Anderson transition for elastic waves in three dimensions
S. E. Skipetrov Y. M. Beltukov























sergey.skipetrov@lpmmc.cnrs.fr
Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France

ybeltukov@gmail.com
Department of Solid State Physics, Ioffe Institute, 194021 St. Petersburg, Russia





We use two different fully vectorial microscopic models featuring nonresonant and resonant scattering, respectively, to demonstrate the Anderson localization transition for elastic waves in three-dimensional (3D) disordered solids. Critical parameters of the transition determined by finite-time and finite-size scaling analyses suggest that the transition belongs to the 3D orthogonal universality class. Similarities and differences between the elastic-wave and light scattering in strongly disordered media are discussed.


Anderson transition for elastic waves in three dimensions
    Y.M. Beltukov
    December 30, 2023
=========================================================





§ INTRODUCTION



Appearance of localized eigenstates and suppression of quantum or wave transport in a strongly disordered medium is a widespread physical phenomenon discovered by Philip Anderson 60 years ago <cit.>. Of special interest is the localization by three-dimensional (3D) disorder that takes place only when the latter is strong enough <cit.>. A critical energy (or frequency) known as a “mobility edge” separates the range of energies (or frequencies) for which the eigenstates are localized from the rest of the spectrum. The localization transition taking place upon crossing a mobility edge is a subject of intense theoretical studies <cit.>. It has been observed in experiments with electrons in doped semiconductors <cit.>, vibrations in elastic networks <cit.>, and cold atoms in random potentials <cit.>. Despite important experimental efforts, however, Anderson localization of light in 3D has not been unambiguously demonstrated up to now <cit.>. In addition, recent calculations show that the vector nature of light may hamper Anderson localization in a model of point scatterers <cit.>, raising a number of important questions. First, the extent to which the conclusion obtained for point scatterers may apply to disordered media composed of scatterers of finite size, is unclear <cit.>. Second, the relevance of results obtained for light to other types of vector waves remains unexplored. In particular, elastic waves (i.e., vibrations) in solids are directly concerned and one may wonder whether the theoretical results of Refs. skip14,bellando14 may put in question the experimental observations of Refs. hu08,cobus16. Once the localization of elastic waves in 3D is firmly established, it is important to determine the universality class of the corresponding Anderson transition and to calculate its critical exponent which experiments are now attempting to measure <cit.>.

Various models of elastic wave systems have been shown to exhibit Anderson localization in 3D but most of them <cit.> relied on the scalar approximation and are therefore not suitable to discuss the role of the vector character of vibrations. Calculations including the latter are scarce <cit.> and predict critical properties that are different from those of scalar waves <cit.>, in contradiction with general but rather formal theoretical arguments that seem to indicate that the vector character of vibrations is unlikely to modify the universality class of the localization transition <cit.>. Another model that takes into account the vector character of vibrations, is the model of a simple fluid with short-range interactions described by a truncated Lennard-Jones potential <cit.>. It yields the same critical exponent of the localization transition for instantaneous normal modes as scalar vibrational models for both real (stable) and imaginary (unstable) frequencies. On the other hand, the vector and scalar percolation problems (also known as rigidity and connectivity percolation, respectively) belong to different universality classes and exhibit different critical exponents <cit.>.

To clarify the role of the vector character of elastic waves in the context of Anderson localization, in this paper we report results obtained for two different models describing strong scattering of elastic waves in 3D with a full account for their vector nature. In the first model, the scattering is nonresonant and the localization transition takes place upon increasing the frequency of vibrations. This model is a direct extension of the scalar model of Ref. beltukov17 and is similar to those employed to study heat transport by phonons in disordered materials <cit.>. The second model describes resonant scattering inducing Anderson localization in a narrow frequency band separated from the rest of the spectrum by two mobility edges, similarly to the scalar model of Ref. skip16. This is typical for disordered samples used in recent experiments on wave localization in elastic networks <cit.>. The critical properties of the first model are deduced from a finite-time scaling analysis whereas the second model is analyzed using a finite-size scaling technique. We show that in both cases and within numerical errors, the critical exponents of localization transitions coincide with those found for scalar waves. This suggests that the Anderson transition of elastic waves in 3D belongs to the orthogonal universality class.



§ A MODEL WITH NONRESONANT SCATTERING





 §.§ Derivation of the model


A model describing nonresonant scattering of elastic waves can be constructed by considering a cubic lattice of N = L^3 identical unit point masses (or atoms) interacting harmonically. The Cartesian components u_m^α (α = x,y,z) of the vector displacement u⃗_m(t) of an atom m from its equilibrium position obey <cit.>

    ü_m^α(t) = - ∑_n β M_mn^αβu_n^β(t),

where M_mn^αβ are elements of a 3N × 3N dynamical matrix M̂. The latter is composed of N 3 × 3 blocks M̂_mn, describing the coupling between Cartesian components of the displacement of the atom m with those of the displacement of the atom n. The dynamical matrix of a mechanically stable system should be positive semidefinite (i.e. its eigenvalues ω^2 should be nonnegative) and hence it can be represented as M̂ = ÂÂ^T, that is

    M_mn^αβ = ∑_kγ A_mk^αγA_nk^βγ.

Disorder is introduced in the model by assuming that the elements A_mn^αβ of off-diagonal matrix blocks (m  n) describing interactions between nearest-neighbor atoms, are real independent, zero-mean Gaussian variables with variances Ω^2. A_mn^αβ = 0 for atoms m and n that are not nearest neighbors. The diagonal blocks A_mm^αβ are obtained by a sum rule A_mm^αβ = -∑_n m A_nm^αβ, which ensures that the total potential energy is invariant upon translation of the system as a whole.
We will measure the frequency ω in units of Ω and the time t in units of 1/Ω from here on.









