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# Stringlet Excitation Model of the Boson Peak

Cunyuan Jiang  School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  Wilczek Quantum Center, Shanghai Jiao Tong University,
Shanghai 200240, China  Matteo Baggioli  School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China  Wilczek Quantum Center,
Shanghai Jiao Tong University, Shanghai 200240, China  Jack F. Douglas

###### Abstract

The boson peak (BP), a low-energy excess in the vibrational density of states
over the phonon Debye contribution, is often identified as being a
characteristic of amorphous solid materials. Despite decades of efforts, its
microscopic origin still remains a mystery and a consensus on its theoretical
derivation has not yet been achieved. Recently, it has been proposed, and
corroborated with simulations, that the BP might stem from intrinsic localized
modes which involve string-like excitations (“stringlets”) having a one-
dimensional (1D) nature. In this work, we build on a theoretical framework
originally proposed by Lund that describes the localized modes as 1D vibrating
strings, but we specify the stringlet size distribution to be exponential, as
observed in independent simulation studies. We show that a generalization of
this framework provides an analytical prediction for the BP frequency
$\omega_{BP}$ in the temperature regime well below the glass transition
temperature in both 2D and 3D amorphous systems. The final result involves no
free parameters and is in quantitative agreement with prior simulation
observations. Additionally, this stringlet theory of the BP naturally
reproduces the softening of the BP frequency upon heating and offers an
analytical explanation for the experimentally observed scaling with the shear
modulus in the glass state and changes in this scaling in simulations of
glass-forming liquids, which are normally performed above the glass-transition
temperature, $T_{g}$. Finally, the theoretical analysis highlights the
existence of a strong damping for the stringlet modes above $T_{g}$, which
leads to a large low-frequency contribution to the 3D vibrational density of
states, observed in both experiments and simulations.

## Introduction

Amorphous solids exhibit a variety of characteristic anomalies in their
vibrational, thermodynamic and transport properties, when compared to the more
familiar case of ordered crystalline matter [1, 2]. Among the different, and
possibly universal [3], anomalous properties of glassy materials, the boson
peak (BP) is probably the most commonly discussed and controversial. This peak
refers to an excess in the vibrational density of states (VDOS) normalized by
the phonon density of states of Debye theory, $g(\omega)/\omega^{d-1}$, where
$d$ is the number of spatial dimensions and $\omega$ the frequency. The BP is
ubiquitously observed in amorphous systems [4] and recently this feature has
also been observed in crystalline systems at finite temperature (e.g., [5,
6]), so that it is unclear whether it arises from structural disorder [7].

Several theoretical models have been proposed to rationalize the BP anomaly in
glasses [8, 9, 10, 11, 12, 13, 14, 15, 16, 7, 17, 18, 19, 20, 21], but a final
verdict has not yet been reached. A common idea is that the BP represents a
signature for the emergence of additional vibrational modes coexisting, but
distinct from phonons. These “excess modes” have been suggested to arise from
both structural disorder or from the anharmonicity in intermolecular
interactions inherent to the liquid state and heated crystals. These commonly
observed vibrational modes are expected to be effectively localized as they
involve a rather limited number of atoms or molecular segments in complex
molecules [14, 15, 22, 23, 24, 25, 26, 27, 28]. At large enough wave-vector,
they also interact with phonons, making them overdamped, i.e., reaching the
Ioffe-Regel (IR) limit [29, 30, 31], leading to hindered thermal transport
because of the consequent strong phonon scattering.

Despite numerous experimental and simulation studies, a fully predictive
theoretical model explaining the existence of these modes and how they exactly
give rise to the observed boson peak remains elusive. Recently, it has been
recognized that, on atomic scales, localized string-like excitations that
involve reversible particle exchange motion in the form of linear polymeric
structures exist in glass-forming liquids. These dynamic structures have been
termed “stringlets” [32] (see Fig.1 for a visual representation from
simulations). Moreover, it has been shown [33, 32] that these stringlets are
the dominant contribution to the boson peak and grow upon heating, leading to
the material softening upon heating. A similar phenomenon was originally
observed in the glassy interfacial dynamics of crystalline Ni nanoparticles in
their “premelted” state, where anharmonic interactions in the crystalline
material are prevalent [33, 34]. The existence of these structures and their
significance for the boson peak have been confirmed in both 2D and 3D
amorphous systems by Hu and Tanaka [35, 36]. Moreover, Betancourt et al. [37,
38] have shown a direct relation between the stringlets and string-like
collective motion that has often been observed in connection with activated
transport occurring on a much longer timescale. A general relation between the
stringlet size and the mean square displacement on the fast $\beta$-relaxation
time, a quantity that has been related to the long time $\alpha$-relaxation
time, has been observed as well [39]. Recently, Hu and Tanaka confirmed and
extended these relations between the fast dynamics and $\alpha$-relaxation
[35, 36]. Additionally, it has been demonstrated that the excess modes seem to
have a predominantly transverse nature, and it has also been noted that such
1D vibrational modes are consistent with the linear frequency dependence of
the light-vibrational coupling parameter above the BP frequency, an important
relation for density of states estimations from low-frequency Raman scattering
data [40] (see also [41]). These observations and their interpretation are
broadly in accord with the idea that topological “defects” can give rise to
the BP in glasses [42, 43], and also with the related suggestion that these
excitations are important for the plasticity and yield strength of glassy
materials [44, 45, 46, 47].

Figure 1: Left panel: A concrete visualization of string-like (“stringlets”)
excitations involving cooperative particle exchange motion, and corresponding
to localized modes having a frequency around the BP value. Figure taken with
permission from [36]. Right panel:  A stringlet atomic configuration of the
Al-Sm metallic glass system [32] at $550$K.

We note that previous models [48, 35, 36] phenomenologically assumed that the
excess modes responsible for the boson peak involve structures with a log-
normal distribution, but the physical basis of this assumption has been
completely unclear. Lund [49] recently sought to describe these excess normal
modes using a continuum solid model in which a random distribution of line
“defects” or “elastic strings” was assumed to exist based on simulation
observations. Lund’s model is an extension of the elastic string model of
Granato and Lücke [50], introduced long ago in the description of the
plasticity in highly defective crystals, where the strings correspond to
pinned segments of dislocation lines in an extended dislocation network. We
note that an even earlier string model of the boson peak was proposed by
Novikov [51]. Lund’s model [49] was able to rationalize many of the anomalous
features of glasses, including the presence of a BP anomaly, Rayleigh sound
attenuation constant, a negative sound dispersion for acoustic phonons with a
minimum near the BP, and the saturation of the IR limit for transverse
phonons. Nevertheless, the application of this model [52] to estimate the
length distribution $p(l)$ of these hypothetical string-like structures, based
on experimental observations of the VDOS in glycerol and silica, unfortunately
led to a string length distribution inconsistent with simulation observations
[32, 35, 36], which appears to universally follow an approximate exponential
form, $p(l)\approx\exp(-l/\lambda)$. We address this inconsistency below.

