Conformal Invariance of Graphene Sheets

TL;DR

This paper demonstrates that iso-height lines at the percolation threshold of suspended graphene sheets are conformally invariant and share properties with SLE curves, revealing a new universality class in critical phenomena.

Contribution

It shows that graphene sheets' iso-height lines are conformally invariant and belong to a new universality class, linking graphene physics with critical phenomena theory.

Findings

Iso-height lines at the percolation threshold are conformally invariant.

These lines share statistical properties with SLE$_{\

Abstract

Suspended graphene sheets exhibit correlated random deformations that can be studied under the framework of rough surfaces with a Hurst (roughness) exponent $0.72 \pm 0.01$. Here, we show that, independent of the temperature, the iso-height lines at the percolation threshold have a well-defined fractal dimension and are conformally invariant, sharing the same statistical properties as Schramm-Loewner evolution (SLE$_{\kappa}$) curves with $\kappa=2.24\pm0.07$. Interestingly, iso-height lines of other rough surfaces are not necessarily conformally invariant even if they have the same Hurst exponent, e.g. random Gaussian surfaces. We have found that the distribution of the modulus of the Fourier coefficients plays an important role on this property. Our results not only introduce a new universality class and place the study of suspended graphene membranes within the theory of critical phenomena, but also provide hints on the long-standing question about the origin of conformal invariance in iso-height lines of rough surfaces.