Eight-vertex model and quantum XYZ model

introduction

• This model includes the dimer, ice and zero-field Ising, F and KDP models as special cases.
• Sutherland 1970 showed that any eight-vertex model transfer matrix commutes with an XYZ Hamiltonian
• Baxter considered (unaware of Sutherland 1970) the problem of the commutation of two eight-vertex transfer matrices

Hamiltonian

• XYZ Hamiltonian

\[\hat H = -\frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}^z - h\sigma_j^{z})\]

related items

• Heisenberg spin chain model
• six-vertex model and Quantum XXZ Hamiltonian
• Yang-Baxter equation
• Dimer model

articles

• Hjalmar Rosengren, Elliptic pfaffians and solvable lattice models, arXiv:1605.02915 [math-ph], May 10 2016, http://arxiv.org/abs/1605.02915
• Moll, Alexander. “Random Partitions and the Quantum Benjamin-Ono Hierarchy.” arXiv:1508.03063 [math-Ph, Physics:nlin], August 12, 2015. http://arxiv.org/abs/1508.03063.
• Rosengren, Hjalmar. “Special Polynomials Related to the Supersymmetric Eight-Vertex Model: A Summary.” arXiv:1503.02833 [math-Ph, Physics:nlin], March 10, 2015. http://arxiv.org/abs/1503.02833.
• Kuniba, Atsuo, Yasuhiro Akutsu, and Miki Wadati. 1986. “Virasoro Algebra, von Neumann Algebra and Critical Eight-Vertex SOS Models.” Journal of the Physical Society of Japan 55 (10): 3285–3288. doi:10.1143/JPSJ.55.3285.
• 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/
• Partition Function of the Eight-Vertex Lattice Model
  • Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972
• http://dx.doi.org/10.1063/1.1665111
  • B. Sutherland, J. Math. Phys. 11:3183-3186 (1970)