Here, we revisit the Lund string model of the boson peak [49] and extend it to
make a parameter-free prediction of $\omega_{BP}$ in both 2D and 3D for
amorphous solids. In contrast to the work of Lund, however, we assume an
exponential form for the string length distribution $p(l)$, at the outset to
be consistent with simulation observations [32, 35, 36], and we further assume
that the average stringlet vibration speed equals the transverse sound
velocity to uniquely fix our model parameters. This model leads an extremely
simple analytic expression for the BP frequency without any fitting
parameters, which appears to be highly predictive in the the low temperature
glass state, where many boson peak measurements are normally performed. In
particular, our predictions are in good agreement with the experimental
results for the BP frequency at $T=0$, with an uncertainty between $1\,\%$ and
$5\,\%$. The remarkably simple model also captures the correct qualitative
trend at high temperatures where these vibrational modes begin to become
overdamped. Interestingly, we are also able to reproduce the softening of the
BP frequency with temperature and to explain the experimentally observed
scaling of $\omega_{BP}$ with the shear modulus $G$ [53]. Our theory suggests
that a strong damping of the stringlet modes is required above $T_{g}$ to
describe simulation observations. As a by-product, the introduction of a
friction term for the string vibrations produces a low-frequency divergence in
the VDOS normalized to the Debye law, $g(\omega)/\omega^{d-1}$, which is
ubiquitously observed in experimental and simulation data at finite
temperature [54, 55] (resembling similar features in anharmonic crystals and
incommensurate structures), and usually not discussed in most of the
traditional theories for the BP (see [56, 57, 58, 59, 60] for some
exceptions).

## The stringlet model

We initially directly follow Lund et al. [49, 52], and model the stringlets as
1D vibrating strings that are anchored at both ends. Similar results follow by
assuming that the strings are free at both ends, since, in both cases, the
vibrations would be characterized by a fundamental wave-vector $k=\pi/l$, with
$l$ the length of the string. As noted before, 1D vibrational modes being
excited by light are consistent with the low-frequency Raman scattering
observations on glasses, where a linear scaling for the light to vibration
coupling coefficient $C(\omega)$ has been found [40]. According to this simple
continuum based model, which is agnostic to the actual physical origin of
these vibrational modes, each stringlet of length $l$ has a corresponding
frequency of vibration given by,

$\omega_{l}=\sqrt{\left(\frac{\pi
v}{l}\right)^{2}-\gamma^{2}}\equiv\sqrt{\omega_{0}^{2}-\gamma^{2}}$ (1)

where $v$ is the speed of propagation of the stringlet wave, and $\gamma$
represents a “friction parameter” (see [49, 52] for details). Eq.(1) is based
on the assumption that only the fundamental frequency of vibration of the
stringlet is relevant, i.e., higher harmonics are neglected. We consider a set
of stringlets whose length follows a distribution denoted as $p(l)$. Based on
this assumption, the stringlet density of states $g_{s}(\omega)$ can readily
be shown to equal,

$g_{s}(\omega_{l})\,d\omega_{l}=(d-1)\,p(l)\,dl$ (2)

where $d$ is the spatial dimension. In three dimensions ($d=3$), the factor of
$d-1=2$ in the r.h.s. of Eq.(2) reflects the ability of the stringlets to
oscillate along two linearly independent directions that are perpendicular to
its equilibrium orientation. Formally, this framework is equivalent to
considering a distribution of damped oscillators with frequency $\omega_{0}$
and damping rate $\gamma$. We can then define the quality factor
$Q=\omega_{0}/\gamma$ according to which $Q\ll 1$ corresponds to underdamped
vibrations and $Q$ of $\mathcal{O}(1)$ to overdamped ones. Notice that a
priori $Q$ depends on the stringlet length $l$ since $\omega_{0}$ exhibits
this property.

We next introduce the new assumption – with respect to [49, 52] – into this
type of quasi-continuum framework that is an exponential distribution for the
stringlet length,

$p(l)=p_{0}\,e^{-l/\lambda}$ (3)

which is motivated directly by simulation observations [32, 35, 36]. Here,
$\lambda$ is the average stringlet length. Notice that this distribution is
different from what assumed in [52], where a localized Gaussian-like
distribution of lengths is considered (see Fig.7 therein). As shown in [32,
35, 36], this assumption is inconsistent with the simulation data. At this
point, using Eq.(1) together with Eq.(2), we can obtain analytically the
stringlet VDOS,

$g_{s}(\omega)=p_{0}\,\pi(d-1)\,v\,\frac{\omega\,e^{-\frac{\pi
v}{\lambda\,\sqrt{\gamma^{2}+\omega^{2}}}}}{\left(\gamma^{2}+\omega^{2}\right)^{3/2}}\,.$
(4)

In the limit of zero damping, well below $T_{g}$, this expression becomes
particularly simple,

$\lim_{\gamma\rightarrow
0}g_{s}(\omega)=p_{0}\,\pi(d-1)\,v\,\frac{\,e^{-\frac{\pi
v}{\lambda\,\omega}}}{\omega^{2}}\,.$ (5)

From Eq.(4), we can see that in two spatial dimensions the Debye normalized
stringlet VDOS $g_{s}(\omega)/\omega$ has a maximum $\omega_{max}$,

$\omega_{max}^{2D}=\frac{1}{3}\sqrt{\pi^{2}v^{2}/\lambda^{2}-9\gamma^{2}},$
(6)

which in the limit of zero damping becomes simply,

$\omega_{max}^{2D}(\gamma=0)=\frac{\pi}{3}\,\frac{v}{\lambda}.$ (7)

In 3D, there is also an analytical maximum in $g_{s}(\omega)/\omega^{2}$, but
the resulting analytic expression is somewhat complicated (see Supplementary
Information, SI). The effect of the friction $\gamma$ is always that of
softening the bare prediction, as shown explicitly for the 2D case in Eq.(6).
The 3D solution simplifies in the zero damping limit, for which we obtain the
concise result,

$\omega_{max}^{3D}(\gamma=0)=\frac{\pi}{4}\,\frac{v}{\lambda}.$ (8)

Finally, we can consider the full vibrational density of states $g(\omega)$,
including the phononic Debye contribution. We then arrive at the simple
relation,

$g(\omega)=g_{\text{Debye}}(\omega)+g_{s}(\omega),$ (9)

where, in 3D for example,

$g_{\text{Debye}}(\omega)=\frac{\omega^{2}}{2\pi^{2}\bar{v}^{3}},\quad\text{and}\quad\frac{1}{\bar{v}^{3}}=\frac{1}{v_{L}^{3}}+\frac{2}{v_{T}^{3}},$
(10)

where $v_{L},v_{T}$ are respectively the speed of longitudinal and tranverse
acoustic phonons. The arguments above indicate that the boson peak frequency
$\omega_{BP}$ coincides with the location of the maximum in the stringlet VDOS
$g_{s}(\omega)$.

