# Effect of Dynamic Ions on Band Structure of Plasmon Excitations

M. Akbari-Moghanjoughi  Faculty of Sciences, Department of Physics, Azarbaijan
Shahid Madani University, 51745-406 Tabriz, Iran

###### Abstract

In this paper we develop a new method to study the plasmon energy band
structure in multispecies plasmas. Using this method, we investigate plasmon
dispersion band structure of different plasma systems with arbitrary
degenerate electron fluid. The linearized Schrödinger-Poisson model is used to
derive appropriate coupled pseudoforce system from which the energy dispersion
structure is calculated. It is shown that the introduction of ion mobility,
beyond the jellium (static ion) model with a wide plasmon energy band gap, can
fundamentally modify the plasmon dispersion character leading to a new form of
low-level energy band, due to the electron-ion band structure mixing. The
effects ionic of charge state and chemical potential of the electron fluid on
the plasmonic band structure indicate many new features and reveal the
fundamental role played by ions in the phonon assisted plasmon excitations in
the electron-ion plasma system. Moreover, our study reveals that ion charge
screening has a significant impact on the plasmon excitations in ion
containing plasmas. The energy band structure of pair plasmas confirm the
unique role of ions on the plasmon excitations in many all plasma
environments. Current research helps to better understand the underlying
mechanisms of collective excitations in charged environment and the important
role of heavy species on the elementary plasmon quasiparticles. The method
developed in this research may also be extended for complex multispecies and
magnetized quantum plasmas as well as to investigation the surface plasmon-
polariton interactions in nanometallic structures.

###### pacs:

52.30.-q,71.10.Ca, 05.30.-d

##  I Introduction

Theories of electron gas excitations are of primary importance in physical
properties of solids kit . Because of inertialess feature of electrons almost
all the electronic properties of matter is influenced by dielectric response
of the electron gas to external perturbations ichimaru1 ; ichimaru2 ;
ichimaru3 . These properties range from optical and mechanical to transport
phenomena wooten ; sarma ; stern2 ; gupta . In crystalline solids due to
complex nature of energy bands the electromagnetic response of the electrons
are fundamentally dependent on the energy dispersion and the structure of
energy bands in a specific direction hwang ; datta . One of the well-known
features caused by free electrons in metals is the existence of sharp plasmon
edge leading to their distinguished optical properties. Plasmons are
elementary collective excitations of electrons with many applications in newly
emerged interdisciplinary fields such as opto- and nanoelectronics haug ;
gardner , plasmonics man1 ; maier  and miniaturized integrated semiconductor
circuit industry markovich  etc. due to their ultrafast terahertz scale
electromagnetic response feature umm ; hu1 ; seeg . Plasmons also find
applications in optical surface engineering due to production of appealing
visual effects on nanocoated materials mlad . On the other hand, electrons
have the first role in many quantum effects in metals and semiconductors such
as the Landau quantization leading to the integer quantum Hall and de Haas–van
Alphen effects ash . Moreover, the role played by electrons in linear and
nonlinear wave phenomenon of complex plasmas is tremendous chen ; krall ;
drazin ; stenf1 ; stenf2 ; stenf3 ; stenf4 . Study of electronic properties of
solid within the free electron model has a limited applications due to the
fact that it ignores interactions between the crystal lattice and the electron
gas. Due to strong ionic coupling in solids, in the jellium model, ions
usually make the neutralizing positive background and do not contribute to the
electronic properties. However, the electrostatic potential of the ionic
lattice is the most important source of interaction between electrons and the
lattice potential. Moreover, manybody theories such as the density functional
theory (DFT) ethan ; white ; filinov ; bonitz  and Hartree-Fock perturbation
lev  take further step in accounting for interactions among electrons in
expense of considerable computational efforts fro ; birdsal ; allen .

In plasma theories, on the other hand, different plasma species are coupled
through the common electromagnetic fields that is caused by these species.
Therefore, the many body interaction effects is the main building block of the
plasma theories. However, in specific oscillation regimes the dynamics effects
due to one or more species can be ignored where these species are considered
as the jellium in the plasma model. For instance, in a plasma model of dust
acoustic excitations both electron and ion dynamics are ignored due to their
inertialess character as compared to dust fluid. Despite the long history of
pioneering developments madelung ; fermi ; hoyle ; chandra ; bohm ; bohm1 ;
bohm2 ; pines ; levine ; klimontovich , recent advancements in quantum kinetic
and (magneto)hydrodynamic theories man2 ; haasbook ; manfredi  have opened new
opportunities to explore the fields of complex dense plasmas with arbitrary
degree of electron degeneracy and ion coupling strength, ranging from strongly
coupled nanometallic compounds and warm dense matter (WDM) glenzer2 ; koenig
up to the astrophysical dense objects such as planetary cores and white dwarf
stars chandrasekhar1 . Application of quantum plasma theories have revealed a
large number of interesting new linear and nonlinear aspects of collective
interactions among different plasma species se ; sten ; ses ; haas1 ; brod1 ;
mark1 ; man3 ; mold2 ; sm ; fhaas ; scripta ; kim ; mannew ; hurst  and has
led do discovery of many new phenomena such as resonant shift and electron
spill-out giu , novel quantum screening seprl ; akbhd2 ; elak ; akbg  and
plasmon-soliton fer , to name a few. Application of linearized Schrödinger-
Poisson system for eigenvalue problem has shown many interesting new aspects
of quantum plasmas due to the unique dual lengthscale character of plasmon
excitations akbquant ; akbheat ; akbint ; akbnew ; akbdirac ; akbgap ;
akbtrans ; akbfano . In current research we would like to extend our previous
results of pseudoforce approach to the plasmon energy band structure in
complex plasma environments. This may lead to a better understanding of
collective effects on electron transport and plasmonic properties of the
fourth state of matter which constitutes at least $90$ percent of the visible
universe.

##  II The Theoretical Model

In order to investigate the energy dispersion relation for one-dimensional
plasmon excitations we use the linearized Schrödinger-Poisson system which is
cast into the following normalized coupled pseudoforce model representing the
eigenvalue problem for plasmon energies in an ensemble of arbitrary degenerate
electron gas in an ambient jellium-like positive background akbnew

$\displaystyle\frac{{d^{2}{\Psi(x)}}}{{d{x^{2}}}}+\Phi(x)+E\Psi(x)=0,$ (1a)
$\displaystyle\frac{{d^{2}{\Phi(x)}}}{{d{x^{2}}}}-{\Psi}(x)=0,$ (1b)

in which $E=(\epsilon-\mu_{0})/E_{p}$ with $E_{p}=\hbar\sqrt{4\pi
e^{2}n_{0}/m_{e}}$ where $n_{0}$ and $\mu_{0}$ are, respectively, the
equilibrium number density and chemical potential of the electron gas.
Moreover, $m_{e}$ is the electron mass and $\epsilon$ is the kinetic energy of
electrons in the band structure. Also, the normalized functions
$\Psi(x)=\psi/\sqrt{n_{0}}$ and $\Phi(x)=e\phi/E_{p}$ represent the local
probability density and electrostatic energy of the gas, so that,
$n(x)=\psi(x)\psi^{*}(x)$ is the local number density functional. Note that in
the Thomas-Fermi approximation the chemical potential remains constant
throughout the gas for linear perturbations. At thermal equilibrium for
isothermal processes at temperature $T$ the electron number density and
chemical potential are connected via a simple equation of state (EoS) elak

${n}=-2{\left({\frac{m_{e}}{{2\pi\beta{\hbar^{2}}}}}\right)^{3/2}}{\rm{L}}{{\rm{i}}_{3/2}}\left[{-{\rm{exp}}\left({\beta\mu_{0}}\right)}\right],\hskip
8.53581pt{P}=-\frac{N}{\beta}{\rm{L}}{{\rm{i}}_{5/2}}\left[{-{\rm{exp}}\left({\beta{\mu_{0}}}\right)}\right],$
(2)

in which $P$ is the statistical pressure satisfying the thermodynamic identity
$n=dP/d\mu_{0}$ and $\beta=1/k_{B}T$. The polylogarithm function
$\rm{Li_{\nu}}$ is given in terms of the Fermi functions

