# Degenerate topological line surface phonons in quasi-1D double helix crystal
SnIP

Bo Peng  TCM Group, Cavendish Laboratory, University of Cambridge, J. J.
Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Shuichi Murakami
Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro-ku,
Tokyo 152-8551, Japan  Tokodai Institute for Element Strategy, Tokyo Institute
of Technology, Nagatsuta, Midori-ku, Yokohama, Kanagawa 226-8503, Japan
Bartomeu Monserrat  TCM Group, Cavendish Laboratory, University of Cambridge,
J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Department of
Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage
Road, Cambridge CB3 0FS, United Kingdom<EMAIL_ADDRESS>Tiantian Zhang
Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro-ku,
Tokyo 152-8551, Japan  Tokodai Institute for Element Strategy, Tokyo Institute
of Technology, Nagatsuta, Midori-ku, Yokohama, Kanagawa 226-8503, Japan
<EMAIL_ADDRESS>
###### Abstract

Degenerate points/lines in the band structures of crystals have become a
staple of the growing number of topological materials. The bulk-boundary
correspondence provides a relation between bulk topology and surface states.
While line degeneracies of bulk excitations have been extensively
characterized, line degeneracies of surface states are not well understood. We
show that SnIP, a quasi-one-dimensional van der Waals material with a double
helix crystal structure, exhibits topological nodal rings/lines in both the
bulk phonon modes and their corresponding surface states. Using a combination
of first-principles calculations, symmetry-based indicator theories and Zak
phase analysis, we find that two neighbouring bulk nodal rings form doubly
degenerate lines in their drumhead-like surface states, which are protected by
the combination of time-reversal symmetry $\mathcal{T}$ and glide mirror
symmetry $\bar{M}_{y}$. Our results indicate that surface degeneracies can be
generically protected by symmetries such as $\mathcal{T}\bar{M}_{y}$, and
phonons provide an ideal platform to explore such degeneracies.

## Introduction

Degeneracies in the bulk energy bands of crystals were intensively studied in
the early days of band theory [1]. Recent breakthroughs have shed new light
into this old topic by associating topological invariants with these
degenerate points or lines [2, 3, 4], and by extending the discussion from
electrons to phonons [5, 6, 7], photons [8], magnons [9], and excitons [10].
Among the degenerate points, twofold degenerate Weyl points always appear in
pairs that can be topologically characterized with opposite Chern numbers [11,
12, 13, 14], and fourfold degenerate Dirac points have a linear dispersion in
both electronic [15, 16, 17, 18, 19, 20] and phononic systems [21, 22]. When
the band crossings are one-dimensional in momentum space, they can form nodal
lines, nodal rings or nodal chains, depending on their shape [23, 24, 25, 26,
27, 28, 29, 30, 31], and these line crossings are protected by symmetries such
as mirror or $\mathcal{PT}$ (where $\mathcal{P}$ is the spatial inversion
symmetry and $\mathcal{T}$ is the time reversal symmetry) [32, 33]. Recent
advances, using both group theoretical analysis and high-throughput
calculations, have enabled a comprehensive understanding of bulk band
degeneracies in both electrons [34, 35, 36] and phonons [22]. These
topological degeneracies exhibit unique physical properties such as a ‘quantum
anomaly’ [37, 38], enabling various applications from spintronics [39] to
topological quantum computation [40].

As a result of the bulk-boundary correspondence, the degenerate bulk points
and lines are associated with topologically protected surface states,
including surface arcs associated with Weyl and Dirac points [41, 42, 43, 44]
and drumhead-like flat bands associated with nodal lines [45, 24]. Similar to
bulk band degeneracies, surface states can also exhibit band crossings in both
semi-infinite slabs and finite slabs (with top and bottom surfaces) [46]. Here
we focus on semi-infinite systems with only one surface. For Weyl points with
Chern numbers of $\pm 1$, the associated helicoid surface states, whose
isoenergy contours are Fermi arcs, form a continuous 2D surface in the energy-
momentum space $E$-k and have no degeneracy [47], as schematically shown in
Figure 1(a). In topological insulators and topological crystalline insulators,
the surface states become degenerate at a single point [48, 49, 50, 51, 52],
forming a surface Dirac cone, as shown schematically in Figure 1(b). In
principle, two surface states can also cross along a line in the surface
Brillouin zone, as shown schematically in Figure 1(c). However, like the bulk
case, these surface line crossings need to be protected by additional
symmetries [47, 53, 54, 55, 56]. While nodal line degeneracies in the bulk
states have been thoroughly explored in a variety of quasiparticle band
structures, the degenerate lines in the surface states have received less
attention. The few examples of surface nodal lines all correspond to
electronic band structures [47, 53, 54, 55, 57, 56] or artificial systems [58,
59]. It is therefore desirable to identify more candidates in other
quasiparticle spectra to fully understand the role of surface degeneracies.

Figure 1: Different types of the surface states in topological materials. (a)
Non-degenerate helicoid surface states, (b) zero-dimensional degenerate Dirac
surface states, and (c) one-dimensional degenerate nodal-line surface states.

Among all the quasiparticles, phonons are of particular interest as they
describe intrinsic ionic vibrations. Moreover, the bosonic nature of phonons
facilitates the exploration of topological phenomena in their entire spectrum.
This is in contrast to fermonic systems (e.g. electronic systems) in which
there is an additional constraint that degeneracies need to occur near the
Fermi level to have an impact on the behaviour of the material. Furthermore,
time-reversal symmetry is hard to break in phononic systems, and consequently
this extra symmetry requirement is automatically satisfied in phonons. These
advantages have led to a recent surge of interest in characterising
topological properties of phonons [5, 6, 7, 21, 60, 61, 62, 63, 64, 22, 65,
66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76], but degenerate lines in surface
phonon states have not yet been reported.

In this work, we propose that SnIP, an inorganic semiconductor with nodal-ring
phonons in the bulk states, exhibits nodal-line surface states in the phonon
spectrum using a combination of first-principles calculations, group theory
analysis and Zak phase analysis. These surface states are doubly degenerate in
a nodal line on the (100) surface. The surface nodal line is protected by the
anti-unitary symmetry $\vartheta=\mathcal{T}\bar{M}_{y}$, which is a
combination of time-reversal symmetry $\mathcal{T}$ and glide mirror symmetry
$\bar{M}_{y}$. With different surface terminations, we can alternatively get
two surface nodal lines on different parts of the (100) surface Brillouin
zone, which can be understood by different Zak phases in the presence of
$\mathcal{T}\bar{M}_{y}$ symmetry. Our results suggest that similar surface
degenerate lines/points will be found in other materials when there are extra
symmetries to protect these surface degeneracies, and we believe that phonons
can be a primary platform to study these degeneracies because of the presence
of time-reversal symmetry and the flexibility to study the degeneracies in the
entire phonon spectrum.