We notice that a similar expression for the BP frequency as in Eqs.(7)-(8) was
suggested before by Granato [61], Hong et al. [54, 62] and Kalampounias et al.
[63] on a more heuristic basis (see also [64, 65]). The length-scale appearing
in Eqs.(7)-(8) has been interpreted in recent work as a “cooperativity length
scale” $\xi$ [54, 62] and there is some evidence that this characteristic
length defined from an equation of the assumed form of Eqs.(7) and (8)
increases with temperature, as observed for the stringlet length $\lambda$ in
simulations [32, 34]. Independently of these experimental observations on the
boson peak, string-like collective motion whose extent grows with temperature
has been inferred from low angle inelastic coherent neutron scattering
measurements [66]. Notice that the numerical prefactor in Eqs.(7)-(8),
defining the characteristic length associated with the boson peak, is
completely unspecified in previous studies, while it is specified exactly in
our theory.

In summary, this simple model exhibits three striking predictions. (I) It
gives an analytical and simple estimate for the BP frequency in both 2D and 3D
systems, Eqs.(7)-(8). These results imply that the BP in a 3D system should
appear at lower energies compared to a 2D system with the same length
distribution and sound speed. (II) If we reasonably assume that the speed of
the stringlet excitation is the same of that of transverse phonons, this model
immediately predicts that $\omega_{BP}\sim G^{1/2}$ in the low temperature
glass state where the stringlet length $\lambda$ is independent of
temperature. (III) At highly elevated temperatures where we may expect the
glass to become a liquid, however, the stringlet theory interestingly predicts
a linear frequency dependence of the VDOS, and thus leads to a $1/\omega$
divergence of the reduced density of states, the density of states normalized
by the Debye density of states. This scaling is consistent with recent
modeling of the density of states so that the stringlet model appears to
recover a physically correct behavior in the high temperature simple liquid as
well as in the low temperature glass states.

Figure 2: The comparison between the analytical theoretical predictions and
the simulation results for the 2D power law (2DPL) model in [35] (left panel)
and the 3D inverse power law model (3DIPL) in [36] (right panel). The black
curve is the analytical theoretical formula, Eq.(4), with zero damping,
$\gamma=0$. The vertical green line locates the experimental value for the BP
frequency. The inset shows the stringlet length distribution from simulations
as reported in [35, 36] together with the best exponential fit using Eq.(3)
(see Supplementary Information for further details).

## Prediction of the boson peak deep in the glass state

We next consider the capacity of the revised string model of the boson peak to
quantitatively account for the frequency dependence of the boson peak. First,
we obtain the value of the parameter $\lambda$ in Eq.(3) by fitting the
distribution of stringlet length obtained from simulations. Then, we take the
value for the velocity of transverse phonons $v_{T}$ which is obtained by
fitting the dispersion relation obtained from simulations. Once these two
parameters are known, and fixed by the simulations, our theory provides a
parameter-free prediction for the BP frequency in both 2D and 3D amorphous
solids, Eqs.(7)-(8). We can then directly compare the predicted value for
$\omega_{BP}$, which we will denote as $\omega_{BP}^{th}$, with the
experimental numbers obtained from the simulation data, which we will label as
$\omega_{BP}^{exp}$.

We start by considering the simulation data for 2D and 3D zero temperature
glasses presented in [35, 36]. In Fig.2, we present two explicit examples
performed using the data for the 2D power law model (2DPL) in [35] and the 3D
inverse power law model (3DIPL) in [36]. By plausibly assuming that the
average velocity of the stringlet propagation coincides with the speed of
transverse sound, we obtain a prediction for the BP of the 2D glass given by
$\omega_{BP}^{th}\approx 1.344$ which has to be compared with the reported
simulation value $\omega_{BP}^{exp}=1.41$. The difference between the two is
less than $5\,\%$.

For the 3D system, the agreement is even superior, showing $\approx 1\,\%$
uncertainty. The predictions of the theory and the compared experimental
values are reported in Table 1.

| $\lambda$ | v_T |  $\omega_{BP}^{th}$ | $\omega_{BP}^{exp}$ | $\Upsilon$
---|---|---|---|---|---
3DIPL [36] | $1.335$ | 4.23 |  $2.487$ | $2.46$ | $1\,\%$
2DPL [35] | $3.717$ | 4.78 |  $1.344$ | $1.41$ | $4.7\,\%$
Table 1: The theoretical prediction versus the experimental value for the BP
frequency using the exponential fit, Eq.(3). The uncertainty $\Upsilon$ is
defined as
$\Upsilon=2|\omega_{BP}^{th}-\omega_{BP}^{exp}|/(\omega_{BP}^{th}+\omega_{BP}^{exp})$.
All quantities presented here are in reduced Lennard-Jones units.

The estimates from the theoretical model are in remarkable agreement, i.e.,
agreement to within $5\,\%$ uncertainty, with the value for the BP frequency
obtained from simulations. It is emphasized that, once the distribution of
length is extracted from the experimental data and the value of the speed of
transverse sound is taken from the simulation dispersion relation, the theory
has no free parameters.

A more careful analysis reveals, however, that the fit to an exponential
becomes rather uncertain for longer strings which are relatively rare in the
simulation sampling, i.e., $l>15$ in reduced LJ units (see insets in Fig.2).
This was already observed in [67], where a combination of a power-law with an
exponential was considered and shown to fit the data better. In order to
investigate this point further, we have performed a second analysis in which
the stringlet length distribution $p(l)$ is not fitted with the exponential
form, Eq.(3), but rather taken as an experimental input for the stringlet
theory of the boson peak. The stringlet VDOS can be derived using Eq.(2), even
if not analytically available this time. The theoretical predictions without
using the exponential fitting function are presented in Fig.3 for the 3DIPL
system in [36] and the 2DPL system in [35]. Since the simulation data for the
stringlet length distribution are quite scattered for $l>15$ (see insets in
Fig.3), the corresponding theoretical predictions for $g_{s}(\omega)$ present
large fluctuations at low frequency, below the BP scale $\omega_{BP}$. In
order to obtain a better estimate from theory for $\omega_{BP}$, we fit the
theoretical data with an empirical log-normal function, as suggested in [48].
Suggestively, this commonly utilized phenomenological fitting function
captures the “shape” of the boson peak rather well, supporting the consistency
of the stringlet theory with experimental observations, , although we are
doubtful about any particular physical significance of the log-normal function
itself. The maximum of the fitted log-normal function for the 3DIPL system
[36], which serves to specify the boson peak frequency following experimental
practice to the value $\omega\approx 1.94$, accords with the simulation value
$\omega_{BP}^{exp}=2.46$ to within a rather large uncertainty of $23\,\%$.
Following the same procedure for the 2DPL system [35], we find $\omega\approx
1.04$ and a corresponding $\approx 30\,\%$ apparent deviation from from the
simulation value $\omega_{BP}^{exp}=1.41$.

