Scaling of phononic transport with connectivity in amorphous solids
Matthieu Wyart
%The propagation of force, heat or sound in amorphous materials %and the fragility in the corresponding liquid appear to be related to the presence of a transition in their vibrational spectrum, the so-called Boson Peak. The effect of coordination on transport is investigated theoretically using random networks of springs as model systems. An effective medium approximation is made to compute the density of states of the vibrational modes, their energy diffusivity (a spectral measure of transport) and their spatial correlations as the network coordination $z$ is varied. Critical behaviors are obtained as $z\rightarrow z_c$ where these networks lose rigidity. A sharp cross-over from a regime where modes are plane-wave-like toward a regime of extended but strongly-scattered modes occurs at some frequency $\omega^*\sim z-z_c$, which does not correspond to the Ioffe-Regel criterion. Above $\omega^*$ both the density of states and the diffusivity are nearly constant. These results agree remarkably with recent numerical observations of repulsive particles near the jamming threshold \cite{ning}. The analysis further predicts that the length scale characterizing the correlation of displacements of the scattered modes decays as $1/\sqrt{\omega}$ with frequency, whereas for $\omega<<\omega^*$ Rayleigh scattering is found with a scattering length $l_s\sim (z-z_c)^3/\omega^4$. It is argued that this description applies to silica glass where it compares well with thermal conductivity data, and to transverse ultrasound propagation in granular matter.
effectivemedium7cond.tex
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\title{Scaling of phononic transport with connectivity in amorphous solids }
\shorttitle{Phononic transport and connectivity} %Insert here a short version of the title if it exceeds 70 characters
\author{Matthieu Wyart}
%\author{F. Author\inst{1,2} \and S. Author\inst{1} \and T. Author\inst{2}}
%\shortauthor{F. Author \etal}

\institute{ Lewis-Sigler Institute, Princeton University, Princeton, NJ 08544;\\
Center for Soft Matter Research, New York University, NY 10003
}
\pacs{ 45.70.-n}{Granular systems}
\pacs{61.43.Fs}{Glasses}
\pacs{83.80.Fg}{Granular solids}

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\newcommand{\wAM}{\omega_{\scriptscriptstyle\rm AM}}
\newcommand{\rhomax}{\rho_{\scriptscriptstyle\rm MAX}}
\newcommand{\phiRCP}{\phi_{\scriptscriptstyle\rm RCP}}
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\abstract{
%The propagation of force, heat  or sound in amorphous materials
%and the fragility in the corresponding liquid appear to be related to the presence of a transition in their vibrational spectrum, the so-called Boson Peak. 
The effect of coordination on transport  is investigated theoretically using random networks of springs as model systems. An effective medium approximation
is made to compute the density of states of the vibrational modes, their energy diffusivity (a spectral measure of transport) and their spatial correlations as the network coordination $z$ is varied. Critical behaviors are obtained as $z\rightarrow z_c$ where these networks lose rigidity. A sharp cross-over from a  regime where modes are plane-wave-like toward a regime of extended but strongly-scattered modes  occurs at some frequency $\omega^*\sim z-z_c$, which does not correspond to the Ioffe-Regel criterion.  Above $\omega^*$ both the density of states and the diffusivity are nearly constant. These results agree remarkably with recent numerical observations of repulsive particles near the jamming threshold \cite{ning}.  The analysis further predicts that the length scale characterizing the correlation of displacements of the scattered modes decays as $1/\sqrt{\omega}$ with frequency, whereas for $\omega<<\omega^*$ Rayleigh scattering is found with a scattering length $l_s\sim (z-z_c)^3/\omega^4$. It is argued that this description applies to silica glass where it compares well with thermal conductivity data, and to transverse ultrasound propagation in granular matter. }

\begin{document} 


%\affiliation{Lewis-Sigler Institute, Princeton University, Princeton, NJ 08544;\\
%Center for Soft Matter Research, New York University, NY 10003 }

%\date{\today}

%\title{Scaling of phononic transport with connectivity in amorphous solids }
%%\title{DRAFT: Role of connectivity on phononic transport in amorphous solids }


%\title{Microscopic relaxation and soft modes in hard sphere systems}




% density of states, the diffusivity, and the spatial correlations of vibrational modes are computed theoretically  for elastic networks close to the minimal coordination -the mean number of contact-  $z_c$ where rigidity is lost.
%The effective medium approximation predicts  that both the density of states and the mode diffusivity  display a plateau above a frequency $\omega^*$ which vanishes linearly near $z_c$, and marks the entry 
%in a regime of strong scattering. These findings agree remarkably with  recent  observations in packing of elastic particles near the jamming threshold \cite{J,ning}. The analysis further predicts that for $\omega>\omega^*$ the correlation %length 
%of the strongly scattered modes vary  as $l_s\sim \omega^{-1/2}$, whereas at low frequency $\omega<\omega^*$ scattering occurs on a length scale $l_s\sim \delta z^3/\omega^2$.



%We use an effective medium approximation to study the transport properties of disordered elastic networks close to the coordination $z_c$ where elasticity is lost. 
%We show that above some frequency scale $\omega^*\sim z-z_c$ a regime of strong scattering appear, where both the density of states and the diffusivity display plateau behaviors, in very good agreement with 
%numerical studies of elastic particles near the jamming threshold \cite{ning,vincenzo}. We predict  the shape of the spatial correlation of these modes, and extract their correlation length $l_s$ of the strongly scattered modes varies as $l_s=%%\omega^{-1/2}$, and is of order $l_s\sim(z-z_c)^{-1/2}$ at the boson peak frequency $\omega^*$.  Below $\omega^*$,  the diffusivity  is found to be $Di(\omega)\sim (z-z_c)^{7/8}/\omega^4$, in agreement with the Rayleigh scattering of %waves with vanishing velocity on non-vanishing disorder.