${\rm{Li}}_{\nu}(-{{\exp[z]}})=-\frac{1}{\Gamma(\nu)}\mathop{\smallint}\limits_{0}^{\infty}\frac{{{x^{\nu-1}}}}{{\exp(x-z)+1}}{\rm{d}}x,\hskip
8.53581pt\nu>0,$ (3)

where $\Gamma$ is the ordinary gamma function. The system (1) has simple
general solution discussed elsewhere akbquant . The linear plasmon dispersion
relation is obtained assuming $\Psi(x)=\Psi_{1}\exp(ikx)$ and
$\Phi(x)=\Phi_{1}\exp(ikx)$ which together with Eq. (1) leads to

$\left({\begin{array}[]{*{20}{c}}{E-{k^{2}}}&1\\\
{-1}&{-{k^{2}}}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{{\Psi_{1}}}\\\
{{\Phi_{1}}}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}0\\\
0\end{array}}\right).$ (4)

which leads to the simple plasmon dispersion $E=(1+k^{4})/k^{2}$ where $E$ and
$k$ are normalized to $E_{p}$ and $k_{p}=\sqrt{2m_{e}E_{p}}/\hbar$,
respectively. The system (1) can be generalized for particle of arbitrary mass
$M$ and charge $Z$

$\displaystyle\gamma\frac{{d^{2}{\Psi(x)}}}{{d{x^{2}}}}-Z\Phi(x)+E\Psi(x)=0,$
(5a) $\displaystyle\frac{{d^{2}{\Phi(x)}}}{{d{x^{2}}}}+Z{\Psi}(x)=0,$ (5b)

where $\gamma=m_{e}/M$ is the fractional mass. The system (5) the energy
dispersion relation $E=(Z^{2}+\gamma k^{4})/k^{2}$ which in the limit
$M=m_{e}$ and $Z=\pm 1$ reduces to the previous dispersion. It is remarked
that the plasmon energy dispersion of electron ($Z=-1$) and positron ($Z=+1$)
gases are identical. The plasmon energy dispersion has two distinct
lengthscales. For $k\gg 1$ it has particle-like character and reduces to
conventional parabolic form $E\simeq\gamma k^{2}$ while for $k\ll 1$ it has
wave-like character and it turns into $E\simeq Z^{2}/k^{2}$. The duallength
scale character of plasmon excitations, which is evidently due to both single-
particle and collective behavior, has shown to produce many fundamental
properties in an unmagnetized electron gas akbheat ; akbint ; akbnew ;
akbdirac ; akbgap ; akbtrans ; akbfano . The dispersion curve has a minimum
value at $k_{m}=\gamma^{1/4}\sqrt{Z}$ and $E_{m}=2Z\sqrt{\gamma}$ and for
$E>E_{m}$ has two characteristic wavenumbers

${k_{\pm}}=\sqrt{\frac{{E\pm\sqrt{{E^{2}}-4\gamma{Z^{2}}}}}{{2\gamma}}},$ (6)

in which $k_{+}$ and $k_{-}$ are particle- and wave-like branches satisfying
the complementarity-like relation $k_{+}k_{-}=|Z|$. For electron or positron
gas we simply have $E_{m}=k_{m}=1$ and $k_{+}=1/k_{-}$.

Figure 1: (a) Variations of electron plasmon energy in terms of the chemical
potential of arbitrary degenerate electron gas for different values of
electron gas temperature. (b) Variations in plasmon energy in terms of
equilibrium electron gas number density. (c) Parametric region in which the
thermal energy of electrons exceed twice the plasmon energy of the electron
gas. (d) Variations of electron number density versus the chemical potential
of the electron gas. The thickness of curves in plots are intended to
charcterize the increasing of the values in varied parameter above each panel.

In Fig. 1(a) we have shown the values of plasmon energies for different values
of the chemical potential of an arbitrary degenerate electron gas for
different value of electron gas temperature. In the completely degenerate gas,
$\mu_{0}\gg 1$, or in dilute classical limit, $\mu_{0}\ll-1$, the effect of
electron gas temperature on plasmon energy is insignificant. It is remarked
that with increase in the chemical potential for fixed temperature the plasmon
energy increases sharply. It is also seen that at room temperature the values
of plasmon energy vanishes for a classical electron gas where the chemical
potential becomes negative. This result is better depicted in Fig. 1(b) where
the variations of plasmon energy is compared in terms of the number density in
logarithmic scale. The plasmon energy increases sharply as the electron number
density increases beyond the degeneracy limit $n_{0}\simeq 10^{18}$cm-3.
Furthermore, Fig. 1(c) the region in temperature-chemical potential for which
the thermal energy in the gas is comparable to twice as much as the plasmon
energy (which is the plasmon gap studied in the following). It is remarked
that for dilute classical electron gas $\mu_{0}<0$ at low temperature there is
small ribbon-shaped area satisfying the condition. However, for degenerate
electron gas such as for semiconductors and metals the thermal energies never
reach the value $2E_{p}$ even for very large electron temperatures. The later
is because the increase in the temperature of the gas also increases its
plasmon energy. However, in the classical region $\mu_{0}<0$ there is region
of high temperature area in which $E_{T}\geq 2E_{p}$. Figure 1(d) shows the
variation of the chemical potential versus the electron number density for
different values of temperature. It ia again remarked that for fully
degenerate gas the effect of temperature becomes insignificant. However, for
the classical electron gas for a fixed chemical potential increase of the
temperature significantly increases the electron number density.

Figure 2: (a) The plasmon energy band struture of plasmon excitations electron
gas of arbitrary degeneracy in Thomas-Fermi approximation in jellium model.
(b) Variations in the electron plasmon band structure with changes in the
normalized chemical potential. (c) Plasmon dispersion structure for heavy
particle and the effects of particle charge state on the band structure. (d)
Effect of atomic number of ion species on the energy band structure of
neutralized ion fluid. The thickness of curves in plots are intended to
charcterize the increasing of the values in varied parameter above each panel.

Figure 2 shows the variations in plasmon energy dispersion curves for various
parameters. Figure 2(a) is the dispersion curve for electron gas with
arbitrary degree of degeneracy. The energy level $E=0$ corresponds to top of
the electron see at zero temperature ($\epsilon=\mu$). At the full degeneracy
limit ($\mu=\epsilon_{F}$ with $\epsilon_{F}$ being the Fermi energy of the
gas) all available electron energies are packed in the area $E\leq 0$ or
$\epsilon<\epsilon_{F}$ ($\epsilon_{F}\simeq 1.59$eV for cesium). The solid
curves indicate the main plasmon band and all excited plasmon energies reside
for $E>2$. The dashed curve, on the other hand, shows the free electron
dispersion. The origin of plasmon band gap is in fact a resonant quantum
scattering between free electron-like ($E=k^{2}$) propagations by their
collective excitation with the characteristic dispersion of $E=1/k^{2}$, shown
as dashed curves in Fig. 1(a). At low temperature where thermal energy of
electrons is low (room temperature thermal energy $\simeq 0.026$meV) only a
small fraction of electron in top of electron Fermi sea may attain kinetic
energies $\epsilon>2E_{p}$ ($E_{p}\simeq 2.9$eV for cesium) in order to reach
the plasmon conduction band. The photon frequencies to excite such electrons
must be at least twice that of the electron plasmon frequency. This small rate
of excitation take place in the form of electron-hole excitations. However, as
the temperature rises more and more electron reach the plasmon band and
plasmonic conduction may be possible. One important feature of the band
structure is the effective mass of electron at the specific scattering
wavenumber $k$. For excitations with $k\gg 1$ (for wavenumbers much larger
compared to plasmon wavenumber of the electron gas) the effective mass
$m_{b}=m^{*}/m_{e}=2(d^{2}E/dk^{2})^{-1}=k^{4}/(k^{4}+3)$ (in the normalized
form) which is definitely positive in the whole plasmon band approaches that
of the free particle limit, i.e. $m^{*}\simeq m_{e}$. On the other hand, for
$k\ll 1$ one obtains $m_{b}\to 0$. Therefore, to distinct regimes of fast
wave-like ($\omega>2\omega_{p}$ and $k<k_{p}$) and slow particle-like
($\omega>2\omega_{p}$ and $k>k_{p}$) electron response to external
perturbations are possible in the plasmon band. Note that in room temperature
there is a very small probability for plasmon excitations at the Fermi surface
of metals due to limiting plasmon occupation number factor
$f(E,\theta)=[\exp(2E/\theta)+1]^{-1}$ ath thermal equilibrium, where
$\theta=T/T_{p}$ with $T_{p}=E_{p}/k_{B}$ being the plasmon temperature.