## Results

### Crystal structure and phonon dispersion

SnIP is an inorganic semiconductor with a double helix crystal structure,
first synthesized in 2016, and can be formed without any templates [77, 78,
79]. The strong covalent bonds along the helices and the weak van der Waals
interactions across the helices combine the electronic properties of inorganic
semiconductors, such as high electron mobility [80], and the mechanical
properties of polymers, such as high flexibility [81]. In addition, by atomic
substitution, a large material family with double helix structures has been
predicted using first-principles calculations, providing a wide range of band
gaps between 1 and 2.5 eV [82]. With a tunable band gap and high mechanical
flexibility, these quasi-one-dimensional semiconductors are extremely
promising for next generation devices ranging from mechanical sensors to
optoelectronics [81, 82]. Moreover, they can serve as a material platform to
realize one-dimensional physical models such as the Su-Schrieffer-Heeger model
[83] and Majorana quantum wires [84, 85].

Bulk SnIP crystallizes in the monoclinic space group $P2/c$ (No. 13) with two
double helices (SnIP)7 in the unit cell with stoichiometry (SnIP)14. The two
double helices, alternatively left- and right-hand twisted, are stacked along
the $a$ direction. Each double helix is formed by an inner helical P chain and
an outer helical SnI chain, as shown in Figure 2(a). The (SnIP)7 unit in each
SnIP chain winding results in a 7/2 helix. From one phosphorus (yellow) atom
to the next, the turning angle is about $\pm 360^{\circ}$/7 followed by a
translation of $c$/7, and the same holds true for the tin (cyan) and iodine
(navy) atoms in the SnI helix [86]. The calculated lattice parameters of $a$ =
7.889 Å, $b$ = 9.768 Å, $c$ = 18.422 Å, and monoclinic distortion $\beta$ =
110.21∘ match well with the measured ones [77, 79].

Figure 2: Structural and vibrational properties of SnIP. (a) Crystal
structure, (b) phonon dispersion and (c) Brillouin zone of SnIP. The orange
region in the phonon dispersion highlighted in (b) corresponds to the
topological features that we focus on. The phonon DOS is also shown in (b).

The calculated phonon dispersion of SnIP is presented in Figure 2(b). It
exhibits wide bandwiths along the helical direction, corresonding to the
$\Gamma$-Y, C-Z, B-A and E-D high-symmetry lines in Figure 2(c), but, by
contrast, phonon bands along other high-symmetry lines, especially those
perpendicular to the helices, have a relatively flat dispersion, a result of
the quasi-one-dimensional nature of this material. This is consistent with the
fact that SnIP has strong covalent bonds along the one-dimensional helices but
only weak van der Waals-like interactions between the double helices. As a
result, SnIP fibers can be macroscopically bent by up to 90∘ without any
significant Raman mode shifts, indicating that SnIP is an ultrasoft inorganic
material and bending the helices has negligible influence on the phononic
properties [81].

There are 42 atoms in the unit cell of SnIP, corresponding to 126 phonon
branches. The calculated atom-projected phonon density of states (DOS) in
Figure 2(b) indicates that the low frequency modes below 4 THz are mainly
vibrations involving the heavier Sn and I atoms, while those between 4-14 THz
are dominated by the motion of the lighter P atoms. There are 14 phosphorus
atoms in the unit cell, leading to 42 P-dominated bands in the phonon spectra,
spanning branches 85 to 126. The phonon modes from 4 to 10 THz (branches
85-108) are more dispersive compared to the almost flat bands between 10 and
14 THz. The two double helices are related to each other by inversion/glide
mirror symmetry, so there always exist two bands with similar eigenvalues and
similar eigenvectors, corresponding to the P atoms from the two separate P
chains, respectively (see the phonon dispersion between 4-8.5 THz in
Supplementary Note 1 of the Supplementary Information).

Hereafter we focus on the frequency range 4-10 THz because the corresponding
phonon bands are relatively ‘clean’ due to the large mass difference between
phosphorus and the other two elements that separates them from the lower
energy bands, and due to the strong intrachain vibrations that make them
highly dispersive. This makes them an ideal playground for exploring
topological features. In addition, these modes in the range 4-10 THz
contribute to nearly half the thermal conductivity along the helix direction
(see the phonon transport properties in Supplementary Note 2 of the
Supplementary Information).

### Nodal ring phonons in SnIP

For the phonon modes between 4 and 10 THz, dominated by the vibrations of the
relatively light P atoms, the phonon bands have a large dispersion along the
helical direction but are almost flat along the perpendicular directions, as
shown in Figure 2(b). The band folding of the seven P atoms in a single
helical P chain [86] leads to several band crossings near the Brillouin zone
boundary, especially in the flat band regions. After a comprehensive search
for band crossing points between bands 85-108 (i.e., 4-10 THz), we find that
the most promising band crossings are formed by phonon branches 92 and 93 with
clean topological features in both bulk and surface states. Marked with # 92
and # 93 in Figure 3(b), these two bands cross along the $\Gamma$-Z high-
symmetry line.

Figure 3: Topological properties of SnIP. (a) Diagnosis process for the single
nodal ring around the Z point using symmetry-based indicator theory for the
lowest 92 bands. (b) Phonon band crossings between branches 92 and 93, and (c)
the nodal rings in the bulk Brillouin zone, as well as their projections on
the (100) surface.

To understand the topological information of the band crossings between
branches 92 and 93, we use symmetry-based indicator theories to analyze the
degeneracies [87, 88, 89, 63, 90, 91]. Because of the existence of a band
crossing along the $\Gamma$-Z high-symmetry line, the compatibility condition
is broken for the lowest 92 bands, as the compatibility relations can only be
satisfied when the band structures have no (non-accidental) band crossings
along any high-symmetry lines or at any high-symmetry points [88]. So we
cannot use the symmetry-based indicator for space group No. 13. Instead, we
need to find a maximal subgroup of SnIP that simultaneously satisfies the
compatibility condition and has a nontrivial symmetry-based indicator. Among
all the subgroups of space group No. 13 (space groups No. 2, No. 3 and No. 7),
No. 2 is the only subgroup which both satisfies the compatibility condition
and has a symmetry-based indicator group: (i) the compatibility condition is
satisfied because the twofold rotation symmetry in No. 13 that protects the
band crossing along $\Gamma$-$Z$ is absent in No. 2; (ii) space group No. 2
has a non-trivial symmetry-based indicator of
$\mathbb{Z}_{2224}\equiv\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}$,
as discussed in detail in Ref. [88]. Therefore we can use $\mathbb{Z}_{2224}$
as indicators to diagnose the topological degeneracy. For space group No. 2,
there are eight high-symmetry points, i.e. high-symmetry points
($q_{a},q_{b},q_{c}$) with $q_{a,b,c}=0,0.5$. We obtain the irreducible
representations (irreps.) at the eight corresponding high-symmetry points
$\Gamma$ (0,0,0), Z (0,0.5,0), B (0,0,0.5), Y (0.5,0,0), C (0.5,0.5,0), D
(0,0.5,0.5), A (0.5,0,0.5), and E (0.5,0.5,0.5), as shown in Figure 3(a). We
then compute the indicator accordingly. The resulting indicator (0101) implies
that there is a single nodal ring around the Z point in SnIP formed by
branches 92 and 93, which is verified by the numerical calculations in Figure
3(c).

