# Homogeneous Electron Liquid in Arbitrary Dimensions:
Exchange and Correlation Using the Singwi-Tosi-Land-Sjölander Approach

L. V. Duc Pham  Institut für Physik, Martin-Luther-Universität Halle-
Wittenberg, 06120 Halle (Saale), Germany  Pascal Sattler  Institut für Physik,
Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), Germany
Miguel A. L. Marques  Institut für Physik, Martin-Luther-Universität Halle-
Wittenberg, 06120 Halle (Saale), Germany  Carlos L. Benavides-Riveros
<EMAIL_ADDRESS>Pitaevskii BEC Center, CNR-INO and Dipartimento
di Fisica, Università di Trento, I-38123 Trento, Italy  Max Planck Institute
for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187, Dresden,
Germany

###### Abstract

The ground states of the homogeneous electron gas and the homogeneous electron
liquid are cornerstones in quantum physics and chemistry. They are archetypal
systems in the regime of slowly varying densities in which the exchange-
correlation energy can be estimated with a myriad of methods. For high
densities, the behaviour the energy is well-known for 1, 2, and 3 dimensions.
Here, we extend this model to arbitrary integer dimensions, and compute its
correlation energy beyond the random phase approximation (RPA), using the
celebrated approach developed by Singwi, Tosi, Land, and Sjölander (STLS),
which is known to be remarkably accurate in the description of the full
electronic density response for $2D$ and $3D$, both in the paramagnetic and
ferromagnetic ground states. For higher dimensions, we compare the results
obtained for the correlation energy using the STLS method with the values
previously obtained using RPA. We found that at high dimensions STLS tends to
be more physical in the sense that the infamous sum rules are better satisfied
by the theory. We furthermore illustrate the importance of the plasmon
contribution to STLS theory.

##  I Introduction

The ground states of the homogeneous electron gas (HEG) and the homogeneous
electron liquid (HEL) have played a prominent role in the modelling and
understanding of a wide range of interacting electronic systems Sommerfeld
(1928); Rajagopal and Kimball (1977); Gori-Giorgi and Ziesche (2002); Huotari
_et al._ (2010); Holzmann  _et al._ (2011); Shepherd  _et al._ (2012); Lewin
_et al._ (2018); Loos and Gill (2016); Zhang and Ceperley (2008); Gontier  _et
al._ (2019); Drummond and Needs (2009); Spink  _et al._ (2013). These systems
are some of the most important models of choice to develop, improve and
benchmark many approximate approaches to the full many-electron problem,
including some of the most popular exchange-correlation functionals of density
functional theory (DFT) Ma and Brueckner (1968); Ceperley and Alder (1980);
Jones (2015); Constantin  _et al._ (2008).

An important question about these models is the dependency of the correlation
energy, and other physical quantities, on $D$, the dimension of the physical
space in which the gas/liquid is embedded. Indeed, there is a wealth of
experimental electronic setups in which one or two of the physical dimensions
are much smaller than the remaining ones. They can thus be modeled as one- or
two-dimensional quantum systems Wagner  _et al._ (2012); Astrakharchik  _et
al._ (2016); Schmidt  _et al._ (2019); Constantin (2016); Gedeon  _et al._
(2022). Furthermore, reduced dimensional systems often exhibit notable
physical properties, ranging from Luttinger physics Voit (1995) to Moiré
superlattices Kennes  _et al._ (2021). More recently, progress in the
fabrication of artificial materials is paving the way for the realization of
non-integer dimensions, as fractal substrates (e.g., Sierpiński carpets of
bulk Cu) confining electron gases Kempkes  _et al._ (2019); van Veen  _et al._
(2016); Fremling  _et al._ (2020). The possibility to circumvent the von
Neumann-Wigner theorem or to produce unconventional topological phases by
engineering (or mimicking) additional synthetic dimensions Hummel  _et al._
(2021); Kanungo  _et al._ (2022); Ghosh  _et al._ (2017); Celi  _et al._
(2014) also highlights the importance of realizing, studying and understanding
interacting fermionic and electronic systems embedded in non-conventional
dimensions.

For the high-density spin-unpolarized HEG in $D=1,2$, and 3, the energy per
electron can be expanded in terms of $r_{s}$, the Wigner-Seitz radius, as
follows Giuliani and Vignale (2005):

$\displaystyle\varepsilon_{D}(r_{s}\rightarrow
0)=\frac{a_{D}}{r_{s}^{2}}-\frac{b_{D}}{r_{s}}+c_{D}\ln
r_{s}+\mathcal{O}(r_{s}^{0})\,.$ (1)

The constants $a_{D}$, $b_{D}$ and $c_{D}$ are independent of $r_{s}$, and
their functional form in terms of the dimension $D$ is well known Loos and
Gill (2016). Quite remarkably, the logarithm that appears in Eq. (1) for the
3D case is due to the long range of the Coulomb repulsion, and cannot be
obtained from standard second-order perturbation theory Bohm and Pines (1953).
In a recent work Schlesier  _et al._ (2020), the HEG was extended to arbitrary
integer dimensions. It was found a very different behavior for $D>3$: the
leading term of the correlation energy does not depend on the logarithm of
$r_{s}$ [as in Eq. (1)], but instead scales polynomially as
$c_{D}/r_{s}^{\gamma_{D}}$, with the exponent $\gamma_{D}=(D-3)/(D-1)$. In the
large-$D$ limit, the value of $c_{D}$ was found to depend linearly with the
dimension. This result was originally obtained within the random-phase
approximation (RPA) by summing all the ring diagrams to infinite order.

While RPA is known to be exact in the limit of the dense gas, includes long-
range interactions automatically, and is applicable to systems where finite-
order many-body perturbation theories break down Ren  _et al._ (2012), it has
well-known deficiencies at the metallic (intermediate) and low densities of
the typical HEL $(1\leq r_{s}\leq 6)$. Quantum Monte-Carlo (QMC) is an option
for those regimes Gubernatis  _et al._ (2016); Spink  _et al._ (2013), but
there are other high-quality approaches such as the celebrated Singwi, Tosi,
Land, and Sjölander (STLS) method that provides results comparable to QMC
Singwi  _et al._ (1968). This method attempts to tackle in an approximate
manner both the exchange and electronic correlations through a local-field
correction. As such, this scheme is often surprisingly accurate in the
description of the full electronic density response and is commonly used to
investigate quasi-one-dimensional Agosti  _et al._ (1998), inhomogeneneuos
Dobson (2009); Kosugi and Matsushita (2017); Dharma-wardana (1981) and warm
dense electron liquids Tanaka (2017); Dornheim  _et al._ (2022, 2018). It has
also inspired the development of new functional methods that explicitly retain
the dynamical and non-local nature of electronic correlations, while properly
accounting for the exchange contribution Dobson  _et al._ (2002); Vilk  _et
al._ (1994); Zantout  _et al._ (2021); Yoshizawa and Takada (2009); Hasegawa
and Shimizu (1975).

The purpose of this paper is to calculate the correlation energy of the HEL
for arbitrary integer dimensions (in particular for $D>3$) with the STLS
method. As expected, the value of this portion of the energy improves
significantly with respect to the RPA result (that tends to over-correlate the
liquid). The remainder of this paper is organized as follows. In Section II we
review the main STLS equations and rewrite them explicitly in arbitrary $D$
dimensions. As a result, we can compute the Lindhard polarizability, the
structure factor, and the so-called local field correction in the Hartree-Fock
approximation, providing explicit formulae for some representative systems. In
Section III we explain how the correlation energy is computed in this scheme,
and discuss the fully polarized case. In Sections V and VI we present and
discuss our numerical results for the exchange and correlation energies, the
pair density, and its Fourier transform. We also discuss the fulfillment of
the compressibility sum rule. Finally, in Section VII we present our main
conclusions. Two Appendices that give further technical details on our
calculations are presented at the end of the text.

##  II STLS Theory in $D$ dimensions

In this section we review the main STLS equations and write them explicitly in
arbitrary $D$ dimensions. We follow the standard notation, namely, $n$ is the
$D$-dimensional particle density, $\Omega$ is the volume occupied by the
electronic liquid, and $q_{F}$ is the usual Fermi wavelength. The STLS theory
departs from RPA (to include short-range correlation between electrons) by
writing the two-particle density distribution
$f_{2}({\mathbf{r}},\mathbf{p},{\mathbf{r}}^{\prime},\mathbf{p}^{\prime},t)$
as follows:

$\displaystyle
f_{2}({\mathbf{r}},\mathbf{p},{\mathbf{r}}^{\prime},\mathbf{p}^{\prime},t)=f({\mathbf{r}},\mathbf{p},t)f({\mathbf{r}}^{\prime},\mathbf{p}^{\prime},t)g({\mathbf{r}}-{\mathbf{r}}^{\prime})\,,$
(2)

where $f({\mathbf{r}},\mathbf{p},t)$ is the one-particle phase-space density
and $g({\mathbf{r}})$ is the equilibrium, static pair distribution function.
Eq. (2) can be seen as an ansatz that terminates the hierarchy that otherwise
would write the two-particle distribution function in terms of the three-
particle distribution function, and so on. This leads to the following
density-density response function Singwi  _et al._ (1968):

$\displaystyle\chi_{D}(\mathbf{q},\omega)=\frac{\chi^{0}_{D}(\mathbf{q},\omega)}{1-\Phi(\mathbf{q})[1-G_{D}(\mathbf{q})]\chi^{0}_{D}(\mathbf{q},\omega)}\,.$
(3)

Here $\chi^{0}_{D}(\mathbf{q},\omega)$ is the Lindhard polarizability, i.e.,
the inhomogeneous non-interacting density response function of an ideal Fermi
gas in $D$ dimensions, $G_{D}(\mathbf{q})$ is the local field correction, and
$\Phi(\mathbf{q})$ is the $D$-dimensional Fourier transform of the Coulomb
potential:

$\displaystyle\Phi(\mathbf{q})=\frac{(4\pi)^{\frac{D-1}{2}}\Gamma\left(\frac{D-1}{2}\right)}{q^{D-1}}\,,$
(4)

where $\Gamma$ denotes the gamma function. The presence of $G_{D}(\mathbf{q})$
in Eq. (3) is the key feature of the STLS equations that gives the “beyond
RPA” flavor to the theory. This local field correction is a direct result of
the short-range correlation between the electrons. In arbitrary dimensions, it
is given by:

$\displaystyle
G_{D}(\mathbf{q})=-\frac{1}{n}\int\mathbf{q}^{\prime}\cdot\mathbf{q}\frac{q^{D-3}}{q^{\prime
D-1}}\left[S_{D}(\mathbf{q}-\mathbf{q}^{\prime})-1\right]\frac{d^{D}\mathbf{q}^{\prime}}{(2\pi)^{D}}\,,$
(5)

with $S_{D}(\mathbf{q})$ being the structure factor. We can simplify this
integral to a two-dimensional one by substituting
$\mathbf{q}-\mathbf{q}^{\prime}=\mathbf{t}$ and using the fact that
$S_{D}(\mathbf{q})=S_{D}(q)$ in homogeneous systems. Afterwards, we rewrite
the integral using $D$-dimensional spherical coordinates, where $\mathbf{q}$
is parallel to the $D^{\text{th}}$-axis, and integrate over all angles except
the angle $\theta$ between $\mathbf{q}$ and $\mathbf{t}$ to obtain:

$\displaystyle G_{D}(\mathbf{q})=-$
$\displaystyle\frac{q^{D-3}}{(2\pi)^{D}n}\frac{2\pi^{\frac{D-1}{2}}}{\Gamma\left(\frac{D-1}{2}\right)}\int_{0}^{\infty}\int_{0}^{\pi}[S_{D}(t)-1]\times$
$\displaystyle\frac{[q^{2}t^{D-1}-qt^{D}\cos\theta](\sin\theta)^{D-2}}{(q^{2}+t^{2}-2qt\cos\theta)^{\tfrac{D-1}{2}}}d\theta
dt\,.$ (6)

Note in passing that in the $2D$ case we must use polar coordinates instead of
the spherical coordinates and obtain:

$\displaystyle G_{2}(\mathbf{q})=-\frac{1}{(2\pi)^{2}n}$
$\displaystyle\int_{0}^{\infty}\int_{0}^{2\pi}[S_{2}(t)-1]\times$
$\displaystyle\frac{qt-t^{2}\cos\theta}{(q^{2}+t^{2}-2qt\cos\theta)^{1/2}}d\theta
dt\,.$ (7)

These equations, together with the equation for the dielectric function,
$1/\epsilon_{D}(\mathbf{q},\omega)=1+\Phi(\mathbf{q})\chi_{D}(\mathbf{q},\omega)$,
lead to an equation for the dielectric function within the STLS theory,
namely:

$\displaystyle\epsilon_{D}(\mathbf{q},\omega)=1-\frac{\Phi(\mathbf{q})\chi^{0}_{D}(\mathbf{q},\omega)}{1+G_{D}(\mathbf{q})\Phi(\mathbf{q})\chi^{0}_{D}(\mathbf{q},\omega)}\,.$
(8)

Finally, the relation between the structure factor and the dielectric function
$\epsilon_{D}(\mathbf{q},\omega)$ (see, for instance, Pines (1999)) can be
easily generalized for arbitrary dimensions:

$\displaystyle S_{D}(\mathbf{q})=-\frac{1}{\pi
n\Phi(\mathbf{q})}\int_{0}^{\infty}{\rm
Im}\left(\frac{1}{\epsilon_{D}(\mathbf{q},\omega)}\right)d\omega\,.$ (9)

We can improve the readability of this equation by separating the
contributions of the single-particle and plasmon excitations Pines and
Nozières (1966):

$\displaystyle S_{D}(\mathbf{q})$ $\displaystyle=-\frac{1}{\pi
n\Phi(\mathbf{q})}\int_{0}^{qv_{F}+\frac{q^{2}}{2}}{\textrm{Im}}\left(\frac{1}{\epsilon_{D}(\mathbf{q},\omega)}\right)d\omega$
$\displaystyle+$
$\displaystyle\frac{1}{\Phi(\mathbf{q})n}\left(\frac{\partial\textrm{Re}\epsilon_{D}(\mathbf{q},\omega)}{\partial\omega}\right)^{-1}\delta(\omega-\omega_{p}(q))\,,$
(10)

where $v_{F}$ is the Fermi velocity and $\omega_{p}(q)$ is the plasmon
dispersion Bernadotte  _et al._ (2013). Equation 10, Eq. 6 and Eq. 8 form the
core of the STLS set of equations. In this framework, they can be evaluated
_self-consistently_. Notice, indeed, that $G_{D}(\mathbf{q})$ and
$S_{D}(\mathbf{q})$ can be written symbolically as
$G_{D}=\mathcal{F}_{1}[S_{D}]$ and $S_{D}=\mathcal{F}_{2}[G_{D}]$, indicating
the existence of the mutual functional relations introduced above Yoshizawa
and Takada (2009).