\maketitle

The temperature dependence of the thermal conductivity in glasses above a few $K^0$ suggests a regime where the scattering of phonons is strong and nearly independent of frequency, as noticed by Kittel  \cite{kittel}. At these temperatures specific heat measurements indicate the presence of excess modes in comparison with the Debye model, the Boson Peak \cite{Phillipsbook}, whose amplitude correlates with other fundamental properties of glasses,  such as the fragility of the  super-cooled liquid \cite{sokolov}. Understanding the nature of the strongly scattered modes leading to an excess density of states  may therefore unravel some of the most mysterious properties of these materials. Several mean-field models have been proposed to explain the presence of such modes \cite{schi1,argentin,mayr} and their effects on transport. Nevertheless, there is yet no simple generic rules %, excepted for under-coordinated covalent glasses \cite{phillips} 
to predict from the microscopic structure of the glass the amplitude of the peak and the frequency at which transport becomes diffusive. Existing mean-field approaches do not explain why, for example,  silica glass presents one of the largest Boson peak, whereas it is almost inexistent for amorphous silicon, although both structure are tetravalent. The  spatial nature of the modes also remains unresolved.  Recently numerics have shown that the vibrational properties of repulsive, short-range particles display scaling behavior when decompressed toward the jamming transition \cite{J}. One observes a frequency scale $\omega^*$ which vanishes with decompression and marks a sharp cross-over toward a regime of strong scattering at larger frequency, where the modes diffusivity, which can be computed numerically via a Kubo formalism \cite{ning,vincenzo}, is nearly constant and small, in consistence with Kittel's conjecture,  and where the density of states displays a plateau. Close to the jamming threshold the cross-over toward strong scattering can occur at a wave-length infinitely larger than the particle size. Thanks to this separation of length scales this system is very suitable to study the spatial structure of the excess mode, and its dependence on the microscopic structure. A variational argument can explain the plateau in the vibrational spectrum, and shows that both the mean coordination and contact strain, averaged on a scale $l^*\sim1/(z-z_c)$ which diverges near threshold, control locally the onset of the excess modes\cite{matthieu1,matthieu2}- referred to as ``anomalous" to underline the difference of these excitations with plane waves. This length does not appear in the static structure, but  characterizes the heterogeneity of the response to a point force \cite{wouter}.  Nevertheless, this variational argument does not yield the spatial correlations of the displacements of the modes, nor their transport ability, or diffusivity. Interestingly, some aspects of the diffusivity can be estimated by making several assumptions on the spatial nature of the modes or their wave-vector decomposition \cite{vincenzo}. However, these assumptions are not trivial and lack theoretical justifications. % Thus a theoretical description of how transport properties are affected by the microscopic structure remains to be built. % and the effect of coordination on transport is therefore not satisfyingly understood.% It is also unclear if the spatial correlation of the strongly scattered modes, which has been measured numerically in several systems \cite{barrat,Leo2}, reflects the scale $l^*$ where heterogeneity are important, or not.  
  In this letter the spatial correlations of the modes and their diffusivity are computed by applying an effective medium approximation, which has been used in similar contexts \cite{thorpe,schi2} and in  electrical resistor networks \cite{kirpatrick}, to random networks of harmonic springs close to the minimal coordination where rigidity is lost and unjamming occurs. Results compare well with properties of sphere packing, and also with measurements of thermal conductivity in silica at high temperature. 

% A variational argument was able to explain the plateau in the density of states, and showed that both average coordination and contact strain, computed on a length $l^*$ which diverges at the transition, are the key determinant of the amplitude and location of the boson peak. $l^*$


% The spatial of excess modes, in all of them this excess stem from an elastic instability encountered as a parameter like the temperature \cite{mayr}, the density \cite{parisi} or the amount of disorder \cite{schirmacher+...} is varied. These models make interesting predictions on the relation between the density of states, transport properties and some scattering measurements \cite{ruocco}. Nevertheless, there is yet no simple rules to predict a priori from the microscopic structure which glasses display a large boson peak and a diminished transport.
% Recently numerics have shown that repulsive, short-range particles display a critical point when decompressed toward the random close packing \cite{ning,vincenzo}. A frequency scale $\omega^*$ which vanishes with compression marks a sharp cross-over toward a regime of strong scattering, where the modes diffusivity is nearly constant and small, in consistence with Kittel's conjecture,  and where the density of states displays a plateau. A variational argument was able to explain the plateau in the density of states, and showed that both average coordination and contact strain, computed on a length $l^*$ which diverges at the transition, are the key determinant of the amplitude and location of the boson peak. $l^*$

% Several mean-field models have been proposed to explain the presence of excess modes, in all of them this excess stem from an elastic instability encountered as a parameter like the temperature \cite{mayr}, the density \cite{parisi} or the amount of disorder \cite{schirmacher+...} is varied. These models make interesting predictions on the relation between the density of states, transport properties and some scattering measurements \cite{ruocco}. Nevertheless, there is yet no simple rules to predict a priori from the microscopic structure which glasses display a large boson peak and a diminished transport. %, or not, suggesting that the key microscopic parameters controlling the amplitude of the peak are not identified. 
%Furthermore, the spatial nature of the excess modes, and the length scales characterizing them, remain unclear. transport and microscopic structure display scaling properties \cite{J,Ning,Vincenzo}. A variational argument was used to  explain the scaling properties of the density of states in this system \cite{matthieu1,matthieu2}, and extended to covalent glasses such as silica \cite{these}, and to some attractive systems \cite{these,ning}. Its main conclusion are that (i) mean coordination and mean contact strain are 
%key determinant of the excess mode  frequency and amplitude. This is true independently of the amount of disorder and applies in crystals as well.  (ii) modes at the peak are characterized by a length scale $l^*$ which depends only on coordination. Fluctuations of coordination need not affect the excess modes frequency if they are localized on length scales shorter than $l^*$, but they must do so if they extend on larger length scale. $l^*$ does not appear in the static structure of the glass, but was observed to characterize the response to a point force \cite{wouter}.  Nevertheless, this variational argument did not make prediction on the transport properties of the anomlaous modes, and it was unclkear if $l^*4 also appear in the spatial correlation of the modes. Here we solve these questions, using...