We next examine in detail the nodal rings formed by bands 92 and 93. We zoom
in the phonon dispersion on the $\Gamma$-B-D-Z plane in Figure 3(b). The band
crossing formed by bands 92 and 93 along the $\Gamma$-Z high-symmetry line
belongs to the single nodal ring around the Z point, carrying a quantized
$\pi$ Berry phase. The nodal ring crosses through the Brillouin zone boundary
around the B-D high-symmetry line, and consequently there are two nodal rings
from two neighbouring Brillouin zones formed around the D point. We also plot
the phonon dispersion around N $(-0.01,0.35,0.51)$, another band crossing
point on the neighbouring nodal ring, as shown in Figure 3(b). Most
interestingly, we find that the two neighbouring nodal rings projected on the
(100) surface have a large overlap, and the two drumhead-like surface states
of the two bulk nodal rings form a doubly degenerate line along
$\bar{\textrm{B}}$-$\bar{\textrm{D}}$ (as discussed below).

### Topological surface states of SnIP

We demonstrate the nodal-line surface states by calculating the surface local
density of states (LDOS) from the imaginary part of the surface Green’s
function [92]. We first study the surface along the (100) direction with the
termination denoted as surface 1 in Figure 4(a)-(e). As shown in Figure 4(a),
along the $\bar{\textrm{D}}$-$\bar{\textrm{Z}}$ and
$\bar{\textrm{Z}}$-$\bar{\Gamma}$ high-symmetry lines, two separate
projections of the bulk band crossings (marked by navy and cyan arrows) belong
to two single nodal rings from two neighbouring Brillouin zones respectively.
The surface states are within the projections of the two bulk nodal rings,
forming two non-degenerate drumhead-like surface states marked by navy and
cyan triangles in Figure 4(a) respectively. The surface states from the two
crossing points along $\bar{\textrm{D}}$-$\bar{\textrm{Z}}$ and
$\bar{\textrm{Z}}$-$\bar{\Gamma}$ merge with each other at $\bar{\textrm{D}}$,
and result in a doubly degenerate surface nodal line along
$\bar{\textrm{B}}$-$\bar{\textrm{D}}$ (green triangle). The surface nodal line
ends at the crossing point of the two projected nodal rings marked by green
arrow in Figure 4(a).

Figure 4: Topological surface states of SnIP. (a) Topological surface states
of surface 1 along the (100) direction, as well as the surface arcs for phonon
frequencies of (b) 6.410 THz, (c) 6.414 THz, (d) 6.415 THz, and (e) 6.416 THz.
(f) Topological surface states of surface 2 along the (100) direction, as well
as the surface arcs for phonon frequencies of (g) 6.400 THz, (h) 6.410 THz,
(i) 6.415 THz, and (j) 6.420 THz.

To have a better view of the doubly degenerate surface states along the
$\bar{\textrm{B}}$-$\bar{\textrm{D}}$ high-symmetry line, we plot the
isofrequency surface arcs for different phonon frequencies in Figure 4(b)-(e).
As shown in Figure 4(b), at the exact frequency of the doubly degenerate
projected bulk band crossings (green arrow) along
$\bar{\textrm{B}}$-$\bar{\textrm{D}}$ at 6.410 THz, the two surface arcs (cyan
triangle) merge at the projected bulk band crossings. For a phonon frequency
of 6.414 THz, the two surface arcs (cyan triangle) move towards each other,
forming two arc crossing points (green triangle) along
$\bar{\textrm{B}}$-$\bar{\textrm{D}}$ in Figure 4(c). At 6.415 THz, the two
arc crossing points move to the $\bar{\textrm{D}}$ point until they merge into
a single arc crossing point [green triangle in Figure 4(d)]. Further
increasing the phonon frequency eventually leads to two separate surface arcs
as the two arcs no longer touch [navy triangle in Figure 4(f)]. The arc
crossing points on the isofrequency plane belong to the doubly degenerate
surface nodal line in the frequency-momentum space [green triangle in Figure
4(a)].

The surface nodal line in SnIP is protected by the combination of time-
reversal symmetry and glide mirror symmetry
$\vartheta=\mathcal{T}\bar{M}_{y}$. For systems with time-reversal symmetry
$\mathcal{T}$ and any nonsymmorphic spatial symmetry $G$, the ‘Kramers-like’
degenerate nodal line appears when $(\mathcal{T}G)^{2}=-1$, either along the
$\mathcal{T}G$ invariant lines or on the $\mathcal{T}G$ invariant planes, and
a degenerate line can thus be protected by the $\mathcal{T}G$ symmetry. This
applies to line degeneracies in both the bulk states [93] and the surface
states [53, 54, 55]. SnIP has glide mirror symmetry $\bar{M}_{y}=\\{M_{y}\ |\
00\frac{1}{2}\\}$, and $(\mathcal{T}\bar{M}_{y})^{2}=\textrm{e}^{{\rm{i}}2\pi
q_{z}}$ ($q_{z}$ in units of the reciprocal lattice vector $2\pi/c$). On the
(100) surface, both $\textit{q}_{z}=0$ and $\textit{q}_{z}=0.5$ are
$\mathcal{T}\bar{M}_{y}$ invariant lines, and
$(\mathcal{T}\bar{M}_{y})^{2}=-1$ only occurs when $\textit{q}_{z}=0.5$,
corresponding to the $\bar{\textrm{B}}$-$\bar{\textrm{D}}$ high-symmetry line.
As a result, the doubly degenerate surface nodal line along
$\bar{\textrm{B}}$-$\bar{\textrm{D}}$ is protected by the
$\mathcal{T}\bar{M}_{y}$ symmetry.

If we change the surface termination (denoted as surface 2) along the (100)
direction, the doubly degenerate surface states along the
$\bar{\textrm{B}}$-$\bar{\textrm{D}}$ high-symmetry line in Figure 4(f) no
longer exist in the frequency range 6.410-6.415 THz, but move to the lower
frequency range 6.384-6.410 THz. The distribution of the surface nodal line
changes to the other side of $\bar{\textrm{B}}$-$\bar{\textrm{D}}$, from the
purple shaded area in Figure 5(a) to that in Figure 5(b).