Figure <ref> shows the density of states (DOS) of the model defined by Eqs.  (<ref>) and (<ref>). It is simply a probability density of a random variable ω, with ω^2 being the eigenvalues of the random matrix M̂. As we will show in Sec. <ref>, Anderson localization takes place for frequencies ω exceeding a critical frequency (mobility edge) ω_c ≃ 12 shown  by a vertical arrow in Fig. <ref>. The DOS vanishes exponentially beyond some ω_max≃ 15 determined by the statistics of the random matrix Â. In this respect, the model is quite particular because it assumes that the properties of the system are dominated by the disorder and neglects the “regular” part of M̂ that would describe the system in the absence of disorder. As a consequence, the low-frequency behavior of the DOS in Fig. <ref> does not show the shape expected in a homogeneous medium DOS(ω) ∝ω^2 (Debye law) for ω→ 0. This is due to vanishing of the Young modulus and, consequently, vanishing of both the rigidity of the lattice and the zero-frequency sound velocity v in our model <cit.>. Then, the Ioffe-Regel criterion k(ω) ℓ(ω) ≲ 1, where k(ω) is the wavenumber and ℓ(ω) is the scattering mean free path, is obeyed already for ω = 0. The model (<ref>) can be made more realistic by defining M̂ as a sum of ÂÂ^T and an additional matrix M̂_0 corresponding to the lattice without disorder. This introduces a non-zero rigidity of the lattice <cit.> and produces a Debye-like behavior of the low-frequency spectrum in Fig. <ref>: DOS(ω) ∝ω^2, which extends from ω = 0 to some finite ω≃ω_IR. Here ω_IR is the Ioffe-Regel frequency obeying k(ω_IR) ℓ(ω_IR) ≃ 1. ω_IR can be made arbitrary small by decreasing the magnitude of M̂_0 and has nothing to do with the mobility edge ω_c that is hardly affected by the introduction of M̂_0. Therefore, the Ioffe-Regel criterion k(ω) ℓ(ω) ≃ 1 is not a good condition of Anderson localization in the model defined by Eqs.  (<ref>) and (<ref>).
A discrepancy between ω_IR and ω_c for vibrational modes has been also noticed for other models of disordered solids <cit.>.


To study the localization transition, we compute the spreading of an initial excitation of the left half of the system [u̇^α_m(0) are taken to be Gaussian random numbers for x_m<0 and 0 elsewhere; all u^α_m(0) = 0] by extending the numerical approach of Ref. beltukov17 to the vector case. The spreading is quantified by a frequency-resolved penetration depth X(ω, t):

    X^2(ω, t) = 1/ϕ_0(ω)∫_0^∞ x ϕ(ω, x,t) dx,

where



ϕ(ω, x,t) = ∑_m E_m(ω, t) δ(x - x_m)
is the 1D energy density,

    E_m(ω, t)    =   1/2∑_αu̇_m^α(ω, t)^2
    
       +   1/2∑_nαβM_mn^αβu_m^α(ω, t)u_n^β(ω, t)

is the energy of a quasimonochromatic excitation localized on the atom m, and ϕ_0(ω)=2E(ω)/L is the average initial energy density in the left half of the system. The windowed Fourier transform of u_m^α(t) is defined as

    u_m^α(ω, t) = 2∫_-τ^τ u_m^α(t-t') W(t')cos(ω t')dt'

with a window function W(t) = (2πτ)^-1/2cos(π t/2τ). We use L = 200 and average X^2(ω,t ) over 10 realizations of disorder.
This relatively small number of realizations turns out to be sufficient to suppress the statistical fluctuations of X^2(ω,t ) thanks to the additional implicit averaging over the transverse profile of the spreading excitation in the (y, z) plane and over the many atoms that are initially excited in the left half of the system. Indeed, for a 3D system of L × L × L point masses, even a single realization of disorder yields X^2(ω,t ) that is effectively averaged over L^2 different linear chains of effective length √(Dt) in diffuse regime or ξ in the regime of Anderson localization. Here D and ξ are the diffusion coefficient and the localization length, respectiveley. The results would be self-averaging in the limit L  →∞, t →∞ but additional configurational averaging is necessary for finite L and t.






 §.§ Finite-time scaling of the penetration depth



Similarly to the scalar case <cit.>, ⟨ X^2(ω, t) ⟩ grows linearly with time for ω < ω_c ≃ 12 but acquires a tendency to saturation at higher frequencies. This identifies ω_c as a mobility edge, which turns out to be shifted to a higher frequency compared to the scalar model. The critical behavior in the vicinity of ω_c can be studied by analyzing a scaling function F(ω, t ) = ρ(ω)^2/3⟨ X^2(ω, t) ⟩/t^2/3, where ρ(ω) is the density of vibrational states <cit.>. Under the hypothesis of one-parameter scaling, F is assumed to depend on a single relevant variable ψ_1 = [t/ρ(ω)]^1/3ν f_1(w) and possibly on an additional irrelevant variable ψ_2 = [t/ρ(ω)]^y/3 f_2(w), where w = (ω-ω_c)/ω_c, ν is the critical exponent of the localization transition, and y < 0. ln F is expanded in power series as a function of ψ_j up to orders n_j, whereas the auxiliary functions f_j are expanded in powers of w up to orders m_j. These expansions are used to fit the numerical data obtained in the vicinity of ω_c for different frequencies ω and different times t. The range of data used for the fits is restricted to ln F within a range lnF̃_c ±Δ, with lnF̃_c being the approximate crossing point of dependencies ln F(ω) obtained for different t. The quality of our numerical data allows us to use Δ as small as Δ = 0.5 for which fits of acceptable quality are obtained with m_1 = 2, n_1 = 1, and m_2 = n_2 = 0, see Fig. <ref>(a) <cit.>.