Figure 2(b) shows the effect of chemical potential on the electron plasmon
dispersion. It is remarked that as the electron gas become dense at constant
temperature $E_{m}$ increases but $k_{m}$ does not change. For a classical
dilute electron gas the chemical potential can become negative and the plasmon
energy can have small values comparable to room temperature thermal energy in
which case the bottom of plasmon conduction band sink into the electron sea.
The later is the reason for why plasmon oscillations of electron gas are
dominant for classical rather than degenerate electron gas (with fixed
jellium-like positive background). Figure 2(c) shows plasmon dispersion curves
for classical gass of charged particle with atomic number $A$ ($\gamma\simeq
1/2Am_{p}$ with $m_{p}$ being the proton mass) and different charge values,
$Z$. The values of $E_{m}\ll 2E_{p}$ and $k_{m}\gg k_{p}$ are due to very
small $\gamma$ values of ions compared to the electron. The in the very high
wavenumber limit the dispersion approaches the particle-like branch $\gamma
k^{2}$. It is remarked that as the charge value increases $k_{m}$ decreases
but $E_{m}$ increases significantly. Such changes in the charge alters the
wave-like branch but does not significantly modify the particle-like branch of
the dispersion. It is concluded that the ions charge state significantly
affects the wave-like behavior of plasmon excitations. On the other hand, Fig.
2(d) shows the effect of changes in the atomic number of ionic gas on the
dispersion pattern. It is seen that in this case the effect is prounanced on
the particle-like branch while the wavelike branch stays almost intact. It is
also remarked that with in crease in the value of $A$, $k_{m}$ increases while
$E_{m}$ decreases as opposed to the variations in Fig. 2(c).

##  III Plasmons in Dynamic-Ion Environments

The model in Sec. III is known as the jellium model in which neutralizing
background ions are static. In this section we will explore how the ion
dynamics even negligible compared to that of electrons can fundamentally
modify the plasmon dispersion character in electron-ion plasma. To this end,
we consider the pseudoforce system

$\displaystyle\frac{{d^{2}{\Psi_{1}(x)}}}{{d{x^{2}}}}+\Phi(x)+E\Psi_{1}(x)=0,$
(7a)
$\displaystyle\gamma\frac{{{d^{2}}{\Psi_{2}}(x)}}{{d{x^{2}}}}-Z\Phi(x)+(E+\mu){\Psi_{2}}(x)=0,$
(7b)
$\displaystyle\frac{{d^{2}{\Phi(x)}}}{{d{x^{2}}}}-{\Psi_{1}}(x)+Z{\Psi_{2}}(x)=0,$
(7c)

where $\Psi_{1}$ and $\Psi_{2}$ respectively denote the probability density of
electrons and ions in the system and we assumed the ions are classical
$\mu_{i}\simeq 0$. Solving the system for plasmon eigenenergy values leads to
the following dispersion relation

$\begin{array}[]{*{20}{c}}{E_{\pm}=\frac{1}{{2{k^{2}}}}\left[{\lambda\pm\sqrt{{\lambda^{2}}+4{k^{4}}\left({{k^{4}}-\lambda-\gamma+1}\right)+4\mu{k^{2}}}}\right]}\\\
{\lambda=\left({1+\gamma}\right){k^{4}}-\mu{k^{2}}+{Z^{2}}+1},\end{array}$ (8)

It is evident that the dispersion relation has two distinct branches due to
dual species gas and reduces to dispersion relation for electron gas in
jellium background for $\gamma=Z=0$. The plasmon dispersion of electron-ion
plasma gives rise to upper and lower excitation bands.

Figure 3: (a) Plasmon energy band for electron-ion plasma which dynamic ions
and arbitrary degenerate electron gas. (b) Phonon assisted plasmon conduction
band in a wider scale for given plasma parameters. (c) Effect of dynamic ion
charge on the plasmon band structure of electron-ion plasma. (d) Effect of the
electron gas chemical potential on the energy dispersion on electron ion
plasmas. The thickness of curves in plots are intended to charcterize the
increasing of the values in varied parameter above each panel.

This is shown in Fig. 3(a) together with the parabolic free electron
dispersion $k2$ and ion plasmon dispersion $E=(Z^{2}+\gamma k^{4})/k^{2}-\mu$
shown as the dashed asymptotic curves. Note that in each plot the point $E=0$
denotes $\epsilon=\mu$ where $\epsilon$ is the normalized kinetic energy of
electrons and $E=-\mu$ denotes the zero kinetic energy point. It is remarkable
that introduction of mobile ions fundamentally alters the plasmon band
structure adding a lower band overlapping the electron sea. It is evident that
the new band originates due to the coupling of free electron and ion plasmon
excitations coupling (crossing of the dashed dispersion curves). Physically,
this is interpreted as the resonant scattering of electrons by the wavelike
branch of ion plasmon excitations. The latter effect is completely different
from the band gap structure caused by the resonant scattering of free
electrons by the lattice potential of static ions in crystalline materials.
Moreover, the main band is slightly lifted so that the height of the band gap
remains almost unchanged. The great resemblance of current band structure to
that in crystal lattices kit  is remarkable. The new band contains a valence-
like ($m_{b}<0$) structure around $k\simeq 1$ as well as two conduction like
($m_{b}>0$) valleys at $k<1.5$ and $k>1.5$. The lower energy band can have a
significant effect on low temperature variation of macroscopic quantities such
as the specific heat and current density. Note that electrons in valence- and
conduction-like band response oppositely to the external field due to the
relation $a=q{\cal E}/m_{e}$ where $a$ and ${\cal E}$ refer to the
acceleration and external electric field respectively. Note also that valence
bands contain $k$ ranges for which the plasmon group-speed
$v_{g}=(1/\hbar)dE/dk$ can be either positive or negative. Due to different
curvature values of the conduction-like valleys the electron at lower $k$
values respond faster compared to those in large $k$ plasmon conduction
electrons. It is also noted that for very low-wavenumber (very large
wavelength) plasmon excitations electrons almost feel free. Because the new
band sinks into the electron sea, the electrons need not to be thermally
excited in order to contribute collectively. However, there is zero-
temperature cut-off wavenumber range $0.5<k<3$ for plasmon excitations with
electron chemical potential value $\mu_{0}=\mu_{0}/2E_{p}=0.5$, $A=3$ and
$Z=2$. Figure 3(b) shows the plasmon dispersion band structure of electron-ion
plasma at a larger wavenumber scale. It is seen that the new energy band
approaches the particle-like branch of ion $E=\gamma k^{2}-\mu$ shown as
dashed curve in Fig. 3(b). Note that in large wavenumber plasmon excitation in
conduction band electrons effective mass equal nearly that of the ions. The
later phonon-coupled plasmon excitations are less effective in the electronic
properties of the plasma compared to the large wavelength plasmon excitations.
Figure 3(c) and 3(d) reveal the effects of ion charge state and elctron fluid
chemical potential on the new band structure in electron-ion plasma. It is
seen that with increase in the ion charge the wavenumber cut-off range
decreases and more and more electrons in the Fermi gas contribute to the
plasmon excitations at lower temperatures. Figure 3(d) on the other hand
reveals that for very low values of $\mu=0.1$ very few electrons contribute to
plasmon excitations while increase of $\mu$ (consequently the degree of
degeneracy) increases the portion of electrons and wavenumbers in the plasmon
excitations. The later is one of the fundamental feature of metal behavior
formation in solids which is beautifully illustrated in current plasmon band
structure of dense electron-ion plasmas.