Figure 5: Zak phase analysis. Zak phase of (a) surface 1 and (b) surface 2
along the (100) direction. The purple and green shaded areas correspond to the
total Zak phase $\gamma=2\pi$ and $\gamma=0$ respectively.

To explain the alternation between the distributions of the surface nodal
line, we need to understand how $\mathcal{T}\bar{M}_{y}$ influences the Zak
phase on the (100) surface Brillouin zone. The Zak phase provides information
of the distribution of the doubly degenerate surface states on the (100)
surface, and can be integrated along the (100) direction,

$\gamma(\textit{q}_{b},\textit{q}_{c})={\rm{i}}\int_{-\pi}^{\pi}\sum_{\nu={\rm
occu}.}\bra{\psi(\textbf{q}\nu)}\frac{\partial}{\partial\textit{q}_{a}}\ket{\psi(\textbf{q}\nu)}d\textit{q}_{a},$
(1)

where $\psi(\textbf{q}\nu)$ is the phonon eigenvector with wave vector q and
band index $\nu$. Note that on the (100) surface Brillouin zone,
$\textit{q}_{b}\parallel\textit{q}_{y}$ and
$\textit{q}_{c}\parallel\textit{q}_{z}$, therefore we use
($\textit{q}_{x}$,$\textit{q}_{y}$,$\textit{q}_{z}$) instead of
($\textit{q}_{a}$,$\textit{q}_{b}$,$\textit{q}_{c}$) to be consistent with
$\mathcal{T}\bar{M}_{y}$. When $\textit{q}_{z}=0.5$,
$(\mathcal{T}\bar{M}_{y})^{2}=-1$ and
$\psi_{1}(-\textit{q}_{x},\textit{q}_{y},0.5)=(\mathcal{T}\bar{M}_{y})\cdot\psi_{2}(\textit{q}_{x},\textit{q}_{y},0.5)$.
Therefore, we can classify the phonon states at constant $\textit{q}_{y}$ and
$\textit{q}_{z}=0.5$ into two subgroups $\\{\psi_{1}(\textbf{q})\\}$ and
$\\{\psi_{2}(\textbf{q})\\}$, i.e., $\\{\psi_{1}(\textbf{q})\\}$ and
$\\{\psi_{2}(\textbf{q})\\}$ are $\mathcal{T}\bar{M}_{y}$ related. Therefore,
for the $\bar{\textrm{B}}$-$\bar{\textrm{D}}$ high-symmetry line
($\textit{q}_{z}=0.5$) on the (100) surface Brillouin zone, the Zak phases for
the two subgroups $\\{\psi_{1}(\textbf{q})\\}$ and
$\\{\psi_{2}(\textbf{q})\\}$ are equal to each other, i.e.
$\gamma_{1}(\textit{q}_{y},0.5)=\gamma_{2}(\textit{q}_{y},0.5)$. Due to
$\mathcal{PT}$,
$\gamma_{1}(\textit{q}_{y},0.5)=\gamma_{2}(\textit{q}_{y},0.5)$ is either 0 or
$\pi$, which depends on the surface termination (i.e. choice of the inversion
center). When
$\gamma_{1}(\textit{q}_{y},0.5)=\gamma_{2}(\textit{q}_{y},0.5)=0$, there are
no surface states as the total Zak phase $\gamma=\gamma_{1}+\gamma_{2}=0$.
When $\gamma_{1}(\textit{q}_{y},0.5)=\gamma_{2}(\textit{q}_{y},0.5)=\pi$, the
total Zak phase $\gamma=2\pi$, leading to doubly degenerate surface states.
For surface 1, the purple shaded line in Figure 5(a) has a total Zak phase of
2$\pi$ with two surface states distributed in this area, while the green
shaded area corresponds to $\gamma=0$ and there are no surface states in this
area. For surface 2, the Zak phase of the purple shaded area and the
colourless area swap, as shown in Figure 5(b), and the distribution of the
surface nodal line switches to the new purple shaded area. It should be
noticed that the standard definition of Zak phase is in terms of modulo
2$\pi$, and a Zak phase of 2$\pi$ is therefore equal to zero under this
definition. However, in our case, a total Zak phase of 2$\pi$ corresponds to
$\gamma_{1}=\gamma_{2}=\pi$, i.e. two $\pi$ Zak phase within each subgroup,
indicating two surface states emerging from each subgroup. On the other hand,
a total Zak phase of zero corresponds to two zero Zak phase within each group,
leading to vanishing surface states.

It is worth mentioning that bending SnIP rods does not break the time-reversal
and glide mirror symmetries on the (100) surface of SnIP. Therefore, the
degenerate topological surface states remain unchanged under bending.

## Discussion

In terms of fundamental physics and applications, topological bulk phonons in
double helical SnIP may lead to non-dissipative surface phonons, which offer a
promising platform for designing thermal devices. The existence of surface
nodal lines also provides opportunities to study surface transport properties
for thermoelectric devices. As another example, the bulk and surface nodal
lines/rings in the phonon spectra result in large phonon DOS, which may also
enhance surface superconductivity. Additionally, because of its quasi-one-
dimensional crystal structure, SnIP could also be a promising material
platform to investigate one-dimensional physical models such as the Su-
Schrieffer-Heeger (SSH) model [83] and Majorana quantum wires [84, 85]. With
its high mechanical flexibility, it is also interesting to investigate the
interplay between flexoelectricity and topology. Our work provides a starting
point to explore all these phenomena and applications.

In summary, we find that double helical SnIP exhibits nodal-ring phonons in
the bulk and nodal-line phonons on the surface based on first-principles
calculations and group theory analysis. Benefiting from a quasi-one-
dimensional crystal structure, the lattice vibrational modes of the intrachain
phosphorus atoms along the helices are highly dispersive. Due to the winding
of the helix P chain, these phonon bands are folded at the Brillouin zone
boundary, forming single nodal rings across the whole Brillouin zone. The
overlap between the two neighbouring nodal rings on the (100) surface leads to
a surface nodal line that is protected by the combination of time-reversal and
glide mirror symmetries $\mathcal{T}\bar{M}_{y}$. We also propose that
different surface terminations result in different distributions of the
surface nodal line, which can be understood by different Zak phases in the
presence of $\mathcal{T}\bar{M}_{y}$. We demonstrate that, similar to the bulk
nodal lines, the surface nodal lines can widely exist when protected by extra
symmetries. Our discoveries may offer opportunities to study the degeneracy of
the topological surface states and their transport properties.