To test the stability of the fit presented in Fig. <ref>(a) and obtain a better estimate of the accuracy of the value of the critical exponent following from it, we repeat the fit for data corresponding to long times t ≥ t_min only.
t_min is a free parameter that we vary from the shortest time t = 100 used in our calculations to t = 1600, which is still considerably shorter than the longest time t = 12800 in Fig. <ref>(a). The best-fit ω_c and ν should not depend on t_min, whereas their statistical errors are likely to grow when t_min increases because a longer t_min implies using a smaller fraction of available numerical data (i.e., only the data corresponding to t ≥ t_min). The independence of ν from t_min is confirmed by Fig. <ref>(b) where we show
the best-fit critical exponent ν as a function of t_min. Averaging over t_min yields ⟨ν⟩ = 1.564 ± 0.009, which coincides, within error bars, with ⟨ν⟩ = 1.57 ± 0.02 found in the scalar version of the same model <cit.>.




§ A MODEL WITH RESONANT SCATTERING



The disordered elastic media in which Anderson localization of vibrations has been observed in experiments <cit.>, were composed of individual scattering units having strong scattering resonances (identical aluminum beads brazed together). The applicability of the nonresonant scattering model considered above to such a medium is questionable and one may wonder whether localization transitions in these two systems belong to the same universality class. To answer this question, we consider a model in which scattering is due to N identical point-like resonant scatterers (resonance frequency ω_0, resonance width Γ_0)
embedded in the infinite space filled with a homogeneous and isotropic elastic medium.



 §.§ Derivation of the model


Assume that the scatterers are randomly distributed in a spherical region of space
of radius R and volume V with an average number density ρ = N/V.
Propagation of an elastic wave in the homogeneous space between the scatterers is described by the elastic wave equation <cit.>

    ρ_0 ω^2 u^α(r⃗, ω)    +   ∑_β, γ, ζ∂/∂ r^β[ c_αβγζ∂/∂ r^γ u^ζ(r⃗, ω) ] 
       =    -f^α(r⃗, ω),

where u⃗(r⃗, ω) is the Fourier transform of the displacement field u⃗(r⃗, t), ρ_0 is the mass density of the medium, c_αβγζ = λ_0
δ_αβδ_γζ+
μ_0 (δ_αγδ_βζ + δ_αζδ_βγ)
is its elasticity tensor, λ_0 and μ_0 are Lamé parameters, f⃗(r⃗, ω) is the Fourier transform of the force field f⃗(r⃗, t), and Greek superscripts run over the Cartesian components of the corresponding vectors: α, β, γ, ζ = x, y, z.

Solutions of Eq. (<ref>) can be classified as compressive (longitudinal) and shear (transverse) waves that travel with velocities c_p = [(λ_0 + 2 μ_0)/ρ_0]^1/2 and c_s = (μ_0/ρ_0)^1/2, respectively <cit.>. A 3 × 3 elastic Green's tensor Ĝ describes the response of the medium to a point-like excitation. Its component G^αη (r⃗, r⃗', ω) gives the (Fourier component of) displacement at location r⃗ in direction α due to a unit force f⃗ acting in direction η at a point r⃗'. It obeys an equation following directly from Eq. (<ref>):

    ρ_0 ω^2  G^αη(r⃗,r⃗',ω)    +   ∑_β, γ, ζ∂/∂ r^β[ c_αβγζ∂/∂ r^γ G^ζη(r⃗,r⃗',ω) ]
    
       =    -δ_αηδ(r⃗-r⃗').

The solution of this equation can be written as <cit.>

    Ĝ(r⃗, r⃗',ω)    =   ω/12 πρ_0 c_p^3{e^i k_p Δ r/k_p Δ r[ 1
    - F̂(k_p Δr⃗) ] .
    
       +   . 2 ( c_p/c_s)^3
    e^i k_s Δ r/k_s Δ r[ 1
    + 1/2F̂(k_s Δr⃗) ] },

where k_p,s = ω/c_p,s, Δr⃗ = r⃗-r⃗', and we defined an auxiliary tensor function F̂(v⃗) = [1 - 3 (v⃗⊗v⃗)/v^2] (1 + 3i/v - 3/v^2).

To describe the multiple scattering of elastic waves by an ensemble of N identical resonant point scatterers at positions {r⃗_m } (m = 1, …, N), we generalize the method developed by Foldy <cit.> and Lax <cit.> for scalar waves. The Cartesian components u_m^α (α = x, y, z) of displacements u⃗_m of the scatterers obey

    u_m^α(ω) = v_m^α(ω) + α_0(ω) ∑_n  m, β G^αβ(r⃗_m, r⃗_n,ω) u_n^β(ω),

where v_m^α(ω) is wave field at r⃗_m due to the force field f⃗(r⃗,ω)  in the absence of scatterers (or the “incident wave” in the language of the scattering theory) and the second term on the right-hand side is a sum of wave fields scattered by all other scatterers. The function α_0(ω) (a “polarizability” in the optical language) describes the response of individual scatterers. For an isotropic scatterer with resonance frequency ω_0 and resonance width Γ_0 ≪ω_0, it can be written as

    α_0(ω) = B (-Γ_0/2)/ω - ω_0 + i Γ_0/2

with a prefactor B = 12 πρ_0 c_p^3/ω [1 + 2 (c_p/c_s)^3] following from the optical theorem.
Equation (<ref>) holds only for infinitesimal displacements u⃗_m that do not modify the positions r⃗_m of scatterers.