%At large wave length sound in sand or heat in glass conducts nicely, and the scattering on disorder is weak. As the wavelength of excitation decreases,
%there is a cross-over toward a regime of strong diffusion, apparent in the thermal conductivity of glasses, and in the coda in sand. This comes with a jump in the density of states, the so-called boson peak. 
%This peak appears to correlate to fundamental properties of glasses, such as their fragility. Various mean-field approach have been proposed to describe this phenomenum. In these descriptions 
 %\cite{schirmacher,gotze,parizi..} the BP is associated with an elastic instability, induced by increasing temperature, disorder, or reducing density. Despite their successes, in particular in terms of relating some aspects of transport to the %excess modes, and predicting some aspects of the high frequency sound, the spatial nature of the modes forming the boson peak is not understood. Furthermore those approaches are not predictive in terms of which glass should have a %large boson peak. Often the crystal structure corresponding to a glass with a large BP has a large Van hove singularity, so the answer to what microscopic structure leads to a large peak must also apply to a crystal. This approach are %nevertheless intrinsically based on disorder, and will therefore not explain that. In the last years it has been observed transport properties,  density of states and microscopic structure display scaling near the jamming transition of elastic %particles. This leads to the hope to relate structure and excess modes. A variational argument was made, which explain the scaling observed. Main predictions: anomalous modes affected by coordination and strain in the contact (ii) length %scale $l^*$, not apparent in the static structure, characterize the spatial averaging that determine the modes property. This nevertheless appear in the response of the system. This analysis was extended to HS, silica and LJ.  Nevertheless %this analysis did not permit to compute the mode diffusivity, and the possible presence of long range correlation in the displacement field. Those properties are derived here, applying an effective medium approximation to elastic networks %near threshold \cite{schirmacher,thorpe,kirpatrick,head}. This allows us to compare the prediction and limit of the mean field approach and the variational argument.  




I consider a system of point particles of unit mass $m\equiv1$, interacting via a lattice  of springs with coordination $z_0$ strictly larger than twice the spatial dimension $d$, where all bonds are assumed to be equivalent, such as in the triangular lattice. Springs are harmonic, unstretched,  of unit length $a\equiv 1$, and have a stiffness $k_0$, whose square root defines our unit frequency (thus $k_0=1$ in our units).  The energy expansion reads  $\delta E=\sum_{\langle ij\rangle} \frac{1}{2} [(\delta {\vec R_i}-\delta {\vec R_j})\cdot {\vec n_{ij}}]^2 +o(\delta R^2)$, where the sum is on all bonds $ij$, whose direction is along the unit vector $\vec n_{ij}$. Springs are randomly removed with a probability $1-p$, such that the coordination of the remaining network is $z=p z_0$. The static elastic properties and the presence of vibrational modes of zero frequency  of such a disordered lattice have been  studied previously \cite{thorpe}.   Here we extend the analysis to compute vibrational properties in the mechanically stable phase. 

The random network is modeled as an effective medium of spring stiffness $k_M(\omega)$.  To determine $k_M(\omega)$, one considers the ``defectuous" network where all bonds have a stiffness $k_M(\omega)$, except for one bond $(ij)=(12)$ at the origin, whose stiffness $k_{12}$ is distributed according to the distribution $P(x)=p\delta(x=1)+(1-p)\delta(x=0)$ of the disordered system. Next one considers a homogeneous oscillatory stress  applied on this system. If the bond $(12)$ were identical to the other bonds, the applied stress would lead to an affine oscillatory response $\delta{\vec R}^a_i(t)\equiv\delta{\vec R}^a_i(\omega) \exp(i\omega t)$, $i=1...N$. Such a response is not a solution in general, as if it is imposed on this network, an unbalanced oscillatory force dipole of amplitude $f_{12}(\omega)=(k_{12}-k_M(\omega)) e(\omega)$ appears on the bond $12$, where $e(\omega)=(\delta{\vec R}_1(\omega)-\delta{\vec R}_2(\omega))\cdot {\vec n_{12}}$ is the contact strain. The solution is obtained by adding the response of this dipole to the affine displacement. This causes  an extra strain $e^*(\omega)$ to appear in the bond (12).  $k_M(\omega)$ is determined self-consistently by imposing that the effective medium respond with the correct average strain to an imposed stress, implying that $\langle e^*(\omega)\rangle=0$, where $\langle X\rangle=\int X(k_{12}) P(k_{12}) dk_{12}$ . In general this requirement leads to a complex solution for $k_M(\omega)$ , reflecting the broken symmetry of translation in the disordered network and the presence of scattering.


