# Theory of sound attenuation in amorphous solids from nonaffine motions

M. Baggioli  Wilczek Quantum Center, School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai 200240, China.  Shanghai Research Center for
Quantum Sciences, Shanghai 201315, China.  A. Zaccone<EMAIL_ADDRESS>,
<EMAIL_ADDRESS>Department of Physics “A. Pontremoli”, University of
Milan, via Celoria 16, 20133 Milan, Italy  Cavendish Laboratory, University of
Cambridge, JJ Thomson Avenue, CB30HE Cambridge, U.K.

(August 28, 2024)

###### Abstract

We present a theoretical derivation of acoustic phonon damping in amorphous
solids based on the nonaffine response formalism for the viscoelasticity of
amorphous solids. The analytical theory takes into account the nonaffine
displacements in transverse waves and is able to predict both the ubiquitous
low-energy diffusive damping $\sim k^{2}$, as well as a novel contribution to
the Rayleigh damping $\sim k^{4}$ at higher wavevectors and the crossover
between the two regimes observed experimentally. The coefficient of the
diffusive term is proportional to the microscopic viscous (Langevin-type)
damping in particle motion (which arises from anharmonicity), and to the
nonaffine correction to the static shear modulus, whereas the Rayleigh damping
emerges in the limit of low anharmonicity, consistent with previous
observations and macroscopic models. Importantly, the $k^{4}$ Rayleigh
contribution derived here does not arise from harmonic disorder or elastic
heterogeneity effects and it is the dominant mechanism for sound attenuation
in amorphous solids as recently suggested by molecular simulations.

##  I Introduction

The mechanism of wave propagation, and, especially, wave attenuation in the
disordered states of matter is a central and open topic in contemporary
physics, with many ramifications in other fields from plasmas to cosmology.
Since the mid 20th century it is well known that wave attenuation in the
acoustic regime of low frequencies and wavelengths is dominated by anharmonic
processes at the lowest wavevectors (the so-called hydrodynamic regime). In
this regime, the sound attenuation, or damping, scales with the wavevector as
$\Gamma\sim k^{2}$. This is a diffusive law which can be derived by using
conservation equations, i.e. with the methods of hydrodynamic theory and
effective field theory Martin  _et al._ (1972).

In the context of crystals, this is also known as Akhiezer damping Akhiezer
(1939), whereas in liquids is known as Brillouin linewidth. The same
phenomenon is known to dominate sound attenuation in amorphous solids, such as
glasses, at low $k$, where it plays an important role in determining the
thermal conductivity Ziman (2001). More recently, the diffusive nature of
vibrational excitations in glasses has been pointed out in numerical
simulations Allen  _et al._ (1999); Shintani and Tanaka (2008); Beltukov  _et
al._ (2013) and used as the starting point for theoretical models of the boson
peak caused by anharmonicity in glasses and crystals Baggioli and Zaccone
(2019, 2020).

In glasses Buchenau  _et al._ (2020), upon going to higher wavevectors, a
crossover from $\sim k^{2}$ to a $\sim k^{4}$ regime is typically observed
experimentally Masciovecchio  _et al._ (2006), where the $\sim k^{4}$ scaling
has been interpreted as Rayleigh-type scattering from random fluctuations of
some (usually macroscopic) quantity. In this sense, the heterogeneous
elasticity theory (HET) has provided a derivation of this Rayleigh type
damping based on the assumption of Gaussian spatial fluctuations of the shear
modulus Schirmacher (2006). This theory, however, is entirely at the continuum
level (its starting point is the elastic modulus, which is a continuum
quantity, while the modulus’ dependence on microphysics, particle
displacements, interactions etc is neglected), hence it does not account for
the microsopic structural order/disorder Milkus and Zaccone (2016) nor for the
underlying microscopic (nonaffine) particle dynamics Caroli and Lemaître
(2019, 2020); Huang  _et al._ (2018).

Importantly, it has been recently showed using simulations Szamel and Flenner
(2021) that the harmonic random fluctuations described by HET are not the only
mechanism behind the ubiquitous Rayleigh term $\sim k^{4}$. Nonaffine motions,
which arise from the dynamics in non-centrosymmetric environments Milkus and
Zaccone (2016), contribute as well to the Rayleigh damping and they are indeed
dominant with respect to the HET contribution, even at $T=0$ Szamel and
Flenner (2021). This recent observation questions the current paradigm for
sound attenuation in amorphous solids and calls for a deeper theoretical
understanding of the dominant nonaffine origin of it.

While the current models of wave damping in amorphous solids are invariably at
the continuum level (or at best effective medium theories DeGiuli  _et al._
(2014)), much progress has been done recently in the microscopic theoretical
description of the viscoelastic response of glasses, through the framework
known as nonaffine response formalism or nonaffine lattice dynamics DiDonna
and Lubensky (2005); Lemaître and Maloney (2006); Maloney and Lemaître (2006);
Zaccone and Scossa-Romano (2011); Saw  _et al._ (2016); Palyulin  _et al._
(2018). This theory takes into account the fact that, upon deforming a
disordered solid, the particles (atoms, molecules) undergo extra displacements
on top of those dictated by the external strain, and these extra
displacements, which are random, are called nonaffine motions. This framework
provides quantitatively accurate predictions of the viscoelastic moduli of
glasses with no fitting parameters in good agreement with simulations Palyulin
_et al._ (2018). It is by now recognized that nonaffine motions play a central
role in determining the dynamics of glasses at the microscopic level.