By grouping all u_m^α in a single vector | u ⟩ = (u_1^x,u_1^y,u_1^z, …, u_N^x,u_N^y,u_N^z)^T (and similarly for v_m^α), we rewrite Eq. (<ref>) in a vector form:

    | u(ω) ⟩ = | v(ω) ⟩ + α_0(ω)/B[ Ĝ(ω) - i 1] | u(ω) ⟩,

where we use the Dirac bra-ket notation. The 3N × 3N “Green's matrix” Ĝ it is composed of N × N blocks Ĝ_mn of size 3 × 3 each, describing coupling between Cartesian components of displacements of scatterers m and n:

    Ĝ_mn(ω)    =   i δ_mn1 + (1 - δ_mn) B  G(r⃗_m, r⃗_n, ω)
    
       =    i δ_mn1 + 1 - δ_mn/1 + 2 (c_p/c_s)^3
       ×   {e^i k_p r_mn/k_p r_mn[ 1
    - F̂(k_p r⃗_mn) ] .
    
       +   . 2 ( c_p/c_s)^3
    e^i k_s r_mn/k_s r_mn[ 1
    + 1/2F̂(k_s r⃗_mn) ] },

where r⃗_mn = r⃗_m - r⃗_n.
The Green's matrix Ĝ is the central object describing all the peculiarities of wave scattering by an ensemble of resonant point scatterers. Equation (<ref>) gives its form for elastic waves; similar expressions for scalar and electromagnetic waves have been derived and analyzed previously <cit.>. We notice that Eq. (<ref>) depends on very few parameters: the frequency ω and the speeds of compressive and shear waves c_p and c_s, respectively.

In general, Eq. (<ref>) is difficult to analyze, but we can simplify it by making use of the strongly resonant nature of scattering. Indeed, the narrow resonance condition Γ_0 ≪ω_0 implies that all the interesting phenomena due to strong scattering can take place only for ω≃ω_0, for which the scattering by individual scatterers is strong. This allows us to replace Ĝ(ω) in Eq. (<ref>) by Ĝ(ω_0). Such an approximation is equivalent to neglecting the times R/c_p,s  that compressive and share waves need to propagate across the disordered system and will allow us to describe slow processes taking place on large time scales τ≫ R/c_p,s, which is the case for the slow decay of localized quasi-modes that we are intended to study.

The matrix Ĝ(ω_0) is a non-Hermitian matrix with complex eigenvalues Λ_n and right and left eigenvectors | R_n ⟩ and ⟨ L_n| obeying

    Ĝ(ω_0) | R_n ⟩ = Λ_n | R_n ⟩, 
          ⟨ L_n | Ĝ(ω_0) = ⟨ L_n | Λ_n, 
          ⟨ L_m | R_n ⟩ = δ_mn.

| R_n ⟩ and ⟨ L_n| form a biorthogonal basis in which any solution | u ⟩ of Eq. (<ref>) with Ĝ(ω) replaced by Ĝ(ω_0) can be represented as

    | u(ω) ⟩ = ∑_n A_n(ω) | R_n ⟩,

where the coefficients A_n are found by substituting the expansion (<ref>) in Eq. (<ref>), applying Eq. (<ref>), multiplying both sides of the resulting equation by ⟨ L_m | from the left, and applying Eq. (<ref>):

    A_n(ω) = (-Γ_0/2)B/α_0(ω) ⟨ L_n | v(ω) ⟩/ω - ω_n + i Γ_n/2

with ω_n = ω_0 - (Γ_0/2) ReΛ_n and Γ_n = Γ_0 ImΛ_n.


The physical meaning of the eigenvectors and eigenvalues of the matrix Ĝ(ω_0) now becomes clear. For a short-pulse excitation | v(t) ⟩∝δ(t), the Fourier transform of | u(ω) ⟩ is

    | u(t) ⟩ = ∑_n A_n(t) | R_n ⟩

with A_n(t) ∝exp(-i ω_n t - Γ_n t/2).
Thus, the eigenvectors | R_n ⟩ correspond to “quasi-modes” of our wave system, where the prefix “quasi-” refers to the fact that, in contrast to the modes of a closed system described by a Hermitian Hamiltonian, the quasi-modes decay in time with decay rates Γ_n. The latter are due to the openness of our wave system: the waves can freely escape from the region of space V occupied by the scatterers, causing the leakage of wave energy and decay of any excitation that was initially created inside V. The concept of quasi-modes is particularly useful and physically meaningful when Γ_n ≪ω_n, which will be the case in the following. Because the eigenvectors and the eigenvalues of the matrix Ĝ(ω_0) correspond to the quasi-modes and their complex frequencies, Ĝ(ω_0) plays the role of an effective non-Hermitian Hamiltonian for elastic waves in the considered system of resonant point scatterers.




