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# Hydrodynamic modes in holographic multiple-axion model

Ling-Zheng Xia1 Wei-Jia Li1 1 Institute of Theoretical Physics, School of
Physics, Dalian University of Technology, Dalian 116024, China.

###### Abstract

In this paper we investigate the shear viscoelasticity and the hydrodynamic
modes in a holographic solid model with several sets of axions that all break
the translations spontaneously on boundary. Comparing with the single-axion
model, the shear modulus is enhanced at high temperatures and the shear
viscosity is always suppressed in the presence of additional axions. However,
the different sets of axions exhibit competitive relationship in determining
the shear modulus at low temperatures. Furthermore, by calculating the black
hole quasi-normal modes, it is found that adding more axions only increases
the amount of diffusive modes. The number of the sound modes always remains
unchanged.

###### Contents

  1. I Introduction
  2. II Holographic action and black hole solution
  3. III Shear viscoelasticity
  4. IV Dispersion relations of hydrodynamic modes
    1. IV.1 Transverse channel
    2. IV.2 Longitudinal channel
  5. V Discussion and outlook
  6. A Verification of the competitive relationship between $\phi^{I}$ and $\chi^{I}$ in determing the shear modulus
  7. B Equations of motion of the linearized perturbations
    1. B.1 Transverse sector
    2. B.2 Longitudinal sector
  8. C Some results for the double-axion model with interaction terms
  9. D A check about the diffusion bound

##  I Introduction

The concepts of symmetry and symmetry breaking play a very important role in
classifying and describing different phases of matter in modern physics
Nicolis  _et al._ (2015). In condensed matter physics, especially for solid
state systems, the spatial translations are ubiquitously broken either
spontaneously or explicitly due to the existence of lattices, density waves,
impurities or some other structures Chaikin and Lubensky (1995); Grüner
(1988).

From the conventional perspective of solid physics, the long-range order
associated with the spontaneous breaking of translations endows the system
with its ability to respond to mechanical deformations elastically. As the
result, the propagating speeds of the associated Goldstone modes (acoustic
phonons) are determined by the elastic properties of the system Landau and
Lifshitz (1970); Chaikin and Lubensky (1995). In some cases, the translations
are not broken in a unique way. A large class of typical examples are the so-
called superlattices that are made from several (twisted or non-twisted)
layers of two-dimensional materials each of which breaks the translations in
its own manner Smith and Mailhiot (1990); Cao  _et al._ (2018). Due to the
more complicated structures, these systems, in general, not only modify
properties of the two-dimensional materials but also enable new features and
applications. However, more complicated structures also present significant
challenges to the current theoretical framework.

During the past two decades, the holographic duality (also known as AdS/CFT
correspondence) has been shown to be a powerful tool in dealing with real-time
dynamics of strongly correlated many-body systems Ammon and Erdmenger (2015);
Zaanen  _et al._ (2015); Hartnoll  _et al._ (2018); Baggioli (2019). In recent
years, this duality has been applied to the realm of solid state physics by
breaking the translations of the boundary system spontaneously Donos and
Gauntlett (2013); Donos (2013); Jokela  _et al._ (2014); Amoretti  _et al._
(2017); Cremonini  _et al._ (2017); Cai  _et al._ (2017); Alberte  _et al._
(2018a); Andrade  _et al._ (2018); Alberte  _et al._ (2018b); Amoretti  _et
al._ (2018a, b); Baggioli and Goutéraux (2023). Many progresses have been made
in the studies of transport properties Ling  _et al._ (2014); Baggioli and
Pujolas (2015, 2017); Goutéraux  _et al._ (2016, 2018); Donos  _et al._
(2018); Amoretti  _et al._ (2019a); Baggioli and Buchel (2019); An  _et al._
(2020); Andrade  _et al._ (2021); Wu  _et al._ (2021); Li  _et al._ (2022);
Liu  _et al._ (2023); Ji  _et al._ (2023); Ahn  _et al._ (2023a); Bai  _et
al._ (2023), quantum chaos Baggioli  _et al._ (2017); Baggioli and Li (2020);
Fu  _et al._ (2022), spectra of collective excitations Donos  _et al._ (2020);
Amoretti  _et al._ (2019b, 2020); Ammon  _et al._ (2019a); Baggioli  _et al._
(2020a); Baggioli and Grieninger (2019); Baggioli  _et al._ (2019, 2020b,
2020c); Baggioli (2020); Wang and Li (2021); Ahn  _et al._ (2023b); Baggioli
_et al._ (2022); Yang and Tian (2023), elastic response Donos and Gauntlett
(2011); Andrade  _et al._ (2019); Baggioli  _et al._ (2020d, e); Pan  _et al._
(2022); Baggioli  _et al._ (2020f, 2024) as well as phase transitions Baggioli
and Frangi (2022); Yang  _et al._ (2023). Among the models involved in these
works, the most economic one is the so-called holographic axion model. It is
constructed by the Einstein gravity coupled with several massless scalars
whose profiles are taken to be linear in the spatial coordinates. This allows
translations to be broken, while the background geometry still remains
homogeneous. For a detailed review of this model, one refers to Baggioli  _et
al._ (2021).

