# A fermion phenomenology of low-temperature strongly-noncrystalline solids

Mihail Turlakov<EMAIL_ADDRESS>
## Abstract

Insisting on the relevance of spin-statistics theorem, I propose that
anomalous low-energy excitations (LEEs) of strongly-noncrystalline solids
(SNSs), observed at low temperatures $T<1K$, are fermions, which are localized
and weakly interacting. This phenomenological theory rationalizes all low-
energy quantum many-body states of SNSs as Goldstone bosons-phonons and half-
integer-spin fermions. Fermi glass of LEEs, rather than statistical ensemble
of isolated two-level systems, appears to be a consistent theory of various
thermal and nonlinear response properties of SNSs. A robust consequence of the
theory is nearly constant-in-energy density of fermion states near Fermi
energy, which in turn implies linear-in-temperature specific heat and
dimensionless frequency-independent sound absorption. Consistent parameters of
Fermi glass are estimated from theoretical scaling arguments as well as
directly from experimental data.

###### Contents

  1. 1 Introduction
  2. 2 Glasson phenomenology and nearly constant-in-energy density of states
    1. 2.1 Low-energy Hamiltonian and energy scales of SNS
    2. 2.2 Single crossover length and weak coupling
  3. 3 Basic results of the phenomenology
    1. 3.1 Basic results for the specific heat and thermal conductivity
    2. 3.2 Universal sound attenuation at $T<1~{}K$
  4. 4 Glasson parameters and open questions
    1. 4.1 Fermi parameters of glassons from experimental data
    2. 4.2 Why fermions, not TLSs, is a consistent description of SNS
    3. 4.3 Open questions and further experiments
  5. 5 Conclusion

##  1 Introduction

Although a lot of theoretical and experimental research on glasses and
strongly noncrystalline solids has been done from the early 1970s, robust
understanding of their low-temperature universal properties at temperatures
$T<1~{}K$ is missing. There are several paradoxes at the heart of the widely-
adopted theoretical picture of the low-temperature properties of strongly
noncrystalline solids (SNSs). This as-of-yet phenomenological picture of
Tunneling Two-Level States (T-TLSs), proposed by Anderson-Halperin-Varma and
Phillips (AHVP)[1, 2], assumed arbitrary low-energy states, which have a local
tunneling nature and yet sufficiently large density of states. These two-level
states (TLSs) are thought paradoxically to remain local T-TLSs, although a
basic estimate gives large interaction energy $E_{int}\sim 10~{}K$, greater
than typical TLS energy $E_{TLS}$, between local T-TLSs at typical distances
$\xi\sim 30\dot{A}$ between tunneling units[3]. The universality of the
dimensionless sound attenuation in a whole group of these different
materials[4] below the temperature $T_{Q}\simeq 1~{}K$ accentuates the deep
puzzles of low-temperature SNSs. In particular, the relation $l_{s}\approx
150\lambda_{s}$ between mean-free path $l_{s}$ and the wavelength
$\lambda_{s}$ of the phonons[5] is almost independent (i.e. universal) from
the detailed chemical and structural nature of SNSs.

Strictly local picture of T-TLSs is problematic for several more reasons in
addition to the neglect of TLS interactions mentioned above. First, two
neighbouring T-TLSs have overlapping quantum wave-functions, and this
consideration is neglected in AHVP phenomenology. Second, long relaxation
times of AHVP theory are due to the long local-tunneling times rather than due
to collective relaxation mechanisms associated with long length-scales and
typical of glassy frustrated systems. Third, AHVP theory postulates many
parameters - the magnitude and nearly constant-in-energy dependence of the
energy density of T-TLSs as well as the distribution of relaxation times - in
order to fit the universality of the sound attenuation[4]. To put the third
reason differently, AHVP model has too many free fitting parameters than a
respectable theory should have in order to explain the remarkable
universality.

I will argue in detail below that low-energy excitations (LEEs), which are
responsible for linear-in-temperature specific heat for $T<1K$ experimentally,
must be described as fermions. These composite fermions are localized due to
quenched disorder and are an alternative basic ansatz to T-TLSs or TLSs of
AHVP phenomenology. At long wavelengths (significantly longer than typical
inter-atomic distances $a$), familiar transverse and longitudinal phonons are
excitations-bosons of any rigid matrix of atoms. Similarly, for any disordered
network of atoms but not under fully understood constraints, fermions appear
as ubiquitously encountered LEEs in SNSs at low temperatures.

In this paper I insist that the quantum mechanical overlaps between localized
LEEs are non-negligible and determine the physical picture qualitatively.
These localized LEEs - fermion quasi-particles - can be called glassons from
the contraction of two words, glass and fermions. I also use the term of
strongly non-crystalline solids (SNSs) intentionally throughout this paper.
Namely, the density of LEEs in SNSs must be sufficiently large, and therefore
quantum-mechanical overlaps and exchanges between LEEs are non-negligible.
Indeed, mutual quantum statistics of LEEs matters only at sufficiently large
density. SNSs include various classes of materials (glasses, amorphous solids,
strongly irradiated crystals, etc.), where the density of LEEs is dense in a
certain sense.

The nearly constant-in-energy density of LEEs, observed in the specific
heat[6] and in the internal sound friction[4], is an important experimental
clue, and the fermion phenomenological theory explains this fact more robustly
and straightforwardly than AHVP theory. Namely, since fermions fill in fully
“Fermi sea” of all energy states below Fermi energy, the density of fermion
states in the energy space just above finite Fermi energy, due to Fermi
surface manifold degeneracy in the momentum space known for Fermi liquids, is
naturally nearly constant as a function of energy.

In the section (2.1), I write down explicitly an effective many-body low-
energy Hamiltonian and discuss the relevant energy scales and coupling
constants. Theoretical scaling arguments of the section (2.2) show that weak
fermion-fermion and fermion-phonon interaction are consistent with the scaling
of phonon, fermion, and fermion-fermion dipole energies as a function of
length scale.

In the part (3.1), I follow up with several calculations of specific heat,
thermal conductivity as well as the corrections to the glassons’ density of
states due to their interactions. The part (3.2) summarizes the calculation of
the internal sound friction $Q^{-1}$ and the reasons for its universality.
While the calculations in (3.1) are straightforward, the calculation in (3.2)
highlights the fact why the ratio $\lambda_{s}/l_{s}$ is frequency and
temperature independent very naturally in the Fermi glass theory of SNSs.

The parameters of localized Fermi liquid, so called Fermi glass[7], are
calculated directly from the experimental data in the part (4.1). This basic
calculation highlights that, by guessing a plausible Fermi energy
$\epsilon_{F}\simeq 10~{}K$ and taking specific heat slope $C_{v}/T$ from
experiments, glasson Fermi parameters are reasonable without any further
fitting parameters. Indeed, the same glasson parameters are also calculated
from theoretical scaling arguments in the part (2.2). The question ”why
fermions, not TLSs?” is discussed in the section (4.2), and the answers to
these questions summarize the most essential and novel elements of the fermion
phenomenology of SNSs. In the part (4.3), I discuss not only open questions
beyond fermion phenomenology but also the motivation for further experimental
probes.

##  2 Glasson phenomenology and nearly constant-in-energy density of states

I start by motivating the Fermi-statistics nature of LEEs. At zero
temperature, spin-statistics theorem tells us that only bosons and fermions
can exist. I reckon that LEEs can only be localized fermions since localized
bosons would hybridize strongly with phonons, extended boson states. Extended
state of a phonon can coexist with a localized state of glasson because they
have different quantum numbers of spin. Phonons have zero spin, while
glassons-LEEs are spin-half quasi-particles. Moreover, weak coupling between
glassons and phonons can be expected on the symmetry grounds, because they are
different species which couple (hybridize) only weakly with each other.

In the phenomenological style of the theories of Fermi liquid of Helium-3 and
Bose liquid of Helium-4 by Lev Landau, I propose that all LEEs in SNSs are two
weakly interacting particle species, propagating bosons-phonons and localized
fermions-glassons. An important difference from the aforementioned theories,
in particular, from liquid Helium-3, is that glasson is a composite fermion,
which possibly does not have a well-recognised and elementary high-energy
fermion “parent” (yet see the last paragraph of the section (4.3) for a
possible “parent” interpretation). For comparison, Landau-Fermi quasi-
particles in Helium-3 do have “parents” as $He^{3}$ atoms, yet phonons have
never had elementary “high-energy parents”.

Low-energy Hamiltonian is postulated in the part (2.1) with plausible
parameters. The ground state of this Hamiltonian is Fermi glass[7], which is
the ground state of Anderson model of localized fermions in the context of
SNSs at zero temperature. By increasing the disorder strength from the limit
of weak or no disorder to the limit of strong disorder, normal Landau-Fermi
liquid can be ‘adiabatically transformed’ to Fermi glass. This ‘adiabatic
transformation’ keeps the same constant-in-energy density of fermion states
near the Fermi energy $\epsilon_{F}$. Theoretical scaling arguments as a
function of the length scale - the part (2.2) - highlight weak-coupling Fermi
glass picture of SNSs as well as the consistency of the same aforementioned
single crossover length scale $\xi$.

Non-negligible quantum overlaps between the same type of (even though
localised) excitations imply the necessity of definitive quantum statistics,
due to the spin-statistics theorem. Virtual exchanges of indistinguishable
particles in the perturbation theory must respect either boson or fermion
commutation relation. Since fermions are localised, Fermi statistics is not
expressed explicitly in the wave-function symmetry under permutations in the
real space, but rather in the virtual exchanges calculated in the perturbation
theory.

Two states of an isolated TLS are equivalent to unoccupied and occupied states
of a single fermion. The same fact can be expressed mathematically because
there is exact mapping between fermion creation and annihilation operators and
Pauli matrices describing a TLS. In this particular sense, in the introduction
I stated that fermions replace TLSs in the context of many-body quantum
states. However it is more important how to describe an ensemble of non-
isolated TLSs. Fermions can be thought as a particular version of an ensemble
of local LEEs, when there are non-negligible quantum-mechanical overlaps
between the localized states.

###  2.1 Low-energy Hamiltonian and energy scales of SNS

There is a clear hierarchy of energy scales in SNSs. The highest energy scale
$\Lambda\sim 10~{}eV\sim 10^{5}~{}K$ (or rather a range of $1-10~{}eV$) is the
chemical inter-molecular and inter-atomic bonding which is also the local
interaction strength. Debye energy $E_{D}\sim 10^{3}~{}K$ is the next energy
scale lower than the chemical bonding energy. Debye energy determines the
upper energy band of any local and phonon-like excitations.

Medium-range order[8] (MRO) length $\xi\sim 30~{}\dot{A}$ defines the energy
scale $E_{int}\sim 10~{}K$ by the relation $E_{int}\simeq\hbar c_{t}/\xi$,
where $c_{t}$ is the transverse sound velocity (for instance, in vitreous
silica $c_{t}\simeq 4\cdot 10^{5}cm/s$). The elastic dipole coupling energy of
local modes $\frac{\Lambda^{2}}{\rho c_{t}^{2}}(1/\xi^{3})$, separated by MRO
distance $\xi$, is also about the energy $E_{int}$ from the relation
$\frac{\Lambda^{2}}{\rho c_{t}^{2}}(1/\xi^{3})\sim(\hbar c_{t})/\xi$[9].

The lowest collective energy scale $E_{Q}$ is given by the quantum crossover
temperature observed as a crossover from linear-in-temperature at $T<1K$ and
cubic-in-temperature at $T>10K$ dependencies of the specific heat as well as
the plateau in the thermal conductivity. To be specific, I define this quantum
(therefore subscript $Q$ in $E_{Q}$) energy scale $E_{Q}=k_{B}T_{Q}=1~{}K$,
which is about a factor of $10^{3}$ lower than Debye energy $E_{D}$.

Effective low-energy Hamiltonian has to describe a ground state and all low-
energy excitations with the transition matrix elements between these different
excitations. The full Hamiltonian is the sum

$\hat{H}_{total}=\hat{H}_{main}+\hat{H}_{int},$ (1)

where $\hat{H}_{main}$ and $\hat{H}_{int}$ are the main and interaction parts
of the Hamiltonian correspondingly. Quantum many-body states are better
expressed in the second quantization notations. The main part $\hat{H}_{main}$
of the Hamiltonian describes glassons and phonons

$\hat{H}_{main}=\sum_{\alpha}\epsilon_{\alpha}\,\hat{c}^{\dagger}_{\alpha}\hat{c}_{\alpha}+\sum_{\alpha\neq\beta}t_{\alpha,\beta}\,\hat{c}^{\dagger}_{\alpha}\hat{c}_{\beta}+\sum_{k}\omega_{k}\hat{b}^{\dagger}_{k}\hat{b}_{k}+\sum_{k}A_{k,k^{\prime}}\hat{b}^{\dagger}_{k}\hat{b}_{k^{\prime}}.$
(2)

The creation and annihilation operators of glassons are
$\hat{c}^{\dagger}_{\alpha}$ and $\hat{c}_{\alpha}$, while the creation and
annihilation operators of phonons are $\hat{b}^{\dagger}_{k}$ and
$\hat{b}_{k}$. Glassons and phonons are well-defined energy resonances of the
same solid medium. The interactions are weak if quasiparticle energies are
small $(\epsilon,\omega)<E_{int}$. Let me call for brevity the proposed here
theory as a glasson-phonon phenomenology (GPP).