We should reiterate here that the crucial aspect of the STLS method is the
appearance of the density-density pair distribution
$g({\mathbf{r}}-{\mathbf{r}}^{\prime})$ in the decoupling of the equation of
motion (2). Despite the crudeness of such a factorization scheme, the
formalism gives excellent correlation energies for both 3D and 2D homogeneous
electron gases. Unfortunately, the factorization also results in a number of
well-known shortcomings, including negative values of
$g({\mathbf{r}}-{\mathbf{r}}^{\prime})$ for sufficiently large values of
$r_{s}$, and the failure to satisfy the compressibility sum rule (i.e., at
long wavelengths the exact screened density response is determined by the
isothermal compressibility). The first problem, the un-physical behavior of
the pair distribution, is counter-balanced by the fact that the exchange-
correlation energy is in reality an integral over $g(r)$: it turns out that
its value is still quite reasonable in the metallic range as it benefits from
an error cancellation Dornheim  _et al._ (2018). As we will see below, this
result, known for the case of 2D and 3D, does also hold for larger dimensions.
To tackle the second problem, Vashista and Singwi Vashishta and Singwi (1972)
provided a correction to the STLS method that established the correct
compressibility sum rule in the metallic-density regime. This is however a
partial solution as both the original STLS method and the Vashista-Singwi
extension under-estimate the exchange energy, as pointed out by Sham Sham
(1973).

###  II.1 Real part of the Lindhard function for $D=5,7$

The expressions for $\chi^{0}_{2}(\mathbf{q},\omega)$ and
$\chi^{0}_{3}(\mathbf{q},\omega)$ are widely known since long ago (see for
instance Giuliani and Vignale (2005); Haug and Koch (2009)), but higher
dimensional expressions are missing in the literature. In Appendix A we detail
their calculation for higher dimensional settings by performing a linear
perturbation from equilibrium. We sketch here the main results. The real part
of $\chi^{0}_{D}(\mathbf{q},\omega)$ can be written explicitly as:

$\displaystyle\mathrm{Re}\chi^{0}_{D}(\mathbf{q},\omega)$
$\displaystyle=\frac{2}{(2\pi)^{D}}\mathcal{P}\int\frac{\Theta(q_{F}-|\mathbf{p}-\tfrac{1}{2}\mathbf{q}|)\Theta(|\mathbf{p}+\tfrac{1}{2}\mathbf{q}|-q_{F})}{\omega-\mathbf{p}\cdot\mathbf{q}}d^{D}\mathbf{p}$
$\displaystyle\quad-\frac{2}{(2\pi)^{D}}\mathcal{P}\int\frac{\Theta(|\mathbf{p}-\tfrac{1}{2}\mathbf{q}|-q_{F})\Theta(q_{F}-|\mathbf{p}+\tfrac{1}{2}\mathbf{q}|)}{\omega-\mathbf{p}\cdot\mathbf{q}}d^{D}\mathbf{p}\,,$
(11)

where $\mathcal{P}$ denotes the principal value. By evaluating these
integrals, it is possible to obtain analytical expressions for specific cases.
For instance, for $D=5$ and $D=7$ one gets:

$\displaystyle\mathrm{Re}\chi^{0}_{5}(\mathbf{q},\omega)=\frac{q_{F}^{3}}{8\pi^{3}}\Bigg{\\{}$
$\displaystyle\frac{1}{96\tilde{q}^{5}}\Bigg{[}\left(\frac{3}{2}\left(\tilde{q}^{2}-2\tilde{\omega}\right)^{4}+24\tilde{q}^{4}-12\tilde{q}^{2}(\tilde{q}^{2}-2\tilde{\omega})^{2}\right)\ln\left|\frac{2\tilde{q}-\tilde{q}^{2}+2\tilde{\omega}}{2\tilde{q}+\tilde{q}^{2}-2\tilde{\omega}}\right|$
$\displaystyle+$
$\displaystyle\left(\frac{3}{2}\left(\tilde{q}^{2}+2\tilde{\omega}\right)^{4}+24\tilde{q}^{4}-12\tilde{q}^{2}(\tilde{q}^{2}+2\tilde{\omega})^{2}\right)\ln\left|\frac{2\tilde{q}-\tilde{q}^{2}-2\tilde{\omega}}{2\tilde{q}+\tilde{q}^{2}+2\tilde{\omega}}\right|+12\tilde{q}^{7}-16\tilde{q}^{5}+144\tilde{q}^{3}\tilde{\omega}^{2}\Bigg{]}-\frac{2}{3}\Bigg{\\}}$
(12)

and

$\displaystyle\mathrm{Re}\chi^{0}_{7}(\mathbf{q},\omega)=\frac{q_{F}^{5}}{368640\pi^{4}\tilde{q}^{5}}\Bigg{\\{}$
$\displaystyle\left[60(16\tilde{q}^{4}+3(\tilde{q}^{2}+2\tilde{\omega})^{4}-12(\tilde{q}^{3}+2\tilde{q}\tilde{\omega})^{2})-\frac{15(\tilde{q}^{2}+2\tilde{\omega})^{6}}{\tilde{q}^{2}}\right]\ln\left|\frac{2\tilde{q}-\tilde{q}^{2}-2\tilde{\omega}}{2\tilde{q}+\tilde{q}^{2}+2\tilde{\omega}}\right|$
$\displaystyle+$
$\displaystyle\left[60(16\tilde{q}^{4}+3(\tilde{q}^{2}-2\tilde{\omega})^{4}-12(\tilde{q}^{3}-2\tilde{q}\tilde{\omega})^{2})-\frac{15(\tilde{q}^{2}-2\tilde{\omega})^{6}}{\tilde{q}^{2}}\right]\ln\left|\frac{2\tilde{q}-\tilde{q}^{2}+2\tilde{\omega}}{2\tilde{q}+\tilde{q}^{2}-2\tilde{\omega}}\right|$
$\displaystyle-$ $\displaystyle
4224\tilde{q}^{5}+1280\tilde{q}^{3}(\tilde{q}^{4}+12\tilde{\omega}^{2})-120\tilde{q}(\tilde{q}^{8}+40\tilde{q}^{4}\tilde{\omega}^{2}+80\tilde{\omega}^{4})\Bigg{\\}}\,.$
(13)

A similar calculation gives a closed expression for the imaginary part:

$\displaystyle\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)=\begin{cases}h(D)\dfrac{1}{\tilde{q}}\left[\left(1-\nu_{-}^{2}\right)^{\frac{D-1}{2}}-\left(1-\nu_{+}^{2}\right)^{\frac{D-1}{2}}\right]&\tilde{\omega}<\left|\tilde{q}-\frac{\tilde{q}^{2}}{2}\right|\text{
and }\tilde{q}<2\\\
0&\tilde{\omega}<\left|\tilde{q}-\frac{\tilde{q}^{2}}{2}\right|\text{ and
}\tilde{q}>2\\\
h(D)\dfrac{1}{\tilde{q}}\left[1-\nu_{-}^{2}\right]^{\tfrac{D-1}{2}}&\left|\tilde{q}-\frac{\tilde{q}^{2}}{2}\right|\leq\tilde{\omega}\leq\tilde{q}+\frac{\tilde{q}^{2}}{2}\\\
0&\tilde{\omega}>\tilde{q}+\frac{\tilde{q}^{2}}{2}\end{cases}$ (14)

where $\tilde{q}=q/q_{F}$, $\tilde{\omega}=\omega/q_{F}^{2}$,
$h(D)=q_{F}^{D-2}\left[2^{D-2}(D-1)\pi^{\tfrac{D-1}{2}}\Gamma\left(\tfrac{D-1}{2}\right)\right]^{-1}$
and $\nu_{\pm}=\tilde{\omega}/\tilde{q}\pm\tilde{q}/2$.

###  II.2 The Hartree-Fock approximation for the first iteration

For the first iteration, the original STLS theory uses the structure factor
obtained from the Hartree-Fock calculation Singwi  _et al._ (1968). We take
the same approach here. Formally, the generalized $D$-dimensional structure
factor is straightforward and reads:

$\displaystyle
S^{\textrm{HF}}_{D}(\mathbf{q})=1-\frac{2}{(2\pi)^{D}\,n}\int_{k,k^{\prime}\leq
q_{F}}\delta(\mathbf{q}-\mathbf{k}^{\prime}+\mathbf{k})\,d^{D}\mathbf{k}^{\prime}d^{D}\mathbf{k}\,.$

Substituting this formal result into Eq. (5) gives the general $D$-dimensional
expression for the local field correction of STLS within the Hartree-Fock
approximation, namely,

$\displaystyle G_{D}^{\textrm{HF}}(\mathbf{q})$
$\displaystyle=\frac{2q^{D-3}}{(2\pi)^{2D}n^{2}}\int_{k,k^{\prime}\leq
q_{F}}\frac{\mathbf{q}\cdot(\mathbf{q}+\mathbf{k}-\mathbf{k}^{\prime})}{|\mathbf{q}+\mathbf{k}-\mathbf{k}^{\prime}|^{D-1}}\,d^{D}\mathbf{k}^{\prime}d^{D}\mathbf{k}\,.$
(15)

This integral can be evaluated by making use of the extracule and intracule
substitutions: $\mathbf{s}=(\mathbf{k}+\mathbf{k}^{\prime})/2\text{ and
}\mathbf{t}=\mathbf{k}-\mathbf{k}^{\prime}$. Then, because the integrand only
depends on $\mathbf{q}$ and $\mathbf{t}$, one can perform the integration over
$\mathbf{s}$. This eventually gives the relevant integration region as the
intersection of two hyperspheres, as discussed in detail in Appendix B, where
we obtain the following expression:

$G_{D}^{\textrm{HF}}(\mathbf{q})=\frac{q^{D-3}}{q_{F}^{D}\sqrt{\pi}}\frac{\Gamma\left(\frac{D}{2}+1\right)}{\Gamma\left(\frac{D-1}{2}\right)}\int_{0}^{2q_{F}}\\!\\!\int_{0}^{\pi}\textrm{I}_{1-\frac{t^{2}}{4q_{F}^{2}}}\\!\\!\left(\frac{D+1}{2},\frac{1}{2}\right)\\\
\times\frac{(q^{2}+qt\cos\theta)t^{D-1}\sin^{D-2}\theta}{(q^{2}+t^{2}+2qt\cos\theta)^{\frac{D-1}{2}}}d\theta
dt\,,$ (16)

with $I_{x}(a,b)$ denoting the regularized incomplete beta function. The
numerical evaluation of the local field correction as expressed in Eq. (16) is
now a much simpler task. In fact, we present the lengthy analytical expression
for $G_{5}^{\textrm{HF}}(\mathbf{q})$ in Appendix B.