To enrich the structure of the boundary system, we generalize the holographic
axion model in this work by introducing more axions in the bulk all of which
break the translations of the dual systems spontaneously. We aim to
investigate the impacts of the additional axions on the viscoelaticity as well
as the spectra of collective excitations on boundary.

The paper is organized as follows: In section II, we introduce the generalized
holographic model with multiple sets of axions and obtain the black hole
solution. In section III, the shear modulus and shear viscosity of the
boundary system are calculated in the framework of linear response. In section
IV,the dispersion relations of collective modes are achieved by computing the
low-lying quasi-normal modes (QNMs) of the black hole. Finally, we conclude in
section V.

##  II Holographic action and black hole solution

In this paper, we generalize the holographic axion model in Baggioli  _et al._
(2021) by including multiple sets of massless scalars. And the action is given
by

$S=\int
d^{4}x\sqrt{-g}\bigg{[}\frac{R}{2}-\Lambda-\mathcal{W}(X_{1},X_{2},\cdots,X_{\mathcal{N}})\bigg{]},$
(1)

where $R$ is the Ricci scalar, $\Lambda$ is the cosmological constant which we
will fix $\Lambda=-3$ such that the AdS radius is a unit, $\mathcal{W}$ is a
general function of
$X_{a}\equiv\dfrac{1}{2}\sum_{I}\partial_{\mu}\Phi^{I}_{a}\partial^{\mu}\Phi^{I}_{a}$,
where $I=x,y$ and $a=1,2,\cdots,{\mathcal{N}}$. Since the scalars have no
mass, the theory is invariant under the global internal shift
$\Phi^{I}_{a}\to\Phi^{I}_{a}+c^{I}_{a}$.

Varying the metric as well as the scalars, we get equations of motion (EOMs)
as follows,

$G_{\mu\nu}-3g_{\mu\nu}+\mathcal{W}g_{\mu\nu}-\sum_{a=1}^{\mathcal{N}}\sum_{I=x,y}\mathcal{W}_{X_{a}}\partial_{\mu}\Phi^{I}_{a}\partial_{\nu}\Phi^{I}_{a}=0,$
(2)
$\partial_{\mu}(\sqrt{-g}\mathcal{W}_{X_{a}}\partial^{\mu}\Phi^{I}_{a})=0,$
(3)

where $G_{\mu\nu}$ is the Einstein tensor and
$\mathcal{W}_{X_{a}}\equiv\partial\mathcal{W}/\partial X_{a}$.

It has been shown in previous studies of holographic axion models that if the
bulk profile of the scalars is imposed as

$\bar{\Phi}^{I}_{a}=\delta_{i}^{I}x^{i},$ (4)

the translational symmetry of the boundary system is broken, while the
homogeneity as well as the isotropy are retained. Then, in the
Eddington–Finkelstein (EF) coordinates, the ansatz on the black hole solution
can be taken as

$ds^{2}=\dfrac{1}{u^{2}}[-f(u)dt^{2}-2dtdu+dx^{2}+dy^{2}],$ (5)

where $u\in[0,u_{h}]$ is the radial coordinate with $u=0$ denoting the AdS
boundary and the black hole horizon is defined by $f(u=u_{h})=0$. From the
Einstein equation, the emblackening factor is given by

$f(u)=u^{3}\int_{u}^{u_{h}}\frac{3-\mathcal{W}(\bar{X}_{1},\bar{X}_{2},\cdots,\bar{X}_{\mathcal{N}})}{z^{4}}\mathrm{d}z,$
(6)

with $\bar{X_{a}}=z^{2}$ representing their background value. The entropy
density can be calculated as $s=2\pi/u_{h}^{2}$ and the Hawking temperature is
given by

$T=-\frac{f^{\prime}(u_{h})}{4\pi}=\frac{3-\mathcal{W}(X_{1}^{h},X_{2}^{h},\cdots,X_{\mathcal{N}}^{h})}{4\pi
u_{h}},\ \text{with}\ X_{a}^{h}=u_{h}^{2}.$ (7)

In the following, we will calculate the excitations around the background. To
do so, we shall first consider the model with two sets of scalars (we will
call it double-axion model in this paper),

$\mathcal{W}(X,Y)=m^{2}V(X)+n^{2}W(Y)+l^{2}Z(X,Y),$ (8)

where $V$, $W$ and $Z$ are general functions. One can easily generalize it
into be extended to the case with more axons. Here, we have denoted $X\equiv
X_{1}\equiv\dfrac{1}{2}\partial_{\mu}\phi^{I}\partial^{\mu}\phi^{I}$ and
$Y\equiv X_{2}\equiv\dfrac{1}{2}\partial_{\mu}\chi^{I}\partial^{\mu}\chi^{I}$,
$m$, $n$, $l$ are coupling parameters and the $Z(X,Y)$ term represents the
interactions between the two sets of scalars.111We find that the interaction
term does not change the behavior of the system qualitatively. For this
reason, we will focus only on the case with $l=0$ in the rest of the main
text. One can read the appendix C for more details about the case with a non-
zero $l$. To perform explicit computation, we impose the following forms for
$V$, $Y$ and $Z$,

