c=-2 conformal field theory in quadratic band touching system

Quadratic band touching with dispersion ε∝k² appears in various condensed matter systems, most notably in AB-stacked bilayer graphene. This non-relativistic dispersion with dynamical exponent z=2 stands in contrast to linear Dirac cones, placing these systems outside relativistic quantum field theory where conformal invariance typically emerges. Nevertheless, I demonstrate that a free fermion model with singular quadratic band touching constitute a conformal quantum critical point with full conformal invariance in ground-state correlations. I analyze both continuum and lattice models and show that the ground-state correlations match those of symplectic fermions, which is a logarithmic conformal field theory with c=-2. I establish correspondence between physical fermion operator ψ_i and weight-zero field θ of symplectic fermion as ψ_i ∝ ∂_iθ. I also uncover exotic anyonic excitations despite the gapless spectrum. These excitations cannot exist individually and exhibit logarithmic spatial correlations. On topologically nontrivial manifolds, moving these excitations along non-contractible loops induces transitions between degenerate ground-states. My findings establish quadratic band touching as a novel route to ground-state conformal symmetry with implications for bilayer graphene and related materials.