##  IV Effect of Ion Charge Screening

Let us now study the effect of charge screening on the plasmon band structure
of electron-ion plasmas. In doing this we consider the following generalized
pseudoforce system

$\displaystyle\frac{{{d^{2}}\Psi_{1}(x)}}{{d{x^{2}}}}+2\xi\frac{{d\Psi_{1}(x)}}{{dx}}+\Phi(x)+{E}\Psi_{1}(x)=0,$
(9a)
$\displaystyle\gamma\frac{{{d^{2}}{\Psi_{2}}(x)}}{{d{x^{2}}}}+2\gamma\xi\frac{{d\Psi_{2}(x)}}{{dx}}-Z\Phi(x)+(E+\mu){\Psi_{2}}(x)=0,$
(9b)
$\displaystyle\frac{{{d^{2}}\Phi(x)}}{{d{x^{2}}}}+2\xi\frac{{d\Phi(x)}}{{dx}}-\Psi_{1}(x)+Z\Psi_{2}(x)=0,$
(9c)

where $\xi=k_{sc}/k_{p}$ with the normalized screening parameter
$\xi^{2}=({E_{p}}/2{n_{0}})\partial
n/\partial\mu=(1/2\theta){\rm{L}}{{\rm{i}}_{1/2}}\left[{-\exp({2\mu}/\theta)}\right]/{\rm{L}}{{\rm{i}}_{3/2}}\left[{-\exp(2{\mu}/\theta)}\right]$
being the one-dimensional screening wavenumber in the finite temperature
Thomas-Fermi model akbnew . The eigenenergy equation for the system (9) may be
found using the transformations $\Psi_{1}(x)=\psi_{1}(x)\exp(-\xi x)$,
$\Psi_{2}(x)=\psi_{2}(x)\exp(-\xi x)$ and $\Phi_{1}(x)=\phi_{1}(x)\exp(-\xi
x)$ and using $\psi_{1}(x)=\psi_{11}\exp(ikx)$,
$\psi_{2}(x)=\psi_{21}\exp(ikx)$ and $\phi(x)=\phi_{1}\exp(ikx)$. Then we find

$\left({\begin{array}[]{*{20}{c}}{E-{k^{2}}-{\xi^{2}}}&0&1\\\
0&{E+\mu-\gamma({k^{2}}+{\xi^{2}})}&{-Z}\\\
{-1}&Z&{-{k^{2}}-{\xi^{2}}}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{{\psi_{11}}}\\\
{{\psi_{21}}}\\\
{{\phi_{1}}}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}0\\\ 0\\\
0\end{array}}\right).$ (10)

The dispersion relation is then given as

$\begin{array}[]{*{20}{c}}{{E_{\pm}}=\frac{1}{{2{K^{2}}}}\left[{\Lambda\pm\sqrt{{\Lambda^{2}}+4{K^{4}}\left({{K^{4}}-\Lambda-\gamma+1}\right)+4\mu{K^{2}}}}\right]}\\\
\begin{array}[]{l}\Lambda=\left({1+\gamma}\right){K^{4}}-\mu{K^{2}}+{Z^{2}}+1,\\\
K=\sqrt{{k^{2}}+{\xi^{2}}}.\end{array}\end{array}$ (11)

It is obvious to find out that all the dispersion relations we have calculated
so far do not depend on the sign of the ion charge. The Eq. (11) reduce to
that of the unscreened jellium model for $\gamma=\xi=Z=0$. Note also that in
the limit $\xi=0$ we retain the dispersion for unscreened electron-ion plasma
(8). It is evident that there are two distinct bands in the dispersion
structure similar to the previous case.

Figure 4: (a) The plasmon energy band structure of electron-ion plasma with
screened ions. (b) The effect of normalized charge screening parameter on the
main plasmon conduction band for given plasma parameters. (c) Effect of
normalized charge screening parameter on lower plasmon conduction band for
high wavelength excitations. (d) Effect of normalized charge screening
parameter on lower plasmon conduction band for small wavelength excitations.
The thickness of curves in plots are intended to charcterize the increasing of
the values in varied parameter above each panel.

Figure 4 shows the energy dispersion of plasmon excitations in screened
electron-ion plasma. Figure 4(a) as compared to Fig. 3(a) reveals that
introduction of charge screening reduces electron population in small
wavenumber range of plasmon conduction band. The details of dispersion curve
variations for each band are shown in Figs. (c)-(d). It is seen from Fig. 4(b)
that the in screened plasma the wave-like branch in main dispersion band is
limited. It is also remarked from Fig. 4(c) that increase in the value of
screening parameter leads to shift of low wavenumber conduction valley to
higher values. It is obvious that such a shift significantly lowers the
plasmon states in the lower energy band which contributes the most to the
plasmon excitations in electron-ion plasmas. Moreover, Fig. 4(d) shows that
the charge screening has insignificant effect on the large wavenumber
conduction valley in the lower plasmon band.

##  V Plasmon Dispersion in Pair Plasmas

For the sake of completeness we would like to investigate the plasmon band
structure in pair plasmas. For instance in a electron-pair-ion plasma the
pseudoforce system reads

$\displaystyle\frac{{d^{2}{\Psi_{1}(x)}}}{{d{x^{2}}}}+\Phi(x)+E\Psi_{1}(x)=0,$
(12a)
$\displaystyle\gamma\frac{{{d^{2}}{\Psi_{2}}(x)}}{{d{x^{2}}}}-Z\Phi(x)+(E+\mu){\Psi_{2}}(x)=0,$
(12b)
$\displaystyle\gamma\frac{{{d^{2}}{\Psi_{3}}(x)}}{{d{x^{2}}}}+Z\Phi(x)+(E+\mu){\Psi_{3}}(x)=0,$
(12c)
$\displaystyle\frac{{d^{2}{\Phi(x)}}}{{d{x^{2}}}}-{\Psi_{1}}(x)+Z{\Psi_{2}}(x)-Z{\Psi_{3}}(x)=0.$
(12d)

The dispersion relation then reads

$\begin{array}[]{*{20}{c}}\begin{array}[]{l}{E_{0}}=\gamma{k^{2}}-\mu\\\
{E_{\pm}}=\frac{1}{{2{k^{2}}}}\left[{\lambda\pm\sqrt{{\lambda^{2}}-4{k^{4}}\left({\gamma{k^{4}}+2{Z^{2}}+\gamma}\right)+4{k^{2}}\left({1+{k^{4}}}\right)}}\right]\end{array}\\\
{\lambda=\left({1+\gamma}\right){k^{4}}-\mu{k^{2}}+2{Z^{2}}+1}.\end{array}$
(13)

It is evident that there are three bands corresponding to three species in the
plasma.

Figure 5: (a) Plasmon energy dispersion of pair-ion electron-ion plasmas. (b)
Plasmon dispersion structure of electron-hole pair with static ions in
intrinsic semiconductor plasmas. Plasmon excitation band structure of
electron-positron plasma (c) without and (d) with dynamic ion species.