## Methods

### Density functional theory calculations

All density functional theory (DFT) calculations are performed using the
Vienna ab-initio simulation package (vasp) [94, 95]. The generalized gradient
approximation (GGA) is used in the Perdew-Burke-Ernzerhof (PBE)
parametrization for the exchange-correlation functional with 14 valence
electrons ($4d^{10}5s^{2}5p^{2}$) for Sn, 7 valence electrons ($5s^{2}5p^{5}$)
for I, and 5 valence electrons ($3s^{2}3p^{3}$) for P. We use a plane-wave
basis with a kinetic energy cutoff of 600 eV and a 7$\times$5$\times$3 k-mesh
for structural relaxation until the energy difference is lower than 10-6 eV
and the Hellman-Feynman force difference is lower than 10-4 eV Å-1. The D2
method of Grimme is used to describe the van der Waals interactions between
the double helices [96].

### Lattice dynamics properties

The Hessian matrix and phonon frequencies are calculated based on the finite
difference method using a 2$\times$2$\times$1 supercell with a
3$\times$3$\times$3 k-mesh. The phonon dispersion is obtained with the phonopy
code [97, 98]. The convergence of the supercell is checked by comparing the
2$\times$2$\times$1 and 1$\times$1$\times$2 supercells in Supplementary Note 1
of the Supplementary Information. The Born effective charges are also computed
using a perturbative approach to account for the splitting between the
longitudinal and transverse optical phonon modes (LO-TO splitting) near the
Brillouin zone center [99], and we find that the LO-TO splitting has a minor
influence on the topological properties of phonons because the topological
features are far away from the zone center.

### Topological properties

To obtain the topological properties of the lattice vibrational modes, we use
WannierTools to calculate the distribution of the band crossing points/lines
in the whole Brillouin zone [92]. The phonon surface states on the (100)
surface are computed using the surface Green’s function.

## Data Availability

The datasets generated during and/or analysed during the current study are
available from the corresponding authors on reasonable request.

## Code Availability

The related codes are available from the corresponding authors on reasonable
request.

## Acknowledgements

B.P. and B.M. acknowledge support from the Winton Programme for the Physics of
Sustainability, and B.M. also acknowledges support from the Gianna
Angelopoulos Programme for Science, Technology, and Innovation. S.M. and T.Z.
acknowledge the supports from JSPS KAKENHI Grants No. JP18H03678 and No.
JP20H04633, Tokodai Institute for Element Strategy (TIES) funded by MEXT
Elements Strategy Initiative to Form Core Research Center. T.Z. also
acknowledge the support by Japan Society for the Promotion of Science (JSPS),
KAKENHI Grant No. JP21K13865. The calculations are performed using resources
provided by the Cambridge Tier-2 system operated by the University of
Cambridge Research Computing Service (http://www.hpc.cam.ac.uk) and funded by
EPSRC Tier-2 capital grant EP/P020259/1, and also with computational support
from the UK Materials and Molecular Modelling Hub, which is partially funded
by EPSRC (EP/P020194), for which access is obtained via the UKCP consortium
and funded by EPSRC grant EP/P022561/1.

## Author Contributions

B.P., B.M. and T.Z. devised the project idea. B.P. performed the first-
principles calculations. S.M. and T.Z. analysed the topological features.
B.P., B.M. and T.Z. prepared the main part of the manuscript. B.P., S.M., B.M.
and T.Z. discussed the results and the ideas for analysis and edited the
manuscript.

## Competing Interests

The Authors declare no Competing Financial or Non-Financial Interests.

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# Predicting Depression and Anxiety: A Multi-Layer Perceptron for Analyzing
the Mental Health Impact of COVID-19

David Fong<EMAIL_ADDRESS>,  Tianshu Chu<EMAIL_ADDRESS>,  Matthew Heflin
<EMAIL_ADDRESS>,  Xiaosi Gu<EMAIL_ADDRESS>and  Oshani
Seneviratne<EMAIL_ADDRESS>
###### Abstract.

We introduce a multi-layer perceptron (MLP) called the COVID-19 Depression and
Anxiety Predictor (CoDAP) to predict mental health trends, particularly
anxiety and depression, during the COVID-19 pandemic. Our method utilizes a
comprehensive dataset, which tracked mental health symptoms weekly over ten
weeks during the initial COVID-19 wave (April to June 2020) in a diverse
cohort of U.S. adults. This period, characterized by a surge in mental health
symptoms and conditions, offers a critical context for our analysis. Our focus
was to extract and analyze patterns of anxiety and depression through a unique
lens of qualitative individual attributes using CoDAP. This model not only
predicts patterns of anxiety and depression during the pandemic but also
unveils key insights into the interplay of demographic factors, behavioral
changes, and social determinants of mental health. These findings contribute
to a more nuanced understanding of the complexity of mental health issues in
times of global health crises, potentially guiding future early interventions.

Machine Learning, Mental Health Trends, Data Analysis, Health Informatics

††copyright: none††ccs: Information systems Extraction, transformation and
loading††ccs: Applied computing Consumer health††ccs: Applied computing Health
care information systems††ccs: Computing methodologies Distributed computing
methodologies††ccs: Applied computing Life and medical sciences††ccs:
Computing methodologies Machine learning††ccs: Social and professional
topics††ccs: Human-centered computing

##  1\. Introduction

Psychiatric disorders have always imposed significant challenges to public
health (Prince et al., 2007), and the advent of the COVID-19 pandemic has
intensified these issues (Talevi et al., 2020; Moreno et al., 2020; Cullen et
al., 2020). The pandemic has led to increased prevalence and severity of
mental health concerns globally, driven by a mix of health fears, social
isolation, economic pressures, and general uncertainty. This environment
presents new challenges and opportunities for understanding and addressing
mental health conditions, in particular depression and anxiety. We employ an
MLP to predict depression and anxiety, which can lead to early intervention
and better management of such potential mental health conditions. Our approach
has the unique ability to process complex data patterns in mental health
datasets, making it particularly suitable for the multifaceted nature of
mental health. Central to our study is a dataset comprising detailed weekly
surveys (Shuster et al., 2021) that offers an invaluable perspective on the
psychological impact of the pandemic. This data was collected at the beginning
of the COVID-19 pandemic (April - June 2020) in the U.S. Even though the
results of this study may not generalize well to pre- and post-pandemic times
or other geographical locations, by analyzing this dataset, we gathered
valuable insights into the mental health trends of periods during stressful
global catastrophes such as the COVID-19 pandemic.

###  1.1. Contribution

We created a machine learning model designed to predict anxiety and depression
levels using qualitative data from an individual during global health crises,
such as the COVID-19 pandemic. We utilized model explainability to reveal
specific qualitative attributes that increase susceptibility to such mental
health issues, which is essential in guiding the creation of targeted mental
health interventions, specifically addressing the needs of the most affected
individuals. Our work aims to advance predictive modeling in mental health and
deepen the understanding of psychological impacts during global health crises,
offering valuable insights for practitioners and researchers.