 §.§ Finite-size scaling of the distribution of Thouless conductance



Similarly to what has been done for scalar waves <cit.> and light <cit.>, the localization transition for elastic waves in an ensemble of resonant point scatterers can be studied by analyzing the eigenvalues Λ_n of the Green's matrix Ĝ(ω_0). This may appear counterintuitive because the spatial localization is a property of eigenvectors | R_n ⟩ and not of eigenvalues, so that it would be logical to analyze the former instead of the latter. There is no doubt that the analysis of the spatial structure of eigenvectors | R_n ⟩ is a more direct way to study Anderson localization. However, one should take into account that such an analysis is much more involved both analytically, because very little is known about the properties of eigenvectors of non-Hermitian matrices, and numerically, because finding the eigenvectors of a large matrix requires considerably more computational resources than finding only the eigenvalues. The last limitation is crucial for us because an accurate characterization of a localization transition, including finding the critical frequency and, especially, the critical exponent, requires averaging over many random configurations of scatterers {r⃗_m } for large numbers of scatterers N ≫ 1, and turns out to be a quite demanding computational task even when only the eigenvalues of Ĝ(ω_0) are computed. Therefore, in the present work we restrict our analysis to the eigenvalues of Ĝ(ω_0) and leave the statistics of eigenvectors for a future study. Nonetheless, we will discuss some properties of eigenvectors in Sec. <ref>, though without attempting a statistical analysis.






In the following, we set c_p/c_s = 2 typical for metals (e.g., aluminum or tin) but also for some other materials, such as, e.g., polystyrene. As discussed in Ref. skip16, the localization transition manifests itself in the statistical properties of the Thouless conductance g = ImΛ/δ(ReΛ), where δ(ReΛ) is the average spacing between real parts of eigenvalues. We compute the probability density p(ln g) by averaging over many independent random configurations {r⃗_m }.
The number of different configurations that we use is adjusted to obtain at least 2 × 10^7 eigenvalues for N ≤ 10^4 and 10^7 eigenvalues for N > 10^4 <cit.>. The small-g part of p(ln g) tends to become independent of the size of the system at a mobility edge and can be used to determine the critical properties of the localization transition. To this end, we follow Ref. skip16 and consider small-q percentiles ln g_q of p(ln g).
The percentiles ln g_q are defined by

    q    =   ∫_-∞^ln g_q p(ln g) d(ln g).



When the density of scatterers is high enough, ln g_q become independent of the system size at critical points where localization transitions take place, as we illustrate in Fig. <ref>(a) for q = 0.02. We find two such points (ReΛ_c ≃ -1 and -2.4 for ρ/k_s^3 = 0.15), with a band of localized states in between.
Collective effects shift the latter with respect to the scattering resonance of individual scatterers corresponding to ReΛ = 0. Although the exact position of this band cannot be predicted based on simple arguments, it is clear that disorder-induced localized states cannot exist far from the resonance frequency ω_0, i.e. for |ReΛ| ≫ 1, because scattering by individual scatterers becomes negligible for |ω - ω_0| ≫Γ_0 and no collective effects can make a collection of such weak scatterers strongly scattering. By construction, our scattering medium becomes weakly scattering and then transparent as the frequency ω is detuned from ω_0 by several Γ_0. It follows then that the number of mobility edges should be even in our model of identical resonant scatterers with a single narrow resonance.
Having two mobility edges near a resonance of individual scatterers is typical for disordered samples used in recent experiments <cit.>, which confirms that our model is relevant for the interpretation of the latter.
One should keep in mind, however, that spherical scatterers used in the experiments of Ref. cobus16 have multiple resonances and that in this case, the link between resonant properties of individual scatterers and those of a large ensemble of them is not simple. For periodic arrangements of spherical scatterers, this problem has been discussed in Ref. kafesaki95 and some arguments of that paper apply to disordered systems as well.

The quality of our numerical data is considerably better around the first critical point ReΛ_c ≃ -1, which we analyze in detail using the finite-size scaling approach. We follow the procedure described in Ref. skip16 (see Ref. slevin14 for a review of the finite-size scaling approach to Anderson localization). ln g_q is expanded in power series as a function of relevant and irrelevant scaling variables ψ_1 = f_1(w)L^1/ν (up to an order n_1) and ψ_2 = f_2(w)L^y (up to an order n_2), where f_j(w), in their turn, are expanded in series in w = (ReΛ - ReΛ_c)/ReΛ_c (up to orders m_j). The numerical data falling within ±Δ of the estimated critical point (ln g_q)_c are fitted using m_1 = n_1 ≤ 4; m_2 = n_2 ≤ 2 with a requirement that the contribution of the irrelevant variable to the fit should not exceed 10% of Δ. For Δ = 2, reasonable fits are obtained with m_1 = n_1 = 3 and m_2 = n_2 = 1, see Fig. <ref>(b).

Among all the fit parameters, the mobility edge ReΛ_c and the critical exponent ν are of main interest for us. Results obtained for very small q suffer from large statistical errors because the limited number of random realizations does not allow to estimate ln g_q reliably. On the other hand, large-q results are not trustworthy because the distributions p(ln g) following from our calculations start to violate the single-parameter scaling hypothesis when ln g is increased. Reliable results correspond to q = 0.001–0.05 shown in Fig. <ref>. Averaging over q yields, in particular, an estimate of the critical exponent ⟨ν⟩  = 1.554 ± 0.085 which is in good agreement with the value 1.55 ± 0.07 following from the same analysis for scalar waves <cit.>. In contrast, the mobility edge ReΛ_c turns out to be shifted to a higher frequency with respect to its value in the scalar model <cit.>, similarly to the model with nonresonant scattering.



