RSOS models

introduction

• restricted solid-on-solid (RSOS) models
• also called as ABF(Andrews-Baxter-Forrester models)
• class of a spin system
• IBF(interaction round a face) model
• vertex counterpart is Belavin's generalization of the 8-vertex model

physical description

• a rough, discrete analogon of a gently fluctutationg surface of a liquid
• neighboring points cannot have heights which differ much from each other
• local energy density is given by the surface energy

height variable

• to each site i, we assign a height variable

critical RSOS model

• A_3 RSOS model = Ising models
• D_4 RSOS model = 3-states Potts model

Pierre Mathieu, Combinatorics of RSOS paths

Every minimal model in conformal field theory can be viewed as the scaling limit of a restricted-solid-on-solid (RSOS) model at criticality. States in irreducible modules of the minimal model M(p',p) can be described combinatorially by paths that represent configurations in the corresponding RSOS model, dubbed RSOS(p',p). These paths are in one-to-one correspondence with tableaux with prescribed hook differences. For p'=2, these are tableaux with successive ranks in a prescribed interval, which are known to be related to the Bressoud paths (whose generating function is the sum side of the Andrews-Gordon identity). We show how the RSOS(2,p) paths can be directly related to these paths. Generalizing this construction, we arrive at a representation of RSOS paths in terms of generalized Bressoud paths (for p>2p'). These new paths have a simple weighting and a natural particle interpretation. This then entails a natural particle spectrum for RSOS paths, which can be interpreted in terms of the kinks and breathers of the restricted sine-Gordon model.

history

• The relevant Bethe ansatz analysis on RSOS models
  • Bazhanov-Reshetikhin JPA 23, 1477 (1992) for ADE
  • Kuniba Nucl. Phys B (1993) for BCFG.
• http://www.google.com/search?hl=en&tbs=tl:1&q=

memo

• MINIMAL MODELS IN CONFORMAL FIELD THEORY AND INTEGRABLE LATTICE MODELS
  
  • Nakanishi Tomoki, Institute of Physics University of Tokyo, 1990

related items

• 5 conformal field theory(CFT)
• minimal models
• six-vertex model and Quantum XXZ Hamiltonian
• Eight-vertex model and quantum XYZ model
• Bethe ansatz for RSOS models

articles

• Jacob, P., and P. Mathieu. “Particles in RSOS Paths.” Journal of Physics A: Mathematical and Theoretical 42, no. 12 (March 27, 2009): 122001. doi:10.1088/1751-8113/42/12/122001.
• Mathieu, Pierre. “Paths and Partitions: Combinatorial Descriptions of the Parafermionic States.” Journal of Mathematical Physics 50, no. 9 (September 1, 2009): 095210. doi:10.1063/1.3157921.
• Konno, Hitoshi. “An Elliptic Algebra and the Fusion RSOS Model.” Communications in Mathematical Physics 195, no. 2 (July 1, 1998): 373–403. doi:10.1007/s002200050394.
• Pearce, Paul A., and Bernard Nienhuis. 1998. “Scaling Limit of RSOS Lattice Models and TBA Equations.” Nuclear Physics B 519 (3) (May 25): 579–596. doi:10.1016/S0550-3213(98)00134-5.
• Gepner, Doron. “Lattice Models and Generalized Rogers Ramanujan Identities.” Physics Letters B 348, no. 3–4 (April 1995): 377–85. doi:10.1016/0370-2693(95)00173-I.
• Zhou, Yu-Kui. 1995. “Further Solutions of Critical ABF RSOS Models.” Journal of Physics A: Mathematical and General 28 (15) (August 7): 4339. doi:10.1088/0305-4470/28/15/014.
• Wu, F. Y. 1992. “Knot Theory and Statistical Mechanics.” Reviews of Modern Physics 64 (4) (October 1): 1099–1131. doi:10.1103/RevModPhys.64.1099.
• Klümper, Andreas, and Paul A. Pearce. 1992. “Conformal Weights of RSOS Lattice Models and Their Fusion Hierarchies.” Physica A: Statistical Mechanics and Its Applications 183 (3) (May 1): 304–350. doi:10.1016/0378-4371(92)90149-K.
• Kuniba, Atsuo, and Tomoki Nakanishi. 1992. “Fusion Rsos Models and Rational Coset Models.” In Quantum Groups, edited by Petr P. Kulish, 303–311. Lecture Notes in Mathematics 1510. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0101196.
• 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.
• E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado Exactly solvable SOS models. Local height probabilities and theta function identities, Nucl. Phys. B 290 (1987), p. 231.
• Huse, David A. 1984. “Exact Exponents for Infinitely Many New Multicritical Points.” Physical Review B 30 (7) (October 1): 3908–3915. doi:10.1103/PhysRevB.30.3908.
• Andrews, George E., R. J. Baxter, and P. J. Forrester. 1984. “Eight-vertex SOS Model and Generalized Rogers-Ramanujan-type Identities.” Journal of Statistical Physics 35 (3-4) (May 1): 193–266. doi:10.1007/BF01014383. http://www.springerlink.com/content/r522x4086p54u438/