In this work, we present a theory of acoustic wave attenuation in amorphous
solids based on nonaffine motions and we analytically predict the
contributions from anharmonicity to the quadratic part of the damping
$\Gamma(k)$. We predict the hydrodynamic diffusive damping at low $k$,
including the prefactor which is related to important physical parameters such
as the Debye frequency, the nonaffine correction to the shear modulus, and the
microscopic friction due to anharmonicity. We provide a useful analytical
expression for the Rayleigh contribution $\Gamma\sim k^{4}$ arising from
nonaffinity and identified in the simulations of Szamel and Flenner (2021) as
the dominant effect on sound damping in amorphous systems. Finally, we also
predict the experimentally observed Masciovecchio  _et al._ (2006) crossover
from diffusive damping to Rayleigh damping at higher $k$ and we give an
estimate of the critical wave-length in terms of the previously mentioned
physical parameters.

##  II Preliminaries

Within standard linear response theory, the time-dependent expectation value
of the stress tensor $\sigma^{ij}(t)$ is given by a linear convolution

$\langle\sigma^{ij}(t)\rangle\simeq\int_{-\infty}^{\infty}\chi_{\sigma\sigma}^{ijkl}(t-t^{\prime})\,\gamma^{kl}(t^{\prime})+\mathcal{O}(\gamma^{2}).$
(1)

with the strain tensor $\gamma^{ij}$, whose kernel is given by the dynamic
response function (sometimes also labelled two-points function, correlator or
Green function) $\chi_{\sigma\sigma}^{ijkl}(t)$, i.e. the stress auto-
correlation function Chaikin and Lubensky (1995). Neglecting dissipative
terms, the zero frequency limit of the Fourier transformed response function
is simply the elastic tensor given in terms of the various elastic constants.
From now on, we will focus only on the shear response ($ij=xy$) and therefore
Latin indices will be omitted to avoid clutter. Upon Fourier transforming, we
have:

$\sigma(\omega)=\chi_{\sigma\sigma}(\omega)\,\gamma(\omega)\,,$ (2)

which is valid in the linear regime, to leading order in the external strain
$\gamma$. Importantly, in general, $\chi_{\sigma\sigma}(\omega)$ is a complex-
valued function whose real and imaginary components encode respectively the
reactive and dissipative parts of the response function and it coincides with
the complex dynamic modulus used in viscoelasticity theory. The real and
imaginary components are related by the Kramers-Kronig relations Toll (1956)
(imposed by causality) and in a simple viscoelastic system, to leading order
in frequency, are given by (see e.g. Ref. Pipkin (2012)):

$\chi_{\sigma\sigma}(\omega)\,=\,\mu\,+\,i\,\omega\,\eta\,+\,\mathcal{O}\left(\omega^{2}\right)\,$
(3)

where $\mu$ is the static shear modulus and $\eta$ the shear viscosity.

Let us now consider a generic operator $\phi$ and its conjugate external field
$\delta h$. We define the dynamic response function $\chi_{\phi\phi}$
associated to such operator as:

$\delta\langle\phi\rangle(\omega,k)=\chi_{\phi\phi}(\omega,k)\,\delta
h(\omega,k)$ (4)

where linear response is assumed as well as time/space translational
invariance. The response function is complex-valued,
$\chi=\chi^{\prime}+i\chi^{\prime\prime}$ and the Kramers-Kronig relation
holds:

$\chi^{\prime}(\omega)=\mathcal{P}\int_{-\infty}^{\infty}\,\frac{\chi^{\prime\prime}(\omega^{\prime})}{\omega^{\prime}-\omega}\,\frac{d\omega^{\prime}}{\pi}.$
(5)

Let us consider the case of transverse phonons and elasticity. Following
standard arguments (cfr. Section 7.3.1 in Ref. Chaikin and Lubensky (1995)),
we obtain:

$\chi_{\textbf{u}_{T}\textbf{u}_{T}}(k,\omega)=\frac{1}{-\rho\,\omega^{2}+\mu\,k^{2}-i\omega\,\eta\,k^{2}}$
(6)

which can be generalized to:

$\chi_{\textbf{u}_{T}\textbf{u}_{T}}(k,\omega)=\frac{1}{\rho}\,\frac{1}{-\omega^{2}+\Omega(k)^{2}-i\omega\,\Gamma(k)}$
(7)

where:

$\Omega(k)^{2}=v_{T}^{2}k^{2}+\dots\,,\qquad\Gamma(k)=D\,k^{2}+\dots$ (8)

and $v_{T}^{2}=\mu/\rho$ and $D=\eta/\rho$ as expected.
Let us go back to (6) and try to expand it at low frequency. We immediately
obtain:

$\chi_{\textbf{u}_{T}\textbf{u}_{T}}(k,\omega)=\frac{1}{k^{2}\,\mu}+i\frac{\eta\,\omega}{k^{2}\,\mu^{2}}+\dots$
(9)

At $\omega=0$, this gives the static susceptibility:

$\chi_{\textbf{u}_{T}\textbf{u}_{T}}(k,\omega=0)=\frac{1}{k^{2}\,\mu}\,,$ (10)

as reported in Eq.6.4.24 in Ref.Chaikin and Lubensky (1995).

Now, comparing (9) with (3), one immediately realizes that the two response
functions are related via:

$\chi_{\sigma_{T}\sigma_{T}}(k,\omega)=\mu^{2}\,k^{2}\,\chi_{\textbf{u}_{T}\textbf{u}_{T}}(k,\omega)$
(11)

which will be explicitly used in the computations which follow (applied to the
nonaffine part of the stress and the displacement).