##  III Energy Contributions

The calculation of the kinetic energy of the $D$-dimensional HEG is as
straightforward as it is for $D=3$. It is given by:

$\displaystyle
E_{\textrm{kin}}=\frac{\alpha_{D}^{2}D}{2(D+2)}\frac{\Upsilon_{2}(\xi)}{r_{s}^{2}}\,,$
(17)

where $\alpha_{D}=2^{(D-1)/D}\Gamma(D/2+1)^{2/D}$,

$\Upsilon_{n}(\xi)=\tfrac{1}{2}\left[(1+\xi)^{(D+n)/D}+(1-\xi)^{(D+n)/D}\right]$

is the spin-scaling function, and $\xi$ is the usual spin-polarization. The
interaction energy per particle is given by the following expression:

$\displaystyle E_{\textrm{int}}=\frac{1}{\Omega}\sum_{\mathbf{q}\neq
0}\frac{\Phi(\mathbf{q})}{2}(S(\mathbf{q})-1)\,.$ (18)

If we introduce the function
$\gamma(r_{s})=-\tfrac{1}{2q_{F}}\int_{0}^{\infty}[S(q)-1]dq$ we can rewrite
the energy of the unpolarized electron gas in Eq. (18) as:

$\displaystyle
E_{\textrm{int}}=-\frac{2^{\tfrac{D-3}{D}}D^{\tfrac{2}{D}}}{\sqrt{\pi}}\Gamma\left(\frac{D-1}{2}\right)\Gamma\left(\frac{D}{2}\right)^{\tfrac{2-D}{D}}\frac{\gamma(r_{s})}{r_{s}}\,.$

By using the adiabatic-connection formula, the total interaction energy per
particle can be computed by using the formula Ferrell (1958):
$\int_{0}^{1}\lambda E_{\textrm{int}}(\lambda r_{s})d\lambda$, where $\lambda$
is a coupling constant that represents the strength of the interaction.
Therefore, the full ground-state energy of the unpolarized electron gas in the
STLS theory can then be written as:

$\displaystyle E_{0}$
$\displaystyle=\frac{\alpha_{D}^{2}D}{2(D+2)}\frac{\Upsilon_{2}(0)}{r_{s}^{2}}-E(D)\frac{1}{r_{s}^{2}}\int_{0}^{r_{s}}\gamma(r_{s}^{\prime})dr_{s}^{\prime}\,,$
(19)

where $E(D)$ is defined as:

$\displaystyle
E(D)=\frac{2^{\tfrac{D-3}{D}}D^{\tfrac{2}{D}}}{\sqrt{\pi}}\Gamma\left(\frac{D-1}{2}\right)\Gamma\left(\frac{D}{2}\right)^{\tfrac{2-D}{D}}\,.$
(20)

To obtain the correlation energy we need to make use of the Hartree-Fock
energy. While the kinetic energy is already calculated, the calculation of the
exchange energy is not trivial. This was already calculated by two of us in
Ref. Schlesier  _et al._ (2020):

$\displaystyle
E_{\textrm{HF}}=\frac{\alpha_{D}^{2}D}{2(D+2)}\frac{\Upsilon_{2}(\xi)}{r_{s}^{2}}-\frac{2\alpha_{D}D}{\pi(D^{2}-1)}\frac{\Upsilon_{1}(\xi)}{r_{s}}\,.$
(21)

Since the correlation energy is defined as the difference between the ground-
state energy and the ground-state Hartree-Fock energy, we get for the
$D$-dimensional gas the following compact formula:

$\displaystyle
E_{\textrm{corr}}=\frac{1}{r_{s}^{2}}\int_{0}^{r_{s}}\left[-E(D)\gamma(r_{s}^{\prime})+\frac{2\alpha_{D}D\Upsilon_{1}(0)}{\pi(D^{2}-1)}\right]dr_{s}^{\prime}\,.$
(22)

This is the formula we are going to use for the calculation of the correlation
energy.

##  IV The fully polarized case

The extension of the above equations for the fully polarized case is quite
straightforward (for $D=3$ see Kumar  _et al._ (2009); Ng and Singwi (1987)).
The relation between the local field correction and the structure factor is
given by:

$\displaystyle
G^{\uparrow\uparrow}_{D}(q)=-\frac{1}{n}\int\frac{(\mathbf{q}^{\prime}\cdot\mathbf{q})q^{D-3}}{q^{\prime
D-1}}[S^{\uparrow\uparrow}_{D}(\mathbf{q}-\mathbf{q}^{\prime})-1]\frac{d^{D}\mathbf{q}^{\prime}}{(2\pi)^{D}}\,,$
(23)

where $S^{\uparrow\uparrow}_{D}(\mathbf{q}-\mathbf{q}^{\prime})$ is the spin-
resolved structure factor. The corresponding density-density response function
is given by:

$\displaystyle\chi^{\uparrow\uparrow}_{D}(\mathbf{q},\omega)=\frac{\chi^{0,\uparrow\uparrow}_{D}(\mathbf{q},\omega)}{1-\Phi(\mathbf{q})[1-G^{\uparrow\uparrow}_{D}(\mathbf{q})]\chi^{0,\uparrow\uparrow}_{D}(\mathbf{q},\omega)}\,.$
(24)

where $\chi^{0,\uparrow\uparrow}_{D}(\mathbf{q},\omega)$ is the polarizability
of the fully polarized non-interacting HEG that can be easily related with the
spinless quantities $\chi^{0}_{D}(\mathbf{q},\omega)$ by a factor of $1/2$
Giuliani and Vignale (2005). In addition, the Fermi wavelength should be re-
scaled $q_{F,\uparrow}=2^{1/D}q_{F}$. Finally, the fluctuation-dissipation
theorem leads to:

$\displaystyle S^{\uparrow\uparrow}_{D}(\mathbf{q})=-\frac{1}{\pi
n}\int_{0}^{\infty}\mathrm{Im}\chi^{\uparrow\uparrow}_{D}(\mathbf{q},\omega)d\omega\,.$
(25)

As a result, in the fully polarized case, the correlation energy is given by:

$\displaystyle
E_{\textrm{corr}}=\frac{1}{r_{s}^{2}}\int_{0}^{r_{s}}\left[-F(D)\gamma(r_{s}^{\prime})+\frac{2\alpha_{D}D\Upsilon_{1}(1)}{\pi(D^{2}-1)}\right]dr_{s}^{\prime}\,.$
(26)

where $F(D)$ is a simple re-scaling of $E(D)$ in Eq. (20):

$\displaystyle F(D)=2^{1/D}E(D)\,.$ (27)

##  V Numerical Results

Starting from the expression for the local field correction in the Hartree-
Fock approximation $G_{D}^{\textrm{HF}}(\mathbf{q})$ presented in section II.2
we calculated $S_{D}(\mathbf{q})$ using Eq. (10). In this case $\omega(p)$ is
obtained from the zero of the dielectric function
$\epsilon(\mathbf{q},\omega)$, which in turn is given by Eq. (8). We then
started the entire self-consistent cycle of the STLS equations. To improve the
convergence of the iterative procedure we applied the following linear mixing:

$\bar{G}_{D}^{i}(\mathbf{q})=G_{D}^{i-1}(\mathbf{q})+[G_{D}^{i}(\mathbf{q})-G_{D}^{i-1}(\mathbf{q})]/a\,,$
(28)

where $i$ is the iteration number and $a$ takes values between 1 and 3.5 (for
the $5D$ and the $7D$ cases we used $a=1.5$). At each iteration, we calculated
the quantity $\gamma(r_{s})$ with the new structure factor
$S_{D}(\mathbf{q})$. After 10 iterations we obtained convergence in
$\gamma(r_{s})$ within 0.1%. The value of $\gamma(r_{s})$ at the end of the
self-consistent calculation is then used to calculate the correlation energy
using Eq. (22).

Figure 1: Correlation energy for HEG in $3D$ and $2D$ expressed as a function
of $r_{s}$. Our results for STLS in the paramagnetic case are compared with
the results of Jonson Jonson (1976) and the results of QMC from Refs. Ceperley
(1978) and Monnier (1972). In the ferromagnetic case our results for STLS are
compared with the results of QMC from Ref.Spink  _et al._ (2013).

We first benchmarked our implementation for $2D$ and $3D$. The results are
presented in Fig. 1, where it can be seen that we recovered the original STLS
results and obtained the celebrated agreement of STLS with the Quantum Monte-
Carlo results Ceperley (1978); Monnier (1972). In the $2D$ case we also
obtained similar results to the ones obtained previously by Jonson Jonson
(1976). The decrease of the magnitude of the correlation energy in the
paramagnetic case is consistent with the Quantum Monte-Carlo results Spink
_et al._ (2013).

(a) HEG in 5D

(b) HEG in 7D

Figure 2: Correlation energy of the unpolarized HEG in $5D$ and $7D$ expressed
as a function of $r_{s}$. Our results for STLS with and without the plasmon
contribution are compared with RPA results from Ref. Schlesier  _et al._
(2020).

Our new results for the HEG in 5D and 7D (for which we were able to compute
analytically the Lindhard polarizability and the local field correction) in
the paramagnetic and the ferromagnetic case are presented in Fig. 2(a), 2(b),
3(a) and 3(b), respectively. Those are compared with the RPA results
previously obtained in Ref. Schlesier  _et al._ (2020) 111We corrected a small
error in the numerical evaluation of the coefficients presented in Table I of
Ref. Schlesier  _et al._ (2020) that however does not affect the main
conclusions of that paper.. It is a general result of our implementation that
the STLS correlation energy decreases (in absolute value) in comparison with
the one from RPA, which confirms that this latter approximation tends to over-
correlate the gas, even in dimensions higher than 3. Yet, for high dimensions
the general error of RPA is smaller, both in the ferromagnetic and the
paramagnetic cases. The correlation energy is smaller in magnitude in the
ferromagnetic case in comparison with the paramagnetic one in all the
dimensions studied, the same behaviour found with the RPA. Furthermore, since
the structure factor of STLS can be easily separated into the pair and the
plasmonic contributions Pines (1999); Bernadotte  _et al._ (2013), we
investigated the plasmon contribution separately by calculating the
correlation energy with and without it. We conclude that the plasmonic
correction is more relevant for intermediate densities at all dimensions: this
term indeed improves the correlation energy on average by 30.9% in the 5D case
and 49.2% in the 7D case.

(a) HEG in 5D

(b) HEG in 7D

Figure 3: Correlation energy of the fully polarized HEG in $5D$ and $7D$
expressed as a function of $r_{s}$. Our results for STLS are compared with the
RPA results from Ref. Schlesier  _et al._ (2020).

For the sake of completeness, we also present in Fig. 4 the values for the
local field correction $G(\mathbf{q})$ we obtained for both paramagnetic and
ferromagnetic gases for $D=3,5,7$ for a selected value of the density (i.e.,
$r_{s}=2$). While the magnitude of the local correction is larger for the
fully polarized gas, the global value diminishes for large dimensions in the
whole domain of $q$.

Finally, to study the quality of the density-density pair distribution in
Figs. 5 and 6 we plot $g(r)$ and its Fourier transform $S(q)$ at the metallic
density $r_{s}=5$ and the large radius $r_{s}=10$ for dimensions $D=3,5$, and
$7$. The behavior of $g(r)$ for the metallic densities (i.e., $r_{s}=5$) is
physical in the sense that it takes positive values for all the analyzed
dimensions, whereas for $r_{s}=10$ an un-physical, though small, negative
value can be observed for small distances. This is in agreement with the 2D
and 3D cases, discussed above; this also shows that the range of validity of
the STLS scheme is limited to the metallic-density regime, even at
unconventional dimensions.

Figure 4: The local field correction $G(\mathbf{q})$ for $r_{s}=2$ in the
$3D$, $5D$ and $7D$ case for the para- and ferromagnetic HEG.

##  VI Sum rules

As already mentioned, one of the known shortcomings of the STLS method is the
failure to satisfy the compressibility sum rule (CSR) which is an exact
property of the HEL: In the long wavelength limit, the static response and
dielectric functions are related to the compressibility of the electronic
system. Indeed, for arbitrary dimensions the following expression holds (see
Appendix C):

$\displaystyle\lim_{q\rightarrow
0}\epsilon_{D}(\mathbf{q},0)=1+\left(\frac{q_{\rm
TF}}{q}\right)^{D-1}\frac{\kappa}{\kappa_{f}}\,,$ (29)

where $\kappa/\kappa_{f}$ is the ratio of the compressibility of the
interacting-free $D$-dimensional electron liquid, and $q_{\rm TF}$ is the
$D$-dimensional Thomas-Fermi wave vector. For the free electron liquid, the
corresponding free compressibility is $\kappa_{f}=Dr_{s}^{2}/n\alpha_{D}^{2}$.
Also by taking the limit $q\rightarrow 0$ at $\omega=0$ in Eq. (8) and
comparing the result with Eq. (29) (see also Appendix C) one obtains:

$\displaystyle\frac{\kappa_{f}}{\kappa}=1-\gamma_{D}\left(\frac{q_{\mathrm{TF}}}{q_{F}}\right)^{D-1}\coloneqq\mathcal{K}_{D}\,,$
(30)

where $\gamma_{D}=-\frac{1}{2q_{\mathrm{F}}}\int^{\infty}_{0}[S_{D}(q)-1]dq$.

Another way to calculate the same compressibility ratio in arbitrary
dimensions is to use (a) the known thermodynamic formula
$\kappa^{-1}=n(\partial P/\partial n)$, where $P$ denotes the pressure, and
(b) the expression of the energy for the free liquid (17). Our result is

$\displaystyle\frac{\kappa_{f}}{\kappa}=\frac{r_{s}^{4}}{\alpha^{2}_{D}D}\left[\frac{(1-D)}{r_{s}}\epsilon^{\prime}_{D}(r_{s})+\epsilon^{\prime\prime}_{D}(r_{s})\right]\coloneqq\bar{\mathcal{K}}_{D}\,.$
(31)

| $3D$ | $5D$ | $7D$
---|---|---|---
| $\mathcal{K}_{3}$ | $\bar{\mathcal{K}}_{3}$ | $\mathcal{K}_{5}$ | $\bar{\mathcal{K}}_{5}$ | $\mathcal{K}_{7}$ | $\bar{\mathcal{K}_{7}}$
$r_{s}=2$ | 0.35 | 0.64 | 0.75 | 0.88 | 0.85 | 0.93
$r_{s}=4$ | -0.39 | 0.25 | 0.46 | 0.73 | 0.69 | 0.85
$r_{s}=6$ | -1.18 | -0.16 | 0.15 | 0.58 | 0.52 | 0.77
Table 1: Compressibility ratios $\mathcal{K}_{D}$ (30) and
$\bar{\mathcal{K}}_{D}$ (31) predicted by the STLS scheme in $3D$, $5D$ and
$7D$.

In practice, Eq. (29) implies that the compressibility ratio
$\kappa/\kappa_{f}$ derived from the thermodynamic expression in Eq. (31)
should be equal to the same ratio derived from the $\lim_{q\rightarrow
0}\epsilon_{D}(\mathbf{q},0)$ of Eq. (8) with $G_{D}(\mathbf{q})$ as given in
Eq. (6). Yet, calculating the compressibility from an approximate expression
of the dielectric function will generally result in a value that differs from
the one obtained as the derivative of the pressure. This is the case of STLS
theory, indeed. As a matter of fact, there are few theoretical settings where
they coincide Sjostrom and Dufty (2013).

