Suspended graphene sheets exhibit correlated random deformations that can be studied under the framework of rough surfaces with a Hurst (roughness) exponent 0.72 ± 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 SchrammLoewner evolution (SLE κ ) curves with κ = 2.24 ± 0.07. 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 isoheight lines of rough surfaces.Rough surfaces are very common in nature and can be found, for instance, in real landscapes 1 and growth surface processes 2 . In many cases, they can be characterised by a Hurst (roughness) exponent that describes the height-height correlations of the surface, and consequently, being called self-affine. Random Gaussian surfaces (RGS) with positive Hurst exponents are examples of rough self-affine surfaces, and their use has become very popular since they are analytically tractable. Recently, it was suggested that iso-height lines in this type of RGS are not conformally invariant 3 , since their statistics is not compatible with the Schramm-Loewner Evolution (SLE) theory 4,4-8 (random curves satisfying SLE statistics are necessarily conformally invariant). However, the fact that iso-height lines of other self-affine rough surfaces, e.g. some grown surfaces 9,10 , follow SLE theory opens the question on which are the main properties that are responsible for conformal invariance.Graphene, consisting of literally a single carbon monolayer, represents the first instance of a truly two-dimensional material (see Fig. 1) [11][12][13] . It owes its stability to the anharmonic coupling between bending and stretching modes, and the resulting deformation in the third dimension 14 . The study of these deformations, often called ripples, is very important because they affect the electronic and mechanical properties. For instance, it has been shown that reducing height fluctuations in graphene samples increases their electronic mobility and electrical conductivity 15 . Since previous numerical studies have shown that the structure of graphene possesses self-affine properties 16 , in this report, we show that iso-height lines, extracted at the percolation threshold, have a well-defined fractal dimension and are conformally invariant (note that to obtain the percolation threshold one starts from the highest point of the membrane and systematically lowers down the height until a cluster, formed by carbon atoms bounded by covalen...

Through molecular mechanics we find that non-covalent interactions modify the fractality of crumpled damaged graphene. Pristine graphene membranes are damaged by adding random vacancies and carbon-hydrogen bonds. Crumpled membranes exhibit a fractal dimension of 2.71 ± 0.02 when all interactions between carbon atoms are considered, and 2.30 ± 0.05 when non-covalent interactions are suppressed. The transition between these two values, obtained by switching on/off the non-covalent interactions of equilibrium configurations, is shown to be reversible and independent on thermalisation. In order to explain this transition, we propose a theoretical model that is compatible with our numerical findings. Finally, we also compare damaged graphene membranes with other crumpled structures, as for instance polymerised membranes and paper sheets, that share similar scaling properties.

We give a partially alternate proof of the reality of the spectrum of the imaginary cubic oscillator in quantum mechanics.

We introduce a different perspective describing electron-phonon interactions in graphene based on curved space hydrodynamics. Interactions of phonons with charge carriers increase the electrical resistivity of the material. Our approach captures the lattice vibrations as curvature changes in the space through which electrons move following hydrodynamic equations. In this picture, inertial corrections to the electronic flow arise naturally effectively producing electron-phonon interactions. The strength of the interaction is controlled by a coupling constant, which is temperature independent. We apply this model to graphene and recover satisfactorily the linear scaling law for the resistivity that is expected at high temperatures. Our findings open up a new perspective of treating electron-phonon interactions in graphene, and also in other materials where electrons can be described by the Fermi liquid theory.

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