Figure 3: The theoretical predictions for the 2DPL system in [35] (left panel)
and the 3DIPL system in [36] (right panel) compared to the simulation results
without using the exponential fit in Eq.(3). The inset shows the stringlet
length distribution obtained from simulations, and used as an input in the
theory. To give a comprehensive view, we plot $p(l)$ in linear scale for the
2DPL system and in log-linear scale for the 3DIPL one. The black symbols are
the theoretical estimates. The thick blue line is a fit of the theoretical
points using a log-normal function. The vertical green line is the simulation
value for the BP frequency.

In summary, we find that using the stringlet simulation data, with no
exponential fit, leads to appreciable uncertainty in the estimation of the
boson peak because of large uncertainties in extracting $p(l)$ for large $l$.
Thus, the theoretical predictions of the boson peak based on the stringlet
model using the exponential distribution and Eqs.(7)-(8) are advised with
respect to quantitatively estimating the boson peak from the simulation data
for $p(l)$.

## Damped dynamics in cooled liquids above the glass transition

Inspired by these positive outcomes in the comparisons of the stringlet model
to simulations deep in the glass state, we next consider simulations for a
simulated Al-Sm metallic glass-forming material in a temperature range above
$T_{g}$ discussed previously in [32] (see also [68]). By using the
experimental values reported in Table 2, see also Fig.S1 in the Supplementary
information (SI), we are again able to provide a parameter-free theoretical
prediction for the BP frequency in the limit of zero damping, which is shown
in the last column of the same table.

Al-Sm [32] | $\lambda$ | v_T |  $\omega_{BP}^{exp}$ |  $\omega_{BP}^{th}$ $(\gamma=0)$
---|---|---|---|---
$T=450$K | $0.930$ | 2151 |  $2.681$ | $7.52$
$T=500$K | $1.018$ | 2110 |  $2.558$ | $6.74$
$T=550$K | $1.132$ | 2043 |  $2.348$ | $5.87$
$T=600$K | $1.269$ | 1968 |  $2.329$ | $5.04$
$T=650$K | $1.460$ | 1851 |  $2.226$ | $4.12$
Table 2: The parameters for the Al-Sm system at finite temperature [32].
$\lambda$ is reported in nanometers, the velocity in m/s and the frequencies
in meV. The last column indicates the theoretical prediction for zero
stringlet damping which has to be compared with the previous column reporting
the corresponding simulation values.

One qualitative finding is evident from Table 2 (see also Fig.S2 in SI).
Simulation observations cannot be described by a theory neglecting stringlet
damping. The theoretical predictions neglecting damping overestimate the
observed values of $\omega_{BP}$ by more than a factor of two. The reason for
this deviation is easy to understand. As opposed to the simulations of Hu and
Tanaka [35, 36], which were performed in the limit of $T=0$ where stringlet
damping can be reasonably neglected, the simulations for the Al-Sm metallic
glass forming material [32] were performed at relatively high temperatures
corresponding to a highly cooled glass-forming metallic glass liquid [32]. At
such temperatures, it is unreasonable to assume $\gamma=0$ even as rough
approximation. The treatment of the boson peak at higher temperatures is
complicated by the fact that the stringlet length, the transverse sound
velocity and the extent of damping described by gamma can all be expected to
vary appreciably with temperature (see Figs. S1 and S2 in the SI for the
temperature variation observed in the Al-Sm metallic glass-forming liquid).
Nevertheless, as we will see below, we can still gain important physical
insights into the variation of the density of states and the boson peak with
temperature by considering the stringlet model with finite damping $\gamma$
and by considering a simplified description for it.

First, for 3D systems, the stringlet model with finite $\gamma$ formally
predicts an interesting feature. In particular, using the stringlet reduced
VDOS presented in Eq.(4), one can derive the low-frequency behavior of the
stringlet VDOS as,

$\frac{g_{s}(\omega)}{\omega^{2}}=\frac{p(0)\,\pi\,(d-1)\,v}{\gamma^{2}}\frac{1}{\omega}+\dots$
(11)

Figure 4: Left panel:  The stringlet vibrational density of states normalized
by Debye law $\omega^{2}$ from Eq.(4). Here, we have fixed $p(0)=v=\lambda=1$
and we move the damping $\gamma\in[10^{-4},0.25]$ from black to red. The
arrows indicate the location of the BP and of the low-frequency $1/\omega$
divergence discussed in Eq.(11). Right panel:  The apparent power law emerging
at low frequency in presence of damping. In this figure, we have fixed
$\lambda=v=p(0)=1$ and dialed the damping $\gamma\in[0,1]$. A Debye term
$g_{Debye}(\omega)=\omega^{2}$ has been added on top of the stringlet
contribution, Eq.(4).

Note also that Eq.(11) implies that the VDOS is linear in frequency at low
energy in the high temperature liquid state so that our stringlet model
exhibits the correct physical behavior in both the low temperature solid and
the high temperature liquid states. This behavior is consistent with the
experimental observations in classical liquids [69, 70, 71] where it has been
rationalized based on the concept of unstable modes [72, 73]. Interestingly,
string-like excitations have been also discussed in glass-forming liquids at
low temperatures [74, 75, 67], so that it would be important to study further
the BP in relation to stringlet dynamics by heating such cooled liquids to
watch how the boson peak evolves in shape and eventually disappears. Notably,
this disappearance of the boson peak upon heating is well-established
experimentally [54, 76, 77, 78, 63, 79]. Similarities to the occurrence of
glassy anomalies in materials undergoing commensurate-incommensurate
transitions [56, 80, 81, 60, 82] and we suggest that the pursuit of this
“analogy” might offer potential insights into the boson peak in glass-forming
liquids.

Additionally, the combination of a Debye term, proportional to $\omega^{2}$,
and a stringlet contribution with finite damping produces an interesting
effect in the total VDOS. In particular, it induces an “apparent power-law”
behavior at low frequency,

$g(\omega)\propto\omega^{\zeta}\qquad\text{with}\qquad 1<\zeta<2\,$ (12)

which is explicitly demonstrated in the right panel of Fig.4. The apparent
power law $\zeta$ gradually moves from the Debye value, $\zeta=2$, for small
damping towards a linear behavior, $\zeta=1$, for very large damping. As a
consequence, for a reasonable value of damping, one expects an apparent power-
law in between these two values. A power $\zeta\approx 4/3$ has been observed
in superionic UO2 heated crystal [83] and a similar power has also been found
experimentally in neutron scattering measurements on a Cu46Zr54 metallic glass
near room temperature [84]. It is tempting to rationalize such an intermediate
power-law using the argument above and ultimately attribute its origin to the
strong stringlet damping emerging at high temperatures. Unfortunately, we are
not aware of a systematic simulation study of this effective power-law in a
wide range of temperatures, which would be necessary to verify our hypothesis.