The numerical results of $\kappa_{f}/\kappa$ in $3D$, $5D$ and $7D$ calculated
both as $\mathcal{K}_{D}$ (30) and $\bar{\mathcal{K}}_{D}$ (31) for some
$r_{s}$ are shown in Table 1. In general, for all dimensions
$\mathcal{K}_{D}\neq\bar{\mathcal{K}}_{D}$. For 3D we obtained the results of
the original STLS paper Singwi  _et al._ (1968). Notice that the STLS
compressibility ratio $\mathcal{K}_{3}$ becomes already negative for
$r_{s}>3$. Yet, interestingly, the larger the underlying dimension the better
the performance of the CSR within the original STLS framework. For instance,
for $r_{s}=4$, $\mathcal{K}_{D}/\bar{\mathcal{K}}_{D}=-1.56,0.63$, and $0.81$
for $3D$, $5D$ and $7D$, respectively.

Figure 5: The density-density pair distribution $g(r)$ of the HEL at $r_{s}=5$
and $10$ for dimensions $D=3,5$ and 7. Figure 6: The structure factor $S(q)$
at $r_{s}=5$ and $10$ for dimensions $D=3,5$ and 7.

There are other well-known sum rules that can be discussed in the context of
STLS Vaishya and Gupta (1973). For instance, in the limit of large
frequencies, one can show that
$\epsilon_{D}(\mathbf{q},\omega)=1-[\omega^{\prime}_{D}(\mathbf{q})/\omega]^{2}$,
where $[\omega^{\prime}_{D}(\mathbf{q})]^{2}=\Phi(\mathbf{q})nq^{2}$ is a
frequency that in the case of $D=3$ is equal to $\omega^{2}_{p}=4\pi n/m$, the
plasma frequency of the liquid. Moreover, by using however the results of
Appendix C and the frequency moment sum rules Vaishya and Gupta (1973), it is
also straightforward to see that the so-called screening requirement is
independent of the dimension of the space:

$\displaystyle\lim_{q\rightarrow 0}[\epsilon_{D}(\mathbf{q},0)]^{-1}=0\,.$
(32)

##  VII Conclusion and outlook

In this paper we have extended the celebrated Singwi, Tosi, Land, and
Sjölander (STLS) scheme for the homogeneous electron liquid (HEL) —whose
accuracy to compute the full electronic density response is comparable to
Quantum Monte-Carlo— to arbitrary integer dimensions. Our main motivation was
to study the quality of the random phase approximation (RPA) results for the
HEL in arbitrary integer dimensions. To that goal, we have provided new
analytical formulae for the real and imaginary parts of the Lindhard
polarizability and for the so-called local field correction of the STLS
theory. This improves the RPA in the density-density response. We have also
discussed the un-physical predictions of the STLS method for large values of
$r_{s}$ in arbitrary dimensions and the incapability of the scheme of
fulfilling the compressibility sum rule. From our results we can conclude that
the algebraic properties of the correlation energy found in Ref. Schlesier
_et al._ (2020) are mainly valid in the high-density limit. Furthermore, in
agreement with what is known in 2 and 3 dimensions, the RPA tends to over-
correlate the gas also in the high dimensions studied here. We have shown the
versatility of STLS to tackle arbitrary dimensions, which can potentially shed
light on the more far-reaching problems of quantum many-body systems embedded
in fractional or synthetic dimensions Kempkes  _et al._ (2019); Pai and Prem
(2019); Fremling  _et al._ (2020); Pal and Saha (2018); Brzezińska  _et al._
(2018). In fact, we have found that the compressibility sum rule behaves
better when the dimension increases. We believe the results of this paper can
be a useful framework to improve our overall comprehension of Coulomb gases,
and to develop a more coherent and unified dimensional approach to the
correlation problem of those systems Constantin (2016); Benavides-Riveros  _et
al._ (2017); Lewin (2022); Bonitz  _et al._ (2020).

###### Acknowledgements.

We thank Robert Schlesier for helpful discussions. We acknowledge financial
support from “BiGmax”, the Max Planck Society’s research network on big-data-
driven materials science, and the European Union’s Horizon Europe Research and
Innovation program under the Marie Skłodowska-Curie grant agreement
n°101065295 (C.L.B.-R.).

##  Appendix A Calculation of the Lindhard polarizability in $D\geq 3$

To compute the Lindhard polarizability for dimensions larger than 3 we start
by writing the one-particle density and the external potential as linear
perturbations from equilibrium:

$\displaystyle f({\mathbf{r}},\mathbf{p},t)$
$\displaystyle=f_{0}(\mathbf{p})+\lambda\,\delta
f({\mathbf{r}},\mathbf{p},t)\,,$ $\displaystyle v({\mathbf{r}},t)$
$\displaystyle=v_{0}({\mathbf{r}})+\lambda\,\delta v({\mathbf{r}},t)\,.$ (33)

After substituting Eq. (33) in Eq. 2 one obtains the following result for the
induced charge density $\rho_{\textrm{ind}}(\mathbf{q},\omega)$:

$\displaystyle\rho_{\textrm{ind}}(\mathbf{q},\omega)=\frac{\chi^{0}_{D}(\mathbf{q},\omega)}{1-\Phi(\mathbf{q})[1-G_{D}(\mathbf{q})]\chi^{0}(\mathbf{q},\omega)}\delta
v(\mathbf{q},\omega)\,,$ (34)

where $\chi^{0}_{D}(\mathbf{q},\omega)$ is given by:

$\displaystyle\chi^{0}_{D}(\mathbf{q},\omega)$
$\displaystyle=-\lim_{\eta\rightarrow
0^{+}}\int\frac{\mathbf{q}\cdot\nabla_{\mathbf{p}}f_{0}(\mathbf{p})}{\omega+i\eta-\mathbf{p}\cdot\mathbf{q}}d^{D}\mathbf{p}$
(35)

By using the Taylor expansion:
$f_{0}(\mathbf{p}\pm\tfrac{1}{2}\mathbf{q})=f_{0}(\mathbf{p})\pm\tfrac{1}{2}\mathbf{q}\cdot\nabla_{\mathbf{p}}f_{0}(\mathbf{p})+\cdots$,
we can then rewrite $\chi^{0}_{D}(\mathbf{q},\omega)$ as:

$\displaystyle\chi^{0}_{D}(\mathbf{q},\omega)\approx\lim_{\eta\rightarrow
0^{+}}\int\frac{f_{0}(\mathbf{p}-\frac{1}{2}\mathbf{q})-f_{0}(\mathbf{p}+\frac{1}{2}\mathbf{q})}{\omega+i\eta-\mathbf{p}\cdot\mathbf{q}}d^{D}\mathbf{p}\,.$
(36)

This is the Lindhard polarizability of a $D$-dimensional Fermi gas Singwi and
Tosi (1981).

At $T=0$ the equilibrium density reads

$\displaystyle
f_{0}(\mathbf{p}\pm\tfrac{1}{2}\mathbf{q})=\frac{2}{(2\pi)^{D}}\Theta\left[\tfrac{1}{2}\left(q_{F}^{2}-\left|\mathbf{p}\pm\tfrac{1}{2}\mathbf{q}\right|^{2}\right)\right]=\frac{2}{(2\pi)^{D}}\Theta(q_{F}-|\mathbf{p}\pm\tfrac{1}{2}\mathbf{q}|)\,.$
(37)

Plugging these equations and the identity $\lim_{\eta\rightarrow
0^{+}}\frac{1}{x-x_{0}\pm i\eta}=\mathcal{P}\frac{1}{x-x_{0}}\mp
i\pi\delta(x-x_{0})$ into Eq. (36) yields:

$\displaystyle\chi^{0}_{D}(\mathbf{q},\omega)$
$\displaystyle=\frac{2}{(2\pi)^{D}}\mathcal{P}\int\frac{\Theta(q_{F}-|\mathbf{p}-\frac{1}{2}\mathbf{q}|)-\Theta(q_{F}-|\mathbf{p}+\frac{1}{2}\mathbf{q}|)}{\omega-\mathbf{p}\cdot\mathbf{q}}d^{D}\mathbf{p}$
$\displaystyle-i\frac{2\pi}{(2\pi)^{D}}\int\left[\Theta\left(q_{F}-\left|\mathbf{p}-\frac{1}{2}\mathbf{q}\right|\right)-\Theta\left(q_{F}-\left|\mathbf{p}+\frac{1}{2}\mathbf{q}\right|\right)\right]\delta\left(\omega-\mathbf{p}\cdot\mathbf{q}\right)d^{D}\mathbf{p}\,.$
(38)

The real part of $\chi^{0}_{D}(\mathbf{q},\omega)$ can then be rewritten as:

$\displaystyle\mathrm{Re}\chi^{0}_{D}(\mathbf{q},\omega)$
$\displaystyle=\frac{2}{(2\pi)^{D}}\mathcal{P}\int\frac{\Theta(q_{F}-|\mathbf{p}-\tfrac{1}{2}\mathbf{q}|)\Theta(|\mathbf{p}+\tfrac{1}{2}\mathbf{q}|-q_{F})}{\omega-\mathbf{p}\cdot\mathbf{q}}d^{D}\mathbf{p}$
$\displaystyle\quad-\frac{2}{(2\pi)^{D}}\mathcal{P}\int\frac{\Theta(|\mathbf{p}-\tfrac{1}{2}\mathbf{q}|-q_{F})\Theta(q_{F}-|\mathbf{p}+\tfrac{1}{2}\mathbf{q}|)}{\omega-\mathbf{p}\cdot\mathbf{q}}d^{D}\mathbf{p}\,.$
(39)

The second integral can be rewritten with the substitution
$\mathbf{p}\rightarrow-\mathbf{p}$ so that the expression of
$\mathrm{Re}\chi^{0}_{D}(\mathbf{q},\omega)$ reduces to just one single
integral:

$\displaystyle\mathrm{Re}\chi^{0}_{D}(\mathbf{q},\omega)=\frac{2}{(2\pi)^{D}}\mathcal{P}\int\Theta(q_{F}-|\mathbf{p}-\tfrac{1}{2}\mathbf{q}|)\Theta(|\mathbf{p}+\tfrac{1}{2}\mathbf{q}|-q_{F})\left(\frac{1}{\omega-\mathbf{p}\cdot\mathbf{q}}-\frac{1}{{\omega+\mathbf{p}\cdot\mathbf{q}}}\right)d^{D}\mathbf{p}\,.$
(40)

Utilizing the following property of the Heaviside step function:
$\Theta(x)=1-\Theta(-x)$ and the symmetry of the integrand under the
transformation $\mathbf{p}\rightarrow-\mathbf{p}$,
$\mathrm{Re}\chi^{0}(\mathbf{q},\omega)$ can be further simplified to:

$\displaystyle\mathrm{Re}\chi^{0}_{D}(\mathbf{q},\omega)=\frac{2}{(2\pi)^{D}}\mathcal{P}\int\Theta(q_{F}-|\mathbf{p}-\tfrac{1}{2}\mathbf{q}|)\left(\frac{1}{\omega-\mathbf{p}\cdot\mathbf{q}}-\frac{1}{{\omega+\mathbf{p}\cdot\mathbf{q}}}\right)d^{D}\mathbf{p}\,.$
(41)

This integral becomes dimensionless by putting
$\mathbf{p}-\frac{\mathbf{q}}{2}=\mathbf{k}$ and introducing the following
substitutions $\tilde{\mathbf{q}}=\mathbf{q}/q_{F}$,
$\tilde{\mathbf{k}}=\mathbf{k}/q_{F}$ and $\tilde{\omega}=\omega/q_{F}^{2}$:

$\displaystyle\mathrm{Re}\chi^{0}_{D}(\mathbf{q},\omega)=\frac{2q_{F}^{D-2}}{(2\pi)^{D}}\mathcal{P}\int\Theta(1-\tilde{k})\left(\frac{1}{\tilde{\omega}-\tilde{\mathbf{q}}\cdot\tilde{\mathbf{k}}-\frac{\tilde{q}^{2}}{2}}-\frac{1}{\tilde{\omega}+\tilde{\mathbf{q}}\cdot\tilde{\mathbf{k}}+\frac{\tilde{q}^{2}}{2}}\right)d^{D}\tilde{\mathbf{k}}\,.$
(42)

To solve this integral we choose $D$-dimensional spherical coordinates such
that $\tilde{\mathbf{q}}$ is parallel to the last axis:

$\displaystyle\mathrm{Re}\chi^{0}_{D}(\mathbf{q},\omega)=\frac{2q_{F}^{D-2}}{(2\pi)^{D}}\frac{2\pi^{\frac{D-1}{2}}}{\Gamma\left(\frac{D-1}{2}\right)}\mathcal{P}\int_{0}^{1}\int_{0}^{\pi}\left(\frac{\tilde{k}^{D-1}\sin^{D-2}\theta}{\tilde{\omega}-\tilde{q}\tilde{k}\cos\theta-\frac{\tilde{q}^{2}}{2}}-\frac{\tilde{k}^{D-1}\sin^{D-2}\theta}{\tilde{\omega}+\tilde{q}\tilde{k}\cos\theta+\frac{\tilde{q}^{2}}{2}}\right)d\theta
d\tilde{k}\,,$ (43)

where $\theta$ is the angle between $\tilde{\mathbf{q}}\text{ and
}\tilde{\mathbf{k}}$. The full evaluation of this integral is quite involved.
We just mention that this gives the known result for $D=3$:

$\displaystyle\mathrm{Re}\chi^{0}_{3}(\mathbf{q},\omega)$
$\displaystyle=\frac{q_{F}}{2\pi^{2}}\Bigg{\\{}-1+\frac{1}{2\tilde{q}}\left[1-\left(\frac{2\tilde{\omega}-\tilde{q}^{2}}{2\tilde{q}}\right)^{2}\right]\ln\left|\frac{2\tilde{q}-\tilde{q}^{2}+2\tilde{\omega}}{2\tilde{q}+\tilde{q}^{2}-2\tilde{\omega}}\right|$
$\displaystyle\quad-\frac{1}{2\tilde{q}}\left[1-\left(\frac{2\tilde{\omega}+\tilde{q}^{2}}{2\tilde{q}}\right)^{2}\right]\ln\left|\frac{2\tilde{q}+\tilde{q}^{2}+2\tilde{\omega}}{2\tilde{q}-\tilde{q}^{2}-2\tilde{\omega}}\right|\Bigg{\\}}\,.$
(44)

We provide the explicit results for $D=5$ in (12) and $D=7$ in (13).

