The vibrational properties of 2D materials are nontrivial due to prominence of both in-plane (IP) and out-of-plane (OP) modes. It has been suggested in 1987 already [1] that 2D polymerized membranes (such as 2D solids) exhibit nontrivial dynamical scaling due to nonlinear coupling between the IP and OP modes, characterized by scaling exponents [2] that are known numerically only [3,4]. In this talk, I will discuss recent results for the static and dynamical roughening of graphene [4] based on the phase-field-crystal (PFC) graphene model [5] and machine-learned molecular dynamics with the neuroevolution potential method [6]. I will also discuss the buckling transition in strained graphene [6] and the roughening in such systems [7]. Finally, I will discuss the recent argument that with rotational invariance the IP-OP coupling is lifted, and roughening becomes that of a harmonic vibrational model [8].


1. D.R. Nelson and L. Peliti, J. Physiq. 48, 1085 (1987)
2. J.A. Aronovitz and T.C. Lubensky, PRL 60, 2634 (1988)
3. K.-I. Mizuochi, H. Nakanishi, and T. Sakaue, EPL 107, 38003 (2014)
4. Enzo Granato, Michelle Greb, K. R. Elder, S. C. Ying , and T. A-N., Phys. Rev. B 105, L201409 (2022)
5. K. R. Elder, C. V. Achim, V. Heinonen, E. Granato, S. C. Ying, and T. A-N., Phys. Rev. Materials 5, 034004 (2021)
6. Z. Fan, Z. Zeng, C. Zhang, Y. Wang, K. Song, H. Dong, Y. Chen, and T. A-N., Phys. Rev. B 104, 104309 (2021)
Enzo Granato, Ken R. Elder, See- Chen Ying, and T.A-N., Phys. Rev. B 107, 035428 (2023)
7. E. Granato, K.R. Elder, S.C. Ying, and T.A-N., Phys. Rev. B 111, 014102 (2025).
8. U. Aseginolaza et al., Nature Phys. 20, 1288 (2024).