##  III Nonaffine motions

In solids, each particle is displaced to a position defined by the macroscopic
strain tensor $\mathbf{F}$, according to
$\mathbf{r}_{i}=\mathbf{F}(\gamma)\cdot\mathbf{r}_{i,0}$. This position is
called the _affine_ position. Due to the structural disorder, since also the
nearest neighbours of particle $i$ are being displaced in the local force
field of interaction, the net force acting on $i$ in the affine position is
not zero, due to the absence of inversion symmetry. This force vector is
denoted as $\mathbf{\Xi}_{i}$ and is called the affine force field since it
represents the forces that trigger the nonaffine displacements Lemaître and
Maloney (2006). It can be shown that the mean squared $\mathbf{\Xi}_{i}$ is
proportional to the mean squared nonaffine displacement, where the nonaffine
displacement is defined as $\delta\mathbf{r}_{NA}$ in
$\mathbf{r}_{i}(\gamma)=\mathbf{F}(\gamma)\cdot\mathbf{r}_{i,0}+\delta\mathbf{r}_{NA}$,
which gives the final position of the particle $i$ in the deformed frame.

An efficient way of representing the nonaffine displacements is as follows,
i.e. by defining Lemaître and Maloney (2006):

$\mathbf{r}_{i}=\mathbf{F}(\gamma)\cdot\mathring{\mathbf{r}}_{i}(\gamma)$ (12)

where the new variable $\mathring{\mathbf{r}}_{i}$ does the book-keeping of
the nonaffine displacements in the undeformed configuration. The ring notation
thus indicates that the particle or material point coordinates are measured in
the undeformed frame. In this formalism, we can write the affine force vector
as follows Lemaître and Maloney (2006):

$\mathbf{\Xi}_{i}=-\frac{\partial
U}{\partial\mathring{\mathbf{r}}_{i}\partial\gamma}\bigg{\rvert}_{\gamma\rightarrow
0}.$ (13)

All this can be summarized into an equation of motion for the displacement
$\mathbf{x}_{i}\equiv\mathring{\mathbf{r}}_{i}(t)-\mathring{\mathbf{r}}_{i}(0)$
of a particle $i$ of mass $m$ of the following form Lemaître and Maloney
(2006):

$m\ddot{\mathbf{x}}_{i}+\zeta\dot{\mathbf{x}}_{i}+\mathbf{H}_{ij}\mathbf{x}_{j}=\mathbf{\Xi}_{i}\gamma,$
(14)

with inertial, dissipative and interaction force terms, respectively, on the
left hand side and the affine-force field on the right side. Here,
$\mathbf{H}_{ij}$ is the Hessian matrix that will be defined shortly below.
For ease of notation, we used deformed-frame coordinates. This equation of
motion, Eq. (14), can be derived from a microscopic reversible particle-bath
Caldeira-Leggett Hamiltonian:

$H=\frac{P^{2}}{2M}+V(P)+\frac{1}{2}\sum_{\alpha=1}^{N}\left[\frac{p_{\alpha}^{2}}{m_{\alpha}}+m_{\alpha}\omega_{\alpha}^{2}\left(X_{\alpha}-\frac{F_{\alpha}(Q)}{m_{\alpha}\omega_{\alpha}^{2}}\right)^{2}\right]$
(15)

where the dynamic variables $\\{P,Q\\}$ with no subscript refer to the
“particle” coupled to a bath of harmonic oscillators (the other particles in
the system) labeled with subscript $\alpha$ (for a derivation see the
Supplementary material of Palyulin  _et al._ (2018) and Ref.Cui  _et al._
(2017)). The particle is dynamically coupled to the $\alpha=1...N$ bath
oscillators via a bilinear coupling with coefficients $c_{\alpha}$ contained
in $F_{\alpha}=c_{\alpha}Q$. The bi-linear coupling between the particle and
the oscillators serves as a model to represent the long-range anharmonic
interactions between the tagged particle and the other particles in the
system. Throughout the literature, the Caldeira-Leggett Hamiltonian has also
been shown to provide a mapping onto anharmonic force-fields of real molecules
in liquids Rognoni  _et al._ (2021). Even though the mapping is only partially
accurate, it is nevertheless an effective approximate way of accounting for
anharmonicity. Importantly, it should also be noted that if the coupling
coefficients $c_{\alpha}$ are all identically zero, then also the viscosity of
the system is identically zero, since the friction $\zeta$ in the Eq. (14) can
be shown to vanish identically, and hence also the low-frequency viscosity
$\eta=G^{\prime\prime}\omega$ will vanish altogether Lemaître and Maloney
(2006). This is because the friction $\zeta(t)$ is proportional to
$\sum_{\alpha}c_{\alpha}^{2}$ as shown in Zwanzig (2001). Also, for better
tractability we restrict our analysis to the Markovian limit, in which $\zeta$
is not a function of time. Finally, in the following, we shall take units in
which $m=1$.