The fermion part of non-interacting $\hat{H}_{main}$ is simply Anderson model
studied widely in the research of single-particle localization. It is natural
to think that indices $\alpha$ and $\beta$ describe states in the real space,
while $k$ and $k^{\prime}$ are the labels in the momentum state. The energies
$\omega_{k}$ describe both, longitudinal $\omega_{k}=c_{l}k$ and transverse
$\omega_{k}=c_{t}k$, phonons. The local fermion energies $\epsilon_{\alpha}$
are distributed in a certain wide range $W\gg E_{Q}$ due to locally different
structure of the blobs. We assume that the disorder is strong for glassons-
fermions, namely that hopping is weak $t\ll W$. Reasonable, but quite
uncertain, estimates are the bandwidth $W\simeq 100~{}K$ and the hopping
parameter $t\leq 0.1-1~{}K$. In reality, both quantities $\epsilon_{\alpha}$
and $t_{\alpha,\beta}$ are likely to be widely distributed (disordered) for
SNSs. The disorder-induced matrix element $A_{k,k^{\prime}}$, which describe
elastic scattering of phonons from disorder, becomes non-vanishing only when
$(k,k^{\prime})\geq\xi^{-1}$.

The interaction part $\hat{H}_{int}$ of the Hamiltonian describes glasson-
glasson, glasson-phonon, and multi-phonon anharmonic interactions. Let us
write down explicitly glasson-phonon interaction of two types

$\hat{H}_{int}^{gl-
ph}=\sum_{\alpha,\beta,k}\sqrt{\omega_{k}}M_{\alpha\beta,k}^{z}\hat{c}_{\alpha}^{\dagger}\hat{c}_{\beta}(\hat{b}^{\dagger}_{-k}+\hat{b}_{k})+\sum_{\alpha,k}\sqrt{\omega_{k}}M_{\alpha,k}^{x}(\hat{c}_{\alpha}^{\dagger}+\hat{c}_{\alpha})(\hat{b}^{\dagger}_{-k}+\hat{b}_{k}),$
(3)

where $M_{\alpha\beta,k}^{z}$ and $M_{\alpha,k}^{x}$ are constants in the
long-wave limit $k\rightarrow 0$. The second term containing
$M_{\alpha,k}^{x}$ is allowed due to the lack of fermion-charge conservation.
These interactions are familiar coupling terms expressed via fermion operators
rather than Pauli matrices $\sigma^{z}$ and $\sigma^{x}$ for TLSs. The main
conjecture of the paper expressed mathematically is that for different indices
$\alpha\neq\beta$ anti-commuting relation
$c_{\alpha}c_{\beta}+c_{\beta}c_{\alpha}=0$ holds rather than commuting
relation assumed in the Standard Model[1, 2]. AHVP proposed that local LEEs,
being equivalent to pseudo-spins, obey bosonic commutation between any local
field operators, describing different local sites, because TLSs are assumed to
be independent.

###  2.2 Single crossover length and weak coupling

Low-energy Hamiltonian, Eqn.(1 \- 2), implies the ground state of Fermi glass.
Yet the important low-energy parameter, which is determined by high-energy
physics and not explicit in the Hamiltonian, is the glassons density in the
ground state. In what follows I estimate the density of glassons, or
equivalently Fermi volume of glassons, by identifying a single crossover
length with the Fermi wavelength of glassons. This crossover length is
determined by matching principal involved energies. The crossover length $\xi$
marks the onset of long-distance universality for $R>\xi$. The same
considerations illustrate the onset of weak coupling of fermion-fermion and
fermion-boson interactions.

Initially it is insightful to compare how different energies, involved in the
problem, scale as a function of length scale $R$ of an amorphous solid blob.
We have the phonon energy $E_{s}=\frac{\hbar c_{t}}{R}$, the kinetic
localization energy (of glasson) $E_{kin}=\frac{\hbar^{2}}{2m_{gl}R^{2}}$, and
the dipole interaction energy $E_{int-dipole}=\frac{\Lambda^{2}}{\rho
c_{t}^{2}R^{3}}$ between two blobs (blocks). For sufficiently large length
scales $R\gg\xi$, we have necessarily the following inequalities

$\frac{\hbar
c_{t}}{R}\gg\frac{\hbar^{2}}{2m_{gl}R^{2}}\gg\frac{\Lambda^{2}}{\rho
c_{t}^{2}R^{3}}.$ (4)

The inequality $E_{s}\gg E_{int-dipole}$ implies weak anharmonic phonon-phonon
coupling due to a phonon energy being larger than dipole interaction between
two blobs, if excited by two different phonons. The inequality $E_{s}\gg
E_{kin}$ implies weak phonon-fermion coupling due to a energy-momentum
mismatch between a phonon and a localized fermion.

The inequality $E_{kin}\gg E_{int-dipole}$ suggests weak fermion-fermion
coupling due to a local fermion excitation energy being larger than dipole
interaction between two fermions-blobs. To put it differently, a localised
fermion is a well-defined local resonance (i.e. excitation). In particular,
the ratio of the glasson kinetic energy and glasson interaction energy is

$\frac{E_{kin}}{E_{int-
dipole}}=\frac{R}{a_{gl}}=\frac{1}{a_{gl}n_{gl}^{1/3}},$ (5)

where $a_{gl}=2\Lambda^{2}m_{gl}/(\hbar^{2}\rho c_{t}^{2})$ and the glasson
density $n_{gl}\simeq 1/R^{3}$ for inter-particle distance $R$. It is clear
that at sufficiently dilute densities $a_{gl}n_{gl}^{1/3}\ll 1$, glassons,
interacting via dipole interaction, are in the regime of the weak interaction
since $E_{kin}/E_{int-dipole}\gg 1$. This situation is opposite to the
situation of electrons interacting via Coulomb interaction at dilute densities
in three dimensions, where $E_{kin}/E_{int-
Coulomb}=a_{B}/R=a_{B}n_{e}^{1/3}\ll 1$ ($a_{B}$ is Bohr radius) is the strong
interaction limit for $n_{e}\rightarrow 0$.

Now let us make a simple, but a far reaching, assumption that there is a
single crossover length scale $\xi$ so that

$\frac{\hbar c_{t}}{\xi}\simeq\frac{\hbar^{2}}{2m_{gl}\xi^{2}},$ (6)
$\frac{\hbar c_{t}}{\xi}\simeq\frac{\Lambda^{2}}{\rho c_{t}^{2}\xi^{3}}.$ (7)

Namely, that all three energies - $E_{s}$, $E_{kin}$, and $E_{int-dipole}$ \-
become roughly the same at the single length scale $\xi$. For $R\gg\xi$, these
energy scales become clearly hierarchically ordered as $E_{s}\gg E_{kin}\gg
E_{int-dipole}$. In the introduction of the part (2.1), we already noticed
that MRO $\xi$ is consistent with the scaling relation of Eqn.7. Appealing to
additional scaling relation of Eqn.6, we can estimate the parameter $m_{gl}$,
which has the dimensionality of physical mass.

Defining a constant $H\equiv\sqrt{\hbar\rho c_{t}}$, which is dependent only
on long-wavelength properties $\rho$ and $c_{t}$, Eqn’s (6, 7) give the
following simple expressions for the length $\xi$ and the mass $m_{gl}$

$\xi=\frac{\Lambda}{c_{t}H},~{}~{}~{}~{}m_{gl}=\frac{\hbar H}{2\Lambda}.$ (8)

Numerical estimates for vitreous silica give $H\simeq 0.9*10^{-10}g/s$,
$\xi\simeq 44\dot{A}$, and $m_{gl}\simeq 5.6*10^{-26}g$. To be concrete, I
used $\Lambda\simeq 10eV$ while more realistic values are only several
$eV^{\prime}s$. The crossover energy is $E_{\xi}\sim 10K$ (order of
magnitude).

The theoretical expressions - Eqn’s (8) - for the length $\xi$ and the mass
$m_{gl}$ are simple and general. The qualitative interpretation seems that,
under some conditions, disorder of chemical bonding can conspire to medium-
range order length $\xi$ and mass parameter $m_{gl}$. These expressions depend
only on four parameters, two experimentally measured macroscopic parameters
$\rho$ and $c_{t}$, chemical bonding energy $\Lambda$, and fundamental
Planck’s constant $\hbar$.

Notice that so far in this section the quantum localisation of a particle was
assumed, but not a particular quantum statistics. In the section 4.1, I will
further assume that localised particles have Fermi statistics.

The simplicity of Eqn’s (8) may appeal to the ubiquity and universality of
LEEs in various disordered solids, but the limitations must be stated.
Sufficient and necessary conditions for the existence of dense LEEs are still
not known very well structurally and chemically. Experimentally not all
disordered solids posses sufficiently large density of LEEs, and it is
important to research further into the microscopic models and constraints on
the applicability of Eqn’s (8). For instance, some large variation of local
density, chemical bonding strengths, and geometry of random-network chemical
bonds, not explicit in Eqn’s (8), seem to be required[10].

In the section (4.1), I calculate the mass $m_{gl}$ and Fermi wave-vector
$k_{F}$ essentially from the experimental data. The consistency of the results
for $m_{gl}$ and $k_{F}$ within a factor of 2-3, derived from experimental
data and from theoretical scaling calculations in this part of the paper, is
definitely encouraging and likely not accidental. Notice that there are only
two ‘free parameters’ $\epsilon_{F}$ and $\Lambda$ in theoretical and
experimental arguments taken together, but even these two parameters can be
reasonably estimated and can vary only within one order of magnitude at most.

##  3 Basic results of the phenomenology

In this part of the paper, I calculate the basic consequences of the glasson-
phonon phenomenology (GPP).

###  3.1 Basic results for the specific heat and thermal conductivity

At the basic level of GPP, core experimental results of SNSs at $T<T_{Q}$ can
be easily understood. Some basic calculations, which follow, are analogous to
the calculations of AHVP model[1, 11], since a localised fermion-glasson can
be mapped exactly into a TLS via the mapping from fermion operators to Pauli
matrices[12].

Firstly, localized glassons for $T\ll\epsilon_{F}$ will exhibit linear-in-
temperature specific heat as any localized Fermi glass and normal Landau-Fermi
liquid[7] does. This contribution is as usual due to the thermally excited
number $n_{F}(T)\sim T/\epsilon_{F}$ of fermions, and this fermion
contribution can exceed phonon contribution to the specific heat being
proportional to $T^{3}$.

Secondly, the phonon thermal conductivity is $\kappa_{ph}(T)\sim
C_{ph}(T)\tau_{k,ph}(T)\sim T^{2}$, due to phonon-glasson scattering. The
thermal conductivity was shown experimentally to be completely dominated by
phonons[13] for $T<T_{Q}$ and to have roughly square-in-temperature
dependence. From the kinetic theory[14] we have for the thermal conductivity
of phonons

$\kappa_{ph}=\frac{1}{3}C_{ph}(T)c_{s}^{2}\tau_{k,ph},$ (9)

where $C_{ph}\sim T^{3}$ is the phonon specific heat for $T\ll T_{D}$ below
Debye energy $T_{D}$, $c_{s}$ is the sound velocity, and $\tau_{k,ph}$ is the
momentum-scattering time of phonons. The inverse scattering time
$\tau_{k,ph}^{-1}$ is proportional to $Q_{eff}n_{F}(T)$, where the quantity
$Q_{eff}$ is the effective glasson-phonon scattering cross-section. Glassons
are point scatterers of phonons, since de Broiglie wavelength of phonons is
much larger $\lambda_{s}\gg\lambda_{F}$ than the analogous quantity for
glassons $\lambda_{F}$. The number of point scatters $n_{F}(T)$ is
proportional to the temperature $T$. Therefore, qualitatively we do get the
quadratic temperature dependence of the phonon thermal conductivity
$\kappa_{ph}(T)\sim C_{ph}(T)\tau_{k,ph}(T)\sim T^{2}$ in the leading order.

Glassons themselves are localised and can contribute to the thermal
conductivity only through thermally activated hopping. We do not calculate
this subleading contribution to the thermal conductivity here.