 §.§ Inverse participation ratio of quasi-modes





As we explained already in Sec. <ref>, a quantitative statistical analysis that would allow for estimating the mobility edges and the critical exponent of the localization transition with a sufficient accuracy based on the localization properties of quasi-modes, would be computationally too demanding. However, one can get a good qualitative understanding of Anderson localization in the model described by the random matrix (<ref>) by considering its eigenvectors | R_n ⟩ obtained for a single random configuration {r⃗_m } of scatterers. The spatial localization of an eigenvector can be quantified by its inverse participation ratio (IPR):

    IPR_n = ∑_m = 1^N( ∑_α = x, y, z| R_n^m, α|^2 )^2/( ∑_m = 1^N ∑_α = x, y, z| R_n^m, α|^2 )^2,

where R_n^m, α denotes the component of the vector | R_n ⟩ on the scatterer m with a polarization parallel to α = x, y, or z. IPR varies from 1/N for a fully extended eigenvector to 1 for an eigenvector localized on a single scatterer. Typically, IPR≃ 1/K for a state localized on K scatterers. Figure <ref> shows the eigenvalues of a random realization of the matrix (<ref>), with the grey scale of a dot showing an eigenvalue determined by the IPR of the corresponding eigenvector. Several comments are in order.


In Fig. <ref>, we clearly see the band of localized states between the mobility edges determined from the finite-size scaling of percentiles in Sec. <ref> [vertical dashed lines in Fig. <ref>(a)]. The spatial localization of quasi-modes corresponding to eigenvalues inside this band becomes stronger as the number of scatterers (and hence the system size R ∝ (N/ρ)^1/3) increases, as one concludes by comparing the panels (a) and (b) of Fig. <ref>, where results are presented for N = 2000 and N = 10000, respectively. At the same time, the decay rates of quasi-modes inside the band decrease exponentially with the system size. Such a scaling with system size is typical for Anderson localization and should be contrasted with the situation taking place for quasi-modes localized on pairs of closely located scatterers and corresponding to the eigenvalues belonging to the long “tail” with large negative ReΛ in Fig. <ref>. Such quasi-modes have IPR≃ 0.5 and are hardly sensitive to the system size, as follows from the comparison of panels (a) and (b) of Fig. <ref>. Their origin is the same as for scalar waves <cit.> and light <cit.>: waves emitted by a pair of closely located scatterers (k_0 r_mn≪ 1) oscillating out of phase interfere destructively and thus cancel each other. The corresponding quasi-mode effectively decouples from the “outside world”, independently of the presence of other scatterers in the latter and their number N. Such “subradiant” localized quasi-modes exist at any scatterer density and number but do not show the scaling behavior associated with Anderson localization.



Figure <ref> shows that the two mobility edges that we clearly see in Fig. <ref>(a) are significantly different in nature. Indeed, whereas the low-frequency mobility edge ReΛ≃ -1 corresponds to a sharp transition from extended quasi-modes with IPR∼ 1/N ≪ 1 to localized quasi-modes with 1/N ≪IPR < 0.5, localized modes are present on the both sides from the high-frequency mobility edge ReΛ≃ -2.4. However, as we discussed above, only the modes on the right from it are genuine Anderson-localized modes. Even though the scaling analysis of Fig. <ref>(a) makes a difference between the two different localization mechanisms and correctly detects a change in scaling behavior at the high-frequency mobility edge, the results presented in Fig. <ref>(a) turn out to be much more noisy near the high-frequency mobility edge than near the low-frequency one. As a result, a scaling analysis of the high-frequency localization transition does not yield conclusive results and we do not present it here.




 §.§ Comparison between elastic and electromagnetic waves


The Green's matrix (<ref>) looks noticeably similar to its optical counterpart describing light scattering <cit.>. In particular, it features near-field terms diverging as 1/r_mn^3 at small r_mn. These terms associated with quasistatic dipole-dipole interactions were found to prevent Anderson localization of light <cit.> but apparently do not play any important role in the elastic case. The mathematical reason for this is that these terms cancel out in Eq. (<ref>)  for k_p,s r_mn≪ 1, leaving us with Ĝ_mn∝ 1/r_mn.
Indeed, the elastic Green's tensor (<ref>) can be separated in contributions of compressive (longitudinal) and shear (transverse) waves that propagate independently of each other only in the far field k_p,sΔ r ≫ 1. In the near field k_p,sΔ r ≪ 1, the corresponding parts of the Green's tensor combine to give

    Ĝ(𝐫, 𝐫',ω)    =   1/8 πρ_0 c_s^2 Δ r{[1 + ( c_s/c_p)^2 ] 1.
    
       +   . [1 - ( c_s/c_p)^2 ] Δ𝐫⊗Δ𝐫/Δ r^2},

which diverges as 1/Δ r only and hence is integrable in 3D. This is very different from the optical case where
Ĝ∝ 1/Δ r^3 and Ĝ_mn∝ 1/r_mn^3 at small distances. The physical reason for elastic waves to behave differently from light stems from the existence of propagating longitudinal waves that get scattered and eventually localized in the same way as scalar waves do. In contrast, propagating waves are transverse in optics, whereas longitudinal fields give rise to dipole-dipole interactions between scatterers. Efficient only at small distances, these interactions open an additional, nonradiative channel of energy transport in sufficiently dense ensembles of scatterers <cit.>.