$V(X)=X^{M},W(Y)=Y^{N},Z(X,Y)=X^{P}Y^{Q}.$ (9)

In the standard quantization, to break the translations on boundary in the
spontaneous manner, we should further require that Baggioli  _et al._ (2021)

$M,\ N,\ P+Q>\frac{5}{2}.$ (10)

In following, we will investigate the shear viscoelasticity as well as the
hydrodynamic excitations by holography in such a specific model.

##  III Shear viscoelasticity

In the linear response theory, the shear viscoelasticity can be extracted from
the retarded Green’s function of the stress tensor which at low frequency and
zero momentum can be expressed as

$\mathcal{G}_{T_{xy}T_{xy}}^{(R)}(\omega,k=0)=G-i\omega\eta+\cdots,$ (11)

where $G$ is the shear elastic modulus and $\eta$ is the shear viscosity.

In holography, this retarded function can be calculated by considering the
metric fluctuation $\delta g_{xy}=\dfrac{1}{u^{2}}h_{xy}e^{-i\omega t}$ and
solving the linearized Einstein equation,

$-h_{xy}\left(2l^{2}uZ_{Y}+2l^{2}uZ_{X}+2m^{2}uV_{X}+2n^{2}uW_{Y}+2i\omega\right)+h_{xy}^{\prime}\left(uf^{\prime}-2f+2iu\omega\right)+ufh_{xy}^{\prime\prime}=0.$
(12)

Near the AdS boundary ($u\rightarrow 0$), its asymptotic behavior is

$h_{xy}=h_{xy}^{(0)}\,(1+\ldots)+h_{xy}^{(3)}\,u^{3}\,(1+\ldots).$ (13)

According to the holographic dictionary, the retarded Green’s function can be
read as

$\mathcal{G}_{T_{xy}T_{xy}}^{(R)}=\frac{3}{2}\,\frac{h_{xy}^{(3)}}{h_{xy}^{(0)}}.$
(14)

Then, the shear modulus and shear viscosity can be achieved respectively from

$G=\lim_{\omega\to
0}\text{Re}\left[\mathcal{G}_{T_{xy}T_{xy}}^{R}(\omega,k=0)\right],$ (15)

and

$\eta=-\lim_{\omega\to
0}\frac{1}{\omega}\text{Im}\left[\mathcal{G}_{T_{xy}T_{xy}}^{R}(\omega,k=0)\right].$
(16)

In FIG. 1, we plot $G$ as a function of $m/T$ for different values of $n/m$.
Comparing with the single-axion model, the rigidity of the system is always
enhanced by the additional scalars, $\chi^{I}$, at high temperatures. This can
be easily explained since we have that

$G\approx\int^{u_{h}}_{0}\frac{m^{2}V_{X}(\bar{X})+n^{2}W_{Y}(\bar{Y})}{z^{2}}\,dz,\quad\text{for}\quad
m,n\ll T.$ (17)

However, at low temperatures, $G$ displays complicated behaviors, depending on
$M<N$ or $M>N$. For $M<N$, we find that $G$ decreases when the impact of
$\chi^{I}$ is enhanced by increasing $n/m$. Then, at some intermediate
temperature, there exists an intersection with the $G$-curve of the single-
axion model. While, for $M>N$, $G$ always increases as the increase of $n/m$.
It looks as if that these two sets of axions exhibit a competition instead of
producing an additive effect at low temperatures. For instance, when $m\gg n$,
the $\phi^{I}$ fields play the dominant pole. If $m\ll n$, the $\chi^{I}$
fields become the most important. And as was shown in the single-axion model
that larger $M$ or $N$ leads to a smaller shear modulus at low temperatures
Alberte  _et al._ (2018b). This explains why the shear modulus in the left
panel of FIG.1 becomes smaller than the single axion model with
$\mathcal{W}=m^{2}X^{5}$ at low temperatures. We have verified this point in
the appendix A.

Figure 1: The shear modulus $G$ as the function of $m/T$ for
$n/m=\\{0,1/2,1,2\\}$ (from black to green). Left: $M=5$ and $N=7$. The
inserted panel shows that intersection points are not overlap. Right: $M=5$
and $N=3$.

The shear viscosity $\eta$, as a function of $m/T$, has been shown in FIG. 2.
It is obvious to see that $\eta$ monotonously decreases as the increase of
$m/T$ and the additional scalars $\chi^{I}$ always suppress its value.

