Hyperuniform isotope engineering of phonon transport: exact scaling laws and a stealthy transparency window in hexagonal boron nitride

Isotopic substitution is one of the few strictly mass-like perturbations available in a crystal: within the Born–Oppenheimer approximation the interatomic force constants are invariant under isotope exchange, so an isotope pattern modifies the dynamical matrix through the mass factors alone. This makes the spatial arrangement of isotopes a design variable independent of the isotopic variance g_2 that has been the sole focus of isotope engineering to date. We generalize the Tamura theory of phonon–isotope scattering to spatially correlated arrangements by introducing the mass-fluctuation structure factor S_\epsilon(k)=g_2 s(k), and show that when s(k) (k/k0)^alpha at small wavenumber the relaxation rate obeys the exact scaling law tau_iso^-1 omega^ d+1+alpha in d dimensions, with closed-form angular prefactor 2/(alpha+2). The hyperuniformity exponent therefore contributes an additive, dimension-independent correction to the Rayleigh exponent, orthogonal to the dimensional baseline. We realize the class II case alpha=1 using sequences with Gaussian-unitary-ensemble spacing statistics—of which the unfolded nontrivial zeros of the Riemann zeta function are the canonical example—mapped onto a one-dimensional 10B/11B modulation of hexagonal boron nitride, obtaining tau_iso^-1 omega^5 in place of the Rayleigh omega^4. We further find numerically that sequences built from zeros at finite height are more rigid than GUE, giving alpha=1.49+/-0.12, and that this exponent is independent of the height of the zeros used: across eight disjoint windows spanning gamma=1.2x10^3 to 6.2x10^7 the fitted trend is -0.138+/-0.075, consistent with zero, so the deeper suppression is available without any tuning. For stealthy (class I) arrangements we prove that the golden-rule rate vanishes identically, not merely asymptotically, for all omega<vK/2, where K is the stealthiness wavenumber. We propose a differential transmission observable Delta-T(omega) between concentration-matched samples that cancels the Casimir and anharmonic backgrounds to first order, identify a feasible observation window of 4–7 THz at liquid-helium temperature, and state the resulting fabrication specification — 1.25 nm modulation pitch over 300 μm — as an explicit design target that exceeds present growth capability.