## A simple physical model for a dimensionless stringlet damping

Here, we introduce a simple physical model of the dimensionless stringlet
damping parameter $\tilde{\gamma}(T)$, defined as the physical damping reduced
by the zero temperature ($\gamma=0$) boson peak frequency, and estimate this
quantity in terms of a readily observable property in both glass-forming and
crystalline materials at elevated temperatures that quantifies the
anharmonicity in interparticle interactions. Our model is based on the
presumption that the apparently universal initial decay of the intermediate
scattering function in both crystalline and glass materials at elevated
temperatures approaching the melting temperature or the glass transition
temperature, respectively, arises from the existence of unstable normal modes
that facilitate the fast relaxation process. These unstable modes in heated
crystalline and glass materials can be calculated from a direct instantaneous
normal mode analysis where it is found that these general classes of materials
appear rather similar from this perspective [85]. Clapa et al. [86] have made
the interesting observation that the ratio of the fraction of localized
unstable modes to stable localized modes in a model binary Lennard-Jones
glass-forming material is rather similar to the ratio of all unstable modes to
stable modes estimated from systematic INM analysis of localized and
delocalized stable and unstable normal modes over a wide range of
temperatures. These findings are relevant to the present analysis since Zhang
et al. [32] suggested the identification of the stringlets with localized
stable modes in the material and that the the relaxational contribution to
fast dynamics was due to the loss of stability of these same modes, where the
relaxation process involves irreversible particle exchange motion rather than
the reversible periodic motion required of stable modes. It is noted that all
modes are stable at zero temperature and this picture of fast dynamics
relaxation simply involves the suggestion that the localized stable modes
found at zero temperature bifurcate into stable and unstable localized modes
fractions at elevated temperature. Mode damping at elevated temperatures is
then a general phenomenon that arises for both the localized stringlet modes
that presumably give rise to the boson peak and structural relaxation on a ps
timescale (“fast relaxation”).

We next translate this qualitative physical picture into a quantitative model
by assuming that the change in the intensity of the normalized intermediate
scattering function, which can be readily estimated from either the
measurement or simulation, directly reflects the fraction of localized stable
modes (stringlets) that have become damped at any given temperature. We
tentatively assume that the dimensionless $\tilde{\gamma}(T)$, i.e., $\gamma$
in Eq. (1) divided by the boson peak frequency at $\gamma=0$, or $T=0$, can be
estimated from the amplitude of the fast beta relaxation process, which is
simply $1$ minus the so-called non-ergodicity parameter describing the
amplitude of the $\alpha$-relaxation process. This estimate of
$\tilde{\gamma}$ has the particular advantage of being readily measured
experimentally.

It is also helpful for our modeling that recent work has shown that the $T$
dependence of the non-ergodicity parameter can be apparently described by a
near-universal function of the mean square particle displacement $\langle
u^{2}\rangle$ at the fast beta relaxation time simply because the non-Gaussian
parameter is inherently small in the fast dynamics regime of relaxation [37,
87]. The model just described leads to a precise estimate of
$\tilde{\gamma}(T)$ given by,

$\tilde{\gamma}(T)=\begin{cases}1-e^{-7.6\,\langle
u^{2}\rangle/\sigma^{2}}\qquad T>T_{o}\\\
0\qquad\qquad\quad\quad\,\,\,\,\,\,\quad T<T_{o}\end{cases}$ (13)

where $\sigma$ the average interatomic distance estimated from the first peak
position of the static structure factor. $T_{o}$ is the temperature at which
$\langle u^{2}\rangle$ extrapolates to zero and it indicates the temperature
at which the glass-formation process “ends” (see Supplementary Information in
[68] for more details). The constant $7.6$ in Eq.(13) derives from the average
interparticle distance of the atoms estimated from the peak of the static
structure expressed in units of sigma and has been validated using MD
simulations in [88]. We notice that $\tilde{\gamma}(T)$ ranges between between
$0$ and $1$, just as the ratio of the fraction of unstable localized and
delocalized modes to their stable counterparts. In particular, the fraction of
unstable modes is likewise known to increase with temperature as anharmonic
interparticle interactions become progressively more prevalent [72], and as
more normal modes corresponding become progressively unstable, i.e., “damped”.
Finally, we note that the expression in Eq.(13) applies quantitatively to
polymer materials having a wide range of structures and a wide range of
crystalline materials.

Figure 5: Left panel:  Model estimate of the dimensionless stringlet damping
$\tilde{\gamma}(T)$ as a function of temperature based on the identification
of the intensity in the fast relaxation process as arising the damping of
localized stable modes identified as being stringlets in previous simulations
[32]. In the inset of the right panel, the data for the mean squared
displacement $\langle u^{2}\rangle$ (in units of $\sigma^{2}$) at the fast
beta relaxation time [37] in the Al-Sm metallic glass system reproduced from
[68]. The horizontal dashed line indicates the critical damping value
$\tilde{\gamma}^{{{*}}}\approx 0.568$ which corresponds to the characteristic
temperature $T^{{{*}}}\approx 848$ K. Right panel: The reduced stringlet
density of states as a function of the frequency normalized by the zero
temperature (or equivalently $\gamma=0$) BP frequency for different
temperatures between $0$ K to $1350$ K (from blue to red). Solid lines are
used to indicate the regimes in which the reduce DOS does not vary much with
temperature. The filled black line is the result at zero temperature or
$\gamma=0$. The dashed black line is the reduced stringlet DOS at
$T^{{{*}}}\approx 848$ K at which the BP simply disappears.

In order to test this model and its predictions we have used the simulation
data for the Al-Sm metallic glass system in [32, 68] for which the temperature
dependence of the mean square displacement is known (see inset in the right
panel of Fig.5). Moreover, we have used the fact that $\sigma\approx
2.8\,\textup{\AA}$ at $500$ K for this material and for simplicity we assume
$\sigma$ is approximately independent of temperature. We plot our estimate of
$\tilde{\gamma}(T)$ in the left panel of Fig.5. We see that
$\tilde{\gamma}(T)$ varies slowly below $\approx 650$ K. It then strongly
increases when the temperature becomes larger, between $650$ K and $1200$ K,
and then plateaus at very high temperatures. Above $\approx 650$ K, $\langle
u^{2}\rangle$ does not vary linearly as for particles subject to harmonic
interparticle interaction, so the large and non-linear variation of $\langle
u^{2}\rangle$ reflected in the strong increase of $\tilde{\gamma}$ provides
clear evidence of large anharmonic interactions in the material where the
boson peak is predicted by this model to become significantly damped. In the
right panel of Fig.5, we show the corresponding reduced stringlet density of
states as a function of the frequency normalized by the $T=0$, or $\gamma=0$,
BP frequency where the dimensionless damping $\tilde{\gamma}$ is modelled
based on Eq.(13).

Evidently, the position of the BP approximately does not vary significantly
with temperature inside the glass state (in Ref.[32], the glass transition
temperature was estimated to be close to $500$ K for the metallic glass under
consideration) and it starts to exhibit appreciable softening only above this
temperature. As noted above, the BP peak disappears at a characteristic
temperature $T^{{{*}}}\approx 848$ K, which corresponds to the analytically
derived critical damping discussed in the previous section. Above $T^{{{*}}}$,
the stringlet excitations are predicted to be fully overdamped and the DOS
then approaches the linear behavior $g_{s}(\omega)\sim\omega$ of a simple
fluid. Notice that at elevated temperatures, above $\approx 1200$ K, the VDOS
does not vary much as a simple consequence of the dimensionless damping
$\tilde{\gamma}$ (left panel of Fig.5) approaching a constant value. All these
features are in good qualitative agreement with experimental and simulation
data. Let us emphasize that the scale at which strong temperature variations
appear should relate to the scale at which strong deviations from the scaling
$\omega_{BP}\sim G^{1/2}$ are observed, suggesting a possible common origin
for these two phenomena.

