Exact Solutions on the Ground States of Ising Models in Magnetic Fields with Frustration on a Diamond Hierarchical Lattice

Abstract

Magnetization processes of Ising models with frustration on diamond hierarchical lattices, which contain vertices with high coordination numbers, are exactly obtained at zero temperature. In antiferromagnetic systems, the magnetization cannot saturate under finite magnetic fields owing to the competition between the antiferromagnetic and Zeeman interactions and the intrinsic long-range nature of hierarchical lattices. For the zero-field classical spin-liquid phase found in [Kobayashi et al., J. Phys. Soc. Jpn. 78, 074004 (2009)], an infinitely small applied magnetic field can induce an infinitely small magnetization, despite Ising models that have discrete energy levels. By examining the structure of the partition function, we obtain the ground state spinconfigurations and clarify the mechanism of the "gapless like behavior".

Explicit formulas for heat kernels on diamond fractals

“…For instance it is known that the Ising model is solvable on them [51]. Recent investigations in the context of random polymers, Ising and Potts models can be found in [4,29,34].…”