Abstract
An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting a regular square lattice, has been developed to investigate the thermodynamics and phase transitions on surface by the methodology of recursive lattice, which we have previously applied to study polymers near a surface. The model retains the advantages of simple formulation and exact calculation of the conventional Bethe-like lattices. An antiferromagnetic Ising model is solved on the surface of this lattice to evaluate thermal properties such as free energy, energy density and entropy, from which we have successfully identified a first-order order–disorder transition other than the spontaneous magnetization, and a secondary transition on the supercooled state indicated by the Kauzmann paradox.
Highlights
The recursive lattice has been a classical methodology in statistical physics for several decades since its invention by Bethe [1] and Husimi [2]
Since this zigzag surface recursive lattice (ZSRL) has coordination of four inside the bulk and alternating two or four on the surface, we hope that it is a good approximation to the regular square lattice with a diagonal boundary
The zigzag structure is taken as a surface assembled by triangle units, and halved Husimi trees are hung on the triangle units to represent the bulk portions
Summary
The recursive lattice has been a classical methodology in statistical physics for several decades since its invention by Bethe [1] and Husimi [2]. We extend to a one-/two-dimensional hybrid recursive lattice to study the phase transitions of Ising model on surface/interface, which is still based on a symmetrical structure [10]. By fractalizing the analogue of a diagonally cut regular square lattice, we can obtain a simple model with partial Husimi trees hung on a zigzag surface. Since this zigzag structure is infinite and homogeneous, it can be handled by recursive calculation [11,12], with approximating partial Husimi trees to be constant statistical contributions, which was derived from previous classical works, the exact calculation of this model appears to be feasible, like other Bethe-like models. An antiferromagnetic Ising model has been solved on the lattice, and we can locate the first-order phase transition and the Kauzmann paradox in the +1 spin system
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