Quantitatively, we can calculate the inverse scattering time
$\tau_{k,ph}^{-1}$ from Fermi Golden rule explicitly for the interaction from
Eqn. (3)

$\frac{1}{\tau_{k,ph}}=\frac{2\pi}{\hbar}|H_{int}^{gl-
ph}|^{2}\delta(E_{f}-E_{i}),$ (10)

where $E_{i}$ and $E_{f}$ are initial and final phonon energies. The first
term in Eqn. (3) gives

$\frac{1}{\tau_{k,ph}^{z}}=\frac{2\pi\omega_{k}}{\hbar}|M_{\alpha\beta,k}^{z}|^{2}\int
d\epsilon_{1}(1-n_{F}(\epsilon_{1}/T))n_{F}(\epsilon_{2}/T)\sim T\omega_{k}$
(11)

where two fermion energies are related by the energy conservation
$\epsilon_{1}-\epsilon_{2}=\hbar\omega_{k}$. The second term in Eqn. (3) gives

$\frac{1}{\tau_{k,ph}^{x}}=\frac{2\pi\omega_{k}}{\hbar}|M_{\alpha\beta,k}^{x}|^{2}(1-2n_{F}(\omega_{k}/T))\sim\omega_{k}tanh\left(\frac{\omega_{k}}{2T}\right)$
(12)

where $\epsilon=E-\epsilon_{F}=\hbar\omega_{k}$ is a fermion energy relative
to the Fermi energy $\epsilon_{F}$. For the thermal phonons such that
$\omega_{k}\sim T$, the dominant scattering mechanism, due to the term
$M_{\alpha\beta,k}^{x}$, is $1/\tau_{k,ph}^{x}\sim T$. Thus, once again, the
phonon thermal conductivity is $\kappa_{ph}(T)\sim
C_{ph}(T)\tau_{k,ph}^{x}(T)\sim T^{2}$.

Thirdly, I calculate the marginal corrections of weak glasson-glasson
interactions to the energy density of states and consequently to the glasson
specific heat and phonon thermal conductivity. Due to dipolar, weak-coupling
but long-range, fermion-fermion interaction, falling as $\alpha_{d-d}/R^{3}$
with the distance $R$, the non-interacting energy density of states
$\nu_{0}(\epsilon)=p_{F}m_{F}/\hbar^{3}=\rho C_{v}(T)/(k_{B}T)$ is suppressed
logarithmically as a function of energy and temperature. Such a ‘dipolar gap’
suppression is due to the same mechanism as Efros-Shklovskii Coulomb gap
mechanism[15]. The corrected density of states $\nu(\epsilon)$ is

$\nu(\epsilon)=\frac{\nu_{0}(\epsilon)}{1+\nu_{0}(\epsilon)\alpha_{d-d}ln(E_{Q}/T)}.$
(13)

Therefore the temperature dependence of the specific heat becomes
$C_{v,gl}\sim T/ln(T_{Q}/T)$ under the condition
$\nu_{0}\alpha_{d-d}ln(E_{Q}/T)\gg 1$. This is again a very crude familiar
argument, and more detailed analysis of the effect of angular dependence of
dipole-dipole interaction is clearly desirable. This correction means that the
specific heat can be approximately written as $\sim T^{1+\alpha}$ with small
$\alpha\ll 1$, an experimental fit in some temperature range, instead of
simple linear-in-temperature $\sim T$ dependence. Since the phonon mean free
path is $l_{s}(T)\sim 1/n_{F}(T)$ due to the scattering of phonons by glassons
of the number $n_{F}(T)\sim\nu(\epsilon)T$, then the corrected thermal
conductivity becomes $\sim T^{2}ln(T_{Q}/T)\approx T^{2-\alpha}$ replacing
$T^{2}$ dependence. These corrections are consistent with experiments[9].

One of the most important and basic properties of glasses, and even glasses at
low temperatures, is long energy relaxation times, for instance, revealed in
the specific heat experiments[16]. At the level of non-interacting Hamiltonian
$\hat{H}_{main}$ (eqn. 2), there is no energy relaxation. Since all types,
glasson-phonon and glasson-glasson, of interactions $\hat{H}_{int}$ are weak
and responsible for thermalization, the relaxation times can remain long. It
is important to investigate in more detail how glasson-phonon and glasson-
glasson induced relaxations are similar and different in Fermi glass versus
TLS glass[17].

###  3.2 Universal sound attenuation at $T<1~{}K$

The internal friction measurements[4] highlight clearly the universality of
LEEs in SNSs. It is particularly intriguing to understand the internal
friction coefficient $Q^{-1}(\omega)$, since this quantity is a direct
measurement of small dimensionless interaction constant. The quantity
$Q^{-1}=1/(2\pi)(\lambda/l)$ is expressesed by the ratio of phonon wavelength
$\lambda(\omega)$ to its mean free path $l(\omega)$ for various frequencies
$\omega$. In the regime $T<T_{Q}$ for SNSs, in a certain range of temperatures
and frequencies, the internal friction is a small quasi-universal quantity of
the order of $10^{-2}-10^{-3}$[4].

It is insightful to make a short excurse into the theory of sound attenuation
by electron Landau-Fermi liquid in simple metals[14]. Similarly to SNSs, in
simple metals the coefficient of sound internal friction is small and
temperature and frequency independent. In particular, this coefficient is
equal to $Q^{-1}_{LFL}\sim c_{s}/v_{F}\sim\sqrt{m_{e}/M_{ion}}$, where $c_{s}$
is the sound velocity and $v_{F}$ is a Fermi velocity for itinerant electrons.
In simple metals, the internal friction is small and is of the order of
$Q^{-1}_{metals}\sim 10^{-2}-10^{-3}$ due to the small ratio $c_{s}/v_{F}\ll
1$. In this case, the smallness of the dimensionless electron-phonon coupling,
expressed by the internal friction $Q^{-1}_{metals}$ is ultimately due to the
small ratio of the electron mass $m_{e}$ to the ion mass $M_{ion}$. It is also
important to highlight that the result $Q^{-1}_{LFL}\sim c_{s}/v_{F}$ is
equally valid independently from the parameter
$\omega_{s}\tau_{F}=(c_{s}/v_{F})(l_{F}/\lambda_{s})$ being larger or smaller
than 1[14].

Following through the kinetic equation calculations[14] or equivalently Fermi
Golden rule under the condition $\omega_{s}\tau_{F,gl}\gg 1$, I get the
expression for the dimensionless absorption

$Q^{-1}_{t}\simeq\pi\frac{\nu(\epsilon_{F})\lambda_{gl-ph}^{2}}{\rho
s^{2}}\frac{c_{t}}{v_{F}}\simeq\pi K_{F-s}\frac{c_{t}}{v_{F}},$ (14)

where the glasson-phonon coupling constant $\lambda_{gl-
ph}^{t}=M^{x,t}_{\alpha\beta}\sqrt{c_{t}}$ from the definition in Eqn.(3). The
dimensionless constant $K_{F-s}$ is given by the expression

$K_{F-s}\equiv\frac{\nu(\epsilon_{F})\lambda_{gl-ph}^{2}}{\rho
c_{t}^{2}}=\frac{\lambda_{gl-
ph}^{2}}{\epsilon_{F}M_{atom}c_{t}^{2}}(k_{F}a)^{3}.$

The question arises how the coupling constant $\lambda_{gl-ph}$ depends on
various parameters of SNSs. I conjecture that $\lambda_{gl-
ph}^{2}\simeq\epsilon_{F}M_{atom}c_{t}^{2}$ scales similarly to the arguments
used in simple metals[14]. Then the coupling constant $\lambda_{gl-ph}$ for
large glasson quasi-particles of size $R\geq\xi\gg a$ is significantly smaller
than the bare coupling coupling constant $\Lambda$ for the local modes.
Numerically, using the numbers $\epsilon_{F}=10~{}K$ and
$M_{atom}c_{t}^{2}\sim 10^{5}~{}K$, the coupling constant is $\lambda_{gl-
ph}\simeq 10^{3}~{}K$. All estimates here are only crude order of magnitude.
Thus $\lambda_{gl-ph}\sim 10^{-2}\Lambda$ is indeed much smaller than the bare
chemical bonding energy $\Lambda$.

Finally the result for the internal friction coefficient of the transverse
phonons in SNSs is

$Q^{-1}_{t}=K_{Q,t}(k_{F}a)^{3}\frac{c_{t}}{v_{F}},$ (15)

where $K_{Q,t}$ is the number of order 1. This result corresponds to the
resonant regime calculation of AHVP theory[11] in the limit $\omega/T\gg 1$.
The internal friction for longitudinal phonons is similarly
$Q^{-1}_{l}\sim(k_{F}a)^{3}$. The ratio of the velocities $c_{s}/v_{F}$ in
SNSs is very roughly of order 1. At the phenomenological level, small constant
$Q^{-1}$ is due to small quantity $(k_{F}a)^{3}$, i.e. small dimensionless
Fermi volume.

##  4 Glasson parameters and open questions

In the part (4.1), I present a basic calculation of glasson Fermi parameters
directly from the linear-temperature slope of the specific heat and
conjectured Fermi energy. In the part (4.2) I summarize the arguments why
universal LEEs are fermions rather than two-level states (TLSs). I conclude
this section of the paper with a tentative discussion of possible microscopic
understanding of the nature of excitations-glassons and experimental tests
(part 4.3).

###  4.1 Fermi parameters of glassons from experimental data

As compared to the theoretical scaling arguments of the part (2.2), we can
deduce some parameters of Fermi glass (or Fermi liquid) essentially from
experimental data directly. Since Fermi glass is ‘adiabatically connected’ to
Landau-Fermi liquid, I will use the standard relations from the Landau-Fermi
liquid theory.

The linear-in-temperature slope of the specific heat capacity is
$(C_{v}/T)\sim p_{F}m_{F}$, while the Fermi energy is
$\epsilon_{F}=p_{F}^{2}/(2m_{F})$. If we take from experiments the temperature
slope of the specific heat $(C_{v}/T)$ and estimate the values of Fermi energy
$\epsilon_{F}$, the above expressions make obvious that we can calculate
separately the glasson Fermi mass $m_{gl}$ and the Fermi momentum $p_{F}=\hbar
k_{F}$. Fermi wave-vector $k_{F}$, or equivalently de Broglie wavelength, has
a meaning, that only the states at and below this wave-vector are occupied in
the momentum space. Due to strong elastic scattering by disorder, glasson at
the energy $\epsilon_{F}$ is localized in real space but it still has a well-
defined energy, which is a linear superposition of multiple-direction waves
with the wavevector $k_{F}$. ‘Fermi glass’ is simply the ensemble of non-
interacting localized fermions[7, 18] filling in the Fermi sea up to the Fermi
energy $\epsilon_{F}$. Since the static strong disorder causes only elastic
scattering, ‘adiabatic continuity’ from normal no-disorder Landau-Fermi liquid
to strongly-disordered Fermi glass as a function of disorder strength makes
quasiparticle momentum not a good quantum number and yet keeps quasi-particle
energy as a good quantum number.

Linear-temperature dependence of the specific heat due the Fermi occupation
number $n_{F}(T)$ becomes very clear when the temperature becomes smaller than
(roughly) one tenth of Fermi energy. Since the linear temperature dependence
of the specific heat in amorphous solids sets in below $1~{}K$, I estimate the
Fermi energy to be $\epsilon_{F}=10~{}K=1.4\cdot 10^{-15}erg$. The
experimental specific heat slope[6] is $C_{v}/T\approx 10~{}erg/(g~{}K^{2})$.
We have the following theoretical expressions for the basic Landau-Fermi
parameters

$k_{F}=\left(6\epsilon_{F}\left(\rho\frac{C_{v}}{T}\right)\right)^{1/3},$ (16)

$m_{gl}=\hbar^{2}\left(\frac{3}{2\epsilon_{F}}\left(\rho\frac{C_{v}}{T}\right)^{2}\right)^{1/3},$
(17)

where $\rho$ is the mass density of a solid ($\rho\simeq 2.2g/cm^{3}$ for
vitreous silica). As an example, for vitreous silica, I estimate the numerical
value of the Fermi wave-vector $k_{F}\approx 2.1*10^{6}cm^{-1}$. For the
reference, for localised glassons, Fermi parameters without direct physical
meaning are the glasson mass $m_{gl}\approx 5*10^{-26}g\sim 10^{-2}m_{He^{3}}$
and Fermi velocity is $v_{F}\simeq 2.3*10^{5}cm/s$.

Importantly, quantum de Broglie wavelength of glassons is $a_{F}\sim
k_{F}^{-1}\sim 50\dot{A}$, much longer than inter-atomic distances of $a\simeq
2.5\dot{A}$ (for the sake of specific numerical value). Since $k_{F}a\ll 1$,
the condition for coarse-graining of blobs of the size $a_{F}$ is satisfied.
Namely, fermions-glassons in SNSs are a consistent concept.

Since glassons are localized, the parameters $m_{gl}$ and Fermi velocity
$v_{F}=p_{F}/m_{gl}$ do not have direct physical meaning. But Fermi volume and
closely related Fermi momentum $p_{F}$ in the presence of only elastic
scattering from disorder, should be measurable and do have physical meaning.

