Exact solution in an external magnetic field of Ising models with three-spin interactions

Abstract

The three&pin interaction Hamiltonian X = --J3Z«»& saba&a» -HZt s; -H'Z& 0& (the s and 0 spins belong to different sublattices) is solved for some two4imensional lattices by a generalized star-triangle transformation. The H' = 0 internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g.,x) are studied. Although the actual critical-point exponents remain the same as for systems with two&pin interaction, the three&pin interaction gives rise to different amplitude and background terms. In particular, the H' = 0 magnetization is nonzero for all finite T, because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singularity, with critical exponent 1af.

Generalized algebraic transformations and exactly solvable classical-quantum models

“…It is therefore valuable to mention few rigorously solved Ising models of this type. The spin-1/2 Ising model with only the triplet interaction is exactly tractable on the triangular lattice (the so-called Baxter-Wu model) [15,16,17,18,19], the union jack lattice [20,21,22] and the diced lattice [23,24]. Among the more general exactly solved Ising models one could also mention the spin-1/2 Ising model on the diced [25] and union jack [26] lattices with the pair and triplet interactions, the spin-1/2 Ising model on the union jack lattice with the triplet interaction and the external magnetic field [27,28,29], as well as, the spin-1/2 Ising model on the kagomé lattice [30] and Cayley tree [31] including the external magnetic field, pair as well as triplet interactions.…”

Spontaneous magnetization of the three-spin Ising model on the Union Jack lattice

“…Considerable recent progress has been made toward the exact solution of the Ising model with pure three-spin interactions. The free energy is now known in closed form for the isotropic triangular lattice (Baxter and Wu 1973, 1974, Baxter 1974, the Union Jack lattice (Hintermann and Merlini 1972) and the dice lattice (Liu and Stanley 1974). For the triangular and the Union Jack lattices, a non-zero spontaneous magnetization exists below the critical point.…”

Strong- and Weak-Universal Critical Behaviour of a Mixed-Spin Ising Model with Triplet Interactions on the Union Jack (Centered Square) Lattice

“…The spin-1/2 Ising model with a three-spin (triplet) interaction on planar lattices belongs to paradigmatic exactly solved models of this type. As a matter of fact, the exact solutions for the spin-1/2 Ising model with the triplet interaction gave rigorous proof for different sets of critical exponents on different planar lattices [2][3][4][5][6][7]. More specifically, the critical exponent α for the specific heat fundamentally differs when this model is defined on centered square lattice (α = 1/2) [2], triangular lattice (α = 2/3) [3][4][5], decorated triangular [6], honeycomb and diced lattices [7] (α ≈ 0, logarithmic singularity).…”

“…As a matter of fact, the exact solutions for the spin-1/2 Ising model with the triplet interaction gave rigorous proof for different sets of critical exponents on different planar lattices [ 2 , 3 , 4 , 5 , 6 , 7 ]. More specifically, the critical exponent for the specific heat fundamentally differs when this model is defined on centered square lattice ( ) [ 2 ], triangular lattice ( ) [ 3 , 4 , 5 ], decorated triangular [ 6 ], honeycomb and diced lattices [ 7 ] ( , logarithmic singularity). In addition, the spin-1/2 Ising model with the triplet interaction on a kagomé lattice [ 8 ] does not display a phase transition at all.…”