The extra strain $e^*(\omega)$ can be expressed in terms of the Green function of the effective homogeneous system \cite{kirpatrick,webman,thorpe}, as is now sketched. Newton's equation for an oscillatory external force field $\delta{\vec F}_i(t)=\delta{\vec F}_i(\omega) \exp(i\omega t)$ leads to $\sum_j A_{ij}(\omega) \cdot \delta {\vec R}_j(\omega)=\delta{\vec F}_i(\omega)$, where: %I use the notation $|\delta R\rangle\equiv \sum_i  \delta {\vec R_i(\omega)} | i\rangle$ for spatial fields, $\delta {\vec R_i(\omega)}$ is the Fourier transform of the displacement $\delta {\vec R_i(t)}$, and:
\be
\label{1}
A_{ij}(\omega)= \delta_{ij}(-\omega^2 +\sum_{\langle l\rangle} k_M  {\vec n_{il}}\otimes {\vec n_{il}}) -\delta_{\langle ij\rangle} k_M  {\vec n_{ij}}\otimes {\vec n_{ij}}
\ee
In Eq.(\ref{1})  $\otimes$ is the tensor product, the sum is made on all neighbors $l$ of $i$, and $\delta_{\langle ij\rangle}=1$ if $i$ and $j$ are neighbors, and zero otherwise. In these notations $A_{ij}$ is a tensor of rank $d$. We denote by $A(\omega)$ the $dN\times dN$ linear application relating displacements to forces field. Its inverse is the Green function $G=A^{-1}$. $G_{ij}$ is the tensor of rank $d$ characterizing the displacement of particle $i$ in response to a force applied in $j$. In  the ''defectuous'' network the additional displacement field $\delta {\vec R_i^*}$, $i=1...N$, generated by the force dipole in the bond $12$ satisfies $\forall i$: 
\begin{eqnarray*}
\label{2}
&&\sum_j A_{ij}  \delta {\vec R}_j^*- (\delta_{i1}-\delta_{i2})(k_{12}-k_M){\vec n_{12}}\otimes{\vec n_{12}}\cdot (\delta {\vec R}_1^*-\delta {\vec R}_2^*) \\
&&=-{\vec n_{12}}(k_M(\omega)-k_{12}) e(\omega) (\delta_{i1}-\delta_{i2})  
\end{eqnarray*}
%%\ba
%\label{2}
%&&[A-(k_{12}-k_M){\vec n_{12}}\otimes{\vec n_{12}}(|1\rangle-|2\rangle)(\langle 1|-\langle 2|)]|\delta R_0\rangle\notag \\
%&&= -{\vec n_{12}}(k_M(\omega)-k_{12}) e(\omega) (|1\rangle-|2\rangle) 
%\ea
After some algebra one gets:
\begin{eqnarray*}
\label{3}
\lefteqn{e^*(\omega)=(\delta{\vec R^*}_1(\omega)-\delta{\vec R^*}_2(\omega))\cdot {\vec n_{12}}} \\
&=& \frac{2 e(\omega)(k_M-k_{12}){\vec n_{12}}\cdot (G_{11}(\omega)-G_{12}(\omega))\cdot {\vec n_{12}}}{1+2(k_{12}-k_M){\vec n_{12}}\cdot (G_{11}(\omega)-G_{12}(\omega))\cdot {\vec n_{12}}}
\end{eqnarray*}
This expression can be simplified  using that $\sum_j  A_{ij} G_{ji}$is the unit tensor of rank $d$, Eq.(\ref{1}), and the assumption that all bonds are equivalent \cite{kirpatrick}. One obtains if $i$ and $j$ are neighbors that ${\vec n_{ij}}\cdot (G_{ii}(\omega)-G_{ij}(\omega))\cdot {\vec n_{ij}}=d(1+ {\vec n_{ij}}\cdot G_{ii}(\omega)\cdot {\vec n_{ij}})/(k_M z_0)$. Applied in Eq.(\ref{3}), and requiring $\langle e_0(\omega)\rangle =0$ leads to:
\be
\label{4}
\langle \frac{k_{12}-k_M}{k_M+(k_{12}-k_M)\frac{2d}{z_0}(1+\omega^2  {\vec n_{12}}\cdot G_{11}(\omega)\cdot {\vec n_{12}})}\rangle=0
\ee
Note that in this expression, ${\vec n_{12}}$ can be replaced by any contact direction  ${\vec n_{ij}}$ and $ G_{11}(\omega)$ by $G_{ii}(\omega)$. 

Taking the static limit  $\omega\rightarrow 0$ in Eq.(\ref{4}) and using $pz_0\equiv z$, one gets $k_M=(z-2d)/(z-2d p)$. Thus as  found by Thorpe and collaborators \cite{thorpe}, the effective stiffness of the network, and therefore all the elastic constants, vanish linearly as $z\rightarrow 2d$,  as found numerically in generic elastic networks \cite{wouter2}. $z=2d$ corresponds to the Maxwell bound where rigidity is lost  \cite{max}, such network are called isostatic. Note that for sphere packing, the bulk modulus does not vanish near the isostatic point, as explained in \cite{these}. For such purely repulsive systems I expect the forthcoming result on the Rayleigh scattering to hold for transverse modes only. 
% observed numerically for the shear modulus of elastic particles near the jamming threshold \cite{J}. A different scaling is observed for the bulk modulus, as  expected \footnote{The scaling  of th e bulk modulus in repulsive particles is not generic of weakly connected medium \cite{these}. This singular behavior is due to the purely repulsive nature of the interactions. For such cases the prediction on the amplitude of the Rayleigh scattering presumably does not hold for longitudinal waves.}. 

