Surface Critical Exponents of the 3D Clean Ising Model and 3D Random Field Ising Model

At continuous phase transitions, power law behavior can give rise to intricate pattern formation reminiscent of the complex patterns that form at the surface of many strongly correlated quantum materials. Such patterns in quantum materials have been shown to exhibit fractal behavior and have scaling laws that are characterized by critical exponents.[1-3] Here, we use simulations to calculate the critical exponents at a free surface of the 3-dimensional Clean Ising Model and of the 3-dimensional Random Field Ising Model. We discuss the difference between the surface patterns and those arising in the bulk. The critical exponents extracted from surface probe data in experimental systems can be compared against these theoretical results, in order to determine whether patterns observed on the surface of a material are confined to that surface or penetrate into the bulk of the material.

[1] Phillabaum et al., Nature Commun. 3, 915 (2012).
[2] Li et al., Nature Commun. 10, 4568 (2019).
[3] Liu et al., Phys. Rev. Lett. 116, 036401 (2016).