The dispersion band structure for (13) is shown in Fig. 5(a). It is remarked
that the new band $E=\gamma k^{2}$ which is the parabolic dispersion of ion
now appears as the new wide band. Due to very large effective mass of
electrons associated with this new conduction band it does not significantly
contribute to electronic properties of electron plasmon excitations. Another
instance of pair plasma is the electron-hole plasma relevant to
semiconductors. The pseudoforce system reads

$\displaystyle\gamma_{e}\frac{{{d^{2}}{\Psi_{e}}(x)}}{{d{x^{2}}}}+\Phi(x)+E{\Psi_{e}}(x)=0,$
(14a)
$\displaystyle\gamma_{h}\frac{{{d^{2}}{\Psi_{h}}(x)}}{{d{x^{2}}}}-\Phi(x)+(E-E_{g}){\Psi_{h}}(x)=0,$
(14b)
$\displaystyle\frac{{d^{2}{\Phi(x)}}}{{d{x^{2}}}}-{\Psi_{e}}(x)+{\Psi_{h}}(x)=0,$
(14c)

where $\Psi_{e}$, $\Psi_{h}$ and $\Psi_{i}$ refer to probability density of
electron, hole and static ion species. Also, $\gamma_{e}=m_{e}/m^{*}_{e}$,
$\gamma_{h}=m_{e}/m^{*}_{h}$ and $E_{g}=\mu_{h}-\mu_{e}$ ae  is the normalized
(to $E_{p}$) gap energy of semiconductor ae . The characteristic eigenenergy
equation in this case follows

$\left({\begin{array}[]{*{20}{c}}{E-{\gamma_{e}}{k^{2}}}&0&1\\\
0&{E-{E_{g}}-{\gamma_{h}}{k^{2}}}&{-1}\\\
{-1}&1&{-{k^{2}}}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{{\psi_{e1}}}\\\
{{\psi_{h1}}}\\\
{{\phi_{1}}}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}0\\\ 0\\\
0\end{array}}\right).$ (15)

Note that the ions have chosen to be static in this case. The corresponding
plasmon energy dispersion reads

$\begin{array}[]{*{20}{c}}{{E_{\pm}}=\frac{1}{{2{k^{2}}}}\left\\{{\lambda\pm\sqrt{{{\left[{{E_{g}}+\left({{\gamma_{h}}-{\gamma_{e}}}\right){k^{2}}}\right]}^{2}}{k^{4}}+4}}\right\\}},\\\
{\lambda=\left({{\gamma_{e}}+{\gamma_{h}}}\right){k^{4}}+{E_{g}}{k^{2}}+2},\end{array}$
(16)

where $\beta_{e}=0.26=1/\gamma_{e}$ and $\beta_{h}=0.39=1/\gamma_{h}$ are the
effective-mass ratios for electrons and holes in the silicon semiconductor,
respectively, which we used in Fig. 5(b). The dispersion curve is shown in the
figure for an electron doped silicon with room temperature number-density of
$n_{0}\simeq 5\times 10^{17}$cm-3 ($E_{p}\simeq 27$meV slightly larger than
the electron thermal energy) consisting of two distinct branches. Note that
the energy values shown above the plot are normalized to plasmon energy. In
fact the unscaled chemical potential value for this density is
$\mu_{e}\simeq-0.1$eV and the unscaled gap energy is $\epsilon_{g}\simeq
1.12$eV. The valence-like plasmon band is not present in this case. It is seen
that thermal excitations are needed to excite electron to plasmon conduction
bands. Furthermore, it is found that the upper plasmon band is far higher to
be reached by electron sea via thermal excitations. The lower conduction band
can be populated mostly with electrons by small thermal agitations so that
doped silicon plasma is N-type in this case. The pool of electron holes is in
negative side of the axis and extends to $E=E_{g}+\mu_{e}$ with $E=0$ denoting
the Fermi energy level which is closer to the conduction band for N-type
semiconductors, as is the case. For the electron-positron plasma we have

$\displaystyle\frac{{{d^{2}}{\Psi_{e}}(x)}}{{d{x^{2}}}}+\Phi(x)+E{\Psi_{e}}(x)=0,$
(17a)
$\displaystyle\frac{{{d^{2}}{\Psi_{p}}(x)}}{{d{x^{2}}}}-\Phi(x)+(E+2\mu_{e}){\Psi_{p}}(x)=0,$
(17b)
$\displaystyle\frac{{d^{2}{\Phi(x)}}}{{d{x^{2}}}}-{\Psi_{e}}(x)+{\Psi_{p}}(x)=0,$
(17c)

where we have used $\mu_{e}+\mu_{p}\simeq 0$. This leads to the dispersion
relation

${E_{\pm}}=({k^{4}}\pm\sqrt{{\mu_{e}^{2}}{k^{4}}+1}+1)/{k^{2}}.$ (18)

The plot of dispersion branches shows that in electron-positron pair plasma
there is significant overlap between the electron sea and the small wavenumber
band available for electron-positron plasmon excitations. This pair plasma is
zero band gap plasmonic system with nearly a free electron parabolic
dispersion conduction band. In the following it may be illustrative to
consider a more realistic electron-positron pair plasma in the presence of a
dynamic ion species. The pseudoforce system in this case is

$\displaystyle\frac{{{d^{2}}{\Psi_{e}}(x)}}{{d{x^{2}}}}+\Phi(x)+E{\Psi_{e}}(x)=0,$
(19a)
$\displaystyle\frac{{{d^{2}}{\Psi_{p}}(x)}}{{d{x^{2}}}}-\Phi(x)+(E+2\mu_{e}){\Psi_{p}}(x)=0,$
(19b)
$\displaystyle\gamma\frac{{{d^{2}}{\Psi_{i}}(x)}}{{d{x^{2}}}}-Z\Phi(x)+(E+\mu_{e}){\Psi_{i}}(x)=0,$
(19c)
$\displaystyle\frac{{d^{2}{\Phi(x)}}}{{d{x^{2}}}}-{\Psi_{e}}(x)+{\Psi_{p}}(x)+Z{\Psi_{i}}(x)=0,$
(19d)

with a long expression for the dispersion solution which is avoided here for
simplicity. The characteristic eigenenergy equation in this case follows

$\left({\begin{array}[]{*{20}{c}}{E-{k^{2}}}&0&0&1\\\
0&{E-{k^{2}}+2{\mu_{e}}}&0&{-1}\\\ 0&0&{E-\gamma{k^{2}}+{\mu_{e}}}&{-Z}\\\
{-1}&1&Z&{-{k^{2}}}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{{\psi_{e1}}}\\\
{{\psi_{p1}}}\\\ {{\psi_{i1}}}\\\
{{\phi_{1}}}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}0\\\ 0\\\ 0\\\
0\end{array}}\right).$ (20)

The dispersion curve is shown in Fig. 5(d). The complex structure shows
multiple band structure with the superposition of asymptotic previously
studied dispersion curves, namely, the electron-positron (Fig. 5(c)) and ion
(Fig. 3(a)). It is remarked that introduction of dynamic ions in this case has
led to double conduction band in the middle and has significantly increased
the electron population in conduction band. Moreover, small wavelength
excitations appear naturally as phonon assisted plasmon excitation in this
system.

##  VI Conclusion

In this research we developed a new theory of plasmon band structure in
plasmas. We used the linearized Schrödinger-Poisson model and the pseudoforce
method to study the energy band structure of plasmon excitations in different
plasmas with arbitrary degree of degeneracy. Our study reveals the fundamental
role played by dynamic ions on the band structure and available energy levels
for electrons in the conduction bands. Many valence and conduction-like
structure is found to be present in the band structure of the electron-ion
plasma with fascinating resemblance to band structure of solids. We also
investigated the effect of charge screening effect on the energy band and
revealed its many important impact on quantum dense plasmas. The significant
effects of ion charge state and electron chemical potential on the plasmon
excitations was revealed and band structure of different pair plasmas were
studied as well.