The Green function can be computed numerically for any lattice, and used to solve the self-consistent Eq.(\ref{4}). In what follows we focus on the low-frequency scaling properties near this critical point. One expects such properties to be universal and not to depend on the details of the lattice. We shall therefore consider a simplified model, where the lattice elasticity is characterized by a single elastic constant $B$, which is taken equal to $k_M$. For the isotropic lattices, such as the hexagonal lattice, two elastic constant are required to describe elasticity. The analysis below is readily extended to this case, and leads to the same scaling behavior. We consider for the Green function:
\be
\label{5}
G_{i1}(\omega)=  \int_0^{k_{max}}\frac{d^dk}{(2\pi)^d}\sum_\alpha \frac{e^{i{\vec k}\cdot{\vec r_i}}}{ k_M(\omega) k^2-\omega^2} \vec{ k}_\alpha\otimes  \vec{ k}_\alpha
\ee
where the Debye cut-off $k_{max}$ is a constant in our units, and is set to one in what follows. ${\vec r_i}$ is the vector going from particle 1 to $i$, and the sum is made on the different polarization $\vec{k}_\alpha$. All spatial directions being equivalent, one obtains:
\begin{eqnarray}
\hbox{ trace}(G_{11}(\omega))/d={\vec n_{12}}\cdot G_{11}(\omega)\cdot {\vec n_{12}}  \\
= \int_0^{1} \frac{d^dk}{(2\pi)^d} \frac{1}{ k_M(\omega) k^2-\omega^2} {} \label{6}
\end{eqnarray}
 Eqs.(\ref{4},\ref{6}) are solved numerically in the three-dimensional case, using  $z_0=12$.  The results for the real and imaginary part of  $k_M\equiv k_1(\omega)-i\Sigma(\omega)$ are shown in Fig.(\ref{f0}) for three different coordination. Qualitatively these results can be obtained by approximating the real part of the integral of Eq.(\ref{6}) by  its large $k$ contribution (obtained by setting $\omega=0$),
and its imaginary part by the singular value of the integral, plus its large $k$ contribution. This is approximately valid for $k_1\geq \Sigma$.
Expanding Eq.(\ref{4}) at low frequency, one finds the following schematic set of equations (where numerical pre-factors are omitted):
\ba
\label{10}
k_1=\delta z-\frac{\omega^2 k_1}{k_1^2+\Sigma^2}\\
\label{11}
\Sigma=\frac{\omega^3}{k_1^{3/2}}+ \frac{\Sigma\omega^2}{k_1^2+\Sigma^2}
\ea
where $\delta z\equiv z-z_c$ characterizes the distance from threshold. These schematic equations lead to a sharp cross-over at $\omega^*=\delta z/2$. This scaling is checked in inset of Fig.(\ref{2}). For smaller frequencies, $k_1\approx \delta z$, as expected from the static analysis, and $\Sigma\approx \omega^3/\delta z^{3/2}<<k_1$. At $\omega^*$, $k_1$ drops by  some numerical factor, and $\Sigma$ increases by a factor of order $1/\sqrt{\delta z}$. For larger frequencies, $\Sigma >> k_1$, and the (middle) term of  Eq.(\ref{11}) corresponding to the singular value of the integral  is not present. This leads to $\Sigma \sim \omega$. 


  \begin{figure}       
  \centering       
  \vspace{0.5cm}                  
   \rotatebox{-0}{\resizebox{8.9cm}{!}{\includegraphics{K_M.eps}}}
   \caption{\label{f0}Logarithmic plot  of the real and imaginary part of the effective stiffness $k_M\equiv k_1-i\Sigma$ {\it vs} frequency for several coordination numbers $\delta z\equiv z-z_c$.  }
   \vspace{0 cm}
\end{figure}

The density of states $D(\omega)$ of the effective medium is readily extracted using $D(\omega)=(2\omega/\pi) \hbox{Im}(\hbox{trace}(G_{11}(\omega))$,
and is shown in Fig.(\ref{f1}). The cross over at $\omega^*$ corresponds to a sharp transition from a Debye behavior $D(\omega)\sim \delta z^{-3/2}\omega^2$ at low frequency toward a plateau at higher frequency, and therefore correspond to the boson peak. The same cross-over is  observed numerically for elastic particles near jamming \cite{J,matthieu2}.
  \begin{figure}       
  \centering       
  \vspace{0.5cm}                  
   \rotatebox{+0}{\resizebox{6.9cm}{!}{\includegraphics{Dwinset2.eps}}}
   \caption{\label{f1}Density of states {\it vs} frequency $\omega$ for five coordination numbers.  Inset: Log-log plot of $\omega^*$, defined as  $D(\omega^*)=0.2$,  {\it vs} $\delta z$. The line has a slope 1. }
   \vspace{-0.cm}
\end{figure}

We next consider the spatial Green function at frequency $\omega$. For large distances $r_i>>1$ one finds:
\be
\label{7}
\hbox{trace}(G_{i1}(\omega))= 3\frac{e^{i\omega r_i/v(\omega)}e^{-r_i/l_s(\omega)}}{4\pi r_i k_M}
\ee
 $l_s(\omega)$ and $v(\omega)$ can be viewed as the scattering length and the  velocity of the corresponding vibrations, and their expression is:
\ba
l_s(\omega)&=&\frac{|k_M|}{\omega |\hbox{Im}(k_M^{1/2})|}\\
v(\omega)&=&\frac{|k_M|}{\hbox{Re}(k_M^{1/2})}\label{22}
\ea
Transport at frequency $\omega$ is characterized by the diffusivity $D_i(\omega)$, whose approximation $ D_i(\omega)= v(\omega)l_s(\omega)/3$ is shown in Fig(\ref{f2}).
This approximation is expected to be accurate  for weak-scattering ($\omega<<\omega^*$) and crude but qualitatively correct for the strongly-scattered modes  ($\omega>>\omega^*$) as long as they are extended, since it enforces the scaling requirement that the diffusivity is the product of a characteristic speed times a characteristic length. A more accurate computation of the diffusivity would require to go beyond the single bond effective medium investigated here.   

