Bethe ansatz for RSOS models

introduction

• see chapter 15 of T-systems and Y-systems in integrable systems for basics

TBA equation

• taken from T-systems and Y-systems in integrable systems
• RSOS model associated with the representation \(W_{s}^{(p)}\) of \(U_q(\hat{\mathfrak{g}})\).
• \(N\) : length of sites
• \(C=(C_{ab})\) : Cartan matrix
• \(L=\ell +h^{\vee}\)

The Bethe equation is the following for the unknowns \(\{u^{(a)}_j \vert \, a \in I, 1 \le j \le n_a \}\):

\begin{equation}\label{ber} \Biggl(\frac{\sinh{\pi\over 2L}\bigl( u^{(a)}_j - \sqrt{-1}{s \over t_p}\delta_{a p}\bigr)} {\sinh{\pi\over 2L}\bigl( u^{(a)}_j + \sqrt{-1}{s \over t_p}\delta_{a p}\bigr)} \Biggr)^N = \Omega_a \prod_{b=1}^r\prod_{k=1}^{n_b} \frac{\sinh{\pi\over 2L}\bigl( u^{(a)}_j - u^{(b)}_k - \sqrt{-1}(\alpha_a \vert \alpha_b)\bigr)} {\sinh{\pi\over 2L}\bigl( u^{(a)}_j - u^{(b)}_k + \sqrt{-1}(\alpha_a \vert \alpha_b)\bigr)}. \end{equation} Here \(n_a=Ns(C^{-1})_{a p}\) as in (3.51) with \((r_i,s_i)=(p,s)\) for all \(i\), and \(\Omega_a\) is a root of unity

related items

• Bethe ansatz
• RSOS models
• Level restricted Q-system and RSOS model
• Fermionic formula for fusion ring

articles

• Babichenko, A. 2004. “From S-matrices to the Thermodynamic Bethe Ansatz.” Nuclear Physics B 697 (3) (October 11): 481–512. doi:10.1016/j.nuclphysb.2004.07.008.
• Dasmahapatra, Srinandan. 1993. “String Hypothesis and Characters of Coset CFTs”. ArXiv e-print hep-th/9305024. http://arxiv.org/abs/hep-th/9305024.
• Kuniba, A. (1993). Thermodynamics of the Uq(Xr(1)) Bethe ansatz system with q a root of unity. Nuclear Physics B, 389(1), 209–244. doi:10.1016/0550-3213(93)90291-V
• Bazhanov, V. V., and N. Reshetikhin. 1990. “Restricted Solid-on-solid Models Connected with Simply Laced Algebras and Conformal Field Theory.” Journal of Physics A: Mathematical and General 23 (9) (May 7): 1477. doi:10.1088/0305-4470/23/9/012.
• Bazhanov, V. V., and N. Yu. Reshetikhin. 1989. “Critical RSOS Models and Conformal Field Theory.” International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Nuclear Physics 4 (1): 115–142. doi:10.1142/S0217751X89000042.
• Reshetikhin, N.Yu., and P.B. Weigmann. 1987. “Towards the Classification of Completely Integrable Quantum Field Theories (the Bethe-Ansatz Associated with Dynkin Diagrams and Their Automorphisms).” Physics Letters B 189 (1–2) (April 30): 125–131. doi:10.1016/0370-2693(87)91282-2.
• Reshetikhin, N. Yu. 1987. “The Spectrum of the Transfer Matrices Connected with Kac-Moody Algebras.” Letters in Mathematical Physics 14 (3) (October 1): 235–246. doi:10.1007/BF00416853.