Applying Fourier transformation $x_{i}(t)\sim\exp{i\omega t}$ and eigenmode
decomposition of the Hessian matrix $\textbf{H}_{ij}$ with respect to the
eigenfrequencies $\nu_{p}$, the above equation motion leads to the linear
response theory, that was developed in Lemaître and Maloney (2006), and to the
following expression analogue of (2),

$\displaystyle\sigma(\omega)$
$\displaystyle=\chi_{\sigma\sigma}(\omega)\,\gamma(\omega)$ (16)
$\displaystyle=\left(\mu_{A}+\sum_{p}\frac{\hat{\Xi}_{p}^{2}}{\omega^{2}-\nu_{p}^{2}-i\omega\,\zeta}\right)\gamma(\omega)$

where $\zeta$ is the (Langevin-type) microscopic damping coefficient for
particle motion, and $\hat{\Xi}_{p}$ is the projection of the $3N$-dimensional
vector $\mathbf{\Xi}$ onto the $3N$-dimensional eigenvector of the Hessian
matrix $|\mathbf{p}\rangle$, i.e. in Dirac’s bra-ket notation,
$\hat{\Xi}_{p}=\langle\mathbf{\Xi}|\mathbf{p}\rangle$. Furthermore, $\mu_{A}$
is the affine shear elastic modulus, and $\nu_{p}$ denotes the $p$-th
eigenfrequency of the solid, i.e. $\nu_{p}^{2}=\lambda_{p}$, where
$\lambda_{p}$ is the $p$-th eigenvalue associated with eigenvector
$|\mathbf{p}\rangle$. We also omitted prefactors with dimension of volume and
assumed that particles masses are all equal to one. The Hessian matrix is
defined as

$\mathbf{H}_{ij}=\frac{\partial
U}{\partial\mathring{\mathbf{r}}_{i}\partial\mathring{\mathbf{r}}_{j}}\bigg{\rvert}_{\gamma\rightarrow
0}=\frac{\partial
U}{\partial\mathbf{r}_{i}\partial\mathbf{r}_{j}}\bigg{\rvert}_{\mathbf{r}\rightarrow\mathbf{r}_{0}}$
(17)

since $\mathring{\mathbf{r}}(\gamma)\rvert_{\gamma\rightarrow
0}=\mathbf{r}_{0}$. It is a $3N\times 3N$ matrix with $p=1,...3N$ eigenvalues
$\lambda_{p}$ and associated eigenvectors $|\mathbf{p}\rangle$. In (16) it is
clear the existence of two contributions, one coming from affine displacements
and encoded in the affine (infinite-frequency) shear modulus $\mu_{A}$, and a
second term in bracket which arises from _nonaffine_ motions and is controlled
by the quantity $\hat{\Xi}_{p}^{2}$, i.e. the square of the affine force field
in the basis of the eigenvectors of the Hessian. Also, the real part of this
second term in bracket represents the (negative) nonaffine contribution to the
shear modulus, and is often denoted as $\mu_{NA}$ (or, with alternative
notation, $G_{NA}$), defined such that

$\mu=\mu_{A}-\mu_{NA}$ (18)

is the total shear modulus containing both affine and nonaffine contributions.
Importantly, the nonaffine term has potentially a dissipative (damping)
component, as we are going to show next. More precisely, the imaginary part of
the second term in the bracket of (16) is non-zero and it gives a finite
contribution to the sound attenuation constant coming entirely from nonaffine
motion.

Hence, we can identify a susceptibility associated with nonaffine motions, as
follows:

$\chi_{NA}(\omega)=\sum_{p}\frac{\hat{\Xi}_{p}^{2}}{\omega^{2}-\nu_{p}^{2}-i\omega\,\zeta}\,.$
(19)

where from now on the $\sigma$ labels will be omitted.
Nonaffinity is not necessarily tied to anharmonicity. On the contrary, the
atomic-scale friction $\zeta$ and the emergent viscosity associated to it
vanish if the Caldeira-Leggett coupling coefficients $c_{\alpha}$ of
$F_{\alpha}=c_{\alpha}Q$ in Eq.(15), which mimic the long-range anharmonic
interactions in the single particle description, are all identically zero. As
we will see later, the Rayleigh term in the damping does not depend on this
coefficient (and therefore on anharmonicity), consistently with previous
approaches from harmonic disorder/heterogeneous elasticity theory (HET) Ganter
and Schirmacher (2010). Nevertheless, and fundamentally, the induced diffusive
($\sim k^{2}$) contribution in the damping would not be present in the absence
of anharmonicity, thus explaining the reason why in the HET harmonic-disorder
approaches Tomaras  _et al._ (2010); Schirmacher (2006) such a term must be
included forcefully “by hand”.

We can now transform the discrete sum over eigenstates in (19) into a
continuous integral over frequency, by introducing the vibrational density of
states (VDOS) of the solid Lemaître and Maloney (2006). We thus obtain Milkus
and Zaccone (2017):

$\chi_{NA}(\omega)=3\,\rho\int_{0}^{\nu_{D}}\frac{g(\nu)\,\xi(\nu)}{\omega^{2}-\nu^{2}-i\,\omega\,\zeta}d\nu$
(20)

where $\rho=N/V$ is the particle density, $\nu_{D}$ is the Debye frequency
and, following Lemaître and Maloney (2006), we have defined

$\xi(\nu)=\langle\hat{\Xi}_{p}^{2}\rangle_{\nu_{p}\in[\nu+d\nu]}$ (21)

where the average is performed for all the projections of $\mathbf{\Xi}$ on
eigenvectors $|\mathbf{p}\rangle$ with eigenfrequency $\nu_{p}\in[\nu+d\nu]$.
Note that in previous literature $\xi(\nu)$ was denoted as $\Gamma(\nu)$
Lemaître and Maloney (2006); Zaccone and Scossa-Romano (2011), not to be
confused with the sound attenuation constant discussed in this work. Note that
in (20) $\omega$ refers to the frequency of the _external_ “mechanical” source
(in linear response language) $x(t)\sim\exp{i\omega t}$, whereas $\nu$ refers
to the microscopic, _internal_ eigenfrequencies of the solid that arise from
diagonalization of the dynamical matrix $\textbf{H}_{ij}$ (cfr. Eq. 42 in
Lemaître and Maloney (2006)).

