Thermodynamic Limit for the Ising Model on the Cayley Tree

TL;DR

This paper demonstrates that finite-sized Ising models on the Cayley tree exhibit magnetic order over a wide temperature range when symmetry is broken, contrasting with the absence of spontaneous magnetization in the thermodynamic limit.

Contribution

It reveals that finite Cayley tree Ising systems can maintain magnetic order at nonzero temperatures when a single spin is fixed, highlighting a key difference from the infinite limit.

Findings

Finite Cayley tree systems show magnetic order at nonzero temperatures when symmetry is broken.

The behavior of the finite Cayley tree model is analogous to the Sierpinski Gasket.

Comparison with 1D chain emphasizes the unique ordering properties of the Cayley tree.

Abstract

While the Ising model on the Cayley tree has no spontaneous magnetization at nonzero temperatures in the thermodynamic limit, we show that finite systems of astronomical sizes remain magnetically ordered in a wide temperature range, if the symmetry is broken by fixing an arbitrary single (bulk or surface) spin. We compare the behavior of the finite size magnetization of this model with that of the Ising model on both the Sierpinski Gasket, and the one-dimensional linear chain. This comparison reveals the analogy of the behavior of the present model with the Sierpinski Gasket case.