27aPS-3 Exact results of an Ising model with odd-spin interactions on a square lattice

Silvia Lacková, Tsuyoshi Horiguchi

Tomoya Horiguchi

Tomoya Horiguchi

Tohru Morita

The diagrammatical approach to the exact solution of two-dimensional Ising model, developed for the system with spin 1/2 and 1, is extended to be applicable to the Ising model of spin S greater than 1/2. The exact solution for the system on the square lattice and other lattices with non-crossing interactions is obtained only when the spin is half an odd integer, and the interaction between two spins s j and s k is expressed as - J j k φ j ( s j )φ k ( s k ) in terms of one paramater J j k and odd functions φ j ( s ) and φ k ( s ) of s satisfying |φ j ( s )|=|φ k ( s )|=1 for s =- S , - S +1,· ··, and S . The exact solution on the honeycomb lattice is obtained for all S , for the cases when the interactions divided by the temperature are expressed in terms of S +1/2 or S parameters.

Jozef Strečka

The mixed spin-1/2 and spin-S Ising model on the Union Jack (centered square) lattice with four different three-spin (triplet) interactions and the uniaxial single-ion anisotropy is exactly solved by establishing a rigorous mapping equivalence with the corresponding zero-field (symmetric) eight-vertex model on a dual square lattice. A rigorous proof of the aforementioned exact mapping equivalence is provided by two independent approaches exploiting either a graph-theoretical or spin representation of the zero-field eight-vertex model. An influence of the interaction anisotropy as well as the uniaxial single-ion anisotropy on phase transitions and critical phenomena is examined in particular. It is shown that the considered model exhibits a strong-universal critical behaviour with constant critical exponents when considering the isotropic model with four equal triplet interactions or the anisotropic model with one triplet interaction differing from the other three. The anisotropic models with two different triplet interactions, which are pairwise equal to each other, contrarily exhibit a weak-universal critical behaviour with critical exponents continuously varying with a relative strength of the triplet interactions as well as the uniaxial single-ion anisotropy. It is evidenced that the variations of critical exponents of the mixed-spin Ising models with the integer-valued spins S differ basically from their counterparts with the half-odd-integer spins S.

Jeremiah H. Barry, T. Tanaka, Mahfuza Khatun +1 more

Simple procedures are first used to obtain exact solutions of highly-localized odd-number Ising correlations on the kagom\'e, square, and honeycomb lattices. To extend these results, a systematic and unifying method is then developed and demonstrated for finding exact solutions of n-site (with n an odd integer) Ising correlations on various planar lattices. The method combines five transformation or mapping theorems and linear-algebraic correlation identities of the triangular Ising model supplemented by a foreknowledge of its spontaneous magnetization and three select triplet correlations. In particular, considering a select seven-site cluster of the triangular Ising model, the knowledge of all its eleven odd-number correlations defined upon this cluster is shown sufficient for determining exactly all honeycomb, decorated-honeycomb, and kagom\'e Ising odd-number correlations upon their correspondingly select 10-, 19-, and 9-site clusters, respectively. The direct applicability of the catenated mapping theorems and relative ease of the calculational procedures are highlighted by the resulting large numbers of multisite correlation solutions (e.g., approximately 80 and 50 for the honeycomb and kagom\'e Ising models, respectively), the large ${\mathit{n}}_{\mathrm{max}}$ values (${\mathit{n}}_{\mathrm{max}}$=9, 9, and 19, respectively, for the honeycomb, kagom\'e, and decorated-honeycomb Ising models), and convenient prescriptions for extracting critical amplitudes. The results also offer examples of correlation degeneracies and other linear-algebraic correlation identities that do not depend explicitly upon the interaction parameters.