We can notice that glasson de Broglie wavelength $a_{F}$ is crudely equal to
MRO length $\xi$, which is estimated as $\frac{\Lambda}{c_{t}H}$ in the
section (2.2). Namely, I guess an important relationship between Fermi
wavevector $k_{F}$ and the crossover length $\xi$, which is $k_{F}\xi\sim 1$.
The existence of a single crossover scale $\xi$, rather than two separate
scales from inequalities (Eqn. 4), might be one of the main criteria for the
convergence to the universality class of SNSs. Thus, LEEs are plausibly
fermions, since glasson de Broglie wavelength from Eqn.(16) is about the same
as the crossover length $\xi$, derived from Eqn.(7).

Using Eqns. 8, we can also write a simple expression for the ratio of the
Fermi energy $E_{F}=p_{F}^{2}/(2m_{F})$ and the crossover energy
$E_{\xi}=\hbar c_{t}/\xi$ from Eqn.(7)

$\frac{E_{F}}{E_{\xi}}=(k_{F}\xi)^{2}.$ (18)

Under the relation $k_{F}\xi\sim 1$, we can easily see that Fermi energy and
the crossover energy to the universal low-temperature regime are about the
same and of the order of $10K$.

###  4.2 Why fermions, not TLSs, is a consistent description of SNS

In this section I argue why localized fermions, not TLSs, is a consistent
framework for LEEs in SNSs.

Firstly, a fundamental theory must explain consistently not only LEEs but also
the ground state. AHVP theory comes short of such a requirement. Although AHVP
model describes successfully the properties of LEEs, it does not describe the
ground state of T-TLSs. In particular, AHVP is silent about why the density of
TLSs is of particular value.

GPP model does describe both, the ground state and LEEs, in a single
framework, even though not microscopically. GPP, similarly to AHVP, describes
LEEs, which explain the specific heat and contribute to the scattering of
phonons (Eqn. 9) and the internal friction coefficient (Eqn. 15). Yet still as
importantly, the ground state of SNSs is described by Fermi volume and Fermi
density of glasson states, which enter into the calculations of the section
(3).

The stability of the ground state of pseudo-spins proposed by AHVP can be
questioned in the view of independent (i.e. bosonic) nature of different TLSs.
The ensemble of anomalous LEEs is only a subsystem of the disordered network
of atoms of SNSs, nevertheless this subsystem must be stable. The stability of
bulk matter was accounted for by Pauli principle[19]. Since, in general, local
LEEs can hybridize, get larger in size, and change their spatial positions,
the stability of the ensemble of bosonic (and dense) LEEs is highly
questionable. The historical works by Lieb[19] and Dyson and Lenard[20] are
strongly suggestive that local LEEs are fermi excitations if these excitations
are not fully “pinned” locally.

Secondly, AHVP model simply uses the constant-in-energy density of states as a
fitting parameter. GPP model has the constant-in-energy density of states near
the Fermi energy due to the Pauli exclusion principle, while the magnitude of
the density of states is derived from the existence of a single crossover
length scale. AHVP model leaves unanswered the question why TLSs can be
considered as completely isolated and have a particular and constant density
of states.

In the discussion of a general many-particle system, it is important to
consider not only the level repulsion but also different symmetries of the
“perturbative” Hamiltonian terms, which hybridize various many-body energy
levels of original Hamiltonian. The symmetries include Fermi-Pauli occupation
constraints as well as the form of various couplings (see Eqn. 3). The
emergence of composite fermions-glassons can mean that some fermionic degrees
of freedom, plentiful at higher energies (for instance, electrons), survive
under mixing effects of disorder and various effective interactions.

Thirdly, AHVP model attributes the energy relaxation to a local tunneling
mechanism rather than collective multi-scale mechanism typical for all
classical and quantum types of glasses. From the early days of theoretical
analysis, it became clear that AVHP model cannot explain all experiments
naturally with a single type of T-TLSs[21]. In the framework of GPP, the
calculations (Eqn’s 8, 15, 16, 17) show that dipolar interactions and Fermi
statistics are important. Fermi glass with dipolar interactions and Fermi
statistics constraints appears to show collective relaxation mechanism[17],
which can be explored in the future work. Various glasses[22, 23], including
electron glasses, do show logarithmic relaxations independently of the nature
of local LEEs, which can be electrons, spins, or TLSs. Fermi glass and TLS
glass seem to belong to the different universality classes[17], and this is
how GPP and AHVP theories might get distinguished.

###  4.3 Open questions and further experiments

Some questions are beyond the initial investigation of this paper. In
particular, a detailed microscopic understanding of glasson excitations is
beyond the scope of the paper. GPP explains the universality of SNSs by the
means of collective formation of composite fermions-glassons and their weak
interactions with each other and with phonons. Fermi volume of glassons is
small $(k_{F}a)^{3}\ll 1$ and determined by MRO length $\xi\simeq k_{F}^{-1}$.
The work, conceptually related to GPP, by Vural-Leggett[24] and more recently
by Shukla[25] aims to derive the universal properties of SNSs microscopically.
In particular, Shukla[25] relates the universality to the universal ratio of
MRO $\xi$ to the ‘closest approach’ inter-atomic distance $a$. The important
role of this small ratio $a/\xi$ was highlighted in my previous work[26] based
on quasi-classical viscoelastic approach to the density-density correlation
function. The current work here starts from fundamentally quantum perspective,
yet the universality of internal friction $Q^{-1}$ relates to the same
structural ratio of $a/\xi\simeq k_{F}a$.

An open question is a detailed relation between a single crossover length
$\xi$ as discussed in (2.2) and medium-range order length as seen in
experiments from the early days of the field[8]. Spectroscopic measurements,
related to this open question, should shed light on the connection between
Boson peak modes ($E_{BP}\sim 4meV\simeq 40K$ for concreteness) and low-energy
modes (T-TLS, glassons, etc.), which was pointed out by several authors[26,
27, 28]. Ioffe-Regel limit for transverse phonons is an important phenomenon
for understanding the properties of SNSs. An important contribution to Ioffe-
Regel condition should come from anharmonic coupling between transverse
phonons and glassons when their momenta match $k_{ph}\sim k_{F}$. This latter
consideration might reveal some relations between close energy scales
$\epsilon_{F}$, $E_{int}$, and $E_{BP}$ to each other.

An important experiment using resonant sound absorption has already been
proposed[29] in order to test the isolated nature of just two energy levels of
TLSs as long as local intra-site transitions are strictly dominant that inter-
site transitions. The linear resonant response of TLSs and localised fermions
seems to be the same under the same assumption. Therefore only the non-linear
response as a function of sound intensity might distinguish between TLSs and
localised fermions[30].

Promising experimental advances in low-temperature glasses in last 10 years
are due to experiments on superconducting circuits encased by aluminium oxide
layers, which are amorphous insulating solids. These experimental works aim to
understand the microscopic nature of LEEs (or TLSs) directly[31] as well as
interactions between LEEs[32, 33]. The latter experiments address electric
dipole moment of LLEs, while this paper considers only elastic dipole moment
of LEEs, mainly due to the associated universal sound absorption by elastic
dipole moments. However it is well-known that LEEs have both, elastic and
electric, dipole moments. One open question is the strength ratio between
elastic and electric dipole moments depending on chemical and physical
structure of SNSs. Another open question is whether glassons, being fermions,
might emerge from “dressed electrons”(see, for instance, [34]). To this end, a
glasson, being a composite fermion, acquires its fermionic nature from an
electron. This composite heavy electron, i.e. glasson, is medium-range
delocalized from local chemical bonds up to length scale $\xi$, yet it is
strongly localized beyond the length $\xi$.

Spectroscopic experiments in the context of superconducting circuits[31] are
promising local probes. These experiments[33] can test whether the spatial
size of local LEEs is of the order of crossover length $\xi$. These
experiments[32] should be able to distinguish between two interacting pseudo-
spins (i.e. TLSs) and two interacting fermions, since the effective form of
fermion-fermion interaction is quite different from spin-spin interaction[30].
It will be necessary to distinguish short-range and long-range (dipole) parts
of the interaction as well as elastic-dipole and electric-dipole parts. But
the advantage of such experiments is that they measure individual interaction
rather than statistically averaged effects of many TLSs.

Non-interacting Fermi glass has a well-defined locus of the Fermi energy in
the momentum space. It remains to be tested and examined whether weakly-
interacting strong-disorder Fermi glass, as conjectured for SNSs here, still
has a detectable locus of the Fermi surface. Nevertheless some spectroscopic
features of the Fermi energy would provide a direct evidence for the existence
of glassons, because such a finding is indeed surprising from the point of
view of the Standard Model[1, 2].

##  5 Conclusion

On the basis of fundamental spin-statistics theorem, I propose that infamous
LEEs in SNSs are localized fermions. These fermions have intrinsic half-
integer spin as a quantum number, and thus they do not hybridize directly with
phonons. Fermions-glassons are stress-field excitations of coarse-grained
isotropic blobs with the size equal or larger than MRO length $\xi$. Localized
and weakly interacting glassons form a Fermi glass in the ground state.
Glassons can be loosely thought as an alternative version of TLSs respecting
quantum statistics, yet with important, experimentally distinguishable,
differences (see 4.2).

The fermion phenomenological theory proposes fermionic LEEs and the ground
state of Fermi glass as an explanation of low-temperature SNSs. Glasson
parameters are calculated theoretically (Eqn. 8) and from experimental data
(Eqns. 16 and 17) and found consistent. The glasson-phonon phenomenology (GPP)
explains (Eqns. 9 and 15) the experimental specific heat due to glassons,
phonon thermal conductivity limited by glasson-phonon scattering as well as
the small internal sound friction coefficient $Q^{-1}\sim(k_{F}a)^{3}$. The
constant-in-energy density of states near the Fermi energy is a natural
consequence of the fermion phenomenology unlike being a free-fitting parameter
for AHVP model.

In the part (4.1), I present a basic calculation of Fermi volume and mass of
glassons directly from the linear-temperature slope of the specific heat and
conjectured Fermi energy. This calculation, which can be considered on its
own, shows simple consistency (Eqn. 16) between plausible value of Fermi
energy and small Fermi volume.

The crossover (as well as medium-range-order) length $\xi$ determines the
small wavevector (radius) $k_{F}$ of glasson Fermi volume due to the condition
$k_{F}\xi\simeq 1$. Weak glasson-glasson and glasson-phonon interaction
constants are also the consequence of the small parameter $a/\xi\ll 1$. The
glasson mass $m_{gl}$ is found to be somewhere between electron $m_{e}$ and
atom $M_{atom}$ masses, $m_{e}<m_{gl}<M_{atom}$.

“Everything should be made as simple as possible, but no simpler.” By
proposing LEEs to be fermions I rely on Occam’s razor principle. The
simplicity of the fermion phenomenology holds promise of tangible progress
beyond the picture of generic collective modes[9, 24, 26].

Acknowledgements

I am grateful for helpful discussions to Ivar Martin (Argonne National Lab)
and Peter B. Littlewood (University of Chicago).

