Ising spins on randomly multi-branched Husimi square lattice: Thermodynamics and phase transition in cross-dimensional range

• The inhomogeneous Husimi lattice with random structure.
• The numerical calculation of the random lattice followed by the exact technique.
• Thermodynamics in cross-dimensional range and the exponential T vs. B transition line.
• Rigorous thermal fluctuations and the entropy singularities in the cross-dimensional range.

An inhomogeneous random recursive lattice is constructed from the multi-branched Husimi square lattice. The number of repeating units connected on one vertex is randomly set to be 2 or 3 with a fixed ratio P2P2 or P3P3 with P2+P3=1P2+P3=1. The lattice is designed to describe complex thermodynamic systems with variable coordinating neighbors, e.g. the asymmetric range around the surface of a bulk system. Classical ferromagnetic spin-1 Ising model is solved on the lattice to achieve an annealed solution via the local exact calculation technique. The model exhibits distinct spontaneous magnetization similar to the deterministic system, with however rigorous thermal fluctuations and significant singularities on the entropy behavior around the critical temperature, indicating a complex superheating frustration in the cross-dimensional range induced by the stochasticity. The critical temperature was found to be exponentially correlated to the structural ratio P with the coefficient fitted as 0.53187, while the ground state energy presents linear correlation to P, implying a well-defined average property according to the structural ratio.