For $\omega<<\omega^*$,  Rayleigh scattering occurs: $D_i(\omega)\sim \delta z^{7/2}/\omega^4$.  At $\omega\sim\omega^*$, there is a sudden decay of the scattering length from $1/\delta z$ toward $1/\sqrt \delta z$  which is of the order  of the wavelength $\lambda\equiv v/\omega$. This sudden decay of the scattering length at  $\omega^*$ implies that this cross-over cannot  be identified from the quasi-plane wave regime $(\omega<<\omega^*)$ using the Ioffe-Regel criterion $l_s=\lambda$. %. Assuming that this is the case, and the scaling of elastic moduli, leads to the correct scaling of $\omega^*$ \cite{vincenzo}.
  For $\omega>>\omega^*$ there is a flat plateau in the diffusivity, again in very good agreement with numerical studies of transport of jammed elastic particles \cite{ning,vincenzo}. This plateau stems from the compensation of two effects: the decrease of the scattering length $l_s(\omega)\sim \omega^{-1/2}$, and the inverse increase of the velocity $v(\omega)\sim \omega^{1/2}$. Note that at the high frequency end of the spectrum a localization edge, rather insensitive  to coordination, is observed numerically \cite{ning} and is not captured by this analysis.


\begin{figure}       
  \centering       
  \vspace{0.5cm}                  
  \rotatebox{-0}{\resizebox{6.9cm}{!}{\includegraphics{bestdiffusivity.eps}}}
  \caption{\label{f2} Log-log plot of the diffusivity $D_i(\omega)$ {\it vs} $\omega$ for several coordination numbers, as indicated in the legend. %To quantify the Rayleigh regime at low frequency, we define the quantity $C(\delta z)= D_i(\omega)*\omega^4$, which does not depend on the choice of $\omega$ for $\omega<<\omega^*$. Inset: log-log plot of $C(\delta z)$ {\it vs} $\delta z$, showing that $C\sim \delta z^{7/2}$.   
  }
  \vspace{0 cm}
\end{figure}


The spatial properties of the strongly scattered normal modes $|e_\omega\rangle$ are deduced from the relation $\langle {\vec e}_\omega(1)\otimes  {\vec e}_\omega(i)\rangle \propto \hbox{Im}(G_{1i}(\omega))\omega/D(\omega)$. Thus Eq.(\ref{7}) implies that at $\omega^*$, the correlation length of the displacement  is of order $\delta z^{-1/2}$, and decreases as $\omega^{-1/2}$ for higher frequencies. 
Scaling arguments for why anomalous modes near $\omega^*$ are characterized by a length of order  $\delta z^{-1/2}$ -which is the value of the wavelength just below $\omega^*$ \cite{Leo2}-  have been given by Vitelli et al. \cite{vincenzo}. 

%Correlations length of order 15 particles size have been observed at the boson peak in numerical silica \cite{barrat}, and they have been observed to decay with frequency in sphere packing \cite{Leo2}.
 Experimentally the dynamical structure factor $S(k,\omega)$ is accessible. In Fig.(\ref{f4}) the quantity  Im($k^2/(k_Mk^2-\omega^2))$, proportional to $\omega S(k,\omega)$ for a harmonic dynamics, is plotted. At threshold, no  peaks are found in the structure factor, which simply consists of a gap at low frequency, which ends approximately when  $\omega^4\sim \Sigma^2 k^4$ leading to  $k\sim \omega^{1/2}$, before reaching a plateau. At larger coordination,  well defined-peaks for  $\omega<\omega^*$ appear in the spectrum, corresponding to weakly-scattered plane waves. At larger frequencies the peak erodes, and the structure factor eventually resembles what is found at the critical point. The intensity gap at small frequency and the presence of a large background of large spatial frequencies have been observed above the Boson peak peak  in various systems such as silica \cite{ruocco} and sphere packing \cite{Leo2}.  
 
I define a dispersion relation by considering  the maximum $k_{max}(\omega)$ of $S(k,\omega)$ at fixed $\omega$. As shown in the lower panel of Fig.(\ref{f4}), this dispersion relations is linear at low frequency, and presents an inflection point near the Boson peak, before increasing as $\omega\sim k^2$. This inflection point leads to a drop in the velocity $\omega/k_{max}$ at the Boson peak frequency as shown in the inset of Fig.(\ref{f4}), a rather subtle effect recently observed in Lennard-Jones simulations  \cite{mossa}.

%Thus, our analysis supports that the Boson peak at $\omega^*$ corresponds to a cross-over between two kinds of modes whose structure factors qualitatively differ. A non-trivial prediction of our analysis and Eq.(\ref{22}) is that as $\omega \rightarrow \omega^*{}^-$, the speed of sound presents a  drop (stemming from the drop of $k_1$ apparent in Fig(\ref{f0})) followed by square root increase characterizing anomalous modes. Such a non-monotonic drop of the velocity  has been observed recently at the Boson peak frequency in a Lennard-Jones liquid \cite{mossa}.


\begin{figure}       
  \centering       
  \vspace{0.5cm}                  
  \rotatebox{0}{\resizebox{8.5cm}{!}{\includegraphics{figSqw.ps}}}
    \vspace{-0.8 cm}
  \caption{ \label{f4} Dynamical structure factor $S(k,\omega)$ times $\omega$ {\it vs}  wave vector $k$ for different frequencies indicated in legend. Upper panel: $\delta z=0.29$. Middle panel: $\delta z= 4*10^{-4}$. No peaks are apparent. Lower panel: Dispersion relation $k_{max}(\omega)$ as defined in the text for $\delta z=0.29$. Inset:  phase velocity $\omega/k_{max}$ {\it vs} frequency $\omega$ for $\delta z=0.29$. The velocity displays a minimum close to the Boson peak frequency $\omega_{BP}=0.09$. 
  %To quantify the Rayleigh regime at low frequency, we define the quantity $C(\delta z)= D_i(\omega)*\omega^4$, which does not depend on the choice of $\omega$ for $\omega<<\omega^*$. Inset: log-log plot of $C(\delta z)$ {\it vs} $\delta z$, showing that $C\sim \delta z^{7/2}$.   
  }
  \vspace{-0.5 cm}
\end{figure}