The imaginary part of the Lindhard polarizability can be computed by using the
same substitutions as in the calculation of
$\mathrm{Re}\chi^{0}_{D}(\mathbf{q},\omega)$:

$\displaystyle\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)=-\frac{q_{F}^{D-2}}{(2\pi)^{D-1}}\int[\Theta(1-\tilde{k})-\Theta(1-|\tilde{\mathbf{q}}+\tilde{\mathbf{k}}|)]\delta\left(\tilde{\omega}-\tilde{\mathbf{q}}\cdot\tilde{\mathbf{k}}-\frac{\tilde{q}^{2}}{2}\right)d^{D}\tilde{\mathbf{k}}\,.$
(45)

Next we want to investigate the symmetry of
$\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)$. By considering
$\mathrm{Im}\chi^{0}_{D}(-\mathbf{q},-\omega)$ and substituting
$\tilde{\mathbf{k}}^{\prime}=\tilde{\mathbf{k}}-\tilde{\mathbf{q}}$ we realize
that
$\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)=-\mathrm{Im}\chi^{0}_{D}(-\mathbf{q},-\omega)$.
This symmetry allows us to restrict ourselves to case where $\omega>0$.
$\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)$ can then be rewritten for
positive $\omega$ as follows:

$\displaystyle\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)=-\frac{q_{F}^{D-2}}{(2\pi)^{D-1}}\int\Theta(1-\tilde{k})\Theta(|\tilde{\mathbf{q}}+\tilde{\mathbf{k}}|-1)\delta\left(\tilde{\omega}-\tilde{\mathbf{q}}\cdot\tilde{\mathbf{k}}-\frac{\tilde{q}^{2}}{2}\right)d^{D}\tilde{\mathbf{k}}\,.$
(46)

There are 2 cases where the integrand is different from 0:

  * •

$\left|\tilde{q}-\frac{\tilde{q}^{2}}{2}\right|\leq\tilde{\omega}\leq\tilde{q}+\frac{\tilde{q}^{2}}{2}$

  * •

$\tilde{\omega}<\tilde{q}-\frac{\tilde{q}^{2}}{2}\,,\ \tilde{q}<2$

The evaluation of both integrals is technically similar. Thus we just want to
discuss the evaluation in the first case. The minimum value of $\tilde{k}$ in
the first case is
$\tilde{k}_{\textrm{min}}=|\tilde{\omega}/\tilde{q}-\tilde{q}/2|$. Performing
the integration over all angles except the angle $\theta$ between
$\tilde{\mathbf{q}}$ and $\tilde{\mathbf{k}}$ and substituting $\cos\theta=t$
leads to:

$\displaystyle\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)=-\frac{q_{F}^{D-2}}{(2\pi)^{D-1}}\frac{2\pi^{\frac{D-1}{2}}}{\Gamma\left(\tfrac{D-1}{2}\right)}\frac{1}{\tilde{q}}\int^{1}_{\left|\tfrac{\tilde{\omega}}{\tilde{q}}-\tfrac{\tilde{q}}{2}\right|}\int^{1}_{-1}\delta\left(\tfrac{\tilde{\omega}}{\tilde{q}\tilde{k}}-\tfrac{\tilde{q}}{2\tilde{k}}-t\right)\tilde{k}^{D-2}(1-t^{2})^{\tfrac{D-3}{2}}dtd\tilde{k}\,.$
(47)

To evaluate it we should put
$\tilde{\omega}=\tilde{q}^{2}/2+\tilde{q}\tilde{k}\cos\theta\Rightarrow-1\leq\tilde{\omega}/\tilde{q}\tilde{k}-\tilde{q}/2\tilde{k}\leq
1$, so that the integral over $t$ can be carried out easily. We obtain in this
case the following result:

$\displaystyle\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)=h(D)\dfrac{1}{\tilde{q}}\left[1-\nu_{-}^{2}\right]^{\tfrac{D-1}{2}}\,,$
(48)

where we define
$h(D)=q_{F}^{D-2}\left[2^{D-2}(D-1)\pi^{\tfrac{D-1}{2}}\Gamma\left(\tfrac{D-1}{2}\right)\right]^{-1}$
and $\nu_{\pm}=\tilde{\omega}/\tilde{q}\pm\tilde{q}/2$.

In the second case, $k_{\textrm{min}}=\sqrt{1-2\omega}$ and we obtain the
following result:

$\displaystyle\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)=h(D)\dfrac{1}{\tilde{q}}\left[\left(1-\nu_{-}^{2}\right)^{\frac{D-1}{2}}-\left(1-\nu_{+}^{2}\right)^{\frac{D-1}{2}}\right]\,.$
(49)

Eventually $\mathrm{Im}\chi^{0}_{D}(\mathbf{q},\omega)$ reads as in Eq. (14).

##  Appendix B The local field correction in the Hartree-Fock approximation

After the substitution $\mathbf{s}=(\mathbf{k}+\mathbf{k}^{\prime})/2\text{
and }\mathbf{t}=\mathbf{k}-\mathbf{k}^{\prime}$ Eq. (15) becomes:

$\displaystyle
G_{D}^{\textrm{HF}}(\mathbf{q})=\frac{2q^{D-3}}{(2\pi)^{2D}n^{2}}\int_{|\mathbf{s}+\frac{\mathbf{t}}{2}|\leq
q_{F}}\int_{|\mathbf{s}-\frac{\mathbf{t}}{2}|\leq
q_{F}}\frac{\mathbf{q}\cdot(\mathbf{q}+\mathbf{t})}{|\mathbf{q}+\mathbf{t}|^{D-1}}d\mathbf{t}d\mathbf{s}\,.$
(50)

Because the integrand only depends on $\mathbf{q}$ and $\mathbf{t}$, we can
perform the integration over $\mathbf{s}$. This gives us the volume of the
integration region and the integral becomes:

$\displaystyle
G_{D}^{\textrm{HF}}(\mathbf{q})=\frac{2q^{D-3}}{(2\pi)^{2D}n^{2}}\int
d\mathbf{t}\frac{\mathbf{q}\cdot(\mathbf{q}+\mathbf{t})}{|\mathbf{q}+\mathbf{t}|^{D-1}}\int
d\mathbf{s}\Theta(q_{F}-|\mathbf{s}+\tfrac{\mathbf{t}}{2}|)\Theta(q_{F}-|\mathbf{s}-\tfrac{\mathbf{t}}{2}|)\,.$
(51)

Thus we need to identify the region in hyperspace in which we are integrating.
The region after the substitution is mathematically defined by
$|\mathbf{s}+\mathbf{t}/2|\leq q_{F}\text{ and }|\mathbf{s}-\mathbf{t}/2|\leq
q_{F}$ and therefore can be seen as the overlap region of two Fermi spheres in
hyperspace (as shown by the colored region in Fig. 7). The centers of the
spheres sit at a distance $|\mathbf{t}|$ from each other, the origin of
$\mathbf{s}$ is the midpoint of the connecting line of two centers.

Figure 7: Integration region of $G_{D}^{\textrm{HF}}(\mathbf{q})$.

The integration over $\mathbf{s}$ is the volume of this overlap region, which
is the combined volumes of two identical hyperspherical caps. The volume of a
$D$-dimensional hyperspherical cap was already calculated by Li Li (2011):

$\displaystyle\mathcal{V}_{D}=\frac{1}{2}V_{D}(r)\textrm{I}_{\sin^{2}\phi}\left(\frac{D+1}{2},\frac{1}{2}\right)\,,$

where $V_{D}(r)$ is the volume of a hypersphere with radius $r$ in
$D$-dimensional space, $\phi$ is the angle between a vector and the positive
$D^{\mathrm{th}}$-axis of the sphere and $\textrm{I}(x,y)$ is the regularized
incomplete beta function. In our case it can easily be verified that
$\textrm{sin}^{2}\phi=1-\frac{t^{2}}{4q_{F}^{2}}$ and
$V_{D}(r)=\pi^{D/2}q_{F}^{D}/\Gamma\left(\tfrac{D}{2}+1\right)$. As a result,
$G^{D}_{\textrm{HF}}(\mathbf{q})$ becomes:

$\displaystyle
G^{\textrm{HF}}_{D}(\mathbf{q})=\frac{2q^{D-3}}{(2\pi)^{2D}n^{2}}\frac{\pi^{D/2}}{\Gamma\left(\frac{D}{2}+1\right)}q_{F}^{D}\int\textrm{I}_{1-\frac{t^{2}}{4q_{F}^{2}}}\left(\frac{D+1}{2},\frac{1}{2}\right)\frac{\mathbf{q}\cdot(\mathbf{q}+\mathbf{t})}{|\mathbf{q}+\mathbf{t}|^{D-1}}d\mathbf{t}\,.$
(52)

By changing to $D$-dimensional spherical coordinates such that $\mathbf{q}$ is
parallel to the $D^{\mathrm{th}}$-axis and using the relation between the
density and the Fermi wavelength $q_{F}$ we obtain Eq. (16). For some cases
this calculation can be done analytically. Here we provide the result for the
$5D$ case:

$\displaystyle G^{\textrm{HF}}_{5}(q)$
$\displaystyle=\frac{-5}{4928}\left(\frac{q}{q_{F}}\right)^{8}+\frac{25}{1232}\left(\frac{q}{q_{F}}\right)^{6}+\frac{1775}{7392}\left(\frac{q}{q_{F}}\right)^{4}-\frac{5}{66}\left(\frac{q}{q_{F}}\right)^{2}+\frac{25}{154}$
$\displaystyle+\left[\frac{-15}{128}\left(\frac{q}{q_{F}}\right)^{5}+\frac{15}{224}\left(\frac{q}{q_{F}}\right)^{3}+\frac{5}{56}\left(\frac{q}{q_{F}}\right)-\frac{25}{154}\left(\frac{q_{F}}{q}\right)\right]\ln\left|\frac{q+2q_{F}}{q-2q_{F}}\right|$
$\displaystyle+\left[\frac{-5}{19712}\left(\frac{q}{q_{F}}\right)^{10}+\frac{5}{896}\left(\frac{q}{q_{F}}\right)^{8}-\frac{15}{224}\left(\frac{q}{q_{F}}\right)^{6}\right]\ln\left|\frac{q^{2}-4q_{F}^{2}}{q^{2}}\right|\,.$
(53)

##  Appendix C Compressibility sum rule

By applying the compressibility sum rule (see, for instance, Giuliani and
Vignale (2005)) one can rewrite the dielectric function in the limit
$q\rightarrow 0$ at $\omega=0$ as follows:

$\displaystyle\lim_{q\rightarrow
0}\epsilon_{D}(q,0)=1+\Phi(q)n^{2}\kappa=1+\frac{(4\pi)^{\frac{D-1}{2}}\Gamma\left(\frac{D-1}{2}\right)}{q^{D-1}}\frac{Dr_{s}^{2}n}{\alpha_{D}^{2}}\frac{\kappa}{\kappa_{f}}\,,$
(54)

where $\kappa_{f}=\frac{Dr_{s}^{2}}{n\alpha_{D}^{2}}$ is the free
compressibility. By defining the $D$-dimensional Thomas-Fermi wavelength as:

$\displaystyle
q_{\mathrm{TF}}=\left(\frac{(4\pi)^{\frac{D-1}{2}}\Gamma\left(\frac{D-1}{2}\right)Dr_{s}^{2}n}{\alpha_{D}^{2}}\right)^{\tfrac{1}{D-1}}$

we obtain Eq. (29). In 3D this amounts to the well-known result

$\lim_{q\rightarrow 0}\epsilon_{3}(\mathbf{q},\omega)=1+\frac{q_{\rm
TF}^{2}}{q^{2}}\frac{\kappa}{\kappa_{f}}\,.$

Finally, by applying the sum rules to the Lindhard polarizability (i.e.,
$\lim_{q\rightarrow 0}\chi_{0}(q,0)=-n^{2}\kappa_{f}\,,$) we obtain the
dielectric function in the STLS theory (Eq. (8)) in the limit $q\rightarrow
0$:

$\displaystyle\lim_{q\rightarrow
0}\epsilon_{D}(\mathbf{q},0)=1+\frac{\left(\tfrac{q_{\mathrm{TF}}}{q}\right)^{D-1}}{1-\gamma_{D}\left(\tfrac{q_{\mathrm{TF}}}{q_{F}}\right)^{D-1}}\,,$
(55)

where $\gamma_{D}=-\frac{1}{2q_{\mathrm{F}}}\int^{\infty}_{0}[S_{D}(q)-1]dq$.
From Eq. 29 one eventually finds:

$\displaystyle\frac{\kappa_{f}}{\kappa}=1-\gamma_{D}\left(\frac{q_{\mathrm{TF}}}{q_{F}}\right)^{D-1}\,.$
(56)