##  VII Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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11institutetext: IDLab, Department of Information Technology
Ghent University - imec
Ghent, Belgium
11email<EMAIL_ADDRESS>
# Disentangling Shape and Pose for Object-Centric Deep Active Inference Models

Stefano Ferraro  Toon Van de Maele  Pietro Mazzaglia
Tim Verbelen  Bart Dhoedt

###### Abstract

Active inference is a first principles approach for understanding the brain in
particular, and sentient agents in general, with the single imperative of
minimizing free energy. As such, it provides a computational account for
modelling artificial intelligent agents, by defining the agent’s generative
model and inferring the model parameters, actions and hidden state beliefs.
However, the exact specification of the generative model and the hidden state
space structure is left to the experimenter, whose design choices influence
the resulting behaviour of the agent. Recently, deep learning methods have
been proposed to learn a hidden state space structure purely from data,
alleviating the experimenter from this tedious design task, but resulting in
an entangled, non-interpreteable state space. In this paper, we hypothesize
that such a learnt, entangled state space does not necessarily yield the best
model in terms of free energy, and that enforcing different factors in the
state space can yield a lower model complexity. In particular, we consider the
problem of 3D object representation, and focus on different instances of the
ShapeNet dataset. We propose a model that factorizes object shape, pose and
category, while still learning a representation for each factor using a deep
neural network. We show that models, with best disentanglement properties,
perform best when adopted by an active agent in reaching preferred
observations.

###### Keywords:

Active Inference Object Perception Deep Learning Disentanglement.

##  1 Introduction

In our daily lives, we manipulate and interact with hundreds of objects
without even thinking. In doing so, we make inferences about an object’s
identity, location in space, 3D structure, look and feel. In short, we learn a
generative model of how objects come about [24]. Robots however still lack
this kind of intuition, and struggle to consistently manipulate a wide variety
of objects [2]. Therefore, in this work, we focus on building object-centric
generative models to equip robots with the ability to reason about shape and
pose of different object categories, and generalize to novel instances of
these categories.

Active inference offers a first principles approach for learning and acting
using a generative model, by minimizing (expected) free energy. Recently, deep
learning techniques were proposed to learn such generative models from high
dimensional sensor data [33, 7, 27], which paves the way to more complex
application areas such as robot perception [14]. In particular, Van de Maele
et al. [16, 18] introduced object-centric, deep active inference models that
enable an agent to infer the pose and identity of a particular object
instance. However, this model was restricted to identify unique object
instances, i.e. “this sugar box versus that particular tomato soup can”,
instead of more general object categories, i.e. “mugs versus bottles”. This
severely limits generalization, as it requires to learn a novel model for each
particular object instance, i.e. for each particular mug.

In this paper, we further extend upon this line of work, by learning object-
centric models not by object instance, but by object category. This allows the
agent to reduce the number of required object-centric models, as well as to
generalize to novel instances of known object categories. Of course, this
requires the agent to not only infer object pose and identity, but also the
different shapes that comprise this category. An important research question
is then how to define and factorize the generative model, i.e. do we need to
explicitly split the different latent factors in our model (i.e. shape and
pose), or can a latent structure be learnt purely from data, and to what
extent is this learnt latent structure factorized?

In the brain, there is also evidence for disentangled representations. For
instance, processing visual inputs in primates consists of two pathways: the
ventral or “what” pathway, which is involved with object identification and
recognition, and the dorsal or “where” pathway, which processes an object’s
spatial location [22]. Similarly, Hawkins et al. hypothesize that cortical
columns in the neocortex represent an object model, capturing their pose in a
local reference frame, encoded by cortical grid cells [8]. This fuels the idea
of treating object pose as a first class citizen when learning an object-
centric generative model.

In this paper, we present a novel method for learning object-centric models
for distinct object categories, that promotes a disentangled representation
for shape and pose. We demonstrate how such models can be used for inferring
actions that move an agent towards a preferred observation. We show that a
better pose-shape disentanglement indeed seems to improve performance, yet
further research in this direction is required. In the remainder of the paper
we first give an overview on related work, after which we present our method.
We present some results on object categories of the ShapeNet database [3], and
conclude the paper with a thorough discussion.

##  2 Related work

Object-centric models. Many techniques have been proposed for representing 3D
objects using deep neural networks, working with 2D renders [5], 3D voxel
representations [32], point clouds [15] or implicit signed distance function
representations [23, 20, 21, 28]. However, none of these take “action” into
account, i.e. there is no agent that can pick its next viewpoint.

Disentangled representations. Disentangling the hidden factors of variation of
a dataset is an long sought feature for representation learning [1]. This can
be encouraged during training by restricting the capacity of the information
bottleneck [9], by penalizing the total correlation of the latent variables
[11, 4], or by matching moments of a factorized prior [13]. It has been shown
that disentangled representations yield better performance on down-stream
tasks, enabling quicker learning using fewer examples [29].

Deep active inference. Parameterizing generative models using deep neural
networks for active inference has been coined “deep active inference” [30].
This enables active inference applications on high-dimensional observations
such as pixel inputs [33, 7, 27]. In this paper, we propose a novel model
which encourages a disentangled latent space, and we compare with other deep
active inference models such as [33] and [17]. For a more extensive review,
see [19].

##  3 Object-Centric Deep Active Inference Models

In active inference, an agent acts and learns in order to minimize an upper
bound on the negative log evidence of its observations, given its generative
model of the world i.e. the free energy. In this section, we first formally
introduce the different generative models considered for our agents for
representing 3D objects. Next we discuss how we instantiate and train these
generative models using deep neural networks, and how we encourage the model
to disentangle shape and pose.

Figure 1: Different generative models for object-centric representations, blue
nodes are observed. (a) A generic POMDP model with a hidden state $s_{t}$ that
is transitioned through actions and which generates the observations. (b) The
hidden state $s$ encodes the appearance of the object, while actions
transition the camera viewpoint $v$ which is assumed to be observable. (c)
Similar as (b), but without access to the camera viewpoint, which in this case
has to be inferred as a separate pose latent variable $p_{t}$.

Generative model. We consider the same setup as [18], in which an agent
receives pixel observations $o$ of a 3D object rendered from a certain camera
viewpoint $v$, and as an action $a$ can move the camera to a novel viewpoint.
The action space is restricted to viewpoints that look at the object, such
that the object is always in the center of the observation.

Figure 1 depicts different possible choices of generative model to equip the
agent with. The first (1a) considers a generic partially observable Markov
decision process (POMDP), in which a hidden state $s_{t}$ encodes all
information at timestep $t$ to generate observation $o_{t}$. Action $a_{t}$
determines together with the current state $s_{t}$ how the model transitions
to a new state $s_{t+1}$. This a model can be implemented as a variational
autoencoder (VAE) [25, 12], as shown in [33, 7]. A second option (1b) is to
exploit the environment setup, and assume we can also observe the camera
viewpoint $v_{t}$. Now the agent needs to infer the object shape $s$ which
stays fixed over time. This resembles the architecture of a generative query
network (GQN), which is trained to predict novel viewpoints of a given a scene
[6, 17]. Finally, in (1c), we propose our model, in which we have the same
structure as (1b), but without access to the ground truth viewpoint. In this
case, the model needs to learn a hidden latent representation of the object
pose in view $p_{t}$. This also allows the model to learn a different pose
representation than a 3D pose in SO(3), which might be more suited. We call
this model a VAEsp, as it is trained in similar vein as (1a), but with a
disentangled shape and pose latent.

Figure 2: The proposed VAEsp architecture consists of three deep neural
networks: an encoder $q_{\phi}$, a transition model $p_{\chi}$, and a decoder
$p_{\psi}$. By swapping the shape latent samples, we enforce the model to
disentangle shape and pose during training.