Let us conclude with a brief discussion about the critical temperature
$T^{{{*}}}\approx 848$. This scale is well above $T_{g}\approx 500$ K, but
close to the onset temperature $T_{A}$ defining the onset of Arrhenius
dynamics for glassy dynamics which has been independent estimated to be around
$900$ K [32]. This temperature exactly corresponds to where the intermediate
scattering function first develops a discernible multi-step decay, and at
which the non-ergodicity parameter becomes measurable. The structural
relaxation time typically becomes on the order of a ps at this characteristic
temperature and non-Arrhenius relaxation, stretched exponential decay, etc.,
also first arise below this temperature. The possibility that all these
phenomena are just downstream consequences of the stringlets acting on ps
timescale is an interesting possibility to consider in the future.

## Outlook

In this work, we showed that a simple theoretical model based on a
distribution of 1D vibrating strings [49] – the stringlet theory of the boson
peak – provides an excellent description of the BP frequency $\omega_{BP}$ for
2D and 3D amorphous systems in the glass state, where the damping of these
excitations is reasonably be assumed to be negligible. In particular, this
theoretical framework gives an analytical, and parameter-free, prediction for
$\omega_{BP}$ that is in quantitative agreement with recent simulation data at
$T=0$ [35, 36]. Moreover, the observed scaling of the BP frequency with the
shear modulus in this type of zero temperature simulation, $\omega_{BP}\sim
G^{1/2}$, follows as an analytic prediction from the stringlet model.

We tentatively explored the stringlet model in the relatively high temperature
regime of glass-forming liquids above $T_{g}$ where some damping of the
stringlet modes can naturally be expected. In this temperature regime, we
addressed the qualitative temperature dependence of the boson peak frequency
and its disappearance at high temperatures, in connection with the dynamic
correlation length introduced phenomenologically in previous work
investigating boson peak measurements [54, 62]. The qualitative softening of
$\omega_{BP}$ upon heating is reproduced by the stringlet model and attributed
mainly to the change of the stringlet length and transverse sound velocity
with temperature. Our qualitative analysis suggests that the stringlet modes
become completely overdamped in the liquid regime at sufficiently high
temperatures. In association with this phenomenon, we observe the emergence of
a density of states that changes to a linear variation with frequency rather
than the quadratic law expected for the Debye theory of solids, a transition
noted in recent studies attempting to define a density of states appropriate
to materials in their liquid state [71, 72, 73, 70]. We finally mention that
recent studies [32, 34, 37, 35] of the boson peak frequency as a function of
temperature show a nearly linear behavior of $\omega_{BP}$ with $G$, rather
than $\propto G^{1/2}$, an observation that offers some clues about the
variation of the boson peak in the temperature regime where the average
stringlet length, the sound velocity and $\gamma$ are all varying in their own
individual ways with temperature. In particular, Zhang et al. [32] indicated
that since the increase of $\langle u^{2}\rangle$ and the occurrence of fast
relaxation first becomes appreciable near the Vogel-Fulcher-Tammann
temperature at which the $\alpha$-relaxation time extrapolates to infinity,
then stringlet damping should also start to arise near this same temperature.
Rossi et al. [89] have observed a deviation of the boson peak frequency from
the Debye frequency at temperatures of about $200$ K below $T_{g}$, roughly
consistent with the arguments of Zhang et al. that softening of the boson peak
occur around $T_{o}$. These and other recent measurements of the pressure
dependence of the boson peak [90, 91], indicating a deviation from the simple
scaling of the boson peak with the transverse sound velocity, also offer some
clues about this more complicated regime for temperatures somewhat below or
above $T_{g}$. Further measurements along this line should be helpful in
understanding the effect of excitation damping on the boson peak.

Our analysis supports the idea that the stringlets correspond to localized
modes, which have long been suggested [32, 35, 36] to be responsible for the
BP anomaly in amorphous systems,. However, the precise nature and origin of
these modes remains an object of continued investigation. This identification
raises questions about how these modes might be related to the multipolar
four-leaf modes [92, 93] that have recently been discussed as being related to
the the emergence of the boson peak. These structures would appear to be much
larger than the stringlets suggesting to us the possibility that that
stringlets may account for their substructure. This possibility also deserves
further investigation.

In order to augment the predictive nature of the stringlet theory, and to
develop it into a full-fledged theory, clearly more work needs to be done.
First, a theoretical model is required for the first-principle prediction of
the stringlet length distribution $p(l)$, without resorting to simulation
data. A promising avenue would be to combine Lund model [49] with the the
well-developed thermodynamic theory of equilibrium polymers discussed in
Refs.[67, 94, 95]. Second, it would be interesting to directly estimate of the
average string length $\lambda$ as a function of temperature, and compare it
with independent estimates of the dynamic correlation length based on
Eqs.(7)-(8), to verify that that this length can be interrelated as
cooperativity length, as proposed heuristically in some experimental studies
[54, 62]. Specifically, Eqs.(7)-(8) are equivalent to the “standard” relation
taken to define the dynamic correlation length by Hong et al. [54, 62] and
others [63], when the parameter $S$ of these experimental studies is taken to
have the specific predicted value, $S=\pi/(d+1)$.

Additionally, we have not considered in the present work the question of the
intensity of the BP and how this relates to the characteristic features of the
stringlets and their size distribution. The universality of the functional
form suggests that the boson peak frequency and the height should be related.
Using Eq.(4) in the limit of negligible damping, $\gamma=0$, one can easily
verify that the intensity of the stringlet contribution at the BP frequency is
given by

$\frac{g_{s}(\omega_{BP})}{g_{Debye}^{T}(\omega_{BP})}\equiv
v_{T}^{d}\frac{g_{s}(\omega_{BP})}{\omega_{BP}^{d-1}}\propto
p(0)\lambda^{d+1},$ (14)

where $g_{Debye}^{T}(\omega_{BP})$ is the Debye contribution arising from the
transverse acoustic modes $\omega^{d-1}/v_{T}^{d}$, up to some neglected
constant numerical factors. It is interesting to notice that, in this
approximation, the intensity is given uniquely in terms of the parameters
appearing in the stringlet size distribution $p(l)$. Also, it is immediate to
verify that the intensity of the BP as in Eq.(14) grows upon heating since the
average length $\lambda$ increases with $T$.

