Enhanced Long-Wavelength Fluctuations and Interaction-Stress Correlations in Active Crystals

Activity can enhance the long-wavelength density fluctuations of a solid beyond the equilibrium fluctuations predicted by the Mermin-Wagner-Hohenberg theorem, but the microscopic origin of this enhancement remains unclear. We study a two-dimensional active Brownian crystal whose longitudinal and transverse displacement covariances scale as $C_{\lambda }^u(q)\sim q^{-3}$ as compared to the passive counterpart with $q^{-2}$ law. This spectrum predicts an MSD-plateau divergence proportional to $L$ in two dimensions, logarithmic in three dimensions, and finite in four dimensions, in close correspondence with recent results reported in Dey et al. [Nat. Commun. 16, 5498 (2025)]. Inverse of the Covariance gives $H_{\lambda }^{\mathrm{c}\mathrm{o}\mathrm{v}}\sim \mathrm{\mid }q{\mathrm{\mid }}^3$ with the corresponding dispersion ${\omega }_{\mathrm{c}\mathrm{o}\mathrm{v},\lambda }\sim q^{3\mathrm{/}2}$, whereas direct mechanical response gives $K_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p},\lambda }\sim q^2$ and acoustic modes with ${\omega }_{0,\lambda }\sim q$. The nonlinear dispersion is therefore covariance-defined rather than mechanical. The mobility and integrated active-force correlations remain non-singular, although a finite-persistence crossover cannot be ruled out completely with the existing data. Microscopically, the longitudinal and transverse Irving-Kirkwood interaction-stress spectra scale as $q^{-1}$. The transverse stress-stress correlation connects well with the anomalous displacement field, indicating that activity anomalously populates ordinary acoustic modes, while the conservative interaction network transmits the resulting fluctuations as long-ranged stress. A three-dimensional active FCC crystal exhibits the same paired low-$q$ trends.