##  2\. Related Work

The exploration of mental health, specifically anxiety and depression, in the
context of the COVID-19 pandemic has been a focus of several studies. Our
research, utilizing an MLP to decode patterns of depression and anxiety,
aligns with these efforts but presents distinct methodologies and insights, as
noted below.

Hueniken et al. (Hueniken et al., 2021) utilized surveys to assess mental
health symptoms, alongside factors like substance use and COVID-19-related
worries, employing principal component analysis. Our work diverges from this
work by focusing on extracting patterns specific to anxiety and depression
using MLPs. Freyhofer et al. (Freyhofer et al., 2021) offer insights into the
impact of coping strategies and external factors on mental health during the
pandemic. In contrast, our research delves into qualitative attributes of
individuals, using a neural network-based approach to uncover their interplay
in mental health disorders. Nemesure et al. (Nemesure et al., 2021) provides
methodological insights that could be compared to our MLP approach. While both
aim at predictive modeling, our emphasis on quantitative data analysis through
an MLP offers a novel perspective. Malik et al. (Malik et al., 2023) discusses
using various machine learning models for mental health analysis during
COVID-19, using a combination of techniques such as decision trees, k-nearest
neighbor, and naive Bayes. Our research contrasts this by predicting patterns
of depression and anxiety, providing an alternate view of mental health
dynamics.

In contrast to the data analysis methods described above, Shukla and
Seneviratne (Shukla and Seneviratne, 2023) implemented a mood prediction
system using personal medical devices that tracked patient physiological data.
They build a decentralized learning machine learning model called
“MentalHealthAI” with federated learning and smart contracts to keep data on
the patient’s end without collecting it in a central server. This provides a
privacy-preserving methodology for analyzing mental health data. While
“MentalHealthAI” (Shukla and Seneviratne, 2023) used a method for predicting
mood through psychological data focusing on mental health as a chronic
disease, our focus in this paper is on predicting depression and anxiety
levels using qualitative survey data collected at the onset of a traumatic
global health crisis such as the COVID-19 pandemic. In a similar vein,
D’Alfonso (D’Alfonso, 2020) focuses on three of the main ways that AI is
applied in mental health: (1) personal sensing or digital phenotyping, (2)
natural language processing of clinical texts and social media content, and
(3) chatbots. These methodologies exemplify a promising future for integrating
AI into digital interventions, especially within web and smartphone
applications, and the authors also underscore the ethical challenges
associated with using AI in mental health care. Our goal is to develop an
application that incorporates these ethical guidelines and advances the
integration of AI in mental health services to assist mental health
professionals.

##  3\. System Design and Implementation

###  3.1. Dataset

Our work utilizes the dataset collected by Shuster et al. (Shuster et al.,
2021), a 10-week survey with a diverse group of participants that examined 308
demographic and psychological features, including anxiety and depression
levels. Participants represented a wide spectrum of diversity, coming from
different ethnic backgrounds, genders, religious beliefs, socioeconomic
statuses, geographical locations, and varying mental health conditions.
Shuster et al. leveraged linear mixed-effects models to examine factors
contributing to longitudinal changes in depression and anxiety and discovered
that gender, age, income level, previous psychiatric diagnosis, informedness
about the pandemic, and marital status correlated with higher overall levels
of anxiety and depression.

Figure 1. Heat map illustrating the relationships among four commonly examined
personal attributes (age, gender, education level, job) and a primary metric
of mental health conditions from the dataset by Shuster et al.. The subtle
variations in correlation strength underscore the challenge of manually
discerning crucial features related to mental health conditions.

Using this same dataset, we created an MLP that predicts depression, i.e., sds
score, as measured by the Self-Rating Depression Scale (Zung, 1965) and
anxiety, i.e., stai-s score, as measured by the state sub-scale of the State-
Trait Anxiety Inventory (Spielberger, 1989), based on patient features. Our
work extends and enhances the understanding gained from the foundational study
that initially collected the data (Shuster et al., 2021). The dataset by
Shuster et al. consists of numeric and text data. The numeric data consisted
of data from numeric or categorical answers, such as age, gender, race,
employment status, mental health disorders, anxiety scores (stai-s), and
depression scores (sds). The textual data consists of user answers that were
not categorical or numerical. One instance of these questions was where
applicants had to describe various images they saw and what emotion they
invoked. In our analysis, we only focused on the numerical data.

###  3.2. Exploratory Data Analysis

Data exploration is crucial for understanding the dataset’s underlying
structure and performing feature engineering. To ensure data integrity and
consistency for our analyses, we followed a meticulous reformatting process
involving several steps, such as selecting complete and consistent participant
records, removing null data entries, and standardizing columns across the ten-
week time points. An essential aspect of this phase was transforming
categorical variables into a numerical format, enhancing their suitability for
processing by our MLP. After our exploratory data analysis, we split our
dataset into a 70-20-10 for training, testing, and validation.

###  3.3. Predictive Model

Mental health data, particularly concerning anxiety and depression, often
involve intricate patterns and interactions among various factors such as
demographic variables, behavioral patterns, and external stressors. Therefore,
our choice of model revolved around deep learning-based MLPs, which are well-
suited for modeling complex, nonlinear relationships within data, providing
more accurate predictions compared to the traditional linear models used in
(Shuster et al., 2021). The MLP autonomously adjusts weights to emphasize
salient features and diminish less important ones. This capability is vital
given the nuanced nature of mental health indicators and the potential for
large datasets to contain a mix of relevant and irrelevant information.

At the heart of our predictive model are two critical scores: the sds score
and the stai-s score, representing depression and anxiety levels,
respectively. Our primary objective was to develop a model that could
accurately forecast these scores based on a comprehensive array of variables
present in the dataset. These scores were derived through a sophisticated
equation incorporating twenty distinct variables. We followed the standard
scoring procedures for each instrument to calculate the stai-s score and the
sds score, as derived by (Shuster et al., 2021).

####  3.3.1. COVID-19 Depression and Anxiety Predictor (CoDAP) Architecture

Figure 2. The CoDAP Architecture

Following our data preparation, we developed an MLP configured to leverage the
entire spectrum of features in the dataset, thus enabling precise predictions
of the sds score and stai-s score. The model architecture, as seen in Figure
2, was designed to accommodate the complex relationships inherent in mental
health data, focusing on capturing subtle patterns and indicators that less
sophisticated models might miss. The input layer is marked by a green node,
which encapsulates a comprehensive set of 287 unique attributes, excluding
target attributes employed to calculate anxiety and depression levels, derived
from the dataset by Shuster et al.. These attributes form the foundational
data that CoDAP will process. Some of these features include age, gender,
race, religion, types of mental health disorders, and other qualitative
features about the participants.