## References

  * Anderson et al. [1972] P. W. Anderson, B. I. Halperin, and C. M. Varma.  Anomalous Low-Temperature Thermal Properties of Glasses and Spin Glasses. _Philos. Mag._ , 1972.
  * Phillips [1972] W. A. Phillips.  Tunneling states in amorphous solids.  _J. Low Temp. Phys. 7, 751_ , 1972.
  * Leggett [1991] A. J. Leggett.  Amorphous materials at low temperatures: why are they so similar? _Physica B 169, 322-327_ , 1991.
  * Pohl et al. [2002] R.O. Pohl, X. Liu, and E. Thompson.  Low-temperature thermal conductivity and acoustic attenuation in amorphous solids.  _Revs. Mod Phys. 74, 991_ , 2002.
  * Freeman and Anderson [1986] J.J. Freeman and A. C. Anderson.  Thermal conductivity of amorphous solids.  _Phys. Rev. B 34, 5684_ , 1986.
  * Zeller and Pohl [1971] R. C. Zeller and R. O. Pohl.  Thermal conductivity and specific heat of noncrystalline solids.  _Phys. Rev. B 4, 2029_ , 1971.
  * Freedman and Hertz [1977] R. Freedman and J. A. Hertz.  Theory of a fermi glass.  _Phys. Rev. B 15, 2384_ , 1977.
  * Elliott [1991] S. R. Elliott.  Medium-range structural order in covalent amorphous solids.  _Nature 354, 445_ , 1991.
  * Yu and Leggett [1988] C. C. Yu and A. J. Leggett.  Low temperature properties of amorphous materials: Through a glass darkly.  _Comments Condens. Matter Phys. 14, 231_ , 1988.
  * Phillips [1987] W.A. Phillips.  Two-level states in glasses.  _Prog. Phys. 50, 1657_ , 1987.
  * Jackle [1972] J. Jackle.  On the ultrasonic attenuation in glasses at low temperatures.  _Z. Phys. 257, 212_, 1972.
  * authorship [as of 2021] Collective authorship.  Wikipedia article about Jordan-Wigner transformation.  _Wikipedia_ , as of 2021.
  * Zaitlin and Anderson [1975] M. P. Zaitlin and A. C. Anderson.  Phonon thermal transport in noncrystalline materialse.  _Phys. Rev B 12, 4475-4486_ , 1975.
  * Abrikosov [original book (1988] A. A. Abrikosov.  Fundamentals of the theory of metals.  _Dover Publications; Reprint edition (October 18, 2017)_ , original book (1988).
  * Efros and Shklovskii [1975] A. L. Efros and B. I. Shklovskii.  Coulomb gap and low temperature conductivity of disordered systems.  _J. Phys. C: Solid State Phys. 8, L49_ , 1975.
  * Loponen et al. [1982] M. T. Loponen, R. C. Dynes, V. Narayanamurti, and J. P. Garno.  Measurements of the time-dependent specific heat of amorphous materials.  _Phys. Rev B 25, 1161_ , 1982.
  * Asban et al. [2017] O. Asban, A. Amir, Y. Imry, and M. Schechter.  The effect of interactions and disorder on the relaxation of two-level systems in amorphous solids.  _Phys. Rev. B 95, 144207_ , 2017.
  * Fleishman and Anderson [1980] L. Fleishman and P. W. Anderson.  Interactions and the anderson transition.  _Phys. Rev. B 21, 2366_ , 1980.
  * Lieb [1976] E. H. Lieb.  The stability of matter.  _Revs. Mod Phys. 48, 553-569_ , 1976.
  * Dyson and Lenard [1967] Freeman J. Dyson and A. Lenard.  Stability of matter. 1.  _J. of Mathematical Physics 8, 423_ , 1967.
  * Black and Halperin [1977] J. L. Black and B. I. Halperin.  Spectral diffusion, phonon echoes, and saturation recovery in glasses at low temperatures.  _Phys. Rev. B 16, 2879_ , 1977.
  * Amir et al. [2012] A. Amir, Y. Oreg, and Y. Imry.  On relaxations and aging of various glasses.  _Proc Natl Acad Sci 109(6), 1850-1855_ , 2012.
  * Mahmood et al. [2021] Fahad Mahmood, Dipanjan Chaudhuri, Sarang Gopalakrishnan, Rahul Nandkishore, and N. P. Armitage.  Observation of a marginal fermi glass.  _Nature Physics 17, 627 - 631_ , 2021.
  * Vural and Leggett [2011] D.C. Vural and A.J. Leggett.  Universal sound absorption in amorphous solids: A theory of elastically coupled generic blocks.  _Journal of Non-Crystaline Solids 357(11), 3528-3537_ , 2011.
  * Shukla [2021] P. Shukla.  Universality of ultrasonic attenuation in amorphous systems at low temperatures.  _https://arxiv.org/abs/2009.00556_ , 2021.
  * Turlakov [2004] M. Turlakov.  Universal sound absorption in low-temperature glasses.  _Phys. Rev. Lett. 93, 035501_ , 2004.
  * Parshin et al. [2007] D. A. Parshin, , H. A. Schober, and V. L. Gurevich.  Vibrational instability, two-level systems, and the boson peak in glasses.  _Phys. Rev. B 76, 064206_ , 2007.
  * Baggioli and Zaccone [2019] M. Baggioli and A. Zaccone.  Universal origin of boson peak vibrational anomalies in ordered crystals and in amorphous materials.  _Phys. Rev. Lett. 122, 145501_ , 2019.
  * Leggett and Vural [2013] Anthony J. Leggett and Dervis Vural.  Tunneling two-level systems model of the low-temperature properties of glasses: Are smoking-gun tests possible?  _Journal of Physical Chemistry 117, 12966 - 12971_ , 2013.
  * Turlakov [2022] M. Turlakov.  Possible experimental tests of the glasson phenomenology of low-temperature glasses.  _unpublished, work in progress_ , 2022.
  * C. et al. [2019] C. Müller C., J. H. Cole, and J. Lisenfeld.  Towards understanding two-level-systems in amorphous solids: insights from quantum circuits.  _Rep. Prog. Phys. 82 124501_ , 2019.
  * et al. [2015] J. Lisenfeld et al.  Observation of directly interacting coherent two-level systems in an amorphous material.  _Nature Comm. 6, Article: 6182_ , 2015.
  * et al. [2014] J. Burnett et al.  Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator.  _Nature Comm. 5, Article: 4119_ , 2014.
  * Agarwal et al. [2013] K. Agarwal, I. Martin, M. D. Lukin, and E. Demler.  Polaronic model of two-level systems in amorphous solids.  _Phys. Rev. B 87, 144201_ , 2013.  doi: 10.1103/PhysRevB.87.144201.  URL https://link.aps.org/doi/10.1103/PhysRevB.87.144201.


# Method of Relative Magnetic Helicity Computation II: Boundary Conditions for
the Vector Potentials

Shangbin Yang 11affiliation: Key Laboratory of Solar Activity, National
Astronomical Observatories, Chinese Academy of Sciences, 100012 Beijing, China
22affiliation: Max-Planck Institute for Solar System Research, 37191
Katlenburg-Lindau, Germany , Jörg Büchner 22affiliation: Max-Planck Institute
for Solar System Research, 37191 Katlenburg-Lindau, Germany , Jean Carlo
Santos 33affiliation: Laboratŕio de Plasmas, Instituto de Física, Universidade
de Brasĺia, Brazil , and Hongqi Zhang 11affiliation: Key Laboratory of Solar
Activity, National Astronomical Observatories, Chinese Academy of Sciences,
100012 Beijing, China

###### Abstract

We have proposed a method to calculate the relative magnetic helicity in a
finite volume as given the magnetic field in the former paper (Yang et al.
Solar Physics, 283, 369, 2013). This method requires that the magnetic flux to
be balanced on all the side boundaries of the considered volume. In this
paper, we propose a scheme to obtain the vector potentials at the boundaries
to remove the above restriction. We also used a theoretical model (Low and
Lou, Astrophys. J. 352, 343, 1990) to test our scheme.

Magnetic helicity

##  1 Introduction

Magnetic helicity is a key geometrical parameter to describe the structure and
evolution of solar coronal magnetic fields ( e.g. Berger, 1999). Magnetic
helicity in a volume $V$ can be determined as

$\centering{H_{\rm M}=\int_{V}\mathbf{A}\cdot\mathbf{B}dV},\@add@centering$
(1)

where A is the vector potential for the magnetic field B in this volume.
Magnetic helicity is conserved in an ideal magneto-plasma (Woltjer, 1958). As
long as the overall magnetic Reynolds number is large, however, it is still
approximately conserved, even in the course of relatively slow magnetic
reconnection (Berger, 1984). The concept of magnetic helicity has successfully
been applied to characterize solar coronal processes, for a recent review
about modeling and observations of photospheric magnetic helicity see, e.g.,
Démoulin and Pariat (2009). Despite of its important role in the dynamical
evolution of solar plasmas, so far only a few attempts have been made to
estimate the helicity of coronal magnetic fields based on observations and
numerical simulations (see, e.g., Thalmann, Inhester, and Wiegelmann, 2011;
Rudenko and Myshyakov, 2011).

Yang et al. (2013) developed a method for an efficient calculation of the
relative magnetic helicity in finite 3D volume which already was applied to a
simulated flaring AR Santos et al. (2011). This method requires the magnetic
flux to be balanced on all the side boundaries of the considered volume. In
this paper, a scheme to remove the restriction has been proposed. In Sec. 2,
we describe the restriction of vector potential in the former paper. In Sec.
3, we present the details of the new scheme to calculate the vector potentials
on the six boundaries. In sec. 4, we use the theoretical model to check our
scheme. The summary and some discussions are given in Sec. 5.

##  2 The former definition of ${\bf A}_{\rm p}$ and A at the boundaries

Let us define a finite three-dimensional (3-D) volume (“box”) in Cartesian
coordinates with a magnetic field ${\bf B}(x,y,z)$ given in this volume. Let
the volume be bounded by $x=[0,l_{x}]$, $y=[0,l_{y}]$, and $z=[0,l_{z}]$.

First one has to provide the values of ${\bf A}_{\rm p}$ and A on all six
boundaries ($x=0,l_{x};y=0,l_{y};z=0,l_{z}$). To take the bottom boundary
($z=0$) for example, we define a new scalar function $\varphi(x,y)$ that
determines the vector potential ${\bf A}_{\rm p}$ of the potential magnetic
field P on this boundary as follows:

${A_{\rm p\it x}=-\frac{{\partial\varphi}}{{\partial y}},\qquad A_{\rm p\it
y}=\frac{{\partial\varphi}}{{\partial x}},\qquad\left.{A_{\rm p\it
z}}\right|_{z=0}=0.}$ (2)

According to the definition of the vector potential, the scalar function
$\varphi(x,y)$ should satisfy the Poisson equation:

$\Delta\varphi(x,y)=B_{z}(x,y,z=0).$ (3)

The value of $\partial\phi/\partial n$ on the four sides of the plane $z=0$ is
set to zero in Equation (3). According to Eq.( 2), $A_{\rm p\it x}$ and
$A_{\rm p\it y}$ will vanish at $y=0,l_{y}$ and at $x=0,l_{x}$, respectively,
on the $z=0$ plane. Thus, the corresponding magnetic flux at the boundary
should also vanish because of Ampère’s law. The values of ${\bf A}_{\rm p}$ on
the other five boundaries could be obtained in a similar way. For the vector
potential A at all boundaries the same values are taken as for ${\bf A}_{\rm
p}$. When the magnetic fluxes at the six boundaries are not zero, we should
calculate the value of vector potentials at the twelve edges of the three-
dimensional (3-D) volume to provide the Neumann boundary for the Poisson
Equation at each side boundary. In next section, we will introduce a scheme to
calculate the vector potentials at the twelve edges.

##  3 new scheme to obtain $\mathbf{A}_{p}$ and $\mathbf{A}$ at the boundaries

For the $\mathbf{A}_{p}$, we define the magnetic flux $\Phi_{i}~{}(i=1,...,6)$
respectively at each side boundary
($z=0;~{}z=l_{z};~{}x=0;~{}x=l_{x};~{}y=0;~{}y=l_{y}$). The integrals of
$\int{\bf A}_{\rm p}\cdot\rm d{\bf l}$ at the twelve edges are defined as
$a_{i}~{}(i=1,...,12)$. The twelve integrals and the corresponding directions
are represented in Fig. 1.

Figure 1: Magnetic flux $\Phi_{i}~{}(i=1,...,6)$ at the six boundaries and the
integrals $a_{i}~{}(i=1,...,12)$ of $\int{\bf A}_{\rm p}\cdot\rm d{\bf l}$ at
the twelve edges.

According to the Ampère’s law, the integral value $a_{i}$ satisfy the
following linear equations

${\rm{TX=B},}$ (4)

where
$\textrm{B}=(\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4},\Phi_{5},\Phi_{6})^{T}$,
$\textrm{X}=(a_{1},a_{2},a_{3},...,a_{12})^{T}$ and T is a matrix of
6$\times$12 , which is equal

$\left[\begin{array}[]{cccccccccccc}1&1&1&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&1&1&1&0&0&0&0\\\ 0&0&0&{-1}&0&0&0&{-1}&1&0&0&1\\\
0&{-1}&0&0&0&{-1}&0&0&0&1&1&0\\\ {-1}&0&0&0&{-1}&0&0&0&{-1}&{-1}&0&0\\\
0&0&{-1}&0&0&0&{-1}&0&0&0&{-1}&{-1}\\\ \end{array}\right].$ (5)

One can check that the six rows-vector in this matrix is not linear
independent because the magnetic field is divergence free and the sum of
$\Phi_{i}$ at the six boundaries is zero. Moreover, the unknown twelve $a_{i}$
are not unique just under the restriction of above six conditions. Hence, we
need to construct twelve independent conditions to obtain the unique solution
for $a_{i}$. We define the new matrix $\hat{\textrm{T}}$ as follows

$\left[\begin{array}[]{ccccccccccccc}1&1&1&1&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&1&1&1&0&0&0&0\\\ 0&0&0&{-1}&0&0&0&{-1}&1&0&0&1\\\
0&{-1}&0&0&0&{-1}&0&0&0&1&1&0\\\ {-1}&0&0&0&{-1}&0&0&0&{-1}&{-1}&0&0\\\
1&0&{-1}&0&0&0&0&0&0&0&0&0\\\ 0&1&0&{-1}&0&0&0&0&0&0&0&0\\\
0&0&1&0&{-1}&0&0&0&0&0&0&0\\\ 0&0&0&1&0&{-1}&0&0&0&0&0&0\\\
0&0&0&0&1&0&{-1}&0&0&0&1&0\\\ 0&0&0&0&0&1&0&{-1}&0&0&0&0\\\
0&0&0&0&0&0&1&0&{-1}&0&0&0\\\ \end{array}\right].$ (6)

One can check that the determinant of $\hat{\textrm{T}}$ is not zero.
According to Cramer rule, the unique solution is existent for the new linear
equation

${\rm{{\hat{T}}X=\hat{B}},}$ (7)

where $\textrm{X}=(a_{1},a_{2},a_{3},...,a_{12})^{T}$ and
$\hat{\textrm{B}}=(\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4},\Phi_{5},0,0,0,0,0,0,0)^{T}$.
Then we can obtain the integrals of $\int{\bf A}_{\rm p}\cdot\rm d{\bf l}$ at
the twelve edges. The corresponding vector potential at the twelve edges could
be obtained by using the following equation:

$\begin{split}{\rm{A}}_{{\rm{px}}}\left({a_{i}}\right)=\frac{{\pi
a_{i}}}{{2L_{x}}}\sin({{\pi x}\mathord{\left/{\vphantom{{\pi
x}{L_{x}}}}\right.\kern-1.2pt}{L_{x}}}),i=1,3,5,7\\\
{\rm{A}}_{{\rm{py}}}\left({a_{i}}\right)=\frac{{\pi
a_{i}}}{{2L_{y}}}\sin({{\pi y}\mathord{\left/{\vphantom{{\pi
y}{L_{y}}}}\right.\kern-1.2pt}{L_{y}}}),i=2,4,6,8\\\
{\rm{A}}_{{\rm{pz}}}\left({a_{i}}\right)=\frac{{\pi
a_{i}}}{{2L_{z}}}\sin({{\pi z}\mathord{\left/{\vphantom{{\pi
z}{L_{z}}}}\right.\kern-1.2pt}{L_{z}}}),i=9,10,11,12\end{split}$ (8)

Note that ${\bf A}_{\rm p}$ at the ends of every edge both are zero according
to the above equation. That is the requirement of Eq. (2). Then we resolve the
Poisson equations to obtain ${\bf A}_{\rm p}$ at the six boundaries. For the
vector potential A at all boundaries the same values are taken as for ${\bf
A}_{\rm p}$. Then we can follow the method of Sec. 2.2 and 2.3 of the former
paper Yang et al. (2013) to calculate the relative magnetic helicity in this
volume.

