ξj k±(ζ, α)
satisfies the reduction to the left
c±[1,n](ζ, α)(qασ31X[2,n])=q(α∓1)σ3
1c±[2,n](ζ, α)(X[2,n])
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11. Creation operators
conjugate transfer matrix
t∗[1,n](ζ, α)(X) = tra(Ta(ζ)qασ3aXTa(ζ)−1)
filling Dirac sea
t∗[1,n](ξl, α)(X[1,j])=2sl−1 · · · s1(qασ3
1 · τ (X[1,j]))
where τ is shift operator
another TQ relations
Q∗±[1,n]
(ζ, α)(X) = ±(1 − q±2(α−S))ζ±(α−S)
×Tr±A
(T∓
A,[1,n](ζ)q±2αDA(X)
)
t∗[1,n](ζ, α)Q∗±[1,n]
(ζ, α)=Q∗±[1,n](q
−1ζ, α)+Q∗±[1,n](qζ, α)
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off diagonal operators
k∗±(ζ, α)(X) = ±(1 − q±2(α±1−S))ζ±(α±1−S)
×Tr±Atra
(σ±a Ta(ζ)T∓
A(ζ)(qα(±2DA+σ3
a)∓2SX))
reduction to the right modulo q exact form
k∗±[1,n](ζ, α)(X[1,n−1]) = k∗±[1,n−1](ζ, α)(X[1,n−1])
+σ±n∆q
(q−q−1
ζ/ξ1−(ζ/ξ1)−1Q∗±[1,n−1](ζ, α)(q
∓2S[1,n−1]X[1,n−1]))
removing ∆q
f∗±[1,n](ζ, α)(X[1,n−1]) = f∗±[1,n−1](ζ, α)(X[1,n−1])
+σ±n κ(ζ/ξn)Q∗±[1,n−1](ζ, α)(q
∓2S[1,n−1]X[1,n−1])
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creation operators
c∗±[1,n](ζ, α)def= f∗±[1,n](qζ, α) + f∗±[1,n](q
−1ζ, α)
−t∗[1,n](ζ, α)f∗±[1,n](ζ, α)
satisfies reduction property in a restricted sense
c∗±[1,n]
(ζ, α)(X[1,j]) (1 ≤ j < l ≤ n)
is regular at ζ = ξl and satisfies
c∗±[1,n]
(ξl, α)(X[1,j]) = c∗±[1,n−1]
(ξl, α)(X[1,j])
(1 ≤ j < l ≤ n − 1)
ConjectureA fermionic basis is created by the creation
operators c∗±[1,n]
(ξl, α)