Figure 2: The shear viscosity $\eta$ as the function of $m/T$ for
$n/m=\\{0,1/2,1,2\\}$ (from black to green). Left: $M=5$ and $N=7$. Right:
$M=5$ and $N=3$.

##  IV Dispersion relations of hydrodynamic modes

Now, we concentrate on the dispersion relations of the hydrodynamic modes in
the boundary system which can be extracted from the low-lying QNMs of black
holes. Then, we need to consider perturbations of the metric and scalar fields
upon the background,

$g_{\mu\nu}=\bar{g}_{\mu\nu}+\delta
g_{\mu\nu},\quad\phi^{I}=\bar{\phi}^{I}+\delta\phi^{I},\quad\chi^{I}=\bar{\chi}^{I}+\delta\chi^{I},$

where the fluctuating fields can be expanded as

$\displaystyle\delta
g_{\mu\nu}=\int\frac{d^{3}k}{(2\pi)^{3}}\,\dfrac{h_{\mu\nu}(\omega,k_{i},u)}{u^{2}}\,e^{-i\omega
t+ik_{i}x^{i}},$ (18)
$\displaystyle\delta\phi^{I}=\int\frac{d^{3}k}{(2\pi)^{3}}\,\delta^{I}_{j}\,\phi_{j}(\omega,k_{i},u)\,e^{-i\omega
t+ik_{i}x^{i}},$
$\displaystyle\delta\chi^{I}=\int\frac{d^{3}k}{(2\pi)^{3}}\,\delta^{I}_{j}\,\chi_{j}(\omega,k_{i},u)e^{-i\omega
t+ik_{i}x^{i}}.$

Since the system possesses a rotational symmetry on the $x-y$ plane, without
loss of generality, one can set the wave vector $k$ to be along the
$y-$direction. Then, the complete set of the fluctuating modes can be divided
two sectors by the parity in the $x-$direction:

$\displaystyle\text{Transverse modes (parity odd):}\quad\\{\delta
g_{xu},\delta g_{tx},\delta g_{xy},\delta\phi^{x},\delta\chi^{x}\\},$ (19)
$\displaystyle\text{Longitudinal modes (parity even):}\quad\\{\delta
g_{uu},\delta g_{tu},\delta g_{yu},\delta g_{tt},\delta g_{ty},\delta
g_{xx},\delta g_{yy},\delta\phi^{y},\delta\chi^{y}\\}.$

Solving the linearized equations for the two sectors by imposing the infalling
boundary conditions at the horizon and the normalizable (sourceless) condition
on the boundary, we achieve the QNM spectra of the black hole for finite
$\omega$ and finite $k$ which correspond to the poles of retarded correlators
for the boundary system. And the low-lying modes give us dispersion relations
of the hydrodynamic modes.

###  IV.1 Transverse channel

For simplicity, we adopt the radial gauge which fixes that $\delta g_{\mu
u}=0$ in the numeric computation. The equations of the transverse modes have
been shown in Appendix B. As the result, we find that there are a pair of
sounds and a purely dissipative mode in the transverse channel whose
dispersion relations at small momenta are given by

$\begin{cases}\text{sound mode}:&\omega=\pm
v_{T}k-i\dfrac{\Gamma_{T}}{2}k^{2}+\cdots,\\\ \text{diffusive
mode}:&\omega=-iD_{T}k^{2}+\cdots,\\\ \end{cases}$ (20)

where $v_{T}$ is the propagating speed of the transverse sounds, $\Gamma_{T}$
the sound attenuation and $D_{T}$ is the diffusivity of the dissipative mode.

From FIG.3-FIG.5, it is obvious to see that, in all cases, the sound speed
grows monotonously as the increase of $m/T$, while the dissipative
coefficients $\Gamma_{T}$ and $D_{T}$ behave in the opposite way. Furthermore,
it has been verified in FIG.6 that the sound speed $v_{T}$, the shear modulus
$G$ and the momentum susceptibility $\chi_{\pi\pi}$ satisfy the following
relation,

$v_{T}=\sqrt{\frac{G}{\chi_{\pi\pi}}},$ (21)

with $\chi_{\pi\pi}=\epsilon+\mathcal{P}=\frac{3}{2}\epsilon$, where
$\epsilon$ denotes the energy density which can be calculated by

$\epsilon=\frac{1}{u_{h}^{3}}\bigg{(}1+m^{2}\frac{u_{h}^{2M}}{2M-3}+n^{2}\frac{u_{h}^{2M}}{2N-3}\bigg{)},$
(22)

and $\mathcal{P}$ is the pressure. Note that (21) matches the prediction of
the elasticity theory of a relativistic conformal solid, and the sounds hence
can be understood as the transverse phonons.222More preciously, the transverse
sounds in a solid at finite temperatures come from a combination of the
transverse momentum and the Goldstone fields associated with the broken
translations. Again, we find intersections of the speed curves for $M<N$ in
the left plot of FIG.3, which is similar to the situation of $G$ discussed in
the previous section.