VAEsp. Our model is parameterized by three deep neural networks: an encoder
$q_{\phi}$, a transition model $p_{\chi}$, and a decoder $p_{\psi}$, as shown
in Figure 2. Observations $o^{i}$ of object instance $i$ are processed by the
encoder $q_{\phi}$, that outputs a belief over a pose latent
$q_{\phi}(p^{i}_{t}|o^{i}_{t})$ and a shape latent
$q_{\phi}(s^{i}_{t}|o^{i}_{t})$. From the pose distribution a sample
$p^{i}_{t}$ is drawn and fed to the transition model $p_{\chi}$, paired with
an action $a_{t}$. The output is a belief
$p_{\chi}(p^{i}_{t+1}|p^{i}_{t},a_{t})$. From the transitioned belief a sample
$p^{i}_{t+1}$ is again drawn which is paired with a shape latent sample
$s^{i}$ and input to the decoder $p_{\psi}(o^{i}_{t}|p^{i}_{t},s^{i})$. The
output of the decoding process is again an image $\hat{o}^{i}_{t+1}$. These
models are jointly trained end-to-end by minimizing free energy, or
equivalently, maximizing the evidence lower bound [18]. More details on the
model architecture and training hyperparameters can be found in Appendix 0.A.

Enforcing disentanglement. In order to encourage the model to encode object
shape features in the shape latent, while encoding object pose in the pose
latent, we only offer the pose latent $p_{t}$ as input to the transition
model, whereas the decoder uses both the shape and pose. Similar to [10], in
order to further disentangle, we randomly swap the shape latent code for two
object instances at train time while keeping the same latent pose, refer to
Figure 2.

##  4 Experiments

We train our model on a subset of the ShapeNet dataset [3]. In particular, we
use renders of 15 instances of the ‘mug’, ‘bottle’,‘bowl’ and ‘can’
categories, train a separate model for each category, and evaluate on unseen
object instances. We compare our VAEsp approach against a VAE model [33] that
has equal amount of latent dimensions, but without a shape and pose split, and
a GQN-like model [17], which has access to the ground truth camera viewpoint.

We evaluate the performance of the three considered generative models. First
we look at the reconstruction and prediction quality of the models for unseen
object instances. Next we investigate how good an agent can move the camera to
match a preferred observation by minimizing expected free energy. Finally, we
investigate the disentanglement of the resulting latent space.

Table 1: One-step prediction errors, averaged over the entire test set. MSE (lower the better) and SSIM (higher the better) are considered.  |  | bottle | bowl | can | mug
---|---|---|---|---|---
MSE $\Downarrow$ | GQN | $0.473\pm 0.0874$ | $0.487\pm 0.141$ | $0.707\pm 0.1029$ | $0.656\pm 0.0918$
| VAE | ${0.471\pm 0.0824}$ | $0.486\pm 0.1487$ | ${0.693\pm 0.1103}$ | $0.646\pm 0.0886$
| VAEsp | $0.480\pm 0.0879$ | ${0.485\pm 0.1486}$ | $0.702\pm 0.1108$ | ${0.626\pm 0.0915}$
SSIM $\Uparrow$ | GQN | $0.748\pm 0.0428$ | $0.814\pm 0.0233$ | $0.868\pm 0.0203$ | $0.824\pm 0.0279$
| VAE | $0.828\pm 0.0238$ | $\mathbf{0.907\pm 0.0178}$ | $0.844\pm 0.0361$ | $\mathbf{0.874\pm 0.0323}$
| VAEsp | $\mathbf{0.854\pm 0.0190}$ | $0.902\pm 0.0291$ | $\mathbf{0.880\pm 0.0176}$ | $0.814\pm 0.0348$

One-step Prediction. First, we evaluate all models on prediction quality over
a test set of 500 observations of unseen objects in unseen poses. We provide
each model with an initial observation which is encoded into a latent state.
Next we sample a random action, predict the next latent state using the
transition model, for which we reconstruct the observation and compare with a
ground truth. We report both pixel-wise mean squared error (MSE) and
structural similarity (SSIM) [31] in Table 1. In terms of MSE results are
comparable for all the proposed architectures. In terms of SSIM however, VAEsp
shows better performance for ‘bottle’ and ‘can’ category. Performance for
‘bowl’ category are comparable to the best performing VAE model. For the ‘mug’
category, the negative gap over the VAE model is consistent. Qualitative
results for all models are shown in Appendix 0.B.

Table 2: MSE for the reached pose through the minimization of expected free energy. For each category, 50 meshes are evaluated, where for each object a random pose is sampled from a different object as preferred pose, and the agent should reach this pose. | bottle | bowl | can | mug
---|---|---|---|---
GQN | $0.0833\pm 0.0580$ | $0.0888\pm 0.0594$ | $0.0806\pm 0.0547$ | $0.1250\pm 0.0681$
VAE | $0.0698\pm 0.0564$ | $\mathbf{0.0795\pm 0.0599}$ | $0.0608\pm 0.0560$ | $0.1247\pm 0.0656$
VAEsp | $\mathbf{0.0557\pm 0.0404}$ | $0.0799\pm 0.0737$ | $\mathbf{0.0487\pm 0.0381}$ | $\mathbf{0.1212\pm 0.0572}$

Reaching preferred viewpoints. Next, we consider an active agent that is
tasked to reach a preferred observation that was provided in advance. To do
so, the agent uses the generative model to encode both the preferred and
initial observation and then uses Monte Carlo sampling to evaluate the
expected free energy for 10000 potential actions, after which the action with
the lowest expected free energy is executed. The expected free energy
formulation is computed as the negative log probability of the latent
representation with respect to the distribution over the preferred state,
acquired through encoding the preferred observation. This is similar to the
setup adopted by Van de Maele et al. [18], with the important difference that
now the preferred observation is an image of a different object instance.

[] []

Figure 3: Two examples of the experiment on reaching preferred viewpoints for
a ‘bottle’ (a) and a ‘mug’ (b). First column shows the target view (top) and
initial view given to the agent (bottom). Next, for the three models we show
the actual reached view (top), versus the imagined expected view of the model
(bottom).

To evaluate the performance, we compute the pixel-wise mean squared error
(MSE) between a render of the target object in the preferred pose, and the
render of the environment after executing the chosen action after the initial
observation. The results are shown in Table 2. VAEsp performs on par with the
other approaches for ‘bowl’ and ‘mug’, but significantly outperforms the GQN
on the the ‘bottle’ and ‘can’ categories, reflected by p-values of $0.009$ and
$0.001$ for these respective objects. The p-values for the comparison with the
VAE are $0.167$ and $0.220$, which are not significant. A qualitative
evaluation is shown in Figure 3. Here we show the preferred target view, the
initial view of the environment, as well as the final views reached by each of
the agents, as well as what each model was imagining. Despite the target view
being from a different object instance, the agent is able to find a matching
viewpoint.

Disentangled latent space. Finally, we evaluate the disentanglement of shape
and pose for the proposed architecture. Given that our VAEsp model outperforms
the other models on ‘bottle‘ and ‘can‘, but not on ‘bowl‘ and ‘mug‘, we
hypothesize that our model is able to better disentangle shape and pose for
the first categories, but not for the latter. To evaluate this, we plot the
distribution of each latent dimension when encoding 50 random shapes in a
fixed pose, versus 50 random poses for a fixed shape, as shown on Figure 4. We
see that indeed the VAEsp model has a much more disentangled latent space for
‘bottle‘ compared to ‘mug‘, which supports our hypothesis. Hence, it will be
interesting to further experiment to find a correlation between latent space
disentanglement and model performance. Moreover, we could work on even better
enforcing disentanglement when training a VAEsp model, for example by adding
additional regularization losses [11, 4]. Also note that the GQN does not
outperform the other models, although this one has access to the ground truth
pose factor. This might be due to the fact that an SO(3) representation of
pose is not optimal for the model to process, and it still encodes (entangled)
pose information in the resulting latent space, as illustrated by violin plots
for GQN models in Appendix 0.C. Figure 5 qualitatively illustrates the shape
and pose disentanglement for our best performing model (bottle). We plot
reconstructions of latent codes consisting of the shape latent of the first
column, combined with the pose latent of the first row.