## References

  * Sommerfeld (1928) A. Sommerfeld, “Zur elektronentheorie der metalle auf grund der fermischen statistik,” Z. Physik 47, 1 (1928).
  * Rajagopal and Kimball (1977) A. K. Rajagopal and J. C. Kimball, “Correlations in a two-dimensional electron system,” Phys. Rev. B 15, 2819 (1977).
  * Gori-Giorgi and Ziesche (2002) P. Gori-Giorgi and P. Ziesche, “Momentum distribution of the uniform electron gas: Improved parametrization and exact limits of the cumulant expansion,” Phys. Rev. B 66, 235116 (2002).
  * Huotari  _et al._ (2010) S. Huotari, J. A. Soininen, T. Pylkkänen, K. Hämäläinen, A. Issolah, A. Titov, J. McMinis, J. Kim, K. Esler, D. M. Ceperley, M. Holzmann, and V. Olevano, “Momentum Distribution and Renormalization Factor in Sodium and the Electron Gas,” Phys. Rev. Lett. 105, 086403 (2010).
  * Holzmann  _et al._ (2011) M. Holzmann, B. Bernu, C. Pierleoni, J. McMinis, D. M. Ceperley, V. Olevano, and L. Delle Site, “Momentum Distribution of the Homogeneous Electron Gas,” Phys. Rev. Lett. 107, 110402 (2011).
  * Shepherd  _et al._ (2012) J. J. Shepherd, G. Booth, A. Grüneis, and A. Alavi, “Full configuration interaction perspective on the homogeneous electron gas,” Phys. Rev. B 85, 081103(R) (2012).
  * Lewin  _et al._ (2018) M. Lewin, E. H. Lieb, and R. Seiringer, “Statistical mechanics of the uniform electron gas,” J. Éc. polytech. Math. 5, 79 (2018).
  * Loos and Gill (2016) P.-F. Loos and P. M. W. Gill, “The uniform electron gas,” WIREs Comput. Mol. Sci. 6, 410 (2016).
  * Zhang and Ceperley (2008) S. Zhang and D. M. Ceperley, “Hartree-Fock Ground State of the Three-Dimensional Electron Gas,” Phys. Rev. Lett. 100, 236404 (2008).
  * Gontier  _et al._ (2019) D. Gontier, C. Hainzl, and M. Lewin, “Lower bound on the Hartree-Fock energy of the electron gas,” Phys. Rev. A 99, 052501 (2019).
  * Drummond and Needs (2009) N. D. Drummond and R. J. Needs, “Phase Diagram of the Low-Density Two-Dimensional Homogeneous Electron Gas,” Phys. Rev. Lett. 102, 126402 (2009).
  * Spink  _et al._ (2013) G. G. Spink, R. J. Needs, and N. D. Drummond, “Quantum Monte Carlo study of the three-dimensional spin-polarized homogeneous electron gas,” Phys. Rev. B 88, 085121 (2013).
  * Ma and Brueckner (1968) S.-K. Ma and K. A. Brueckner, “Correlation Energy of an Electron Gas with a Slowly Varying High Density,” Phys. Rev. 165, 18 (1968).
  * Ceperley and Alder (1980) D. M. Ceperley and B. J. Alder, “Ground State of the Electron Gas by a Stochastic Method,” Phys. Rev. Lett. 45, 566 (1980).
  * Jones (2015) R. O. Jones, “Density functional theory: Its origins, rise to prominence, and future,” Rev. Mod. Phys. 87, 897 (2015).
  * Constantin  _et al._ (2008) L. A. Constantin, J. M. Pitarke, J. F. Dobson, A. Garcia-Lekue, and J. P. Perdew, “High-Level Correlated Approach to the Jellium Surface Energy, without Uniform-Gas Input,” Phys. Rev. Lett. 100, 036401 (2008).
  * Wagner  _et al._ (2012) L. O. Wagner, E. M. Stoudenmire, K. Burke, and S. R. White, “Reference electronic structure calculations in one dimension,” Phys. Chem. Chem. Phys. 14, 8581 (2012).
  * Astrakharchik  _et al._ (2016) G. E. Astrakharchik, K. V. Krutitsky, M. Lewenstein, and F. Mazzanti, “One-dimensional Bose gas in optical lattices of arbitrary strength,” Phys. Rev. A 93, 021605(R) (2016).
  * Schmidt  _et al._ (2019) J. Schmidt, C. L. Benavides-Riveros, and M. A. L. Marques, “Machine Learning the Physical Nonlocal Exchange-Correlation Functional of Density-Functional Theory,” J. Phys. Chem. Lett. 10, 6425 (2019).
  * Constantin (2016) L. A. Constantin, “Simple effective interaction for dimensional crossover,” Phys. Rev. B 93, 121104(R) (2016).
  * Gedeon  _et al._ (2022) J. Gedeon, J. Schmidt, M. J. P. Hodgson, J. Wetherell, C. L. Benavides-Riveros, and M. A. L. Marques, “Machine learning the derivative discontinuity of density-functional theory,” Mach. Learn.: Sci. Technol. 3, 015011 (2022).
  * Voit (1995) J. Voit, “One-dimensional Fermi liquids,” Rep. Progr. Phys. 58, 977 (1995).
  * Kennes  _et al._ (2021) D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J. Millis, J. Hone, C. R. Dean, D. N. Basov, A. N. Pasupathy, and A. Rubio, “Moiré heterostructures as a condensed-matter quantum simulator,” Nat. Phys. 17, 155 (2021).
  * Kempkes  _et al._ (2019) S. N. Kempkes, M. R. Slot, S. E. Freeney, S. J. M. Zevenhuizen, D. Vanmaekelbergh, I. Swart, and C. M. Smith, “Design and characterization of electrons in a fractal geometry,” Nature Phys. 15, 127 (2019).
  * van Veen  _et al._ (2016) E. van Veen, S. Yuan, M. I. Katsnelson, M. Polini, and A. Tomadin, “Quantum transport in Sierpinski carpets,” Phys. Rev. B 93, 115428 (2016).
  * Fremling  _et al._ (2020) M. Fremling, M. van Hooft, C. M. Smith, and L. Fritz, “Existence of robust edge currents in Sierpiński fractals,” Phys. Rev. Research 2, 013044 (2020).
  * Hummel  _et al._ (2021) F. Hummel, M. T. Eiles, and P. Schmelcher, “Synthetic Dimension-Induced Conical Intersections in Rydberg Molecules,” Phys. Rev. Lett. 127, 023003 (2021).
  * Kanungo  _et al._ (2022) S. K. Kanungo, J. D. Whalen, Y. Lu, M. Yuan, S. Dasgupta, F. B. Dunning, K. R. A. Hazzard, and T. C. Killian, “Realizing topological edge states with Rydberg-atom synthetic dimensions,” Nat. Commun. 13, 972 (2022).
  * Ghosh  _et al._ (2017) S. K. Ghosh, S. Greschner, U. K. Yadav, T. Mishra, M. Rizzi, and V. B. Shenoy, “Unconventional phases of attractive fermi gases in synthetic hall ribbons,” Phys. Rev. A 95, 063612 (2017).
  * Celi  _et al._ (2014) A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliūnas, and M. Lewenstein, “Synthetic Gauge Fields in Synthetic Dimensions,” Phys. Rev. Lett. 112, 043001 (2014).
  * Giuliani and Vignale (2005) G. Giuliani and G. Vignale,  _Quantum Theory of the Electron Liquid_ (Cambridge University Press, 2005).
  * Bohm and Pines (1953) D. Bohm and D. Pines, “A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas,” Phys. Rev. 92, 609 (1953).
  * Schlesier  _et al._ (2020) R. Schlesier, C. L. Benavides-Riveros, and Miguel A. L. Marques, “Homogeneous electron gas in arbitrary dimensions,” Phys. Rev. B 102, 035123 (2020).
  * Ren  _et al._ (2012) X. Ren, P. Rinke, C. Joas, and M. Scheffler, “Random-phase approximation and its applications in computational chemistry and materials science,” J. Mater. Sci. 47, 7447 (2012).
  * Gubernatis  _et al._ (2016) J. Gubernatis, N. Kawashima, and P. Werner,  _Quantum Monte Carlo Methods: Algorithms for Lattice Models_ (Cambridge University Press, 2016).
  * Singwi  _et al._ (1968) K. S. Singwi, M. P. Tosi, R. H. Land, and A. Sjölander, “Electron correlations at metallic densities,” Phys. Rev. 176, 589 (1968).
  * Agosti  _et al._ (1998) D. Agosti, F. Pederiva, E. Lipparini, and K. Takayanagi, “Ground-state properties and density response of quasi-one-dimensional electron systems,” Phys. Rev. B 57, 14869 (1998).
  * Dobson (2009) J. F. Dobson, “Inhomogeneous STLS theory and TDCDFT,” Phys. Chem. Chem. Phys. 11, 4528 (2009).
  * Kosugi and Matsushita (2017) T. Kosugi and Y. Matsushita, “Quantum Singwi-Tosi-Land-Sjölander approach for interacting inhomogeneous systems under electromagnetic fields: Comparison with exact results,” J. Chem. Phys. 147, 114105 (2017).
  * Dharma-wardana (1981) M. W. C. Dharma-wardana, “The dynamic response of a non-uniform distribution of electrons,” J. Phys. C: Solid State Phys. 14, L167 (1981).
  * Tanaka (2017) S. Tanaka, “Improved equation of state for finite-temperature spin-polarized electron liquids on the basis of Singwi–Tosi–Land–Sjölander approximation,” Contrib. Plasma Phys. 57, 126 (2017).
  * Dornheim  _et al._ (2022) T. Dornheim, J. Vorberger, Z. A. Moldabekov, and P. Tolias, “Spin-resolved density response of the warm dense electron gas,” Phys. Rev. Research 4, 033018 (2022).
  * Dornheim  _et al._ (2018) T. Dornheim, S. Groth, and M. Bonitz, “The uniform electron gas at warm dense matter conditions,” Phys. Rep. 744, 1 (2018).
  * Dobson  _et al._ (2002) J. F. Dobson, J. Wang, and T. Gould, “Correlation energies of inhomogeneous many-electron systems,” Phys. Rev. B 66, 081108(R) (2002).
  * Vilk  _et al._ (1994) Y. M. Vilk, L. Chen, and A. M. S. Tremblay, “Theory of spin and charge fluctuations in the Hubbard model,” Phys. Rev. B 49, 13267 (1994).
  * Zantout  _et al._ (2021) K. Zantout, S. Backes, and R. Valentí, “Two-Particle Self-Consistent Method for the Multi-Orbital Hubbard Model,” Ann. Phys. 533, 2000399 (2021).
  * Yoshizawa and Takada (2009) K. Yoshizawa and Y. Takada, “New general scheme for improving accuracy in implementing self-consistent iterative calculations: illustration in the STLS theory,” J. Phys.: Condens. Matter. 21, 064204 (2009).
  * Hasegawa and Shimizu (1975) T. Hasegawa and M. Shimizu, “Electron Correlations at Metallic Densities, II. Quantum Mechanical Expression of Dielectric Function with Wigner Distribution Function,” J. Phys. Soc. Japan 38, 965 (1975).
  * Pines (1999) D. Pines,  _Elementary Excitations in Solids_ (CRC Press, 1999).
  * Pines and Nozières (1966) D. Pines and P. Nozières,  _The Theory of Quantum Liquids: Normal Fermi Liquids_ (CRC Press, 1966).
  * Bernadotte  _et al._ (2013) S. Bernadotte, F. Evers, and C. R. Jacob, “Plasmons in molecules,” J. Phys. Chem. C 117, 1863 (2013).
  * Vashishta and Singwi (1972) P. Vashishta and K. S. Singwi, “Electron Correlations at Metallic Densities. V,” Phys. Rev. B 6, 875–887 (1972).
  * Sham (1973) L. J. Sham, “Exchange and Correlation in the Electron Gas,” Phys. Rev. B 7, 4357 (1973).
  * Haug and Koch (2009) H. Haug and S. W. Koch,  _Quantum Theory of the Optical and Electronic Properties of Semiconductors_, 5th ed. (World Scientific Publishing Co., 2009).
  * Ferrell (1958) R. A. Ferrell, “Rigorous Validity Criterion for Testing Approximations to the Electron Gas Correlation Energy,” Phys. Rev. Lett. 1, 443 (1958).
  * Kumar  _et al._ (2009) K. Kumar, V. Garg, and R. K. Moudgil, “Spin-resolved correlations and ground state of a three-dimensional electron gas: Spin-polarization effects,” Phys. Rev. B 79, 115304 (2009).
  * Ng and Singwi (1987) T. K. Ng and K. S. Singwi, “Arbitrarily polarized model Fermi liquid,” Phys. Rev. B 35, 6683 (1987).
  * Jonson (1976) M. Jonson, “Electron correlations in inversion layers,” J. Phys. C: Solid State Phys. 9, 3055 (1976).
  * Ceperley (1978) D. Ceperley, “Ground state of the fermion one-component plasma: A monte carlo study in two and three dimensions,” Phys. Rev. B 18, 3126 (1978).
  * Monnier (1972) R. Monnier, “Monte Carlo Approach to the Correlation Energy of the Electron Gas,” Phys. Rev. A 6, 393 (1972).
  * Note (1) We corrected a small error in the numerical evaluation of the coefficients presented in Table I of Ref. Schlesier  _et al._ (2020) that however does not affect the main conclusions of that paper.
  * Sjostrom and Dufty (2013) T. Sjostrom and J. Dufty, “Uniform electron gas at finite temperatures,” Phys. Rev. B 88, 115123 (2013).
  * Vaishya and Gupta (1973) J. S. Vaishya and A. K. Gupta, “Dielectric Response of the Electron Liquid in Generalized Random-Phase Approximation: A Critical Analysis,” Phys. Rev. B 7, 4300 (1973).
  * Pai and Prem (2019) S. Pai and A. Prem, “Topological states on fractal lattices,” Phys. Rev. B 100, 155135 (2019).
  * Pal and Saha (2018) B. Pal and K. Saha, “Flat bands in fractal-like geometry,” Phys. Rev. B 97, 195101 (2018).
  * Brzezińska  _et al._ (2018) M. Brzezińska, A. M. Cook, and T. Neupert, “Topology in the Sierpiński-Hofstadter problem,” Phys. Rev. B 98, 205116 (2018).
  * Benavides-Riveros  _et al._ (2017) C. L. Benavides-Riveros, N. N. Lathiotakis, and M. A. L. Marques, “Towards a formal definition of static and dynamic electronic correlations,” Phys. Chem. Chem. Phys. 19, 12655 (2017).
  * Lewin (2022) M. Lewin, “Coulomb and Riesz gases: The known and the unknown,” J. Math. Phys. 63, 061101 (2022).
  * Bonitz  _et al._ (2020) M. Bonitz, T. Dornheim, Zh. A. Moldabekov, S. Zhang, P. Hamann, H. Kählert, A. Filinov, K. Ramakrishna, and J. Vorberger, “Ab initio simulation of warm dense matter,” Phys. Plasmas 27, 042710 (2020).
  * Singwi and Tosi (1981) K. S. Singwi and M. P. Tosi, “Correlations in electron liquids,” Solid State Phys. 36, 177 (1981).
  * Li (2011) S. Li, “Concise Formulas for the Area and Volume of a Hyperspherical Cap,” Asian J. Math. Stat. 4, 66 (2011).