In Ref.Zaccone and Scossa-Romano (2011) it was shown that, for amorphous
solids in $d$ space dimensions,

$\langle\mathbf{\Xi}|\mathbf{p}\rangle\langle\mathbf{p}|\mathbf{\Xi}\rangle=d\,\kappa\,R_{0}^{2}\,\lambda_{p}\sum_{\alpha}B_{\alpha,xyxy}$
(22)

where $\alpha=x,y,z$ and $B_{\alpha,xyxy}$ are coefficients that originate
from angular averaging over bond orientation vectors. The scaling with
$\lambda_{p}$ has been verified in numerical simulations of different
disordered and glassy systems in Milkus and Zaccone (2017); Palyulin  _et al._
(2018). Note that
$\mu_{NA}=\frac{1}{V}\sum_{p}\frac{\langle\mathbf{\Xi}|\mathbf{p}\rangle\langle\mathbf{p}|\mathbf{\Xi}\rangle}{\lambda_{p}}$
Lemaître and Maloney (2006); Zaccone and Scossa-Romano (2011).

The above equation therefore implies that
$\xi(\nu)\propto\lambda_{p}\propto\nu^{2}$ and consequently

$\chi_{NA}(\omega)=c\int_{0}^{\nu_{D}}\frac{\nu^{4}}{\omega^{2}-\nu^{2}-i\omega\,\zeta}\,d\nu$
(23)

where we used the Debye VDOS, $g(\nu)\sim\nu^{2}$, and hid all the pre-factors
into an overall parameter $c$ with units Pa$\times s^{3}$ whose physical
interpretation will be clearer in the following. The scaling
$\xi(\nu)g(\nu)\sim\nu^{4}$ has been successfully verified in the simulations
of Szamel and Flenner (2021).

##  IV Sound propagation

Now that we have found the stress-stress auto-correlation function, we can use
it to obtain the transverse displacement auto-correlation function
$\chi_{\textbf{u}_{T}\textbf{u}_{T}}(\omega,k)$ which in the rest of the
manuscript will be indicated as $C(\omega,k)$, and that can be written as
Ziman (1969); Baldi  _et al._ (2011); Xu  _et al._ (2008); Chaikin and
Lubensky (1995):

$C(\omega,k)\,=\,\frac{1}{m}\,\frac{1}{-z^{2}+E(k)^{2}-\Sigma(\omega,k)}$ (24)

where $z\equiv\omega+i\epsilon$ is the complex-valued frequency, $E(k)$ is the
bare phonon energy $E(k)^{2}=v^{2}k^{2}$ (with $v$ the Born speed of
transverse sound in this case), $\Sigma(\omega,k)$ the self-energy and $m$ the
mass density. Whether in the numerator one has $1$ or $2z$ depends on the
arbitrary choice of normalization of the bosonic operators in the Hamiltonian
Khomskii (2010), here we follow the convention of Ziman Ziman (1969). By
setting the self-energy to zero, and solving for the poles of the Green
function in (24), we obtain $\omega=\pm v\,k$, the dispersion relation of the
acoustic phonons in which the effects of nonaffinity are neglected. Again,
explicit indices are not shown and we consider only the transverse phonons
correlation function which corresponds to the shear stress response discussed
before. Upon decomposing the self-energy into its real and imaginary parts,

$\Sigma(\omega,k)=\Sigma^{\prime}(\omega,k)+i\,\Sigma^{\prime\prime}(\omega,k)\,,$
(25)

then the dispersion relation of the sound mode can be obtained as a solution
of the following equation:

$\omega^{2}=v^{2}k^{2}-\Sigma^{\prime}(\omega,k)-i\,\Sigma^{\prime\prime}(\omega,k)$
(26)

which must be compared with the general form:

$\omega^{2}=\Omega(k)^{2}-i\,\omega\,\Gamma(k).$ (27)

where $\Gamma(k)$ is the sound attenuation Chaikin and Lubensky (1995).
A simple comparison implies

$\Omega(k)^{2}=v^{2}k^{2}-\Sigma^{\prime}(vk,k)\,,\qquad\Gamma(k)=\frac{1}{vk}\Sigma^{\prime\prime}(vk,k)\,,$
(28)

where we have substituted $\omega\rightarrow vk$ assuming the limit of small
wave-vector.
In other words, the expressions in (28) are valid only in the region where
$\mathrm{Re}\,\omega=vk$. Equation (28) is in perfect agreement with the
discussion in Szamel and Flenner (2021) and will serve as starting point to
derive the sound damping based on the nonaffine formalism in the next
sections.

###  IV.1 The correction to the shear modulus and the nonaffine dynamic
response function

Finally, as a consistency check of our analysis, we can study the nonaffine
correction to the sound speed given by:

$v^{\prime
2}=v^{2}-\frac{\Sigma^{\prime}(vk,k)}{k^{2}}=\frac{\mu_{A}-\mu_{NA}}{m},$ (29)

and check that it correctly reaches a constant value in the limit
$k\rightarrow 0$. For that to be true, we need that:

$\lim_{k\rightarrow 0}\Sigma^{\prime}(vk,k)\sim k^{2}\,.$ (30)

Now:

$\mathrm{Re}\,C(vk,k)\,=\,-\frac{1}{m}\,\frac{\Sigma^{\prime}(vk,k)}{\Sigma^{\prime}(vk,k)^{2}+\Sigma^{\prime\prime}(vk,k)^{2}}$
(31)

and then:

