Resolution of the Nematic-to-Smectic A Universality Problem: A Computational Study of the Quasiperiodic Metal–Insulator Mapping

The nematic-to-smectic A (N-SmA) phase transition exhibits continuously varying critical exponents depending on molecular structure, a phenomenon that has remained unexplained for over four decades despite extensive experimental and theoretical effort. Here we show computationally that the N-SmA transition maps exactly onto the Aubry–André–Harper (AAH) metal–insulator transition at the self-dual critical point V = 2J, using only the de Gennes free energy, lattice discretization, and the generic incommensurability of smectic layer spacing and molecular length. At this critical point, the energy spectrum is a Cantor set with Hausdorff dimension Ds = 1/2, yielding the correlation length exponent ν = 2/3 and continuously varying heat capacity exponent α from 0 (3D-XY) to 2/3 (fully decoupled layers). Setting the quasiperiodic frequency to 1/ϕ (golden ratio, justified by maximal incommensurability) produces a zero-free-parameter formula α(r) = (2/3)((r−rc)/(1−rc))4 with rc = 1−1/ϕ4 =0.8541. This formula fits 11 experimental compounds spanning 40 years of published calorimetry with RMS = 0.033, reduced χ2 = 0.47, and all points within 2σ.