Anderson localization of phonons in dimension $d=1,2,3$ : finite-size properties of the Inverse Participation Ratios of eigenstates

We study by exact diagonalization the localization properties of phonons in mass-disordered harmonic crystals of dimension $d=1,2,3$. We focus on the behavior of the typical Inverse Participation Ratio $Y_2(\omega ,L)$ as a function of the frequency $\omega$ and of the linear length $L$ of the disordered samples. In dimensions $d=1$ and $d=2$, we find that the low-frequency part $\omega \to 0$ of the spectrum satisfies the following finite-size scaling $L{Y}_2(\omega ,L)=F_{d=1}(L^{1\mathrm{/}2}\omega )$ in dimension $d=1$ and $L^2{Y}_2(\omega ,L)=F_{d=2}((\ln L{)}^{1\mathrm{/}2}\omega )$ in dimension $d=2$, with the following conclusions (i) an eigenstate of any fixed frequency $\omega$ becomes localized in the limit $L\to +\mathrm{\infty }$ (ii) a given disordered sample of size $L^d$ contains a number $N_{deloc}(L)$ of delocalized states growing as $N_{deloc}(L)\sim L^{1\mathrm{/}2}$ in $d=1$ and as $N_{deloc}(L)\sim L^2/(\ln L)$in$d=2$\mathrm{.}Indimension$d=3$, we find a localization-delocalization transition at some finite critical frequency ${\omega }_c(W)>0$ (that depends on the disorder strength $W$). Our data are compatible with the finite-size scaling $L^{D(2)} Y_2(\omega,L)=F_{d=3}(L^{1/\nu} (\omega-\omega_c))$withthevalues$D(2) \simeq 1.3$and$\nu \simeq 1.57$ corresponding to the universality class of the localization transition for the Anderson tight-binding electronic model in dimension $d=3$.