$\mathrm{Re}\,\chi_{NA}(vk,k)\,\sim-\,\frac{k^{2}\,\Sigma^{\prime}(vk,k)}{\Sigma^{\prime}(vk,k)^{2}+\Sigma^{\prime\prime}(vk,k)^{2}}$
(32)

where all the dimensionful parameters are omitted since not relevant for the
purpose of this analysis.
From (23), we see that in the limit of zero frequency (and therefore
equivalently momentum), the real part of the nonaffine dynamic response
function goes to a negative constant,
$\mathrm{Re}\,\chi_{NA}(vk,k)\,\sim\,-\beta$, with $\beta>0$. Therefore, we
have that:

$\lim_{k\rightarrow
0}\,\frac{k^{2}\,\Sigma^{\prime}(vk,k)}{\Sigma^{\prime}(vk,k)^{2}+\Sigma^{\prime\prime}(vk,k)^{2}}\sim\beta>0$
(33)

Now, we know that $\Sigma^{\prime\prime}\sim k^{3}$, in the regime of small
frequency/momentum, meaning that:

$\lim_{k\rightarrow
0}\frac{k^{2}\,\Sigma^{\prime}(vk,k)}{\Sigma^{\prime}(vk,k)^{2}+k^{6}}\sim\beta>0$
(34)

and therefore, expanding at small wave-vector, we get $\Sigma^{\prime}\sim
k^{2}/\beta>0$ which indeed gives a constant, and importantly a _negative_
correction to the speed of sound and the static shear modulus (see (29)), in
agreement with previous numerical calculations Milkus and Zaccone (2017);
Palyulin  _et al._ (2018); Szamel and Flenner (2021) and theoretical analysis
Lemaître and Maloney (2006); Maloney and Lemaître (2006); Zaccone and Scossa-
Romano (2011).

###  IV.2 The physical meaning of the $c$ parameter

The previous analysis is helpful to re-write the dimensionful parameter $c$
appearing in the main text in terms of more transparent physical quantities.
By restoring all the factors in the above formulas, one obtains that at low
energy:

$\mathrm{Re}\,\chi_{NA}(vk,k)\,=\,-\,\frac{\mu_{NA}^{2}\,k^{2}}{m}\,\frac{1}{\Sigma^{\prime}(vk,k)}\,.$
(35)

Moreover, by using (29), we can rewrite the real part of the self-energy as:

$\Sigma^{\prime}(vk,k)\,=\,\frac{\mu_{NA}}{m}\,k^{2}\,+\,\dots$ (36)

(36) together with (35) imply:

$\mathrm{Re}\,\chi_{NA}(vk,k)\,=\,-\,\mu_{NA}$ (37)

which allows us to fix $c$ as:

$c\,=\,\frac{3\,\mu_{NA}}{\nu_{D}^{3}}\,.$ (38)

The result above will be useful to express the sound attenuation constant in
terms of fundamental physical quantities.

###  IV.3 Sound attenuation

By taking the imaginary part of the auto-correlation function in (24), we
obtain:

$\mathrm{Im}\,C(vk,k)\,=\frac{1}{m}\,\,\frac{\Sigma^{\prime\prime}(vk,k)}{\Sigma^{\prime\prime}(vk,k)^{2}+\Sigma^{\prime}(vk,k)^{2}}$
(39)

where we also used the low-energy expression $\omega=vk$.
As demonstrated in Chaikin and Lubensky (1995) and in (11), a simple relation
between the stress auto-correlation function $\chi_{\sigma\sigma}$ and the
displacements auto-correlation function $\chi_{uu}$ exists. Restricting it to
the nonaffine components, that yields:

$\chi^{NA}_{\sigma\sigma}(\omega,k)=\mu_{NA}^{2}k^{2}\chi^{NA}_{uu}(\omega,k)=\mu_{NA}^{2}\,k^{2}C_{NA}(\omega,k).$
(40)

This (recall that the affine part of $\chi(\omega)$ is independent of $\omega$
Lemaître and Maloney (2006)) finally implies:

$\mathrm{Im}\,\chi_{NA}(vk,k)\,=\,\frac{\mu_{NA}^{2}\,k^{2}}{m}\,\frac{\Sigma^{\prime\prime}(vk,k)}{\Sigma^{\prime\prime}(vk,k)^{2}+\Sigma^{\prime}(vk,k)^{2}}.$
(41)

Now, using (28) together with (18), one can immediately verify that the real
part of the self-energy at low momentum is given by
$\Sigma^{\prime}(vk,k)=\mu_{NA}/m\,k^{2}$. Hence, using
$\Sigma^{\prime\prime}=vk\Gamma$, we get:

$\mathrm{Im}\,\chi_{NA}(vk,k)\,=\,\frac{\Gamma(k)\,k\,\mu_{NA}^{2}\,m\,v}{k^{2}\mu_{NA}^{2}+\Gamma(k)^{2}m^{2}v^{2}}.$
(42)

Now, let us assume that $\Gamma$ scales faster than $k$ (as observed in all
experimental and simulations results), therefore at low $k$ we have:

$\mathrm{Im}\,\chi_{NA}(vk,k)\,=m\,v\,\frac{\Gamma(k)}{k}\,.$ (43)

This clearly implies that a linear in $k$ term in the imaginary part of the
self-energy will produce a diffusive $\sim k^{2}$ in the damping, while a term
$k^{3}$ will produce the Rayleigh damping $\sim k^{4}$, as we are going now to
verify by direct evaluation of the nonaffine theory developed above.

