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!