Careful consideration was given to the MLP network’s architecture to balance
predictive power and model generalizability. It is a fundamental requirement
for an MLP to have at least one hidden layer, which enables the model to
perform non-linear transformations and learn complex relationships in the
input data. During the exploratory phase of our research, we conducted a
series of experiments to determine the optimal number of hidden layers. Our
empirical investigations revealed that a two-layered hidden structure provided
the most robust performance.

In the CODAP model architecture, each layer comprises 100 neurons, illustrated
as blue nodes. These neurons are strongly connected, with each neuron in one
layer forming connections to every neuron in the subsequent layer. The initial
hidden layer acts as a feature transformer, mapping raw inputs into an
intermediate space, and the second hidden layer builds upon this transformed
feature space, enabling the MLP to integrate these initial patterns into
higher-order representations. To enhance the network’s ability to model non-
linear relationships, a characteristic essential for capturing the
complexities of mental health data, the Rectified Linear Unit (ReLU)
activation function was applied within these hidden layers.

The output layer, indicated by red nodes, does not directly output the final
sds score and stai-s score. Instead, it produces an intermediary set of 20
features to calculate these mental health scores. The output includes
estimates for a series of scores: $STAI_{1}$ through $STAI_{20}$ and $SDS_{1}$
through $SDS_{20}$. Each set comprises 20 distinct scores. The stai-s scores
gauge levels of anxiety, ranging from one (indicating minimal presence) to
four (indicating a high degree), based on responses to various questions like
“I feel calm,” “I feel secure,” and others along these lines.

Similarly, the sds scores measure aspects of depression on the same scale of
one to four, with questions geared towards identifying symptoms of depression,
such as changes in weight or eating habits, exemplified by statements like “I
notice I am losing weight” and “I eat as much as I used to.”

The actual computation of the sds score and stai-s score, symbolized by yellow
nodes, is performed using specific formulas derived from (Shuster et al.,
2021). The output features include predictions of responses to each item
within the questionnaires, which are then used to calculate the total scores
for each scale.

The Adam optimization algorithm was employed to refine the network’s
predictive accuracy. This optimizer iteratively adjusts the network’s weights
to minimize the discrepancy, quantified by the Mean Squared Error (MSE),
between CoDAP’s predictions and the actual observed values from the dataset.

####  3.3.2. Cross Fold Validation

Algorithm 1 K-fold Cross Validation with the CODAP model

1:

2:$data$: the dataset for cross-validation

3:$k$: the number of folds

4:$model$: the MLP model described in Figure 2

5:

6:$AveragePerformance$: the average performance metric across all folds

7:

8:$k\\_folds\leftarrow EquallyDivide(data)$

9:$OverallPerformance\leftarrow 0$

10:for $i\leftarrow 1$ to $k$ do

11: $ValidationSet\leftarrow k\\_folds(i)$

12: $TrainingSet\leftarrow k\\_folds(allfoldsexcepti)$

13: $model.train(TrainingSet)$

14: $Performance\leftarrow model.evaluate(ValidationSet)$

15: $OverallPerformance\leftarrow OverallPerformance+Performance$

16:end for

17:$AveragePerformance\leftarrow OverallPerformance/k$

18:Output $AveragePerformance$

A critical aspect of our methodology was the validation and optimization of
our model. We utilized the K-fold cross-validation technique as a robust
strategy to counteract overfitting and assure our model’s reliability and
generalizability across different subsets of the data demonstrated in
Algorithm 1. This method involved partitioning the dataset into several folds,
systematically using each as a validation set while training the model on the
remaining data. This allowed us to rigorously test and refine the model,
ensuring its accuracy and robustness. This is crucial in mental health
prediction, where overfitting to a particular subset of data could lead to
misleading results. Also, since each fold serves as a validation set at some
point, cross-validation reduces the likelihood of the model’s performance
being biased by a particular random split of training and test data. This is
especially important when dealing with complex and potentially heterogeneous
datasets, such as those involving mental health symptoms during a pandemic.
Additionally, this method maximizes the data utilization, as the available
data was limited. K-fold cross-validation allows for the efficient use of
data, as each observation is used for training and validation at some point.

####  3.3.3. Hyperparameter Tuning

For k-fold cross-validation, we opted for $k=5$ to strike a balance between
obtaining a stable estimate of model performance and maintaining efficient
computational processing. Additionally, we experimented with various
hyperparameter tuning configurations of hidden layers, neurons per layer, and
learning rates, each assessing the impact on model performance. This process
was crucial for identifying the optimal parameters to produce the most
accurate and reliable predictions. As a result, the hyperparameters specified
for our model, CoDAP, are as follows: learning rate of 0.01, 100 epochs of
training, ReLU activation function, Adam optimization algorithm, and the MSE
as the loss function.

LR | 0.1 | 0.05 | 0.01 | 0.005 | 0.001
---|---|---|---|---|---
MSE | 2.8502 | 1.5652 | 0.5239 | 0.5630 | 0.5939
Table 1. MSE with Different Learning Rates

Nodes | 10 | 50 | 100 | 150 | 200
---|---|---|---|---|---
MSE | 0.8576 | 0.5474 | 0.5420 | 0.5713 | 0.6234
Table 2. MSE with Different Layers (lr = 0.01)

Upon finalizing the hyperparameters, we trained CoDAP using the entire
training set, which incorporated all the folds from our cross-validation
process. The model thus trained was then employed to predict and compare
results with the designated testing set. This approach, encompassing multiple
rounds of training, hyperparameter adjustments, and validation, contributed
significantly to the robustness and reliability of CoDAP for predictive
analysis.

###  3.4. Explaining the Model Results

To explain the CoDAP model results, we employed Local Interpretable Model-
agnostic Explanations (LIME) (Ribeiro et al., 2016) for approximating the
model with a simpler, interpretable model around the vicinity of the
prediction that needs to be explained. LIME is particularly useful because it
is model-agnostic.

Figure 3. LIME stai-s score (Anxiety) Result Figure 4. LIME sds score
(Depression) Result

Because LIME is designed to explain model predictions for individual instances
rather than offering a comprehensive interpretation of the entire model, we
created an objective overview of the most important features. We first
normalized the values for each of our features. Then, we leveraged LIME’s
local interpretation, dividing individual features into categories specified
by LIME and averaging importance scores. Afterward, we carried out a series of
ten iterative steps. We generated LIME reports in each iteration that focused
separately on two specific scores: stai-s score and sds score, based on our
test dataset. After the ten iterations, we grouped the same features and
averaged their importance scores. This process helped us understand how each
category of features influenced the stai-s score and sds score across all
iterations. We then created a graphical representation illustrating the most
important categories corresponding to both stai-s score and sds score. In this
step, a uniform color scheme was adopted for categories belonging to the same
feature to improve the readability of the graph. We selected the top 40 most
important features from there and grouped items within the same category. This
approach helped discern the important features that influence a particular
prediction to explain the predictive capabilities of our model.