§ CONCLUSIONS



We studied the Anderson localization transition in two different models of elastic wave scattering in 3D: a model with nonresonant scattering, where the localization transition takes place upon increasing the frequency of a wave, and a model with resonant scattering, where a narrow band of localized states is separated from the rest of the spectrum by two mobility edges. We find that for both models, the vector character of waves shifts the localization transitions to higher frequencies with respect to the scalar case. At the same time, the critical exponent ν coincides with its value in the scalar case within the accuracy of our analysis, suggesting that the universality class of the transition does not change. Our best estimates of ν are 1.564 ± 0.009 and 1.554 ± 0.085 for the two models, respectively. They agree with the value ν≃ 1.571 in the 3D orthogonal universality class <cit.>. Hence, our results suggest that despite their vector character, elastic waves exhibit a disorder-induced localization transition of the same orthogonal universality class as the spinless electrons in a 3D disordered potential.


This work was funded by the Agence Nationale de la Recherche (project ANR-14-CE26-0032 LOVE). A part of the computations presented in this paper were performed using the Froggy platform of the CIMENT infrastructure (https://ciment.ujf-grenoble.fr), which is supported by the Rhone-Alpes region (grant CPER0713 CIRA) and the Equip@Meso project (reference ANR-10-EQPX-29-01) of the programme Investissements d'Avenir supervised by the Agence Nationale de la Recherche. Y.M.B. acknowledges support of the Council of the President of the Russian Federation for Support of Young Scientists and Scientific Schools (project no. SP-3299.2016.1).


99

anderson58
P.W. Anderson,
Absence of Diffusion in Certain Random Lattices,
Phys. Rev. 109, 1492–1505 (1958).

abrahams10
50 Years of Anderson Localization, edited by E. Abrahams (Singapore, World Scientific, 2010).

abrahams79
E. Abrahams, P.W. Anderson, D.C. Licciardello, and T.V. Ramakrishnan,
Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions,
Phys. Rev. Lett. 42, 673–676 (1979).

evers08
F. Evers and A.D. Mirlin,
Anderson Transitions,
Rev. Mod. Phys. 80, 1355 (2008).

rosenbaum80
T.F. Rosenbaum, K. Andres, G.A. Thomas, and R.N. Bhatt,
Sharp Metal-Insulator Transition in a Random Solid,
Phys. Rev. Lett. 45, 1723 (1980).

paalanen82
M.A. Paalanen, T.F. Rosenbaum, G.A. Thomas, and R.N. Bhatt,
Stress Tuning of the Metal-Insulator Transition at Millikelvin Temperatures,
Phys. Rev. Lett. 48, 1284 (1982).

hu08
H. Hu, A. Strybulevych, J.H. Page, S.E. Skipetrov, and B.A. van Tiggelen,
Localization of Ultrasound in a Three-Dimensional Elastic Network,
Nature Phys. 4, 945 (2008).

cobus16
L.A. Cobus, S.E. Skipetrov, A. Aubry, B.A. van Tiggelen, A. Derode, and J.H. Page,
Anderson Mobility Gap Probed by Dynamic Coherent Backscattering,
Phys. Rev. Lett. 116, 193901 (2016).

chabe08
J. Chabé, G. Lemarié, B. Grémaud, D. Delande, P. Szriftgiser, and J.C. Garreau,
Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves,
Phys. Rev. Lett. 101, 255702 (2008)

jendr12
F. Jendrzejewski, A. Bernard, K. Müller, P. Cheinet, V. Josse, M. Piraud, L. Pezzé,	L. Sanchez-Palencia, A. Aspect and P. Bouyer,
Three-Dimensional Localization of Ultracold Atoms in an Optical Disordered Potential,
Nature Phys. 8, 398 (2012).

vanderbeek12
T. van der Beek, P. Barthelemy, P.M. Johnson, D.S. Wiersma, and A. Lagendijk,
Light Transport through Disordered Layers of Dense Gallium Arsenide Submicron Particles,
Phys. Rev. B 85, 115401 (2012).

sperling16
T. Sperling, L. Schertel, M. Ackermann, G.J. Aubry, C.M. Aegerter, and G. Maret,
Can 3D Light Localization be Reached in 'White Paint'?
New J. Phys. 18, 013039 (2016).

skip16njp
S.E. Skipetrov and J.H. Page,
Red Light for Anderson Localization,
New J. Phys. 18, 021001 (2016).

skip14
S.E. Skipetrov and I.M. Sokolov,
Absence of Anderson Localization of Light in a Random Ensemble of Point Scatterers,
Phys. Rev. Lett. 112, 023905 (2014).

bellando14
L. Bellando, A. Gero, E. Akkermans, and R. Kaiser,
Cooperative Effects and Disorder: A Scaling Analysis of the Spectrum of the Effective Atomic Hamiltonian,
Phys. Rev. A 90, 063822 (2014).

escalante17
J.M. Escalante and S.E. Skipetrov,
Longitudinal Optical Fields in Light Scattering from Dielectric Spheres and Anderson Localization of Light,
Ann. Phys. (Berlin) 529, 1700039 (2017).

sheng94
P. Sheng, M. Zhou, and Z.Q. Zhang,
Phonon Transport in Strong-Scattering Media,
Phys. Rev. Lett. 72, 234 (1994).

chaudhuri10
A. Chaudhuri, A. Kundu, D. Roy, A. Dhar, J.L. Lebowitz, and H.Spohn,
Heat Transport and Phonon Localization in Mass-Disordered Harmonic Crystals,
Phys. Rev. B 81, 064301 (2010).