Interestingly, the stringlet dynamics shares many similarities with the
behavior of defects and anharmonic modes in crystals [61, 96] (See [97] for a
direct proof of the stringlet anharmonicity in relation to the BP in glasses.)
and the role of weakly-dispersing soft optical modes, which has been
identified in several instances as the origin for the BP anomaly in
crystalline materials with no structural disorder [98, 81, 80, 56, 6, 5, 99,
100, 57]. A similarity of this kind is also reflected by the fact that the
string model of Lund was inspired by Granato and Lücke string model of crystal
plasticity [50]. A unifying picture based on defect dynamics would be highly
desirable and possible related also to the roton-like excitations in liquids
and glasses [101, 102, 103], as reviewed in [104]. Part of the dispersion
curve with “softer” phonons (like rotons in liquid helium) have been suggested
to be related to the BP anomaly [105, 106] (see also [107]), and might be
connected with the dynamics of stringlets discussed in this work.

Finally, our work revives the long-standing and vigorously debated question of
whether the structures responsible for the BP are structural defects or
dynamical excitations deriving from the anharmonicity of intermolecular
interactions. The evidence would appear to favor the latter interpretation,
explaining why the boson peak should also be expected in crystalline materials
at elevated temperatures where anharmonicity in intermolecular interactions
becomes prevalent as in cooled liquids. Recent simulations [83] have indeed
confirmed that the boson peak also arises in crystalline materials at elevated
temperatures, but below the melting temperature, where anharmonic
intermolecular interactions start to become prevalent.

## Data availability

The datasets generated and analysed during the current study are available
upon reasonable request by contacting the corresponding authors.

## Code availability

The code that supports the findings of this study is available upon reasonable
request by contacting the corresponding authors.

## Acknowledgements

We would like to thank A. Zaccone, J. Zhang and M. A. Ramos for many
discussions about the BP and the vibrational dynamics of amorphous solids. We
are grateful to Y. Wang. and W. Xu for useful comments on a preliminary
version of this draft. We thank P. Jacobson and C. Soles for helpful comments.
We would like to thank H. Zhang for the picture in the right panel of Fig.1
and several fruitful discussions. CJ and MB acknowledge the support of the
Shanghai Municipal Science and Technology Major Project (Grant
No.2019SHZDZX01). MB acknowledges the sponsorship from the Yangyang
Development Fund.

## Author contributions

M.B. and J.F.D conceived the idea of this work; C.Y. and M.B. performed the
theoretical computations and numerical analysis; M.B. and J.F.D. wrote the
manuscript.

## Competing interests

The authors declare that no competing interests exist.

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Supplementary Information

In this Supplementary Information (SI), we:

  * •

provide more details about the analysis of the boson peak at elevated
temperatures in the Al-Sm system of [32];

  * •

present the derivation of the prediction of the critical damping at which the
boson peak disappears within the stringlet model of the BP at finite
temperature;

  * •

provide further details about the exponential fits for the stringlet size
distribution $p(l)$ presented in the main text and discuss possible
improvements.

## A Simple Model of the Boson Peak at Finite Temperature and the Al-Sm
metallic glass-forming material

In this Section, we provide additional details regarding the analysis on the
Al-Sm metallic glass system at elevated temperature [32] for which there is
data for the stringlet length distribution, the speed of transverse sound, and
the BP frequency at different temperatures between $450$ K and $650$ K. The
stringlet length distribution and the average stringlet length $\lambda$ are
shown for different temperatures in the left panel of Fig.S1. Moreover, we
obtained the values for the density $\rho$ and the shear modulus $G$ from [32,
68] and derived the transverse speed of sound using the defining relation:

$v_{T}^{2}=\frac{G}{\rho}.$ (S1)

The data for $v_{T}$ required for our purposes are summarized in Table 2 and
shown in the right panel of Fig.S1.

Figure S1: Left panel: The distribution of the stringlet lengths for the
finite temperature Al-Sm system in [32] and the fits to the exponential form,
Eq.(3) in the main text. The inset shows the dependence of the average
stringlet length $\lambda$ as a function of temperature. Right panel: The
speed of transverse sound, extracted from the simulation data using Eq.(S1),
as a function of temperature. The values of $\lambda$ and $v_{T}$ are reported
explicitly in the main text in Table 2.

Figure S2: Left panel: The phenomenological damping parameter used in the
theoretical predictions at high temperature, above $T_{g}$, for the Al-Sm
system [32]. The colors match those used in the right panel. Right panel: The
comparison between the theoretical predictions and the simulation results for
the finite temperature Al-Sm system in [32]. The black symbols are the
simulation data. The orange line indicates the theoretical results in absence
of stringlet damping, $\gamma=0$. The pink and red line indicate the
theoretical results in presence of damping, as explained in the main text.

Some insight in the damping parameter can be obtained by adopting idealized
assumptions about its temperature dependence and comparing to the observed
behavior of how the boson peak varies with temperature. As a first crude
approximation, we take $\gamma$ to be a constant (pink curve in the left panel
of Fig.S2) and not excessively large in magnitude, i.e., $\gamma$ smaller than
$\omega_{0}$ in Eq.(1) in the main text. We see that this option does not
allow for a reasonable estimate of the BP frequency; see the pink curve in the
right panel of Fig.S2. As it might have been anticipated, damping must be
temperature dependent and relatively large in magnitude at temperatures well
above $T_{g}$. Next, we consider for the sake of argument a temperature
dependent $\gamma$ that is larger in amplitude but that decreases with
temperature (see red curve in the left panel of Fig.S2), leads to a better
accord between the stringlet model and simulation estimates of the boson peak,
see red line in the right panel of Fig.S2, but the agreement is still
inadequate. Moreover, we find it hard to envision a microscopic explanation
for why the damping should decrease with temperature. This is clearly a non-
linear dynamics many-body phenomenon. A damping that is larger in amplitude
and that grows with temperature would then appear to be the choice for the
variation of $\gamma$ but it poses an immediate difficulty. Heuristically,
this problem has a simple origin. A large damping is necessary to soften the
stringlet frequency and consequently the predicted value for $\omega_{BP}$,
which, in absence of damping, is too large compared to the simulation
observations. Nevertheless, a very large damping in Eq.(1) in the main text
clearly makes the stringlet frequency uncertain when $\gamma$, along with the
stringlet length and the transverse sound velocity, all depend on temperature.
Eventually, a too large damping would make the stringlet frequency in Eq.(1)
in the main text complex valued, which is obviously an issue within the model
considered. We leave the task of more quantitatively understanding the
temperature dependence of the boson peak (and its intensity) above $T_{g}$ for
future work.

