Post on 18-Mar-2021
Exact Solution of a Relativistic Quantum Toda Chain
Yupeng Wang
Institute of Physics, CAS
2017-07-14, MATRIX
Collaborators : J. Cao, W.L. Yang, K. ShiX. Zhang……
JPA 50(2017)124003, JHEP 09 (2015)212
I. Introduction
II. The local vacuum
III. The Bethe Ansatz solution
IV. The case at roots of unit
IV. Concluding Remarks & Perspective
Outline
I. Introduction
The Toda chain model is an important integrablemodel possessing unique properties
(1) Infinite dimensional; (2) Without U(1) symmetry
Many persons contributed to its solution
Sutherland 1978, Gutzwiller 1981, Skyanin 1985, Pasquier & Gaudin 1992……
I. Introduction
The Relativistic Quantum Toda Chain
Ruijsenaars, Suris, Nikrasov, Kundu, Huang, etc
Related to the Seiberg-Witten theory& quantization in Calabi-Yau manifold.
I. Introduction
The integrability
Lax matrix
Yang-Baxter
form a Weyl algebra
I. Introduction
Monodromymatrix
YBE
Transfer Matrix
Hamiltonian
II. The Local Vacuum
Gauge matrices
Transformation
Takhtajan & Faddeev, 1979Cao et al, 2003
The local vacuum
II. The Local Vacuum
III. The Bethe Ansatz Solution
The reference state
Bethe state
Eigenstates
III. The Bethe Ansatz Solution
III. The Bethe Ansatz Solution
Asymptotic behavior defines
Bethe Ansatz equations :
Eigenvalue of Hamiltonian
T-Q relation
BAE
III. The Bethe Ansatz Solution
IV. The case at roots of unit
The Quantum Tau-2 model Bazhanov & Strogonov 90
Relativistic Toda
Pakuliak & Sergeev 01 &……
IV. The case at roots of unit
IV. The case at roots of unit
The conserved (Zp) charges
The quantum determinant
Average & quantum determinant
C-number
p-times fused transfer matrix
Tarasov 92
The fusion identitiesFusion identities
The inhomogeneous T-Q relation
IV. The case of roots of unit
IV. Concluding Remarks & Perspective
•The relativistic quantum Toda chain can be solveld via algebraic Bethe Ansatz.
•This approach can be used also in other similar integrable models.
•In the classical limit , we readily recover Sklyanin and Pasquier & Gaudin’s result.
•At roots of unit the ODBA can be applied.
Thanks!