The nature of the extra diffusive mode in the transverse channel is
mysterious. It is irrelevant to the conserved momentum which is already
coupled to the transverse Goldstone fields and gives rise to the sound pair.
Its origin should be attributed to the broken global shift symmetries due to
the scalar profiles (4) in the bulk Donos  _et al._ (2019). In all cases, the
dimensionless quantity $D_{T}T$ decreases monotonously when lowering the
temperature. Moreover, we have also checked if it obeys the lower bounds on
diffusion, i.e. $D\geq v_{B}^{2}\,\tau_{L}$ with the butterfly velocity
$v_{B}$ and the Lyapunov time $\tau_{L}$, which was firstly proposed in Blake
and Donos (2017). In our model, it is found that $D_{T}\ll
v_{B}^{2}\,\tau_{L}$. Therefore, the bound is strongly violated. We present
this result in Appendix D.

Figure 3: The transverse sound velocity $v_{T}$ as the function of $m/T$ for
$n/m=\\{0,1/2,1,2\\}$ (black, red, blue, green). $\mathbf{Left}$:
$M=5,N=7$.$\mathbf{Right}$: $M=5,N=3$.

Figure 4:  The transverse sound attenuation $\Gamma_{T}$ as the function of
$m/T$ for $n/m=\\{0,1/2,1,2\\}$ (black, red, blue, green). Left: $M=5,N=7$.
Right: $M=5,N=3$.

Figure 5:  The transverse diffusivity $D_{T}$ as the function of $m/T$ for
$n/m=\\{1/2,1,2,3\\}$ (red, blue, green, orange). Left: $M=5,N=7$. Right:
$M=5,N=3$. Figure 6: Comparison of the transverse sound speed extracted from
the QNMs (dots) and from the analytic relation (21) (lines). Blue: $n/m=1.0$
and $m/T=2.0$. Yellow: $n/m=0.5$ and $m/T=0.5$. We set $M=5$ and $N=7$.

###  IV.2 Longitudinal channel

Now, we turn to study the hydrodynamic modes in the longitudinal channel. The
linearized equations of the longitudinal modes have also been shown in
Appendix B. The dispersion relations of the hydrodynamic modes at small
momenta are given by

$\begin{cases}\text{sound mode}:&\omega=\pm
v_{L}k-i\dfrac{\Gamma_{L}}{2}k^{2}+\cdots,\\\ \text{diffusive mode
``1"}:&\omega=-iD_{\phi}k^{2}+\cdots,\\\ \text{diffusive mode
``2"}:&\omega=-iD_{\chi}k^{2}+\cdots,\\\ \end{cases}$ (23)

where there is again an extra diffusive mode comparing with the case of
single-axion model whose transport coefficient is denoted as $D_{\chi}$.

The sound speed and attenuation have been shown in FIG.7 and FIG.8. One can
check that the following relation about the two sound speeds

$v_{L}^{2}-v_{T}^{2}=\frac{1}{2},$ (24)

is always ensured due to the conformal symmetry. Then, the speed curve of the
longitudinal sound just shifts upwards by $1/2$ with respect to the one of the
transverse sound. In addition, the sound attenuation $\Gamma_{L}$ always gets
suppressed in the presence of the $\chi^{y}$ fields.

The existence of the diffusive mode “1” in the longitudinal channel has
already been revealed in the hydrodynamics of solids long time ago Zippelius
_et al._ (1980). In the non-relativistic case, it can be understood as point
defect transport process that is not caused by the compressibility of the
crystal. And its relativistic version which originates from the coupling
between the energy density and the longitudinal Goldstone field at finite
temperatures has also been investigated in the relativistc hydrodynamics,
holography as well as effective field theories in recent years Delacrétaz  _et
al._ (2017); Ammon  _et al._ (2019b). In the presence of the $\chi^{y}$ field,
an extra diffusive mode denoted by “2” appears due to the coupling of the two
longitudinal Goldstone fields corresponding to the bulk fields $\phi^{y}$ and
$\chi^{y}$. Unlike the transverse diffusion, both of the two diffusive modes
obey the proposed diffusion bound.

Figure 7: The sound phonon velocity $v_{L}$ as the function of $m/T$ for
$n/m=\\{0,1/2,1,2\\}$ (black, red, blue, green). When $m/T\to 0$,
$v_{L}^{2}=1/2$ is satisfied. $\mathbf{Left}$: $M=5,N=7$. $\mathbf{Right}$:
$M=5,N=3$.

Figure 8: The longitudinal phonon attenuation $\Gamma_{L}$ as the function of
$m/T$ for $n/m=\\{0,1/2,1,2\\}$ (black, red, blue, green). $\mathbf{Left}$:
$M=5,N=7$. $\mathbf{Right}$: $M=5,N=3$.

Figure 9: $D_{\phi}\,T$ as the function of $m/T$ for $n/m=\\{0,1/2,2,3\\}$
(black, red, green, orange). $\mathbf{Left}$: $M=5,N=7$. $\mathbf{Right}$:
$M=5,N=3$.