We now recall (20) and perform the integral analytically, obtaining:

$\displaystyle\chi_{NA}(\omega)=$ $\displaystyle
c\int_{0}^{\nu_{D}}\frac{\nu^{4}}{\omega^{2}-\nu^{2}-i\,\omega\,\zeta}d\nu=$
$\displaystyle-\frac{c\,\nu_{D}^{3}}{3}-c\,\nu_{D}\,\omega\,(\omega-i\zeta)$
$\displaystyle+c\,\omega^{3/2}(\omega-i\zeta)^{3/2}\tanh^{-1}\left(\frac{\nu_{D}}{\sqrt{\omega(\omega-i\zeta)}}\right)\,.$
(44)

Let us first discuss the behaviour at the lowest order in frequency. In that
limit, the above integral becomes:

$\chi_{NA}(\omega)=-\frac{c\,\nu_{D}^{3}}{3}+i\,\zeta\,c\,\nu_{D}\,\omega+\dots$
(45)

implying that:

$\Gamma(k)=\,c\,\frac{\zeta\,\nu_{D}}{m}\,k^{2}\,.$ (46)

This result can be simplified further by using (38), leading to:

$\Gamma(k)=\,\,\frac{3\,\zeta\,\mu_{NA}}{m\,\nu_{D}^{2}}\,k^{2}\,.$ (47)

This is an important result: it shows that the coefficient for diffusive sound
damping is proportional to the microscopic Langevin friction $\zeta$, which in
turn is related to anharmonicity, and it is also proportional to $\mu_{NA}$,
meaning that in a perfect centrosymmetric crystal at zero temperature (hence
with $\mu_{NA}=0$), the sound damping is identically zero. This is consistent
with the expectation that sound damping is present only in anharmonic crystals
with defects and/or thermal fluctuations (which also cause nonaffinity Das
_et al._ (2010); Ganguly  _et al._ (2013); Squire  _et al._ (1969)), and in
amorphous solids.

Despite the dependence of the leading quadratic term in Eq.(47) on the
frictional parameter $\zeta$ might appear surprising, it is not. Indeed, it is
well known that the diffusive damping in solids is proportional to the
viscosity $\eta$ (see for example Chaikin and Lubensky (1995)),
$\Gamma\sim\eta k^{2}$. The latter grows with the internal friction parameter
$\zeta$ leading to $\Gamma\sim\zeta\,k^{2}$ as derived in Eq.(47). From a
physical perspective, it is expected that a larger sound attenuation
corresponds to larger internal friction, as mathematically derived in Eq.(47).

Let us emphasize, that the presence of a diffusive attenuation constant
$\Gamma\sim Dk^{2}$ for the collective excitations does not indicate
dissipation of energy and/or momentum which are perfectly conserved quantities
in the full system. This term is “dissipative” in the sense that it breaks
time-reversal invariance (reversibility) and it is connected with entropy
production. This is consistent with the discussion that the Reader can find in
Brazhkin and Trachenko (2014).

Hence, we find a contribution from nonaffine motions to the ubiquitous
_diffusive_ damping, which has been observed in countless experimental and
simulation studies of glasses, liquids and supercooled liquids Rufflé  _et
al._ (2006); Shintani and Tanaka (2008); Hansen and McDonald (2006).
Importantly, the microscopic theory explains that the nonaffine contribution
to the diffusion coefficient $D$ in $\Gamma_{\textrm{diff}}=Dk^{2}$ is
proportional to the microscopic damping coefficient for particle motion,
$D\propto\zeta$, which clarifies the close connection of the diffusive damping
with anharmonicity of particle motion. The Langevin-type damping $\zeta$ can
be shown, through particle-bath models of the Caldeira-Leggett type Zwanzig
(1973), to arise from long-range interparticle interactions, which in
condensed matter states are due to the long-range anharmonic part of the
interparticle potential Palyulin  _et al._ (2018); Rognoni  _et al._ (2021).
The most general derivation yields a non-Markovian friction coefficient
$\zeta(t)$ which, however, reduces to a Markovian, time-independent
coefficient $\zeta$, just as the one used here, when the coupling coefficients
between each particle and the other oscillators are all the same Zwanzig
(1973); Zwanzig and Mountain (1965). This suffices for the sake of our
analysis and the more complex non-Markovian case is left for future studies.

In general, the expansion of the expression in (44) is complicated and not
particularly illuminating, nevertheless it becomes rather simple in the limit
of small microscopic friction, $\zeta\ll 1$. The imaginary part of the
$\tanh^{-1}$ in the regime of $\omega\gg\zeta$ and $\omega\ll\nu_{D}$ (small
friction and low frequency) is approximately constant and equal to $-\pi/2$,
leading to:

$\begin{split}\mathrm{Im}\chi_{NA}(\omega)=c\,\nu_{D}\,\omega\,\zeta+\,c\,\frac{\pi}{2}\,\omega^{3}+\dots{}\end{split}$
(48)

This result implies a smooth crossover between a diffusive damping at low
wave-vector, $\Gamma(k)\sim k^{2}$, and a Rayleigh one, $\Gamma(k)\sim k^{4}$,
at larger wave-vector:

$\Gamma(k)=\Gamma_{2}k^{2}+\Gamma_{4}k^{4}\,.$ (49)

Importantly, the weight of the diffusive term at low wave-vector is controlled
by the size of the microscopic Langevin friction $\zeta$:

$\frac{\Gamma_{2}}{\Gamma_{4}}=\frac{2}{\pi}\frac{\nu_{D}\,\zeta}{v^{2}}$ (50)

and it vanishes in absence of microscopic friction, $\zeta=0$.