%\begin{figure}[ht]
%\centering
%\subfigure{
%\includegraphics[scale=0.35]{Sqw00004bis.eps}
%\label{fig:subfig1}}
%\vspace{-0cm}
%\subfigure{
%\includegraphics[scale=0.35]{Sqw029bis.eps}
%\label{fig:subfig2}}
%\label{fig:subfigureExample}
%\caption[Optional caption for list of figures]{\label{f4} Dynamical structure factor $S(k,\omega)$ times $\omega$ {\it vs}  wave vector $k$ for different frequencies indicated in legend. Upper panel: $\delta z= 4*10^{-4}$. No peaks are apparent. Lower panel: $\delta z=0.29$.}
%\end{figure}

%\begin{figure}       
%  \centering       
%  \vspace{0.5cm}                  
%  \rotatebox{-0}{\resizebox{8.9cm}{!}{\includegraphics{S(k,w)joint.eps}}}
%  \caption{\label{f4} Dynamical structure factor $S(k,\omega)$ times $\omega$ {\it vs}  $k$ for different frequencies indicated in legend. Upper panel: $\delta z= 4*10^{-4}$. No peaks are apparent. Lower panel: $\delta z=0.29$.}
%  \vspace{0 cm}
%\end{figure}

{\it Applications to amorphous solids}: Numerical observations show that when friction is added, the scaling relations valid for the vibrational spectrum of purely repulsive particles still hold \cite{wim}. Thus the results on transport derived here are expected to hold in granular matter as well, at least in the limit where dissipative effects  \cite{jia0} do not dominate sound propagation. In small systems of grains,  low-frequency ultrasounds propagate ballistically, whereas propagation is diffusive at high-frequency \cite{Jia}. According to the present analysis the cross-over wavelength between these regimes should decrease with coordination. The coordination depends on the system preparation and the friction coefficient, and can be measured independently by comparing transverse and longitudinal speed of sound \cite{bruno,jacob,roux}. 


Covalent glasses are also well suited to study the effect of coordination on transport.  Practically important examples are tetravalent covalent networks where the joints linking tetrahedra have a small bending force constant such as in silica glass (where the Si-O-Si angle is easily deformable) or germanium oxide. If these joints are are assumed to be freely rotating and only the other covalent interactions are taken into account (the so-called RUM model \cite{silica1}), such networks turns out to be isostatic and display a flat density of states up to zero frequency  \cite{silica2}, in consistence with the present analysis and previous arguments \cite{these,matsi}. When the small but finite bending force constant among tetrahedra is taken into account, the plateau of anomalous modes in the spectrum is shifted and appears above 1Thz  \cite{these,matsi}, corresponding to the Boson peak frequency. Most of the spectrum thus consists of anomalous modes which must display a constant diffusivity according to the present analysis. Assuming a constant density of states and diffusivity up to some maximal frequency $\omega_{max}$ enables to compute the thermal conductivity $\kappa(T)=\int_{0}^{\infty} {D}(\omega) C(\omega) {D_i}(\omega) d\omega\propto \int_{0}^{\omega_{max}}  C(\omega) d\omega$ where $C(\omega)\propto (\omega/T)^2 \exp({\bar h}\omega/kT)/( \exp({\bar h}\omega/kT)-1)^2$ is the heat capacity of the modes at frequency $\omega$. Fig.(\ref{silica}) shows that such an estimate captures accurately the thermal conductivity for several decades above its plateau. In this high-temperature estimate, I have neglected the contribution of plane-wave like modes at low frequency. This is a good approximation since the contribution of these modes can be read at lower temperature where only those modes affect transport and is small. The plane waves contribution is of order of the thermal conductivity plateau where the cross-over occurs, which is significantly smaller than the high temperature conductivity.  This analysis thus supports that anomalous modes are responsible for transport at room temperature.
Note that in tetravalent structures with strong bending force constant at the joints, such as in amorphous silicon where the same covalent bonds characterize the joints and the tetrahedra themselves, the effective coordination is high, and a Boson peak is hardly detectable, as expected.
The present results are expected to apply in simple molecular liquids where long-range interactions are present as well \cite{these,xu}. 

%
%The present results are expected to apply in simple molecular liquids where long-range interactions are present as well. Disorder in bond strength can be taken into account via a variational argument \cite{these}, leading to the estimation of an effective coordination which enables e.g. to compute the Boson peak frequency for Lennard-Jones liquids \cite{xu}. Such an effective coordination can be plugged in the present formalism. 


\begin{figure}       
  \centering       
  \vspace{0.5cm}                  
  \rotatebox{-0}{\resizebox{6.9cm}{!}{\includegraphics{conductivitysilica.eps}}}
  \caption{\label{silica} Log-log plot of the thermal conductivity in silica glass against temperature in arbitrary units. Observations taken from Freeman and Anderson \cite{freeman} appear in crosses. Full line: prediction for a flat spectrum of anomalous modes with constant diffusivity.  %To quantify the Rayleigh regime at low frequency, we define the quantity $C(\delta z)= D_i(\omega)*\omega^4$, which does not depend on the choice of $\omega$ for $\omega<<\omega^*$. Inset: log-log plot of $C(\delta z)$ {\it vs} $\delta z$, showing that $C\sim \delta z^{7/2}$.   
  }
  \vspace{0 cm}
\end{figure}

{\it Comparison with the variational argument of \cite{matthieu1}}:
In \cite{matthieu1} a variational argument guessing the nature of the modes leading to a plateau in the density of states was able to explain the abrupt change in this quantity at some frequency $\omega^*\sim \delta z$.
The corresponding anomalous modes are not built by softly distorting translational modes like plane waves are. The view that the Boson peak is a cross-over between two distinct types of excitations is reflected by the sharp cross-overs in the density of states and the transport properties at $\omega^*$ found here, supporting that new objects suddenly appear in the spectrum at this frequency. Although anomalous modes also can be characterized by a length scale and a velocity, their spatial structure  qualitatively differs from plane waves as appears in their dynamical structure factor, flat  with a gap at small frequencies rather than peak-like as for plane waves.