[VAEsp bottle]
[VAEsp mug]

Figure 4: Violin plots representing the distribution over the latent dimension
when keeping either the pose or shape fixed. For the bottle model (a) the pose
latent dimensions (0-7) vary when only varying the pose, whereas the shape
latent dimensions (8-23) don’t vary with the pose. For the mug model (b) we
see the shape and pose latent are much more entangled. Figure 5: Qualitative
experimentation for the bottle category. Images are reconstructed from the
different pairings of the pose latent and shape latent of the first row and
column respectively.

##  5 Conclusion

In this paper, we proposed a novel deep active inference model for learning
object-centric representations of object categories. In particular, we
encourage the model to have a disentangled pose and shape latent code. We show
that the better our model disentangles shape and pose, the better the results
are on prediction, reconstruction as well as action selection towards a
preferred observation. As future work, we will further our study on the impact
of disentanglement, and how to better enforce disentanglement in our model. We
believe that this line of work is important for robotic manipulation tasks,
i.e. where a robot learns to pick up a cup by the handle, and can then
generalize to pick up any cup by reaching to the handle.

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##  Appendix 0.A Model and training details

This paper compares three generative models for representing the shape and
pose of an object. Each of the models has a latent distribution of $24$
dimensions, parameterized as a Gaussian distribution and has a similar amount
of total trainable parameters.

VAE: The VAE baseline is a traditional variational autoencoder. The encoder
consists of 6 convolutional layers with a kernel size of 3, a stride of 2 and
padding of 1. The features for each layer are doubled every time, starting
with 4 for the first layer. After each convolution, a LeakyReLU activation
function is applied to the data. Finally, two linear layers are used on the
flattened output from the convolutional pipeline, to directly predict the mean
and log variance of the latent distribution. The decoder architecture is a
mirrored version of the encoder. It consists of 6 convolutional layers with
kernel size 3, padding 1 and stride 1. The layers have 32, 8, 16, 32 and 64
output features respectively. After each layer the LeakyReLU activation
function is applied. The data is doubled in spatial resolution before each
such layer through bi-linear upsampling, yielding a 120 by 120 image as final
output. A transition model is used to predict the expected latent after
applying an action. This model is parameterized through a fully connected
neural network, consisting of three linear layers, where the output features
are 64, 128 and 128 respectively. The input is the concatenation of a latent
sample, and a 7D representation of the action (coordinate and orientation
quaternion). The output of this layer is then again through two linear layers
transformed in the predicted mean and log variance of the latent distribution.
This model has $474.737$ trainable parameters.

GQN: The GQN baseline only consists of an encoder and a decoder. As the model
is conditioned on the absolute pose of the next viewpoint, there is no need
for a transition model. The encoder is parameterized exactly the same as the
encoder of the VAE baseline. The decoder is now conditioned on both a latent
sample and the 7D representation of the absolute viewpoint (coordinate and
orientation quaternion). These are first concatenated and transformed through
a linear layer with 128 output features. This is then used as a latent code
for the decoder, which is parameterized the same as the decoder used in the
VAE baseline. In total, the GQN has $361.281$ trainable parameters.

VAEsp: Similar to the VAE baseline, the VAEsp consists of an encoder, decoder
and transition model. The encoder is also a convolutional neural network,
parameterized the same as the encoder of the VAE, except that instead of two
linear layers predicting the parameters of the latent distribution, this model
contains 4 linear layers. Two linear layers with 16 output features are used
to predict the mean and log variance of the shape latent distribution, and two
linear layers with 8 output features are used to predict the mean and log
variance of the pose latent distribution. In the decoder, a sample from the
pose and shape latent distributions are concatenated and decoded through a
convolutional neural network, parameterized exactly the same as the decoder
from the VAE baseline. The transition model, only transitions the pose latent,
as we make the assumption that the object shape does not change over time. The
transition model is parameterized the same as the transition model of the VAE,
with the exception that the input is the concatenation of the 8D pose latent
vector and the 7D action, in contrast to the 24D latent in the VAE. The VAEsp
model has $464.449$ trainable parameters.

All models are trained using a constrained loss, where Lagrangian optimizers
are used to weigh the separate terms [26]. During training, we tuned the
reconstruction tolerance for each object empirically. Respectively to
’bottle’, ’bowl’, ’can’ and ’mug’ categories, MSE tolerances are: 350, 250,
280 and 520. Regularization terms are considered for each latent element. For
all models, the Adam optimizer was used to minimize the objective.

##  Appendix 0.B Additional qualitative results

[bottle] [bowl]
[can] [mug]

Figure 6: One-step prediction for different object categories.

##  Appendix 0.C Latent disentanglement

In Figures 7, 8, 9 and 10, we show the distribution over the latent values
when encoding observation where a single input feature changes. The blue
violin plots represent the distribution over the latent values for
observations where the shape is kept fixed, and renders from different poses
are fed through the encoder. The orange violin plots represent the
distribution over the latent values for observations where the pose is kept
fixed, and renders from different shapes within the object class are encoded
through the encoder models.

In these figures, we can clearly see that the encoding learnt by the VAE is
not disentangled for any of the objects as the latent dimensions vary for both
the fixed shape and pose cases. With the GQN, we would expect that the latent
dimensions would remain static for the fixed shape case, as the pose is an
explicit external signal for the decoder, however we can see that for a fixed
shape, the variation over the latent value still varies a lot, in similar
fashion as for the fixed pose. We conclude that the encoding of the GQN is
also not disentangled. For the VAEsp model, we can see that in Figures 7 and
8, the first eight dimensions are used for the encoding of the pose, as the
orange violins are much denser distributed for the fixed pose case. However,
in Figures 9 and 10, we see that the model still shows a lot of variety for
the latent codes describing the non-varying feature of the input. This result
also strokes with our other experiments where for these objects both
reconstruction as well as the move to perform worse.

In this paper, we investigated the disentanglement for the different
considered object classes. We see that our approach does not yield a
disentangled representation each time. Further investigation and research will
focus on better enforcing this disentanglement.

[VAE bottle]
[GQN bottle]
[VAEsp bottle]

Figure 7: Distribution of the latent values for the different models (VAE, GQN
and VAEsp) for objects from the “bottle” class. In this experiment, 50 renders
from a fixed object shape with a varying pose (fixed shape, marked in blue)
are encoded. The orange violin plots represent the distribution over the
latent values for 50 renders from the same object pose, with a varying object
shape.

[VAE can]
[GQN can]
[VAEsp can]

Figure 8: Distribution of the latent values for the different models (VAE, GQN
and VAEsp) for objects from the “can” class. In this experiment, 50 renders
from a fixed object shape with a varying pose (fixed shape, marked in blue)
are encoded. The orange violin plots represent the distribution over the
latent values for 50 renders from the same object pose, with a varying object
shape.

[VAE mug]
[GQN mug]
[VAEsp mug]

Figure 9: Distribution of the latent values for the different models (VAE, GQN
and VAEsp) for objects from the “mug” class. In this experiment, 50 renders
from a fixed object shape with a varying pose (fixed shape, marked in blue)
are encoded. The orange violin plots represent the distribution over the
latent values for 50 renders from the same object pose, with a varying object
shape.

[VAE bowl]
[GQN bowl]
[VAEsp bowl]

Figure 10: Distribution of the latent values for the different models (VAE,
GQN and VAEsp) for objects from the “bowl” class. In this experiment, 50
renders from a fixed object shape with a varying pose (fixed shape, marked in
blue) are encoded. The orange violin plots represent the distribution over the
latent values for 50 renders from the same object pose, with a varying object
shape.