As a general expectation, however, we infer from our model that a large
variation of $\gamma$ in comparison to $\omega_{0}$, and possibly a situation
in which $\gamma>\omega_{0}$ and $\omega_{l}$ in Eq.(1) in the main text
becomes complex, are required to describe the simulated boson peak
observations of the metallic glass forming liquid at large temperature. This
indicates that the stringlets dynamics ultimately become completely overdamped
in the high temperature liquid regime and the present formalism requires
generalization. In particular, we have assumed that the effects of damping
enter only into the renormalization of the stringlet energy $\omega_{l}$ in
Eq.(1) in the main text. This is perhaps highly questionable at high
temperatures, since the lifetime of the stringlets vibrations is short so that
the imaginary part of their frequency $\omega_{l}$ cannot then be neglected.
As already explained, this assumption is clearly not valid if $\gamma$ is
comparable or larger than $\omega_{0}$ in Eq.(1) in the main text. Among the
other possible limitations of the stringlet model in this high temperature
regime, we have also made the bold assumption that the stringlet damping is
independent of the stringlet length $l$. Again, we emphasize that these and
other issues with the stringlet model will require a thorough examination in
the future.

## Estimate of the critical damping in the stringlet model at finite
temperatures

Here, we derive the expression for the critical damping at which the BP
disappears in the stringlet model of the BP. For simplicity, we do this only
for a 3D system. The computation for the 2D system can be easily performed
following similar steps.

Let us start by the reduced stringlet density of states in three dimensions
derived in the main text, Eq.(4),

$\frac{g_{s}(\omega)}{\omega^{2}}=2\pi\,p_{0}\,v\,\frac{e^{-\frac{\pi\,v}{\lambda\sqrt{\gamma^{2}+\omega^{2}}}}}{\omega\left(\gamma^{2}+\omega^{2}\right)^{3/2}}.$
(S2)

By defining $\omega^{2}\equiv y$, the maximum of such a function, which
corresponds to the location of the BP, is given by the solution of the
following equation

$\pi\,v\,y\sqrt{\gamma^{2}+y}-\lambda\left(\gamma^{4}+4y^{2}+5\gamma^{2}y\right)\,=\,0.$
(S3)

Despite the solution to this equation is complicated, one can easily derive
that such a solution involves the square root of the following quantity:

$\mathcal{F}(v,\lambda,\gamma)\equiv-432\gamma^{4}\lambda^{4}-4\pi^{4}v^{4}+207\pi^{2}\gamma^{2}\lambda^{2}v^{2}\,.$
(S4)

Whenever $\mathcal{F}(v,\lambda,\gamma)<0$, then, the solution becomes complex
and therefore the function in Eq.(S2) does not show a maximum for real values
of the frequency $\omega$. In order to have the solution real, and therefore a
well-defined boson peak, the stringlet damping has to obey the following
inequality

$\gamma<0.466\frac{v}{\lambda}.$ (S5)

This depends explicitly on the value of the stringlet speed and average
length. However, we can obtain a universal result by considering a
dimensionless stringlet damping $\tilde{\gamma}$ defined as the ratio between
the stringlet damping and the BP frequency at $\gamma=0$, or equivalently at
zero/low temperature,

$\omega^{(0)}_{BP}\equiv\frac{\pi}{4}\,\frac{v}{\lambda}.$ (S6)

In doing so, we obtain a universal prediction for the disappearance of the BP
within this simple model. Whenever the dimensionless damping reaches the
critical value:

$\gamma^{{{*}}}\equiv\left(\frac{\gamma}{\omega^{(0)}_{BP}}\right)^{{{*}}}=0.568\,,$
(S7)

the BP then disappears as it is absorbed in the low-frequency $1/\omega$ tail
shown in the left panel of Fig.4 in the main text. This is a prediction of our
theoretical model which can be converted in a temperature scale once a
modelling of the dimensionless stringlet damping is proposed. Interestingly,
if we think of $\omega_{BP}$ as the average frequency of vibration of the
stringlets, this critical value is close to the onset of overdamped dynamics
in which the oscillations of the stringlets are rapidly damped and decay
exponentially.

## Details on the exponential fit for the stringlet length distribution and
possible corrections

Figure S3: The exponential fits for the stringlet size distribution $p(l)$ for
the 2D power law (2DPL) model in [35] (left panel) and the 3D inverse power
law model (3DIPL) in [36] (right panel). The black dashed lines represent the
best exponential fits presented as insets in Fig.2 in the main text and used
for the BP estimates.

In this section, we provide a more detailed analysis of the fits for the
stringlet size distribution $p(l)$ for the 2D power law (2DPL) model in [35]
and the 3D inverse power law model (3DIPL) in [36] presented as insets in
Fig.2 in the main text. In the main text, the results have been shown in a
logarithmic scale which enhances the deviation from the real values for large
stringlet size $l\gtrapprox 10$. For completeness, in Fig.S3 we show the
results of the fits in both normal and logarithmic scale. The best exponential
fits, used in the main text, are shown with black dashed lines in Fig.S3. In
the logarithmic scale, it is clear that they are not very accurate for large
stringlet sizes, $l\gtrapprox 10$, specially for the 3D system (3DIPL). It is
nevertheless evident that the data for $p(l)$ themselves are very scattered
and uncertain for large stringlet size as well. If one tries to improve by
hand the exponential fit using the logarithmic plot, as done for the blue
dashed lines in Fig.S3, the final results will be actually worse. This is
evident in the normal scale, where the blue dashed lines show strong
deviations from the actual data for short stringlet sizes, $l\lessapprox 10$.

More in general, as already envisaged in [67], it is clear that the stringlet
distribution is not a perfect exponential but it presents clear deviations for
large stringlet size. A better representation would be to combine an
exponential form with a power-law correction, as done in [67],

$p(l)=p_{0}\,l^{-\theta}\,e^{-l/\lambda}.$ (S8)

It is straightforward to re-derive the stringlet VDOS using this distribution.
This yields to

$\tilde{g}_{s}(\omega)=\frac{(d-1)\pi^{1-\theta}p_{0}\,\omega\left(\frac{v}{\sqrt{\gamma^{2}+\omega^{2}}}\right)^{3-\theta}e^{-\frac{\pi
v}{\lambda\sqrt{\gamma^{2}+\omega^{2}}}}}{v^{2}}$ (S9)

which reduces to Eq.(4) in the main text when $\theta=0$. For zero damping,
$\gamma=0$, this is simply

$\lim_{\gamma\rightarrow
0}\tilde{g}_{s}(\omega)=(d-1)\pi^{1-\theta}p_{0}\,v^{1-\theta}\omega^{\theta-2}e^{-\frac{\pi
v}{\lambda\omega}}.$ (S10)

In 3D, and in the limit of zero damping, the reduced stringlet VDOS
$\tilde{g}_{s}(\omega)/\omega^{2}$, displays a maximum at:

$\tilde{\omega}_{max}^{3D}(\gamma=0)=\frac{\pi}{(4-\theta)}\,\frac{v}{\lambda}$
(S11)

which is a simple generalization of Eq.(8) in the main text. Given that the
accuracy of the data for $p(l)$ for large $l$ is quite low, and that the
simple exponential fit gives already excellent results for the BP frequency in
the glass state, we defer a complete analysis involving different stringlet
size distributions $p(l)$ for future studies.

Finally, notice that this corrected stringlet distribution, Eq.(S8), does not
affect the prediction of the stringlet model of a $1/\omega$ divergence of the
reduced 3D VDOS in the limit of finite damping and small frequency presented
in the main text.