##  4 Testing the scheme

For testing the new scheme to obtain the vector potentials at the boundaries,
we use the axisymmetric nonlinear force-free fields of Low and Lou (1990). We
used the model labeled $P_{1,1}$ with $l=0.3$ and $\Phi=\pi/2$ in the notation
of their paper. We calculated the magnetic field on a uniform grid of
$64\times 64\times 64$. The pixel size in the calculation is assumed to be 1.

We calculate the magnetic fluxes $\Phi_{0}$ at the six boundaries and
substitute it to the Eq. (7) to obtain the integral $a_{i}$ at the twelve
edges of the 3D volume. Then we substitute $a_{i}$ into Eq. (8) respectively
to get the boundary value for resolving the Poisson equation in Eq. (3) at the
six boundaries. After we attain ${\bf A}_{\rm p}$ at the six boundaries, we
could also calculate the magnetic flux $\Phi$ according to the relation
between the vector potential and the magnetic field: ${\bf B}\cdot\hat{\rm
n}=\nabla\times{\bf A}_{\rm p}\cdot\hat{\rm n}$. Table. 1 represents the final
result after we apply the the above scheme. It can be found that the
calculated magnetic fluxes at the six boundaries by using our scheme
respectively coincide well with the original value from the theoretical model.
Note that the total magnetic flux of the theoretical model is not exact zero.
However, it is required that the total magnetic flux is exact zero when
resolving the linear equation Eq. (7), which cause the total magnetic fluxes
of $\oint{\bf A}_{\rm p}\cdot\rm d{\bf l}$ and $\Phi$ are different with that
of $\Phi_{0}$. On the other hand, the numerical errors when resolving the
Poisson equation are also unavoidable, which will also introduce the
difference for the total magnetic flux as well.

Table 1: Testing using the new scheme to a theoretical model. side boundary | $\Phi_{0}^{\tablenotemark{a}}$ | $\oint{\bf A}_{\rm p}\cdot\rm d{\bf l}^{\tablenotemark{b}}$ | $\Phi^{\tablenotemark{c}}$
---|---|---|---
$z=0$ | -3615.81 | -3615.81 | -3487.1068
$z=l_{z}$ | 1461.13 | 1461.13 | 1490.2739
$x=0$ | -1471.95 | -1471.95 | -1563.7657
$x=l_{x}$ | -1471.95 | -1471.96 | -1564.9280
$y=0$ | 4006.42 | 4006.42 | 4087.8426
$y=l_{y}$ | 1068.27 | 1092.17 | 1066.6782
Total flux | -23.89 | 0.00012 | 28.99

##  5 Summary

In this paper, we propose a new scheme to calculate the vector potential at
the boundaries to remove the restrictions in the former paper Yang et al.
(2013). In principle, now we can calculate the relative magnetic helicity of
any magnetic field structure in Cartesian coordinates. In the observations, we
could use force-free extrapolation method to obtain the three-dimensional
magnetic structure to analyze the evolution of relative magnetic helicity. On
the other hand, we can also use a sequence of magnetograms to estimate the
accumulated magnetic helicity in the solar corona (Ref. Démoulin and Pariat,
2009). It will be very interesting to compare the two types of accumulated
magnetic helicity and analyze the correlation between magnetic helicity and
solar eruption (e.g. Jing et al., 2012). In the simulations, we could also
calculate the relative magnetic helicity directly based the known magnetic
field structure to understand how the magnetic helicity plays an important
role in solar reconnection and dynamos.

This study is supported by grants 10733020, 10921303, 41174153,11173033
11178016 and 11103038 of National Natural Science Foundation of China,
2011CB811400 of National Basic Research Program of China, a sandwich-PhD grant
of the Max-Planck Society and the Max-Planck Society Interinstitutional
Research Initiative Turbulent transport and ion heating, reconnection and
electron acceleration in solar and fusion plasmas of Project No. MIF-IF-A-
AERO8047.The authors also like to thank the Supercomputing Center of Chinese
Academy of Sciences (SCCAS) for the allocation of computing time.

## References

  * Berger (1984) Berger, M. A., Field, G., B.: 1984, J. Fluid Mech. 147, 133\.
  * Berger (1999) Berger, M. A.: 1999, Plasma Phys. Contr. Fusion 41, 167.
  * Boulmezaoud (1999) Boulmezaoud, T. Z.: 1999, Étude des champs de Beltrami dans des domaines de R3 bornś et non bornś et applications en astrophysique”, Ph.D. thesis, Univ. Paris VI.
  * Démoulin and Pariat (2009) Démoulin, P., Pariat, E.: 2009, Adv. Space Res. 43, 1013\.
  * Jing et al. (2012) Jing, J., Park, S., Liu, C., Lee, J., Wiegelmann, T., Xu, Y., Deng, N., & Wang, H. M. : 2012, Astrophys. J. 752, L9
  * Low and Lou (1990) Low, B. C., Lou, Y.Q.: 1990, Astrophys. J. 352, 343.
  * Rudenko and Myshyakov (2011) Rudenko, G. V., Myshyakov, I. I.: 2011, Solar phys. 270, 165\.
  * Santos, Büchner, and Otto (2011) Santos, J. C., Büchner, J., Otto, A.: 2011, Astron. Astrophys. 535, A111.
  * Seehafer, Kuzanyan, and Pipin (2003) Seehafer, N., Gellert, M., Kuzanyan, K. M., Pipin, V. V.: 2003, Adv. Space Res. 32, 1819.
  * Thalmann, Inhester, Wiegelmann (2011) Thalmann, J. K., Inhester, B., Wiegelmann, T.: 2011, Solar Phys. 272, 243.
  * Valori, Démoulin and Pariat (2012) Valori, G., Démoulin, P., Pariat, E.: 2012, Solar Phys. 278, 347.
  * Woltjer (1958) Woltjer, L. 1958, Proc. Natl Acad. Sci. USA, 44, 480
  * Santos et al. (2011) Santos, J. C., Büchner, J., & Otto, A. 2011, A&A, 535, A111
  * Yang et al. (2013) Yang, S., Büchner, J. , Santos, J. C., & Zhang, H.: 2013, Solar Physics,283, 369.
  * Yang, Büchner, and Zhang (2009a) Yang, S., Büchner, J., Zhang, H.: 2009, Astrophys. J. Lett. 695, L25.
  * Yang, Büchner, and Zhang (2009b) Yang, S., Zhang, H., Büchner, J.: 2009, Astron. Astrophys. 502, 333.
  * Zhang (2006) Zhang, H.: 2006, Astrophys. Space Sci. 305, 211.
  * Zhang, Flyer, and Low (2006) Zhang, M., Flyer, M., Low, B.: 2006, Astrophys. J. 644, 575\.


# On the required radio resources for ultra-reliable communication in highly
interfered scenarios

Gilberto Berardinelli, Ramoni Adeogun Gilberto Berardinelli and Ramoni Adeogun
are with the Department of Electronic Systems, Aalborg University, Denmark
(e-mail:{gb<EMAIL_ADDRESS>
###### Abstract

Future wireless systems are expected to support mission-critical services
demanding higher and higher reliability. In this letter, we dimension the
radio resources needed to achieve a given failure probability target for
ultra-reliable wireless systems in high interference conditions, assuming a
protocol with frequency hopping combined with packet repetitions. We resort to
packet erasure channel models and derive the minimum amount of resource units
in the case of receiver with and without collision resolution capability, as
well as the number of packet repetitions needed for achieving the failure
probability target. Analytical results are numerically validated, and can be
used as benchmark for realistic system simulations.

###### Index Terms:

Ultra-reliable communication, channel hopping, subnetworks.

##  I Introduction

The increasing utilization of wireless for mission-critical applications has
stressed the importance of ensuring reliable over-the-air communication [1].
The 5th generation (5G) radio access technology, standardized by the third
generation partnership project (3GPP), has introduced the concept of ultra-
reliable low latency communication (URLLC), targeting reliability levels,
defined as packet error rates, in the order of $10^{-5}-10^{-6}$ [2]. The
trend towards wireless support of critical services is expected to continue in
the coming years with 6th Generation (6G) radios, targeting more and more
demanding reliability and availability requirements [3].

For example, the recent concept of 6G _in-X subnetworks_ envisions short-range
radio cells to be installed inside entities like industrial robots, production
modules, vehicles or even human bodies, for critical control applications [4].
Specifically, in-X subnetworks may replace hard real time industrial
communication protocols such as PROFINET Isochronous Real Time (IRT) and
Ethernet Powerlink, characterized by strictly periodic transmissions with
deterministic cycle times [5]. In these systems, a certain packet transmission
by e.g., a sensor is expected to reach the receiving node within a bounded
time, for the sake of preserving the stability of the control loop. Since in-X
subnetworks can become very dense, such as those installed in robots operating
in a matrix production mode, interference is a major threat hindering the
reliability of the communication.

In our previous work, we have proposed a transmission protocol featuring a
combination of packet repetitions (a.k.a. repetition code) and frequency
channel hopping, as a viable approach for achieving ultra-reliable
communication in dense subnetwork scenarios characterized by a large number of
uncoordinated persistent transmitters concurring for the same radio resources
[6]. In particular, each packet transmission is repeated a number of times
over extremely short time intervals, in order to ensure ultra-reliable
communication with sub-ms latencies in dense scenarios. Channel hopping is a
well-known approach for harvesting frequency and interfering diversity; our
protocol is inspired to the main operational principle of the IEEE 802.15.1
standard [7], as well as the $\rm{K}$-repetition mode, proposed by the 3GPP as
a grant-free alternative to traditional grant-based access [8].

The performance of our proposed protocol has been evaluated via system level
simulations in an industrial scenario (e.g.,[6, 9]), highlighting the complex
trade-offs between resource utilization, number of repetitions and deployment
characteristics. Though system level simulations are a powerful approach for
realistic performance analysis, their results are necessarily conditioned to
specific assumptions in terms of operational scenario and radio propagation
model (e.g., an indoor factory propagation scenario in our analysis). This
makes difficult to generalize the learnings and distinguish the protocol
aspects from the combined effects of radio channel and deployment
characteristics.

In this letter, we perform an analysis of our frequency-hopped protocol with
focus on the relationship between resource utilization, packet repetitions,
and high reliability, for the case of a large number of persistent radio
devices sharing the same resource grid. We resort to erasure channel models to
derive analytically the minimum amount of radio resources for supporting a
target failure probability, and the corresponding required number of packet
repetitions. The usage of packet erasure channel models is a very common
approach for the basic analysis of wireless communication protocols (see,
e.g., the works on coded slotted ALOHA and coded random access in [10, 11]),
as they allow to isolate the pure protocol aspects from the radio propagation
and modelling parts. Our aim is to obtain closed form expressions for the
number of resource units ensuring a given failure probability target, to be
used as a benchmark for coming system level analysis of 6G frequency-hopped
systems for ultra-reliable communication. The validity of our analytical
results is proved via a numerical study.

##  II Setting up the scene

We consider $d+1$ devices connected to one or more access points (APs). We
assume one out of the $d+1$ devices to be our device of interest, while the
other $d$ devices are potential interferers. This can be the case, for
example, of a sensor in a subnetwork reporting periodic measurements to its
serving APs, with the other $d$ devices belonging to different subnetworks.
The reference scenario is pictorially depicted in Fig. 1. We assume for
simplicity that the devices adhere to a common clock and are frame aligned,
but their operations are not coordinated as each subnetwork operates
independently.

