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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 $Y_2(ω,L)$ as a function of the frequency $ω$ and of the linear length $L$ of the disordered samples. In dimensions $d=1$ and $d=2$, we find that the low-frequency part $ω\to 0$ of the spectrum satisfies the following finite-size scaling $L Y_2(ω,L)=F_{d=1}(L^{1/2} ω)$ in dimension $d=1$ and $L^2 Y_2(ω,L)=F_{d=2}((\ln L)^{1/2} ω)$ in dimension $d=2$, with the following conclusions (i) an eigenstate of any fixed frequency $ω$ becomes localized in the limit $L \to +\infty$ (ii) a given disordered sample of size $L^d$ contains a number $N_{deloc}(L)$ of delocalized states growing as $N_{deloc}(L)\sim L^{1/2}$ in $d=1$ and as $N_{deloc}(L)\sim L^2/(\ln L)$ in $d=2$. In dimension $d=3$, we find a localization-delocalization transition at some finite critical frequency $ω_c(W)>0$ (that depends on the disorder strength $W$). Our data are compatible with the finite-size scaling $L^{D(2)} Y_2(ω,L)=F_{d=3}(L^{1/ν} (ω-ω_c))$ with the values $D(2) \simeq 1.3$ and $ν\simeq 1.57$ corresponding to the universality class of the localization transition for the Anderson tight-binding electronic model in dimension $d=3$.