Bethe ansatz for RSOS models
imported>Pythagoras0님의 2013년 6월 28일 (금) 10:17 판
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
articles
- http://dx.doi.org/10.1016/j.nuclphysb.2004.07.008
- 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. 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.