11institutetext: Zhejiang University
11email: {jiachenlei, zhongjieba<EMAIL_ADDRESS>
22institutetext: Microsoft, Redmond, WA, USA
22email: {shuama, sai.vemprala<EMAIL_ADDRESS>

# Masked Autoencoders for Egocentric Video Understanding @ Ego4D Challenge
2022

Jiachen Lei 11 Shuang Ma 22 Zhongjie Ba 11 Sai Vemprala 22 Ashish Kapoor 22
Kui Ren 11

###### Abstract

In this report, we present our approach and empirical results of applying
masked autoencoders in two egocentric video understanding tasks, namely,
Object State Change Classification and PNR Temporal Localization, of Ego4D
Challenge 2022. As team ”TheSSVL”, we ranked 2nd place in both tasks. Our code
will be made available.111https://github.com/jasonrayshd/egomotion

###### Keywords:

Masked autoencoders, Ego4d Challenge

##  1 Masked Autoencoder

MAE[5], proposed by Kaiming, etc, has now drawn the wide interest of
researchers in self-supervised learning due to its pretraining efficiency and
generalization ability in various downstream tasks. Recently, two parallel
works[9][3] called VideoMAE are proposed to extend MAE from image to video
domain. In this report, we demonstrate that merely pretraining the model on
3rd-person view datasets (e.g. Kinetics 400[7]) under the settings of VideoMAE
can achieve state-of-the-art performance on egocentric video understanding
tasks including Ego4d object state change classification and Ego4d PNR
temporal localization. This demonstrates the great representation learning and
generalization ability of VideoMAE in self-supervised video pretraining.

##  2 Downstream-Task Formulation

Following Ego4d official baseline setting on object state change
classification (OSCC) and PNR temporal localization tasks[4], we simply
formulate OSCC and temporal localization tasks as a binary and a multi-class
classification problem respectively. We finetune our model on these two tasks
under a multi-objective optimization setting. Therefore, our model is composed
of a shared backbone and two separate linear heads each used for OSCC or
temporal localization. Besides, our optimization objective can be represented
by equation 1, where $\theta$ denotes model parameters. $\lambda 1$ and
$\lambda 2$ are constants and are both set to 1 in all our experiments.

$\displaystyle\arg\min_{\theta}\ \ L=\arg\min_{\theta}\ \ \lambda 1\times
L_{oscc}+\lambda 2\times L_{tl}$ (1)

Instead of predicting index of the PNR frame only when a state change occurs
in given input frames as in [4], We treat the temporal localization task as a
$N+1$ classification problem where N denotes the number of input frames.
Therefore, the classification head for temporal localization is obliged to
predict whether state change occurred in given input frames and occurred in
which frame if do. We believe this helps smooth the model training process and
improve model performance by providing sufficient supervised information.

##  3 Experiments

###  3.1 Implementation details

In our experiments, we use ViT as our backbone of which weights are
initialized from VideoMAE[9] pretrained on Kinetics-400. Specifically, we use
ViT-L pretrained for 1600 epochs in OSCC and ViT-B pretrained for 800 epochs
in temporal localization. Pretraining details and model checkpoints can be
found in [9]. We will show that even with weights obtained on 3rd-person view
datasets, VideoMAE shows great generalization ability on egocentric downstream
tasks and surpass most existing methods both on OSCC and temporal localization
tasks.

When finetuning on OSCC and temporal localization tasks, we sample 16 frames
from each clip as input and set the training epochs to 100. More finetuning
details are listed in Table 3.

During inference, we sample 3 clips out of 16 frames from each test video via
spatially random cropping. The spatial dimension of obtained clips is 224x224.
The final results are the average of model raw logits on these 3 clips. We
submit the test result of the model checkpoint that performs best on the
validation set.

###  3.2 Results

For a more comprehensive comparison on OSCC and temporal localization tasks,
we cite results both from the leaderboard and publicly available technical
reports of other teams. As shown in Table 1 and Table 2, by simply pretraining
on Kinetics 400 under the settings of VideoMAE, we ranked 2nd place in both
tasks.

It is worth noting that in PNR temporal localization task, always selecting
the frame corresponding to 0.45 times the video duration achieves an absolute
temporal error of 0.67, as stated in [2]. This indicates methods including
ours struggle to precisely predict the PNR frame.

Table 1: Results on Ego4d challenge 2022 Object State Change classification task. Each method marked by * is the latest submissions of the team and the results are different from the one in technical report  Method | Extra data | Accuracy $\uparrow$
---|---|---
Always positive | - | 47.7
Uniandes-Google[2] | K600 | 68
TarHeels[6] | IN-21K | 72
Egocentric VLP[8] | IN-21K+BC+EClip | 73.7
SViT*[1] | N/A | 75
TarHeels*[6] | N/A | 76
PearRepublic | N/A | 76
ohohoh | N/A | 80
Ours | K400 | 77.7

However, team ”ohohoh” achieves a better result and surpasses other methods by
a large margin. In addition to model performance, We suppose that this gap
might also root in the frame sampling strategy. Specifically, when adopting
random sampling strategy following official codes, the PNR frame in each clip
is not guaranteed to be sampled both in training and inference time.
Therefore, it brings noise into the model training process and introduces an
absolute time shift away from the ground-truth PNR frame in both training and
inference time.

Table 2: Results on Ego4d challenge 2022 PNR temporal localization task. Two simple baseline methods are presented, including always predicting center frame as the PNR frame and always predicting frame at 0.45 times total video duration as the PNR frame Method | Extra data | Absolute temporal error$\downarrow$
---|---|---
Always Center Frame | - | 1.06
Always at 0.45 duration | - | 0.67
Uniandes-Google[2] | K600 | 0.66
icego | N/A | 0.66
SViT[1] | K400, Ego4D, 100DOH | 0.660
Egocentric VLP[8] | IN-21K+BC+EClip | 0.666
ohohoh | N/A | 0.56
Ours | K400 | 0.654

During training, at first, each 30FPS 8-second clip is uniformly trimmed to
between 5 and 8 seconds and the required number of frames (e.g. 16 frames)
will then be sampled from it. If we denote the number of frames in a trimmed
clip as $S$, its probability density function can be written as:

$\displaystyle p(S)=\left\\{\begin{array}[]{ll}\dfrac{1}{90}\ \ if\
150<=S<=240\\\ \\\ \ 0\ \ \ else\end{array}\right.$ (5)

Among all the sampled frames, the frame that is closest to the ground-truth
PNR frame is taken as a pseudo-PNR frame and used as a hard label when
computing per-frame classification loss. if we sample 16 frames from each
trimmed clip as model input, then, the expectation of absolute time shift away
from the ground-truth PNR frame is $E(\frac{S}{16*30*2})~{}=0.2\ s$. This non-
negligible shift has a negative impact on model performance and will lead to a
loss of precision. The same is true during inference time.

Table 3: Finetuning setting. We follow the linear scaling rule: $lr=base\\_lr\times batchsize/256$ while warmup learning rate is not affected. Hyperparameters | Value
---|---
Optimizer | AdamW
Base learning rate | 5e-4
Weight decay | 0.05
Adam $\beta$ | (0.9, 0.999)
Batch size | 8 or 32
Learning rate sched | Cosine decay
Training epochs | 100
Warmup learning rate | 1e-6
Warmup epochs | 5
Layer-wise lr decay | 0.75
Patch size | 16x16
Tubelet size | 2
Loss weight | (1, 1)
Input size | 224x224
Augmentation | RandAugment(9, 0.5)
Drop path | 0.1
Mixup | 0.8
Cutmix | 1.0
Label smoothing | 0.1

##  4 Conclusions

Although we did not achieve the best performance on OSCC and temporal
localization tasks in Ego4d Challenge 2022, we believe that, by paying much
more attention to downstream task formulation and optimization, models that
are pretrained on egocentric datasets under the settings of VideoMAE will
further improve state-of-the-art performance on various Ego4d tasks. In the
meantime, we will go a step further in egocentric video learning based on
masked autoencoding and are glad to share our work later.

## References

  * [1] Ben-Avraham, E., Herzig, R., Mangalam, K., Bar, A., Rohrbach, A., Karlinsky, L., Darrell, T., Globerson, A.: Structured video tokens@ ego4d pnr temporal localization challenge 2022. arXiv preprint arXiv:2206.07689 (2022)
  * [2] Escobar, M., Daza, L., González, C., Pont-Tuset, J., Arbeláez, P.: Video swin transformers for egocentric video understanding@ ego4d challenges 2022\. arXiv preprint arXiv:2207.11329 (2022)
  * [3] Feichtenhofer, C., Fan, H., Li, Y., He, K.: Masked autoencoders as spatiotemporal learners. arXiv preprint arXiv:2205.09113 (2022)
  * [4] Grauman, K., Westbury, A., Byrne, E., Chavis, Z., Furnari, A., Girdhar, R., Hamburger, J., Jiang, H., Liu, M., Liu, X., et al.: Ego4d: Around the world in 3,000 hours of egocentric video. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 18995–19012 (2022)
  * [5] He, K., Chen, X., Xie, S., Li, Y., Dollár, P., Girshick, R.: Masked autoencoders are scalable vision learners. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 16000–16009 (2022)
  * [6] Islam, M.M., Bertasius, G.: Object state change classification in egocentric videos using the divided space-time attention mechanism. arXiv preprint arXiv:2207.11814 (2022)
  * [7] Kay, W., Carreira, J., Simonyan, K., Zhang, B., Hillier, C., Vijayanarasimhan, S., Viola, F., Green, T., Back, T., Natsev, P., et al.: The kinetics human action video dataset. arXiv preprint arXiv:1705.06950 (2017)
  * [8] Lin, K.Q., Wang, A.J., Soldan, M., Wray, M., Yan, R., Xu, E.Z., Gao, D., Tu, R., Zhao, W., Kong, W., et al.: Egocentric video-language pretraining@ ego4d challenge 2022. arXiv preprint arXiv:2207.01622 (2022)
  * [9] Tong, Z., Song, Y., Wang, J., Wang, L.: Videomae: Masked autoencoders are data-efficient learners for self-supervised video pre-training. arXiv preprint arXiv:2203.12602 (2022)


#  Android Based Low Cost Sitting Posture Monitoring System

Nusrat Binta Nizam$1$,Tohfatul Jinan$2$, Wahida Binte Naz Aurthy$3$, Md. Rakib
Hossen$4$ and Jahid Ferdous$5$ Department of Biomedical Engineering,
Bangladesh University of Engineering and Technology
Dhaka-1205, Bangladesh
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Back pain is one of the leading causes of disability-adjusted life year
globally and the most common cause of low back pain is poor sitting posture.
There are several actions that can be adopted proactively to avoid poor
sitting posture induced back pain including behavioral change, regular
exercise, and use of an ergonomic chair. However, these are either expensive
and/or difficult to execute for prolonged periods. Sitting posture monitoring
systems continuously observe the sitting pattern of a person in real-time and
give feedback/alert poor sitting posture is observed. In this study, a real-
time posture monitoring system has been designed and a functional prototype
has been developed using simple electrical elements and an android
application. Appropriate position of the sensor in the spine to measure the
degree of bending and the threshold sensor values for good posture sittings
have been determined based on the results from healthy volunteers of different
ages and height. The android application continuously monitors the degree of
bending and provides vibration when the bending reaches the threshold of bad
posture or the duration of the sitting crosses the clinically recommended time
limit to prevent prolonged sitting. Positive user feedback has been received
in terms of comfortability, effectiveness, and satisfaction levels. The
manufacturing cost of the developed monitoring system is minimal compared to
the available expensive systems in the market and the cost would further go
down if it is produced in bulk. This device efficiently monitors the sitting
posture pattern to prevent back pain and within the affordable price range for
the people from middle to under-developed countries.

###### Index Terms:

Back pain; sitting posture monitoring system; android application; spine.