In particular, by averaging the influence scores generated by LIME, we
identified key features that significantly impact the model’s predictions.
Notably, certain features exerted either a positive or negative influence on
the model’s anxiety and depression scores. We started by recognizing the
category using the legend and finding the corresponding y-axis score to
interpret the graph. The x-axis displays the overall importance score from
LIME, where negative values indicate a negative correlation with the anxiety
or depression scores, and positive values suggest a positive correlation with
the anxiety or depression scores.

Among the most influential features were su_crave10 and su_crave3, which gauge
the craving intensity for substances like Xanax and marijuana. Both of these
features showed a positive correlation with anxiety, aligning with the
understanding that individuals with heightened cravings for addictive
substances tend to exhibit increased stress levels. This can be correlated
with two alternative explanations due to the COVID-19 pandemic: 1) supply for
these drugs was disrupted, leading to increased mood symptoms; 2) people who
were taking these already had more mental health issues to start with;
therefore, it is not surprising they craved more for drugs during the pandemic
and had more depression/anxiety.

Another notable feature, Covid_emo3, measures the fear associated with
COVID-19. Interestingly, this feature displayed a negative correlation with
anxiety and depression at lower levels, suggesting that individuals with
diminished COVID-19 fears might experience less overall stress and,
subsequently, fewer mood symptoms. This observation is consistent with the
premise that fear of COVID-19 is a primary stressor during this period.
Similarly, additional features, such as Covid_emo4, demonstrated parallel
trends, reinforcing our model’s ability to capture the nuanced interplay
between various factors and their impact on anxiety and depression.

##  4\. Results

Our exploratory data analysis revealed noteworthy patterns and correlations.
We observed trends in mental health scores across different demographic groups
and during various pandemic phases. For instance, our findings (see Figure 1)
indicate that across all participants and pandemic phases, younger people
typically exhibited higher levels of depression and anxiety. Moreover, there
was a negative correlation between income and the levels of depression and
anxiety, with individuals of lower income often experiencing more pronounced
mental health challenges during the pandemic period. These insights were
instrumental in shaping the design and focus of CoDAP, providing a solid
foundation for predictive analysis.

Since predicting the level of anxiety and depression can be framed as a
regression problem, we utilized the MSE, which was particularly suitable for
this task. The CODAP model demonstrated high accuracy in predicting the sds
and stai-s scores, indicating its effectiveness in capturing the complex
relationships within the data. The use of MSE as a loss function was
particularly beneficial, given its sensitivity to large errors and
computational efficiency. The MSE also provided a standard metric to compare
the performance of different models and configurations. Table 3 presents the
MSE in the five rounds of the cross-validation process.

Round | 1 | 2 | 3 | 4 | 5 | Average
---|---|---|---|---|---|---
MSE | 0.5064 | 0.4924 | 0.5720 | 0.5401 | 0.5393 | 0.5300
Table 3. MSE for Cross-Validation

Throughout the training and k-fold cross-validation process, the CODAP model
predicts 20 features each for calculating sds score and stai-s score. Despite
the limitation of a relatively small data set, the model’s predictive
performance is promising, with an average MSE of approximately 0.53 across
five rounds of cross-validation for the feature values utilized in computing
the sds score and stai-s score. In the dataset, the feature values were
assigned positive or negative weights based on their contributions to
increasing or decreasing levels of anxiety and depression. The outcome
features sds score and stai-s score were normalized to range from 1 to 4, with
1 denoting the minimum level and 4 representing the maximum level. Also, the
cross-validation outcomes affirm the model’s reliability and underline its
robustness, with MSE values demonstrating consistency within the range of 0.49
to 0.57 across different training rounds. This consistency is crucial because
it suggests that the model is stable and reliable across different subsets of
data. Although a smaller MSE would be preferred, given the primary task of the
MLP is to understand which features are important predictors of mental health
state and the range and the variance of the outcome features, i.e., the target
variables, we deem these average MSE of 0.53 to be a good metric for the task
at hand.

##  5\. Conclusion and Future Work

Mental health has become an increasingly important area in healthcare, and
this trend is anticipated to persist, if not escalate, in the future. As such,
being able to predict mental health symptoms is critical for us to monitor and
intervene, given the rising level of global health crises, including and
beyond COVID-19. Our exploratory data analysis, data reformatting, and robust
model validation techniques contribute to the validity of our research
findings. The model’s performance suggests a promising foundation, even though
it is based on a limited data set. By focusing on the quantitative attributes
of individuals (e.g., demographics, COVID-specific behaviors, etc.) and the
time elapsed since the event, our research provides valuable insights into the
dynamics of mental health during such crises. Our model can be tailored and
refined with additional data and feedback. This ongoing improvement is
particularly important in a rapidly changing scenario like a pandemic, where
initial models may need refinement as more data becomes available and the
nature of the mental health challenges evolves. However, since the CoDAP’s
training dataset was primarily collected during the COVID-19 pandemic (Shuster
et al., 2021), the results may not generalize well to pre- and post-pandemic
conditions. Nevertheless, we believe the CODAP model can be adapted and
applied to different contexts, providing valuable tools for understanding and
predicting mental health responses to various environmental and chronic
stressors.

Our vision for CoDAP is to harness the model’s predictive power by integrating
it into a user-friendly application that will capture baseline mental health
measures, including but not limited to age, income, and craving — features our
model identified as having significant predictive importance. The application
will then be able to forecast future changes in mental health status based on
these baseline measures and ongoing user interaction. Consequently, while
direct measures of depression and anxiety are integral components of the
initial assessment, the predictive model will be essential for providing
longitudinal insights and anticipating future mental health trends for users.
This approach aims to offer users not only a snapshot of their current mental
health status but also a proactive tool for monitoring and managing their
mental health over time. To achieve this, we plan to utilize the Auto-
Regressive Integrated Moving Average (ARIMA) model (Nelson, 1998) for time
series prediction. Such longitudinal analysis is critical for understanding
the dynamics of mental health and identifying patterns or changes that may
require intervention.

In conclusion, the ultimate goals of this work are to enhance understanding of
mental health dynamics, using the predictive power of MLPs that analyze
complex psychological data and extend to other techniques such as time-series
analysis and transfer learning to gauge individuals’ mental health conditions
in wide-ranging scenarios.

Resources: The GitHub repository with our code and documentation is available
at https://github.com/block-iot/MentalHealth.

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