Figure 10: $D_{\chi}\,T$ as the function of $m/T$ for $n/m=\\{1/2,1,2,3\\}$
(red, blue, green, orange). $\mathbf{Left}$: $M=5,N=7$. $\mathbf{Right}$:
$M=5,N=3$.

##  V Discussion and outlook

In this paper, we investigate the shear viscoelasticity as well as the
hydrodynamic modes in holographic multiple-axion models where the translations
are broken spontaneously. Our results show that adding more sets of axions in
the bulk, the shear modulus is enhanced at high temperatures and shear
viscosity is always suppressed. However, different sets of axions exhibit
competitive relationship in determining the shear modulus at low temperatures.
Furthermore, adding more bulk axions will not increase the amount of sound
pairs, but just increases diffusive modes on boundary. Although we have so far
demonstrated the second point above only in the double-axion model, our
conclusion can be easily extended to cases with more axions. As shown in
FIG.11, two diffusive modes emerge in the tranverse channel of the triple-
axion model.

Nevertheless, there are still many open questions. For instance, what are the
physical natures of the extra diffusive modes in the multiple-axion model?
This can be studied by using the method in Arean  _et al._ (2021). What is the
hydrodynamic description of the multiple-axion models? This may be constructed
by following Armas and Jain (2020); Baggioli and Goutéraux (2023). How do the
additional axions affect the mechanical stability of the system in the non-
linear elastic regime Baggioli  _et al._ (2024)? We leave these for future
work.

Figure 11: Two transverse diffusive mode emerges in the triple-axion model
with $\mathcal{W}=m^{2}(X^{6}+Y^{5}+U^{4})$.

### Acknowledgments

We would like to thank M. Baggioli, S.-Q. Lan, L. Li, H.-T. Sun and X.-J. Wang
for helpful discussions. This work is supported by the National Natural
Science Foundation of China (NSFC) under Grant No.12275038, the Fundamental
Research Funds for the Central Universities No.DUT23BK053, and DUT Innovation
and entrepreneurship program for undergraduate students No.20231014110345.

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##  Appendix A Verification of the competitive relationship between $\phi^{I}$
and $\chi^{I}$ in determing the shear modulus

Here, we verify the competitive relationship between the two sets of axions in
$G$ at low temperatures which is mentioned in Section III. For $m\gg n$, the
contribution from $\chi^{I}$ fields is just a small correction. Then, the
result is expected to be not very different from the single-axion model.
Therefore, we only need to consider the case of $m\ll n$. In FIG.12, we plot
the $G$-curves for the double-axion model with
$\mathcal{W}(X,Y)=m^{2}X^{5}+n^{2}Y^{7}$ by varying the value of $n$. It is
obvious to see that when $n/m=100$, the result is very close to that in the
single-axion model with $\mathcal{W}(X)=m^{2}X^{7}$ at low temperatures which
implies that the behavior of $G$ in the double-axion model is solely
controlled by the $Y$-term in this regime when $m\ll n$.

Figure 12: The shear elastic modulus $G$ as the function of $m/T$ for the
double-axion model with $\mathcal{W}(X,Y)=m^{2}X^{M}+n^{2}Y^{N}$, $n/m$ =
$\\{1,2,100\\}$ (blue, green, red) and the single-axion model with
$\mathcal{W}(X)=m^{2}X^{7}$ denoted by the dashed line.