mohthus10
C. Monthus and T. Garel,
Anderson Localization of Phonons in Dimension d=1,2,3: Finite-Size Properties of the Inverse Participation Ratios of Eigenstates,
Phys. Rev. B 81, 224208 (2010).

pinski12
S.D. Pinski, W. Schirmacher, T. Whall, and R.A. Römer,
Localization-Delocalization Transition for Disordered Cubic Harmonic Lattices,
J. Phys.: Condens. Matter 24 405401 (2012).

pinski12a
S.D. Pinski, W. Schirmacher, and R.A. Römer,
Anderson Universality in a Model of Disordered Phonons,
EPL 97 16007 (2012).

amir13
A. Amir, J.J. Krich, V. Vitelli, Y. Oreg, and Y. Imry,
Emergent Percolation Length and Localization in Random Elastic Networks,
Phys. Rev. X 3, 021017 (2013).

beltukov17
Y.M. Beltukov and S.E. Skipetrov,
Finite-Time Scaling at the Anderson Transition for Vibrations in Solids,
Phys. Rev. B 96, 174209 (2017).

ludlam03
J.J. Ludlam, S.N. Taraskin, and S.R. Elliott,
Disorder-Induced Vibrational Localization,
Phys. Rev. B 67, 132203 (2003).

sepehrinia08
R. Sepehrinia, M.R. Rahimi Tabar, and M. Sahimi,
Numerical Simulation of the Localization of Elastic Waves in Two- and Three-Dimensional Heterogeneous Media,
Phys. Rev. B 78, 024207 (2008).

john83
S. John, H. Sompolinsky, and M.J. Stephen,
Localization in a Disordered Elastic medium Near Two Dimensions,
Phys. Rev. B 27, 5592 (1983).

photiadis17
D.M. Photiadis,
Elastic Wave Transport in Disordered, Isotropic Media: A Supersymmetric Sigma Model,
Ann. Phys. (Berlin) 529, 1600353 (2017).

huang09
B.J. Huang and T.M. Wu,
Localization-Delocalization Transition in Hessian Matrices of Topologically Disordered Systems,
Phys. Rev. E 79, 041105 (2009).

jacobs96
D.J. Jacobs and M.F. Thorpe,
Generic Rigidity Percolation: The Pebble Game,
Phys. Rev. Lett. 75, 4051 (1995).

skip16
S.E. Skipetrov,
Finite-Size Scaling Analysis of Localization Transition for Scalar Waves in a Three-Dimensional Ensemble of Resonant Point Scatterers,
Phys. Rev. B 94, 064202 (2016).

beltukov13
Y.M. Beltukov, V.I. Kozub, D.A. Parshin,
Ioffe-Regel Criterion and Diffusion of Vibrations in Random Lattices,
Phys. Rev. B 87, 134203 (2013).

taraskin99
S.N. Taraskin and S.R. Elliott,
Low-Frequency Vibrational Excitations in Vitreous Silica: the IoffeRegel Limit,
J. Phys.: Condens. Matter 11, A219 (1999).

landau
L.D. Landau and E.M. Lifshitz,
Theory of Elasticity (Pergamon Press, Oxford, 1970).


ben81
A. Ben-Menahem and S.J. Singh,
Seismic Waves and Sources (Springer-Verlag, New York, 1981).

snieder02
R. Snieder,
General Theory of Elastic Wave Scattering,
in Scattering and Inverse Scattering in Pure and Applied Science,
edited by R. Pike and P. Sabatier (Academic Press, San Diego, 2002), pp. 528–542.

foldy45
L.L. Foldy,
The Multiple Scattering of Waves: I. General Theory of Isotropic Scattering by Randomly Distributed Scatterers,
Phys. Rev. 67, 107 (1945).

lax51
M. Lax,
Multiple Scattering of Waves,
Rev. Mod. Phys. 23, 287 (1951).

skip15
S.E. Skipetrov and I.M. Sokolov,
Magnetic-Field-Driven Localization of Light in a Cold-Atom Gas,
Phys. Rev. Lett. 114, 053902 (2015).

skip18
S.E. Skipetrov,
Localization Transition for Light Scattering by Atoms in an External Magnetic Field,
arXiv:1805.04314.

kafesaki95
M. Kafesaki and E.N. Economou,
Interpretation of the Band-Structure Results for Elastic and Acoustic Waves by Analogy with the LCAO Approach,
Phys. Rev. B 52, 13317 (1995).

slevin14
K. Slevin and T. Ohtsuki,
Critical Exponent for the Anderson Transition in the Three-Dimensional Orthogonal Universality Class,
New J. Phys. 16, 015012 (2014).

nieuwen94
Th.M. Nieuwenhuizen, A.L. Burin, Yu. Kagan, and G.V. Shlyapnikov,
Light Propagation in a Solid with Resonant Atoms at Random Positions,
Phys. Lett. A 184, 360–365 (1994).

note1
We attempted fits using m_1, n_1 up to 3 and m_2, n_2 up to 2 but found that increasing m_1, n_1 or introducing the irrelevant variable by setting m_2, n_2 > 0  improves the quality of fits only marginally while increasing the uncertainties of the values of best-fit parameters.

note2
The actual number of independent random configurations {r⃗_m } that we use is
3400 (for N = 2000),
1716 (N=4000),
1126 (N=6000),
862 (N=8000),
689 (N=10000),
300 (N=12000),
255 (N=14000), and
209 (N=16000).