##  I Introduction

Proper sitting posture refers to the position in which natural spinal
curvature and arm, shoulder, hip, and knee position are properly maintained
and aligned. Incorrect posture gradually decreases the flexibility of the
spine causing pain in neck, shoulder and back. Furthermore, incorrect sitting
posture has detrimental effects on digestion and respiration. Back pain is
recognized as a common health issue and 75-85% of adults deal with back pain
at least once in their lifetime [1]. In 2016, low back pain and neck pain
combinedly were the $4^{th}$ leading cause of disability-adjusted life year
(DALY) globally [2]. The World Health Organization (WHO) reported that 9.5
million people in South-East Asia were affected by back and neck pain. A
descriptive cross-sectional study conducted among school teachers from
different school in Dhaka showed 46% of school respondents have reported lower
back pain [3]. Lack of knowledge of correct posture is assumed to be the
primary cause of backache, yet a study that 60.8% of physiotherapists in
Bangladesh have low back pain [4]. A cross-sectional study on Ready-Made
Garment employees (RMG) of Bangladesh have reported 24.7% of female worker
respondents had lower back pain and 23.% of female worker respondents had neck
pain [5]. As many people spend most of the time sitting down in the workplace
or home, bad alignment of sitting causes several problems such as bad support
on muscles, ligaments, tendons and unhealthy spine [6].
There are several treatment methods for these abnormalities and depending on
the severity, doctor advices to stay active and/or bed rest, prescribe
medication [7], physical therapies [8], exercises [9], surgery [10], spinal
traction therapy [11] etc. On the other hand, as a proactive practice, good
sitting posture improves the body’s circulatory and digestive system,
increases self-confidence, decreases the risk of abnormal wearing of the joint
surfaces, improves core and scapular strength etc [12]. Therefore, it is an
excellent decision to make a habit of good sitting posture by monitoring as it
brings many benefits for the human body without any unnecessary side-effects.
There has been a lot of effort to prevent back pain using eTextile sensor [13]
and flex sensor to monitor sitting posture based ergonomic design. Moreover,
there are some approaches based on machine learning and accelerometers [14]
[15]. Most of these are very useful but complex, difficult to operate and very
expensive.
The objective of this project is to develop a relatively inexpensive,
comfortable and user-friendly system to monitor sitting posture in real-time.
An elementary portable device connected to a smartphone via bluetooth using a
flex sensor is represented. The user has to set the range in which he feels
comfortable and receives a vibration in the motor when the angle crosses the
safe range of good posture.

##  II Design Methods

###  II-A Overall Design

The proposed design contains two parts: a circuitry part and a mobile
application (app) part (Fig. 1). The circuitry part mainly determines the
sitting posture pattern and sends signal to the app. The app monitors the
duration of sitting and gives feedback to the user it the bending reaches the
safety threshold limits.

Figure 1: Block diagram of the system.

###  II-B Instrumentation

####  II-B1 Materials

Detachable flex sensor (SEN-08606), Vibration Motor (Grove), Bluetooth (HC-05)
device are used along with an Arduino-nano (Atmel ATmega 328) to operate
and/or control. All of these electrical devices are assembled on a printed
circuit board (PCB). A back supporting belt (Royal Posture) is used that holds
the whole circuitry part. A 12V rechargeable LIPO battery with average
lifespan of 150-250 cycles is used as a power supply. The overall weight of
the device including the battery is 340 gm.

####  II-B2 Circuit Design

The circuit components are implanted on a 4-inch x 4-inch PCB and 4-inch-long
flex sensor whereas the battery and the vibration motor are attached to the
sensor belt (Fig. 2). The circuit contains a voltage regulator system that
converts the 12V to 5V to ensure the proper power supply. Bluetooth transfers
the sensor value to the android app to set the threshold value based on the
user comfortability and continuously monitors the posture position. The
correlation between the sensor value and posture angle had been developed to
convert sensor value corresponding to the user’s postural angle. The vibration
motor feedback system alerts the user to get back to the initial set postural
position once the sitting posture angle crosses the threshold of good sitting
posture.

Figure 2: Circuit connections in proteus software.

####  II-B3 Mobile APP Design

The home page of the mobile app named SIPO app has two options i.e., “Connect
to Arduino” and “Instructions”. Instruction window illustrates a tutorial
about the proper sitting process for the user (Fig. 3). The user has to
connect the app with the Arduino in PCB through the connect option. After
connecting the device, the user has to set a threshold value from the sensor
reading on the app screen according to his comfortable position to monitor
posture at a fixed position continuously. The timer feature is for counting
the total duration of sitting.

Figure 3: Pages of android application, where, (1) Home page, (2) Instruction
page, (3) Connection page, and (4) Monitoring and timer tab.

###  II-C Study Design

An experimental study is done on healthy volunteers having a range of age,
weight, height and gender to find the best position for the sensor placement,
to calibrate the sensor value with posture angle and to get user feedback of
the system to ensure the participant’s safety and effectiveness.

####  II-C1 Settings

A marker is used to observe and track the volunteers bending status when he is
advised to bend forward and backward freely from a normal sitting position and
a camera is used to capture the position for the angle measurement with
respect to the initial position (Fig. 4). The mobile app (SIPO) needs to be
installed and the circuitry part is used to observe the corresponding sensor
value of the posture angle. ImageJ software/ angle meter app is used to
calculate the angle from the images. It is important to make sure that there
are enough spaces so that the user can easily move forward and backward while
sitting.

Figure 4: Posture bending angle measurement.

####  II-C2 Data Collection from Participants

Four positions are selected to specify the best location for the sensor
placement based on the average human anatomical position to best track the
bending movement. They are- (a) upper thoracic (6 cm below neck), (b) lower
thoracic (12 cm below neck), (c) upper lumbar (20 cm below neck), and (d)
lower lumbar (26 cm below neck). In order to develop the relationship between
the sensor value and posture angle, participants are advised to change their
posture angle by moving forward and backward six times (three times each side
covering a wide range of posture angle) from sitting at a normal position. The
position of the sensor is an important factor as it varies with the height of
the person. There are in total 9 participants, age ranging from 21 to 50
($\pm{7.91}$), height and weight ranging from 5’ to 5’11” ($\pm{0.29}$) and 45
kg to 75 kg respectively.

###  II-D User Feedback

User feedback form has been filled by the user to evaluate the different
aspects of the device from end-user point of view (comfortability, preference,
effectiveness, satisfaction).

##  III Results Analysis

###  III-A Position Variability Analysis

For the analysis of the sensor position variability, the collected data of
four different positions are studied at five different selected angles ( 75∘,
80∘, 90∘, 100∘, 115∘). For a fixed position and angle value, the sensor values
of 9 subjects are averaged (Fig. 5). In the lower thoracic position the range
of sensor values for different angles is higher, which shows better
sensitivity of sensor and wide range of sensor values at this position. For
this reason, the lower thoracic position is selected for the placement of the
sensor.

Figure 5: Determination of the sensor position for the best sensitivity.

###  III-B Threshold Range Calculation

Bending angle range of 90∘ \- 110∘ has been reported as the safe zone for good
posture whereas 90∘ \- 95∘ is the range of normal sitting posture (Fig. 6)
[16]. Therefore, in order to develop the corresponding sensor value for good
posture position, a model has been developed for lower thoracic position and
fitted with a $3^{rd}$ degree of a polynomial equation (Eqn. 1). So, sensor
value beyond the thresholds of safety regions generates signal to the
vibration motor and alert the user to change the sitting position.

$SV=0.0003(A)^{3}-0.0605(A)^{2}+4.8789(A)+345.23$ (1)

where SV = sensor value and A = angles in degrees.

Figure 6: Calibration between the sensor value and bending angle.

##  IV Discussion

The survey on the users’ feedback gets 7.6/10 in terms of comfortability,
9.2/10 in terms of preference, 9.6/10 effectiveness and 8.0/10 in terms of
overall satisfaction based on the average of the given marks by the users in
user feedback form. The feedback also indicates that the user will be able to
use this device for 5-6 hours a day.
Continuous monitoring of a posture position through the developed app makes
the device more used-friendly, comfortable, portable along with low-cost
compared to the other devices available in the market. The available devices
in the market cost about 150 USD - 600 USD, whereas the cost of SIPO is only
about 42 USD. The manufacturing cost can be reduced by mass production thus
making it an affordable solution for those who prefer to continuously monitor
their sitting posture and achieve a healthy spine.
Sensitivity and accuracy of the system can be upgraded in future by controlled
movement of bending during the experiment using more sophisticated camera and
automatic calculation of the bending angle. More data from a large population
having wide variation of height, weight, and age can be used to calibrate the
sensor value with the bending angle. Moreover, the apps can be updated by
adopting artificial intelligence and determine the initial set position and
threshold values based on the user inputs (weight, age, gender, and height).

##  V Conclusion

The primary goal of this project is to provide an effective solution to
maintain a healthy sitting posture that is user-friendly, wearable, safe,
inexpensive and non-invasive. The developed functional prototype demonstrates
a cost-effective way for real-time monitoring of a posture position. Due to
light-weight and portability, this wearable product is one of the best
solutions for the desk-workers, students and travelers who spend a significant
amount of time in sitting position.

##  VI Acknowledgment

This work has been conducted with the help of Department of Biomedical
Engineering, Bangladesh University of Engineering and Technology. The authors
would like to thank the volunteers for their cooperation. Special thanks to
Navid Ibtehaj Nizam, Assistant Professor, Department of Biomedical
Engineering, Bangladesh University of Engineering and Technology.

## References

  * [1] Y. Tanaka, “Epidemiology of low back pain,” _Clin. calcium_ , vol. 15, no. 3, pp. 35–38, 2005.
  * [2] S. I. Hay, A. A. Abajobir, K. H. Abate, C. Abbafati, K. M. Abbas, F. Abd-Allah, R. S. Abdulkader, A. M. Abdulle, T. A. Abebo, S. F. Abera _et al._ , “Global, regional, and national disability-adjusted life-years (DALYs) for 333 diseases and injuries and healthy life expectancy (HALE) for 195 countries and territories, 1990–2016: a systematic analysis for the global burden of disease study 2016,” _The Lancet_ , vol. 390, no. 10100, pp. 1260–1344, 2017.
  * [3] J. Akter, M. Kamrujjaman, and A. Maleque, “Prevalence and factors associated with low back pain among school teachers residing in northern dhaka city,” 2018\.
  * [4] R. Mondal, R. C. Sarker, S. Akter, P. C. Banik, and S. K. Baroi, “Prevalence of low back pain and its associated factors among physiotherapists in dhaka city of bangladesh in 2016,” _J. of Occupational Health and Epidemiology_ , vol. 7, no. 2, pp. 70–74, 2018.
  * [5] M. D. Hossain, A. Aftab, M. H. Al Imam, I. Mahmud, I. A. Chowdhury, R. I. Kabir, and M. Sarker, “Prevalence of work related musculoskeletal disorders (WMSDs) and ergonomic risk assessment among readymade garment workers of bangladesh: A cross sectional study,” _PloS one_ , vol. 13, no. 7, p. e0200122, 2018.
  * [6] _Southern California Orthopedic Institute - in Alliance with UCLA Health, Regents of The University of California_ , 2020 (accessed July 18, 2020). [Online]. Available: https://www.scoi.com/services/physical-therapy/importance-good-posture
  * [7] B. Koes, M. Van Tulder, and S. Thomas, “Diagnosis and treatment of low back pain,” _Bmj_ , vol. 332, no. 7555, pp. 1430–1434, 2006.
  * [8] J. Rajfur, M. Pasternok, K. Rajfur, K. Walewicz, B. Fras, B. Bolach, R. Dymarek, J. Rosinczuk, T. Halski, and J. Taradaj, “Efficacy of selected electrical therapies on chronic low back pain: a comparative clinical pilot study,” _Med. Sci. monitor: Int. Med. J. of Exp. and Clin. Res._ , vol. 23, p. 85, 2017.
  * [9] Z. Abdollahzade, A. Shadmehr, K. Malmir, and N. Ghotbi, “Effects of 4 week postural corrective exercise on correcting forward head posture,” _J. of Modern Rehabil._ , pp. 85–92, 2017.
  * [10] K.-J. Cho, Y.-T. Kim, S.-h. Shin, and S.-I. Suk, “Surgical treatment of adult degenerative scoliosis,” _Asian Spine J._ , vol. 8, no. 3, p. 371, 2014.
  * [11] H. Oh, S. Choi, S. Lee, J. Choi, and K. Lee, “The impact of manual spinal traction therapy on the pain and oswestry disability index of patients with chronic back pain,” _J. of Physical Therapy Sci._ , vol. 30, no. 12, pp. 1455–1457, 2018.
  * [12] _12 Benefits of Good Posture — and How to Maintain It_ , 2020 (accessed July 18, 2020). [Online]. Available: https://www.healthline.com/health/fitness-exercise/posture-benefits
  * [13] W. Xu, Z. Li, M.-C. Huang, N. Amini, and M. Sarrafzadeh, “ecushion: An etextile device for sitting posture monitoring,” in _2011 Int. Conf. on Body Sensor Netw._ IEEE, 2011, pp. 194–199.
  * [14] J. Roh, H.-j. Park, K. J. Lee, J. Hyeong, S. Kim, and B. Lee, “Sitting posture monitoring system based on a low-cost load cell using machine learning,” _Sensors_ , vol. 18, no. 1, p. 208, 2018.
  * [15] W. Y. Wong and M. S. Wong, “Detecting spinal posture change in sitting positions with tri-axial accelerometers,” _Gait & Posture_, vol. 27, no. 1, pp. 168–171, 2008.
  * [16] _Proper Sitting Posture at a Computer (According to Experts)-Ergonomic Trends_ , 2020 (accessed July 18, 2020). [Online]. Available: http://ergonomictrends.com/proper-sitting-posture-computer-experts/