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 ±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.

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 ±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.

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 ±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.

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 ±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.