Possible new phase transition in the 3D Ising Model associated with boundary percolation

TL;DR

This paper proposes a new boundary percolation transition in the 3D Ising model, linking it to a phase transition in the dual gauge theory with unique critical behavior.

Contribution

It introduces boundary percolation as a relevant phenomenon in the 3D Ising model and connects it to a spin-glass transition in the dual gauge theory, revealing novel critical exponents.

Findings

Boundary percolation occurs at about 13% minority spins.

A phase transition in the dual gauge theory matches the boundary percolation transition.

Critical exponent ν ≈ 1.3 indicates a weak specific heat singularity.

Abstract

In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13\% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent $\nu \sim 1.3$ is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent $\alpha \sim -1.9$. The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different $\nu$ exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there are two different correlation lengths present.