Devices operate over the same radio resources, where each transmission happens
over a frame consisting of a grid of $N_{RU}=pq$ resource units, with $p$
frequency channels and $q$ time slots. Obviously, the wireless system designer
must dimension the size of a resource unit according to the expected desired
signal strength, data packet size and spectrum availability. Each device maps
then a data packet in a resource unit, and transmits it $n$ times by selecting
for each repetition a different resource unit, for example according to a
pseudo-random pattern. It is assumed that each repetition happens over a
different time slot and frequency channel, and therefore the constraint $1\leq
n\leq{\rm{min}}(p,q)$ holds.

An example of such transmission is depicted in Fig. 2, where the patterns of
two non-interfering devices are shown. The pseudo-random channel hopping is a
well-known approach in wireless communications, and has been shown to provide
significant reliability gain thanks to the possibility of exploiting time,
frequency and interference diversity [12]. In a real system the hopping
pattern can change at every transmission, in order to achieve robustness to
jammers and malicious interferers [13].

In the next sections, we derive the minimum amount of resources needed for
achieving a given transmission failure probability target, considering packet
erasure channel models.

##  III Resource dimensioning with collisions as disruptive events

We first consider a classic packet erasure channel model, where collisions are
disruptive events, and transmission is always successful in the case of at
least one collision-free packet repetition [14]. In spite of its simplicity,
we believe that the packet erasure channel model is well-suited for the basic
analysis of an interference-limited in-X subnetwork scenario. This is because
in short range subnetworks the receive signal strength of the desired
transmitter is expected to be high; moreover, the AP can use techniques such
as power control and link adaptation to counteract fading and ensure correct
reception for interference-free transmissions [4]. This means, transmission
failures are only caused by collisions. By using a packet erasure channel
model, the $d$ interfering devices are to be intended as those devices whose
proximity generate disruptive collisions when operating over the same resource
units as the target device. In the following, we use the subscript $0$ to
denote the no collision case for the derivation of the failure probability.

Let us consider the transmission of a given packet repetition by the target
device. Since an interfering device is transmitting over $n$ of the $pq$
resource units, the probability of the target device having no collision with
its transmissions is given by $\frac{pq-n}{pq}$. The probability of having no
collision for a given packet repetition with any of the $d$ interfering
devices is then ${P_{0}}=\left(\frac{pq-n}{pq}\right)^{d}$.

The complement probability $(1-P_{0})$ represents then the probability of
having at least one collision at a given packet repetition of the target
device. The probability of transmission failure, i.e., having at least one
collision for each of the $n$ packet repetitions, is then given by

$P_{{f,0}}=\left(1-{P_{0}}\right)^{n}=\left(1-\left(1-\frac{n}{N_{RU}}\right)^{d}\right)^{n}.$
(1)

The amount of required resource unit for a given failure probability target
$P^{(\rm{T})}_{{f}}$ set by the system designer can be then calculated by
setting $P_{f,0}=P^{(\rm{T})}_{{f}}$ and solving (1) for $N_{RU}$, leading to

${N}_{RU}=\left\lceil{\frac{n}{1-\left(1-{P^{({\rm{T}})}_{{f}}}^{\frac{1}{n}}\right)^{\frac{1}{d}}}}\right\rceil,$
(2)

with $\left\lceil\cdot\right\rceil$ denoting the ceiling operator.

Figure 1: Reference scenario, with a number of devices served by their own AP,
interfering the transmissions of the target device. Figure 2: Example of
transmission pattern over a frame resource grid. The colors refer to the
transmissions of two non-interfering devices.

Since ${N}_{RU}=pq$, the wireless system designer can engineer the split of
the resource grid in the time and the frequency domains. Latency-limited
services may use a low number $q$ of time slots, while expanding the frequency
dimension by increasing the number of frequency channels (calculated as
$p={N}_{RU}/q$). Conversely, in the case of limited available spectrum
resources, the number $p$ of frequency channels must necessarily be reduced
and the number $q$ of time slots increased (calculated as $q={N}_{RU}/p$);
this comes at the expense of latency. Also, please observe that the size of a
resource unit has to be dimensioned according to the packet size and the
achievable signal-to-noise ratio (SNR) at the receiver to ensure correct
reception. An analysis of the time/frequency resources needed for achieving a
target failure probability as a function of the achievable SNR for the target
device is left for future work.

######  Theorem III.1.

In case the receiver is unable to resolve collisions, for a large number of
interfering devices the minimum amount of resource units $N^{min}_{RU}$ for
achieving $P^{({\rm{T}})}_{{f}}$ can be approximated as

$N^{min}_{RU}\approx\left\lceil\frac{{\rm{ln}}\left(P^{({\rm{T}})}_{{f}}\right)}{\left[\left(\frac{1}{2}\right)^{\frac{1}{d}}-1\right]{\rm{ln}}\left(2\right)}\right\rceil.$
(3)

###### Proof.

Let us generalize (2) in the continuous domain as a function of $n$, i.e.,
$N_{RU}=f(n)$, $n\in\mathbb{R}$, $n\geq 1$. $f(n)$ is a convex function, and
attains its minimum at the zero of its derivative. As $f(n)$ is in the form of
a quotient $\frac{u}{v}$, its derivative with respect to $n$ will be zero if
$vu^{\prime}-uv^{\prime}=0$, i.e.,

$1-\left(1-\left(P^{({\rm{T}})}_{{f}}\right)^{\frac{1}{n}}\right)^{\frac{1}{d}}+\\\
+\frac{{\rm{ln}}\left(P^{({\rm{T}})}_{{f}}\right)\left(P^{({\rm{T}})}_{{f}}\right)^{\frac{1}{n}}}{dn}\left(1-\left(P^{({\rm{T}})}_{{f}}\right)^{\frac{1}{n}}\right)^{\frac{1}{d}-1}=0.$
(4)

By defining $h=\frac{1}{d}$, and dividing both terms of the equality by $h$,
(4) can be rewritten as

$-\frac{\left(1-\left(P^{({\rm{T}})}_{{f}}\right)^{\frac{1}{n}}\right)^{h}-1}{h}+\\\
+\frac{{\rm{ln}}\left(P^{({\rm{T}})}_{{f}}\right)\left(P^{({\rm{T}})}_{{f}}\right)^{\frac{1}{n}}}{n}\left(1-\left(P^{({\rm{T}})}_{{f}}\right)^{\frac{1}{n}}\right)^{h-1}=0.$
(5)

For large $d$, we have $h\rightarrow 0$, and the leftmost term
$\frac{\left(1-\left(P^{({\rm{T}})}_{{f}}\right)^{\frac{1}{n}}\right)^{h}-1}{h}\rightarrow{\rm{ln}}\left(1-\left(P^{({\rm{T}})}_{{f}}\right)^{\frac{1}{n}}\right)$.
(5) can be then expressed as

$y\rm{ln}\left(y\right)-\left(1-y\right)\rm{ln}\left(1-y\right)=0$ (6)

where $y=\left(P^{({\rm{T}})}_{{f}}\right)^{\frac{1}{n}}$. (6) has three
zeros, i.e., $y=0,\frac{1}{2},1$, with $\frac{1}{2}$ being the solution of
interest. We therefore obtain

$n=\frac{1}{{\rm{log}}_{P^{({\rm{T}})}_{{f}}}\left(\frac{1}{2}\right)}=-\frac{{\rm{ln}}\left(P^{({\rm{T}})}_{{f}}\right)}{{\rm{ln}}\left(2\right)}.$
(7)

This represents the minimum number of packet repetition for achieving
$P^{({\rm{T}})}_{{f}}$; interestingly, it does not depend on the number of
devices $d$. By applying (7) in (2), we obtain the result in (3). ∎

Since for large $d$, $1/d\rightarrow 0$, we can resort to a Maclaurin
expansion of the type $y^{x}\approx 1+x{\rm{ln}}(y)$, and obtain
$\left(\frac{1}{2}\right)^{1/d}\approx
1+\frac{1}{d}\rm{ln}\left(\frac{1}{2}\right)$. Therefore:

$N^{min}_{RU}\approx-\left\lceil\frac{d}{{\rm{ln}}^{2}\left(2\right)}{\rm{ln}}\left(P^{({\rm{T}})}_{{f}}\right)\right\rceil=-\left\lceil
2.0814\cdot d\cdot{\rm{ln}}\left(P^{({\rm{T}})}_{{f}}\right)\right\rceil.$ (8)

The minimum amount of radio resource units needed for achieving a failure
probability $P^{({\rm{T}})}_{{f}}$ has then a logarithmic relationship with
the failure probability itself, while it scales linearly with the number of
devices.

##  IV Dimensioning with resolvable collisions

Let us assume now that our receiver is able to resolve collisions up to a
given extent, thanks to e.g., the usage of successive cancellation receivers
and multi-antenna processing [15]. In this case, collisions are not disruptive
events. Let us denote the number of colliding devices at a given resource unit
as $N_{\text{c}}$. When deriving the failure probability, the subscript
$c=0,\cdots,N^{\rm{max}}_{\text{c}}$, is used to denote the number of
resolvable collisions at a given packet repetition, with
$N^{\rm{max}}_{\text{c}}$ being the maximum number of resolvable collisions.

The probability that a packet transmission of the target device collides with
one of the $n$ transmissions of a given interfering device is given by
$\frac{n}{N_{RU}}$. The probability of having $N_{\text{c}}=c$ collisions in
the presence of $d$ interfering devices, with $0\leq c\leq d$, can be then
characterized by the probability mass function of a binomial distribution,
i.e.,

$P(N_{\text{c}}=c)=\binom{d}{c}\left(\frac{n}{N_{RU}}\right)^{c}\left(1-\frac{n}{N_{RU}}\right)^{d-c}.$
(9)

We are then softening the definition of traditional packet erasure channel
models, and assuming that a transmission failure happens when all packet
repetitions of the target device experience more than
$N^{\rm{max}}_{\text{c}}$ collisions, thanks to the capability of the receiver
to resolve at most $N^{\rm{max}}_{\text{c}}$ collisions. The corresponding
failure probability can be then expressed as

$P_{f,N^{\rm{max}}_{\text{c}}}=\left(1-\sum^{N^{\rm{max}}_{\text{c}}}_{c=0}P(N_{\text{c}}=c)\right)^{n}.$
(10)

Note that (10) leads to (1) for $N^{\rm{max}}_{\text{c}}=0$. The minimum
number of radio resource units needed for achieving a certain failure
probability target $P^{(\rm{T})}_{{f}}$ can be calculated from (10) by setting
$P_{f,N^{\rm{max}}_{\text{c}}}=P^{(\rm{T})}_{{f}}$ and solving it numerically
for $N_{RU}$.

###  IV-A Case of a single resolvable collision

For the case $N^{\rm{max}}_{\text{c}}=1$, it is possible to derive an explicit
expression for $N_{RU}$, as well as for its minimum $N^{min}_{RU}$.

######  Theorem IV.1.

In case the receiver is able to resolve at most one collision, for a large
number of interfering devices the amount of resource units $N_{RU}$ for
achieving a target failure probability $P^{(\rm{T})}_{{f}}$ can be
approximated as

$N_{RU}\approx-\left\lceil\frac{nd}{W_{-1}\left(\frac{{P^{(\rm{T})}_{{f}}}^{\frac{1}{n}}-1}{\rm{e}}\right)+1}\right\rceil.$
(11)

where $W_{-1}\left(\cdot\right)$ denotes the negative branch of the Lambert W
function [16].

###### Proof.

For $N^{\rm{max}}_{\text{c}}=1$, by setting $P^{(\rm{T})}_{{f}}=P_{f,1}$, (10)
reads

$P^{(\rm{T})}_{{f}}=\left(1-\left(1-\frac{n}{N_{RU}}\right)^{d}-\frac{nd}{N_{RU}}\left(1-\frac{n}{N_{RU}}\right)^{d-1}\right)^{n}.$
(12)

Let us generalize (12) in the continuous domain of the $N_{RU}$ variable,
i.e., $N_{RU}\in\mathbb{R}$. Note that, if
$f(N_{RU})=\left(1-\frac{n}{N_{RU}}\right)^{d}$, then
$f^{\prime}(N_{RU})=\frac{nd}{N^{2}_{RU}}\left(1-\frac{n}{N_{RU}}\right)^{d-1}$.
We therefore obtain
$f(N_{RU})+N_{RU}f^{\prime}(N_{RU})=1-{P^{(\rm{T})}_{{f}}}^{\frac{1}{n}}$. By
observing that
$f(N_{RU})+N_{RU}f^{\prime}(N_{RU})=[N_{RU}f(N_{RU})]^{\prime}$, it must hold
that its anti-derivative is of the form $N_{RU}f(N_{RU})=aN_{RU}+b$, with
$a=1-{P^{(\rm{T})}_{{f}}}^{\frac{1}{n}}$, and $b$ a constant.