If the  variational procedure of \cite{matthieu1} and the effective medium theory developed here lead to consistent results, these approaches are also complementary. The variational argument suggested that  anomalous modes would lead to poor transport \cite{these,vincenzo}, but did not enable to compute their dynamical structure factor  explicitly nor the correlation length $l_s\sim 1/\sqrt {\delta z}$ of their displacement field, as was done here. One the other hand, the present approach does not yield the length scale $l^*\sim 1/\delta z$ on which fluctuations in the microscopic structure affect the anomalous modes \cite{matthieu1} and the response to a local perturbation \cite{wouter}. This may be a generic limitation of mean field approaches.


{\it Comparison with previous mean field approaches}:
Previous mean-field analysis of the Boson peak, see e.g. \cite{schi1,mayr,argentin},  have also connected this phenomenon to the presence of an elastic instability. Nevertheless the nature of the instability in those cases is singularly different from the present study, for example they find that the density of states does vanish at zero frequency at the instability threshold. This discrepancy  underlines that two parameters, and two frequency scales, must be included to describe excess modes even in simple amorphous solids.  In particular, the present analysis has not included the effect of compression on the vibrational spectrum, or equivalently  the presence of a finite strain in the contacts.   This is  a good approximation for covalent glasses and granular matter at limited pressures, but compression plays an important role e.g. for colloids near the glass transition where it even determines the microscopic structure of the glass \cite{brito}. For any fixed coordination $z>2d$, increasing compression shifts anomalous modes toward lower frequencies and eventually leads to an elastic instability as shown in \cite{matthieu2}, for which the density of states does  vanish at zero frequency. Near this instability, the spectrum displays two decoupled characteristic frequency: a Boson peak frequency where the first anomalous modes appear and a higher frequency $\omega^*$ above which the density of states displays a plateau \cite{matthieu2}, and where the present analysis is expected to hold.  These two characteristic frequencies are identical only when pressure is negligible, in general they can be different, reflecting that two parameters, coordination and pressure, strongly affect the vibrational spectrum. Previous mean-field analysis \cite{schi1,mayr,argentin} are presumably well suited to describe transport properties at low-frequency near the instability driven by compression. Including compression to the present effective medium approach will assess this view, and may provide a  unified description of  transport  applicable to a broad range of amorphous solids.


 
%A few remarks are in order. The impressive match between the predictions of this mean field approach and observations in repulsive particles near jamming is presumably due to the weakness of fluctuations of coordination in this system.
%Dynamical arguments have been given to justify these small fluctuations \cite{matthieu2}, but it is likely that in some molecular glasses like silica  fluctuations in coordination are very small at equilibrium as well, since making defects in the tetravalent structure is very costly. The previous coordination-based variational argument could explain well the density of states in silica \cite{these}, so I expect the present results on transport to apply in this system. It is yet unclear if some glasses are strongly affected by fluctuations\cite{phillips2}. 
%.  Using springs networks as a model system of amorphous solids, an effective medium approximation was used to show that the Ioffe-Regel frequency
%where transport becomes diffusive is controlled by the network coordination, and vanishes linearly as the coordination decays toward a threshold value where rigidity is lost.  At higher frequency both the diffusivity and the density of states  are predicted to depend very weakly on frequency, in remarkable agreement with observations in assemblies of repulsive particles.  
%Plane-wave like modes at low frequency display Rayleigh scattering, whose amplitude depends strongly on coordination. Above the Ioffe-Regel limit, 
%strongly scattered normal modes have a spatial correlation $l_s\sim \sqrt \omega$. This length scale differs from the length $l^*$, previously obtained from a variational argument \cite{matthieu1} and observed numerically in the response to a point force \cite{wouter}. Thus the length scale at which fluctuations in the microscopic structure affect the modes is not extracted from the present approach, which may be a general limitation of mean field aproximations. 

%
%How generic are these results? First, the analysis has focused on the role of coordination, and has neglected the presence of strain in the contacts, always presents at finite compression.  Compression affects the low-frequency end of the spectrum, in particular it shifts the boson peak toward lower frequencies, and must be taken into account to describe quantitatively the density of states of The role of strain on transport remains thus to be elucidated, and a mean-field approach including this effect would therefore be very valuable. Second, the mean field approximation is presumably powerful when fluctuations of coordination are small,  which is the case near the jamming threshold as fluctuations are not favored by the dynamics \cite{matthieu2}. Clearly in other glasses, like silica, the coordination is quite homogeneous as defects in the tetravalent structure are energetically costly. It remains to be seen if some glasses largely departs from this assumption and require a specific treatment \cite{phillips2}.






%
%\begin{figure}[ht]
%\begin{minipage}[b]{0.5\linewidth}
%\centering
%\includegraphics[scale=0.20]{S(q,w)00004.eps}
%\caption{default}
%\label{fig:figure1}
%\end{minipage}
%\hspace{0.0cm}
%\begin{minipage}[b]{0.5\linewidth}
%\centering
%\includegraphics[scale=0.20]{S(q,w)029.eps}
%\caption{default}
%\label{fig:figure2}
%\end{minipage}
%\end{figure}



%\begin{acknowledgments}
It is a pleasure to thank V. Vitelli for discussions stimulating this work, A. Liu, S. Nagel and N. Xu for discussions, and D. Huse and M. Muller for comments on the manuscript.
%\end{acknowledgments}




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