##  Appendix B Equations of motion of the linearized perturbations

###  B.1 Transverse sector

In the radial gauge, $h_{xu}=0$, the EOMs of the transverse modes are given by

$\displaystyle 0=$
$\displaystyle-2l^{2}uZ_{Y}\chi_{x}^{\prime}-2l^{2}uZ_{X}\phi_{x}^{\prime}-uh_{tx}^{\prime\prime}+2h_{tx}^{\prime}+ikuh_{xy}^{\prime}-2m^{2}uV_{X}\phi_{x}^{\prime}-2n^{2}uW_{Y}\chi_{x}^{\prime},$
(25) $\displaystyle 0=$ $\displaystyle\
2l^{2}u^{2}h_{xy}V_{Y}+2l^{2}u^{2}h_{xy}V_{X}-2l^{2}h_{xy}V-4l^{2}u^{2}h_{xy}Z_{Y}-4l^{2}u^{2}h_{xy}Z_{X}+2l^{2}h_{xy}Z+$
$\displaystyle
2il^{2}ku^{2}\chi_{x}Z_{Y}+2il^{2}ku^{2}\phi_{x}Z_{X}+u^{2}f^{\prime}h_{xy}^{\prime}+u^{2}fh_{xy}^{\prime\prime}-2ufh_{xy}^{\prime}+iku^{2}h_{tx}^{\prime}-2ikuh_{tx}+$
$\displaystyle 2iu^{2}\omega
h_{xy}^{\prime}-2m^{2}u^{2}h_{xy}V_{X}-2n^{2}u^{2}h_{xy}W_{Y}-2iu\omega
h_{xy}+2ikm^{2}u^{2}\phi_{x}V_{X}+2ikn^{2}u^{2}\chi_{x}W_{Y},$ (26)
$\displaystyle 0=$ $\displaystyle\
2l^{2}u^{2}fZ_{Y}\chi_{x}^{\prime}+2l^{2}u^{2}fZ_{X}\phi_{x}^{\prime}+2l^{2}h_{tx}V+2l^{2}u^{2}h_{tx}Z_{Y}+2l^{2}u^{2}h_{tx}Z_{X}-2l^{2}h_{tx}Z+$
$\displaystyle
2il^{2}u^{2}\omega\chi_{x}Z_{Y}+2il^{2}u^{2}\omega\phi_{x}Z_{X}-iku^{2}fh_{xy}^{\prime}+2m^{2}u^{2}fV_{X}\phi_{x}^{\prime}+2n^{2}u^{2}fW_{Y}\chi_{x}^{\prime}-iu^{2}\omega
h_{tx}^{\prime}+$ $\displaystyle
k^{2}u^{2}h_{tx}+2m^{2}u^{2}h_{tx}V_{X}+2n^{2}u^{2}h_{tx}W_{Y}+ku^{2}\omega
h_{xy}+2im^{2}u^{2}\omega\phi_{x}V_{X}+2in^{2}u^{2}\omega\chi_{x}W_{Y},$ (27)
$\displaystyle 0=$ $\displaystyle\
l^{2}uZ_{X}f^{\prime}\phi_{x}^{\prime}+l^{2}ufZ_{X}\phi_{x}^{\prime\prime}-2l^{2}fZ_{X}\phi_{x}^{\prime}+2l^{2}u^{2}fZ_{XY}\phi_{x}^{\prime}+2l^{2}u^{2}fZ_{XX}\phi_{x}^{\prime}+l^{2}uZ_{X}h_{tx}^{\prime}-$
$\displaystyle
2l^{2}h_{tx}Z_{X}+2l^{2}u^{2}h_{tx}Z_{XY}+2l^{2}u^{2}h_{tx}Z_{XX}-il^{2}kuh_{xy}Z_{X}-l^{2}k^{2}u\phi_{x}Z_{X}+2il^{2}u\omega
Z_{X}\phi_{x}^{\prime}-$ $\displaystyle
2il^{2}\omega\phi_{x}Z_{X}+2il^{2}u^{2}\omega\phi_{x}Z_{XY}+2il^{2}u^{2}\omega\phi_{x}Z_{XX}+m^{2}uf^{\prime}V_{X}\phi_{x}^{\prime}+2m^{2}u^{2}fV_{X}^{\prime}\phi_{x}^{\prime}+$
$\displaystyle
m^{2}ufV_{X}\phi_{x}^{\prime\prime}-2m^{2}fV_{X}\phi_{x}^{\prime}+m^{2}uh_{tx}^{\prime}V_{X}+2m^{2}u^{2}h_{tx}V_{X}^{\prime}-2m^{2}h_{tx}V_{X}-ikm^{2}uh_{xy}V_{X}-$
$\displaystyle
k^{2}m^{2}u\phi_{x}V_{X}+2im^{2}u^{2}\omega\phi_{x}V_{X}^{\prime}+2im^{2}u\omega
V_{X}\phi_{x}^{\prime}-2im^{2}\omega\phi_{x}V_{X},$ (28) $\displaystyle 0=$
$\displaystyle
l^{2}uZ_{Y}f^{\prime}\chi_{x}^{\prime}+l^{2}ufZ_{Y}\chi_{x}^{\prime\prime}-2l^{2}fZ_{Y}\chi_{x}^{\prime}+2l^{2}u^{2}fZ_{YY}\chi_{x}^{\prime}+2l^{2}u^{2}fZ_{XY}\chi_{x}^{\prime}+l^{2}uZ_{Y}h_{tx}^{\prime}-$
$\displaystyle
2l^{2}h_{tx}Z_{Y}+2l^{2}u^{2}h_{tx}Z_{YY}+2l^{2}u^{2}h_{tx}Z_{XY}-il^{2}kuh_{xy}Z_{Y}-l^{2}k^{2}u\chi_{x}Z_{Y}+2il^{2}u\omega
Z_{Y}\chi_{x}^{\prime}-$ $\displaystyle
2il^{2}\omega\chi_{x}Z_{Y}+2il^{2}u^{2}\omega\chi_{x}Z_{YY}+2il^{2}u^{2}\omega\chi_{x}Z_{XY}+n^{2}uf^{\prime}W_{Y}\chi_{x}^{\prime}+2n^{2}u^{2}fW_{Y}^{\prime}\chi_{x}^{\prime}+$
$\displaystyle
n^{2}ufW_{Y}\chi_{x}^{\prime\prime}-2n^{2}fW_{Y}\chi_{x}^{\prime}+n^{2}uh_{tx}^{\prime}W_{Y}+2n^{2}u^{2}h_{tx}W_{Y}^{\prime}-2n^{2}h_{tx}W_{Y}-ikn^{2}uh_{xy}W_{Y}-$
$\displaystyle
k^{2}n^{2}u\chi_{x}W_{Y}+2in^{2}u^{2}\omega\chi_{x}W_{Y}^{\prime}+2in^{2}u\omega
W_{Y}\chi_{x}^{\prime}-2in^{2}\omega\chi_{x}W_{Y},$ (29)