Figure 1: The imaginary part of the nonaffine stress autocorrelation function
as a function of the frequency $\omega$. The different colors represent
different values of the microscopic damping for particle motion, $\zeta$. The
dashed line guides the eye towards the Rayleigh damping scaling. The different
regimes of sound damping $\Gamma(k)$ predicted by the theory (via Eq.(42)) are
indicated. The dimensionful parameter $c$ is set to unity.

The behaviour of the imaginary part of the nonaffine stress correlation
function is shown in Fig. 1 for different strengths of the microscopic damping
$\zeta$. The full function displays a smooth crossover from a diffusive regime
($\mathrm{Im}\chi\sim\omega$) to a Rayleigh regime
($\mathrm{Im}\chi\sim\omega^{3}$), whose location is controlled by the value
of $\zeta$. The larger the $\zeta$, the more extended (and therefore more
important) the diffusive regime. At very low values of the microscopic
damping, the function is well approximated by (48) and the diffusive regime is
pushed to very low frequency, whereas the Rayleigh one becomes more
predominant.

For completeness, the low frequency behaviour of the nonaffine stress
autocorrelation function is shown in Fig.2 for different values of the damping
$\zeta$. Interestingly, the correction to the shear modulus at finite (but
low) frequency decreases with the damping $\zeta$, while the sound attenuation
constant increases with it.

Figure 2: Real and imaginary parts of the nonaffine stress autocorrelation
function at low frequency. Different colors represent different values of the
microscopic damping $\zeta$. The dimensionful coefficient $c$ is set to unity.

From our analytic expression, we can also derive, in the small damping limit
$\zeta\ll 1$, the contribution from nonaffinity to the Rayleigh $\sim k^{4}$
term in the sound attenuation, the prefactor of which is given by:

$\Gamma_{4}=\frac{3\pi}{2}\,\frac{v^{2}\,\mu_{NA}}{m\,\nu_{D}^{3}}$ (51)

where $\mu_{NA}$ is the nonaffine part of the shear modulus and $m$ is the
mass density. Interestingly, this contribution does not vanish in the limit
$\zeta\rightarrow 0$ and it is therefore present even at $T=0$, as directly
showed in the simulations of Ref.Szamel and Flenner (2021).

Finally, in the limit of small microscopic friction $\zeta$, our theory
provides an analytic estimate of the crossover point $k_{*}$ between the
quadratic and quartic scaling in (49) given by:

$k_{*}^{2}\,=\,\frac{2\,\nu_{D}\,\zeta}{\pi\,v^{2}}\,.$ (52)

As evident from this expression, upon increasing the temperature, the
microscopic friction parameter $\zeta$ grows and the quadratic diffusive
regime extends up to larger values of the wave-vector $k$. This behaviour is
apparent in Fig.1.

##  V Discussion

In summary, we presented an analytical theory of sound attenuation in
amorphous solids starting from single particle motion and taking into account
the inherently nonaffine dynamics. We derived in closed form the contributions
from nonaffinity to both the hydrodynamic diffusive term $\sim\Gamma_{2}k^{2}$
(47) and the Rayleigh term $\sim\Gamma_{4}k^{4}$ (51), with the prefactors
$\Gamma_{2},\Gamma_{4}$ expressed in explicit form as functions of important
physical parameters. The Rayleigh attenuation due to nonaffine motions, (51),
survives in the limit of zero temperature, or equivalently zero microscopic
friction $\zeta=0$, and it has been recently shown via simulations to be the
dominant process of sound attenuation in amorphous solids Szamel and Flenner
(2021), while the harmonic disorder/HET contribution to the same scaling plays
a comparatively very minor role. Furthermore, our theory analytically predicts
the crossover from diffusive to Rayleigh damping at larger wavevector (52),
which was trivially obtained before only from ad-hoc combinations of continuum
HET and anharmonic theory, and neglecting the microscopic nonaffine dynamics
Tomaras  _et al._ (2010). Crucially, Refs. Tomaras  _et al._ (2010);
Schirmacher (2006) are theories limited to the continuum level because their
input are the fluctuations of the elastic moduli, whereas the micro-physics of
the elastic moduli (i.e. how they are related to atomic-level displacements,
bonding and atomic-level structure) is completely neglected. Here, instead we
have did consider the microphysics of the moduli, i.e. their full eigenmode
decomposition into microscopic atomic vibrations, and, crucially the atomic
nonaffine displacements which prove key to demonstrate the origin of sound
attenuation at the microscopic level.

The analytical prediction of Rayleigh damping from nonaffinity is fully
supported by recent numerical data Szamel and Flenner (2021), which also prove
its fundamental importance and dominance, with respect to the effects of
harmonic disorder/HET, in the determination of sound attenuation in amorphous
solids. Additionally, our main result in (47), namely the low-$k$ sound
attenuation constant being quadratic in the frequency (or wave-vector) and
increasing with the microscopic damping $\zeta$, is confirmed by numerical
simulations in Saitoh and Mizuno (2021) where the internal friction is
parametrized in terms of the inelasticity.

###### Acknowledgements.

We thank Giorgio Frangi, Bingyu Cui and Dmitry Parshin for reading an early
version of the manuscript and for providing useful comments. We thank Lijin
Wang, Grzegorz Szamel and Elijah Flenner for correspondence and useful
comments which helped us to improve our manuscript. A.Z. acknowledges
financial support from US Army Research Office, contract nr. W911NF-19-2-0055.
M.B. acknowledges the support of the Shanghai Municipal Science and Technology
Major Project (Grant No.2019SHZDZX01).

##  DATA AVAILABILITY

The data that supports the findings of this study are available within the
article.

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