For large $d$, we can use the approximation
$\left(1-\frac{n}{N_{RU}}\right)^{d}\approx e^{-\frac{nd}{N_{RU}}}$, and (12)
can be therefore expressed as

$N_{RU}e^{-\frac{nd}{N_{RU}}}=aN_{RU}+b.$ (13)

By setting $w=1/N_{RU}$, (13) can be reformulated as $e^{-ndw}-bw-a=0$. Such
equation can be solved by using Lambert W functions [17, p.4], leading to

$w=-\frac{1}{nd}{\rm{ln}}\left(\frac{b}{nd}W\left(\frac{nde^{\frac{and}{b}}}{b}\right)\right),$
(14)

where $W(x)$ denotes the multivariate Lambert W function. Since for such
function it holds that ${\rm{ln}}(W(x))={\rm{ln}}(x)-W(x)$ [17, p.2], (14) can
be rewritten as

$w=-\frac{1}{nd}\left({\rm{ln}}\left(\frac{b}{nd}\right)+{\rm{ln}}\left(\frac{nde^{\frac{and}{b}}}{b}\right)-W\left(\frac{nde^{\frac{and}{b}}}{b}\right)\right)\\\
=-\frac{and-bW\left(\frac{nde^{\frac{and}{b}}}{b}\right)}{ndb}$ (15)

and therefore

$N_{RU}\approx-\frac{ndb}{and-bW\left(\frac{nde^{\frac{and}{b}}}{b}\right)}.$
(16)

By definition, $W(x)$ has a single real solution equal to $-1$ only at
$x=-\frac{1}{e}$. By setting $\frac{nde^{\frac{and}{b}}}{b}=-\frac{1}{e}$ and
$q=nd/b$, we obtain an equation of the form $qe^{aq}+1/e=0$, which is again
solvable by using Lambert W functions [17, p.3], leading to

$b=\frac{nda}{W\left(-\frac{a}{e}\right)}.$ (17)

For $-\frac{1}{e}\leq x<0$, $W(x)$ is two-valued in the positive and negative
branch [16], i.e., $W_{0}(a/e)$ and $W_{-1}(a/e)$, respectively, with the
negative branch solution ensuring positive values for $N_{RU}$ in (14). By
inserting (17) in (14), and applying the ceiling operator, we obtain (11). ∎

######  Theorem IV.2.

In case the receiver is able to resolve at most one collision, with a large
number $d$ of interfering devices the minimum amount of resource units
$N^{min}_{RU}$ for achieving a target failure probability
$P^{({\rm{T}})}_{{f}}$ can be approximated as

$N^{min}_{RU}\approx-\left\lceil 0.7502\cdot
d\cdot{\rm{ln}}\left(P^{(\rm{T})}_{{f}}\right)\right\rceil$ (18)

###### Proof.

By setting $z=\frac{\left(P^{(\rm{T})}_{{f}}\right)^{\frac{1}{n}}-1}{\rm{e}}$,
we obtain
$n=\frac{{\rm{ln}}\left(P^{({\rm{T}})}_{{f}}\right)}{{\rm{ln}}\left(1+ez\right)}$.
In the continuous domain, (11) becomes then:

$N_{RU}\approx\frac{-d\cdot{\rm{ln}}\left(P^{({\rm{T}})}_{{f}}\right)}{{\rm{ln}}\left(1+ez\right)\left[W_{-1}\left(z\right)+1\right]}.$
(19)

Since $P^{({\rm{T}})}_{{f}}\ll 1$, the numerator of (19) is a positive number,
and (19) is minimized when the denominator
$g(z)={\rm{ln}}\left(1+ez\right)\left[W_{-1}\left(z\right)+1\right]$ is
maximized. $g(z)$ is a convex function, with a maximum at $z\approx-0.2798$.
We therefore obtain that $N_{RU}$ is minimized when

$n\approx-0.6995\cdot{\rm{ln}}\left(P^{({\rm{T}})}_{{f}}\right).$ (20)

By applying (20) in (11), we obtain (18). ∎

$N^{min}_{RU}$ has also a linear dependency with the number of devices, and a
logarithmic dependency with the target failure probability. By comparing (18)
with (8), we can observe that the capability of the receiver to resolve one
collision leads to a reduction of the minimum amount of resource units of a
factor of $\sim\times 2.77$ for achieving the same target failure probability.

##  V Validation of analytical results

We validate our analytical results with simulation studies, assuming a failure
probability target $P^{({\rm{T}})}_{{f}}=10^{-6}$. Analytical results are
obtained with (2) and (11), for $N^{\rm{max}}_{\text{c}}=0$ and
$N^{\rm{max}}_{\text{c}}=1$, respectively, and the analytical minima with (3)
and (18). For the cases where analytical results are not available, i.e.,
$N^{\rm{max}}_{\text{c}}=2,3$, we resort to numerical results, obtained by
numerical inversion of (10) with respect to $N_{RU}$. Simulation results are
obtained as follows: we first set a given number of packet repetitions $n$,
with $n=2,\cdots 26$; we then set a value for $N_{RU}$, and generate up to 20
million samples of transmission patterns for the target and interfering
devices; we calculate the empirical empirical failure probability when up to
$N^{\rm{max}}_{\text{c}}=0,\cdots,3$ collisions are resolvable; we repeat the
pattern generation for a large set of $N_{RU}$ values, and save, for the given
$n$, the minimum $N_{RU}$ value that allows to achieve the failure probability
target. Note that, for each $N_{RU}$, the resource grid is generated by
selecting arbitrarily $p$ and $q$ such that $p,q\geq n$ and $pq=N_{RU}$;
patterns of both target and interfering devices are then generated by randomly
selecting $n$ resource units in the grid, with uniform probability.

Fig. 3 shows the amount of radio resource units needed to support the failure
probability target with $d=100$ interfering devices. Simulation results show a
perfect match with the analytical and numerical ones. When the receiver has no
collision solving capabilities, a large number of packet repetitions is needed
in order to have the highest chance of a collision-free transmission. The
capability of resolving collisions leads to a lower amount of needed resource
units, but also to a lower number of packet repetitions for approaching the
minimum, reasonably translating to major transmit power reduction. Observe
that, while for $N^{max}_{\text{c}}=0$ the number of required resource units
becomes soon fairly insensitive to the number of packet repetitions, the
$N^{max}_{\text{c}}\geq 1$ cases exhibit the intuitive trade-off between
resource usage and generated interference, i.e., one needs a certain number of
repetitions to improve resilience to collisions, but when such number
increases, performance is affected by the higher likelihood of collisions
themselves. This stresses the need of an accurate setting of the $n$ parameter
in a practical system.

Fig. 4 shows the accuracy of the analytical results in (8) and (18) as a
function of number of interfering devices $d$. Since the values of $n$ that
lead to minimum resource usage in Fig. 3 (orange circles) are independent from
the number of interfering devices, they can be used for the analysis in Fig.
4. For $N^{\rm{max}}_{\text{c}}=0$, the analytical results are compared to the
values obtained with (2), using the value $n$ taken from the result in Fig. 3.
With $N^{\rm{max}}_{\text{c}}=1$, the analytical results are compared to the
numerical results obtained by numerical searches for solutions of (12), with
$n$ also taken from the result in Fig. 3. Though results in (8) and (18) were
obtained in the limit for large $d$, the match with the numerical results is
nearly perfect also for a low number of interfering devices.

Figure 3: Number of resource units needed for supporting a target failure
probability of $10^{-6}$, for 100 interfering devices. Figure 4: Accuracy of
analytical results for minimum number of radio resource elements, assuming a
target failure probability of $10^{-6}$.

##  VI Conclusions

In this letter, we have studied the problem of dimensioning radio resources
for a frequency-hopped radio system targeting ultra-reliable communication in
the presence of a large number of persistent interferers. We have derived a
closed form expression for the required number of repetitions and the minimum
amount of resource units needed to achieve a failure probability target,
considering the cases of collisions as disruptive events, as well as receiver
tolerance to single collision events. The number of packet repetitions
ensuring minimum radio resource usage results to be independent from the
number of interfering devices; while the minimum number of resource elements
grows linearly with the number of devices, and has a logarithmic dependency
with the failure probability target. Numerical results have proved the
accuracy of the derived analytical results.

We believe that our results can provide concrete insights to wireless
engineers for dimensioning frequency-hopped wireless protocols for high
reliability, and can be used as benchmark for system simulation studies based
on realistic models of radio propagation and receiver capabilities.

## Acknowledgements

This research is partially supported by the HORIZON-JU-SNS-2022-STREAM-B-01-03
6G-SHINE project (Grant Agreement No. 101095738).

## References

  * [1] N. A. Mohammed, A. M. Mansoor, and R. B. Ahmad, “Mission-critical machine-type communication: An overview and perspectives towards 5G,” _IEEE Access_ , vol. 7, pp. 127 198–127 216, 2019.
  * [2] Y. Liu, Y. Deng, M. Elkashlan, A. Nallanathan, and G. K. Karagiannidis, “Analyzing grant-free access for URLLC service,” _IEEE Journal on Selected Areas in Communications_ , vol. 39, no. 3, pp. 741–755, 2020.
  * [3] H. Viswanathan and P. E. Mogensen, “Communications in the 6G era,” _IEEE Access_ , vol. 8, pp. 57 063–57 074, 2020.
  * [4] G. Berardinelli, P. Baracca, R. O. Adeogun, S. R. Khosravirad, F. Schaich, K. Upadhya, D. Li, T. Tao, H. Viswanathan, and P. Mogensen, “Extreme communication in 6G: Vision and challenges for ‘in-X’ subnetworks,” _IEEE Open Journal of the Communications Society_ , vol. 2, pp. 2516–2535, 2021.
  * [5] M. Schumacher, J. Jasperneite, and K. Weber, “A new approach for increasing the performance of the industrial Ethernet system PROFINET,” in _2008 IEEE International Workshop on Factory Communication Systems_. IEEE, 2008, pp. 159–167.
  * [6] R. Adeogun, G. Berardinelli, P. E. Mogensen, I. Rodriguez, and M. Razzaghpour, “Towards 6G in-X subnetworks with sub-millisecond communication cycles and extreme reliability,” _IEEE Access_ , vol. 8, pp. 110 172–110 188, 2020.
  * [7] Z. Aalam, S. Vhatkar, and B. Mishra, “IEEE 802.15. 1 Simulation and BER analysis under the Interference,” in _2nd International Conference and workshop on Emerging Trends in Technology (ICWET), Proceedings published by International Journal of Computer Applications®(IJCA)_. Citeseer, 2011, pp. 24–28.
  * [8] T.-K. Le, U. Salim, and F. Kaltenberger, “Enhancing URLLC uplink Configured-grant transmissions,” in _2021 IEEE 93rd Vehicular Technology Conference (VTC2021-Spring)_. IEEE, 2021, pp. 1–5.
  * [9] R. Adeogun, G. Berardinelli, I. Rodriguez, and P. Mogensen, “Distributed dynamic channel allocation in 6G in-X subnetworks for industrial automation,” in _2020 IEEE Globecom Workshops (GC Wkshps_. IEEE, 2020, pp. 1–6.
  * [10] Z. Sun, Y. Xie, J. Yuan, and T. Yang, “Coded slotted ALOHA for Erasure Channels: Design and Throughput Analysis,” _IEEE Transactions on Communications_ , vol. 65, no. 11, pp. 4817–4830, 2017.
  * [11] E. Paolini, C. Stefanovic, G. Liva, and P. Popovski, “Coded random access: Applying codes on graphs to design random access protocols,” _IEEE Communications Magazine_ , vol. 53, no. 6, pp. 144–150, 2015.
  * [12] Z. Kostic, I. Maric, and X. Wang, “Fundamentals of dynamic frequency hopping in cellular systems,” _IEEE Journal on selected areas in communications_ , vol. 19, no. 11, pp. 2254–2266, 2001.
  * [13] L. Chiarello, P. Baracca, K. Upadhya, S. R. Khosravirad, and T. Wild, “Jamming Detection with Subcarrier Blanking for 5G and beyond in Industry 4.0 scenarios,” in _2021 IEEE 32nd Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC)_. IEEE, 2021, pp. 758–764.
  * [14] J. Massey and P. Mathys, “The collision channel without feedback,” _IEEE Transactions on Information Theory_ , vol. 31, no. 2, pp. 192–204, 1985.
  * [15] D. Tse and P. Viswanath, _Fundamentals of wireless communication_. Cambridge University Press, 2005.
  * [16] R. M. Corless, G. H. Gonnet, D. E. Hare, D. J. Jeffrey, and D. E. Knuth, “On the LambertW function,” _Advances in Computational mathematics_ , vol. 5, no. 1, pp. 329–359, 1996.
  * [17] S. Edwards, “Extension of Algebraic Solutions Using the Lambert W Function,” _arXiv preprint arXiv:1902.08910_ , 2019.