Critical Ripples and Dirac Fermions in Crystalline Membranes

Abstract

Crystalline membranes hosting Dirac fermions, with graphene as the paradigmatic example, combine two low-energy sectors with sharply different dynamics: nonrelativistic flexural phonons and relativistic-like Dirac quasiparticles. We develop the low-energy field theory of this coupled system at charge neutrality and determine how this dynamical mismatch controls the coupling between the two sectors. In the long-wavelength flat phase, rotational symmetry ties the renormalization of the leading local scalar strain--density coupling to the scale-dependent bending rigidity, causing its dimensionless strength to decrease logarithmically. At the same time, flexural modes become parametrically slower than the Dirac fermions, so the resulting fermionic feedback vanishes as a power law.The flat phase is therefore stable against this perturbation. The physics changes when elastic interactions or electronic softening destabilize the membrane at a finite wavelength, selecting a ripple pattern formed by modes at $\pm\mathbf{Q}$. For an isolated pair of ordering wavevectors, provided that commensurability-induced phase pinning is irrelevant, the transition is governed by the bosonic Wilson--Fisher fixed point, while the Dirac fermions remain spectators. A genuinely hybrid electronic--structural critical point arises instead when symmetry permits a mass-type Dirac bilinear to share the ripple's momentum and quantum numbers, including horizontal-reflection parity. The transition is then described by the chiral-XY Gross--Neveu--Yukawa universality class. Using the known one-loop critical exponents, we characterize this transition, determine the induced secondary elastic distortion, and show that the fermionic and bosonic velocities lock in the isotropic continuum limit.