where the prime denotes the partial derivative with respect to the radial
coordinate $u$ and the subscripts ‘$X$’ and ‘$Y$’ represent $\partial/\partial
X$ and $\partial/\partial Y$.

###  B.2 Longitudinal sector

For the longitudinal sector, we introduce the symmetric combination
$h_{xs}=(h_{xx}+h_{yy})/2$ and the anti-symmetric combination
$h_{xa}=(h_{xx}-h_{yy})/2$ to simplify the equations. In the the radial gauge,
$h_{uu}=h_{tu}=h_{yu}=0$, we have that

$\displaystyle 0=$ $\displaystyle\ h_{xs}^{\prime\prime},$ (30) $\displaystyle
0=$ $\displaystyle\
2l^{2}uZ_{Y}\chi_{y}^{\prime}+2l^{2}uZ_{X}\phi_{y}^{\prime}+uh_{ty}^{\prime\prime}-2h_{ty}^{\prime}+ikuh_{xa}^{\prime}+ikuh_{xs}^{\prime}+2m^{2}uV_{X}\phi_{y}^{\prime}+2n^{2}uW_{Y}\chi_{y}^{\prime},$
(31) $\displaystyle 0=$
$\displaystyle-l^{2}ufZh_{xs}^{\prime}+2l^{2}u^{2}fh_{xs}Z_{Y}+2l^{2}u^{2}fh_{xs}Z_{X}-2il^{2}ku^{2}f\chi_{y}Z_{Y}-2il^{2}ku^{2}f\phi_{y}Z_{X}-$
$\displaystyle il^{2}kuh_{ty}Z-il^{2}u\omega
h_{xs}Z-2ufh_{tt}^{\prime}+6fh_{tt}+ikufh_{ty}+k^{2}u^{2}fh_{xa}-m^{2}ufVh_{xs}^{\prime}-$
$\displaystyle
n^{2}ufWh_{xs}^{\prime}+3ufh_{xs}^{\prime}+uf^{2}h_{xs}^{\prime}+k^{2}u^{2}fh_{xs}+2m^{2}u^{2}fh_{xs}V_{X}+2n^{2}u^{2}fh_{xs}W_{Y}+$
$\displaystyle iu\omega
fh_{xs}-2ikm^{2}u^{2}f\phi_{y}V_{X}-2ikn^{2}u^{2}f\chi_{y}W_{Y}-k^{2}u^{2}h_{tt}+2iu\omega
h_{tt}-ikm^{2}uh_{ty}V-$ $\displaystyle ikn^{2}uh_{ty}W-2ku^{2}\omega
h_{ty}+3ikuh_{ty}-im^{2}u\omega h_{xs}V-in^{2}u\omega
h_{xs}W-2u^{2}\omega^{2}h_{xs}+3iu\omega h_{xs},$ (32) $\displaystyle 0=$
$\displaystyle\
2l^{2}u^{2}fZ_{Y}\chi_{y}^{\prime}+2l^{2}u^{2}fZ_{X}\phi_{y}^{\prime}+2l^{2}u^{2}h_{ty}Z_{Y}+2l^{2}u^{2}h_{ty}Z_{X}+2il^{2}u^{2}\omega\chi_{y}Z_{Y}+2il^{2}u^{2}\omega\phi_{y}Z_{X}$
$\displaystyle+iku^{2}fh_{xa}^{\prime}+iku^{2}fh_{xs}^{\prime}+2m^{2}u^{2}fV_{X}\phi_{y}^{\prime}+2n^{2}u^{2}fW_{Y}\chi_{y}^{\prime}-iku^{2}h_{tt}^{\prime}+2ikuh_{tt}-iu^{2}\omega
h_{ty}^{\prime}+$ $\displaystyle
2m^{2}u^{2}h_{ty}V_{X}+2n^{2}u^{2}h_{ty}W_{Y}-ku^{2}\omega h_{xa}-ku^{2}\omega
h_{xs}+2im^{2}u^{2}\omega\phi_{y}V_{X}+2in^{2}u^{2}\omega\chi_{y}W_{Y},$ (33)
$\displaystyle 0=$
$\displaystyle-2l^{2}u^{2}h_{xa}Z_{Y}-2l^{2}u^{2}h_{xa}Z_{X}-2l^{2}u^{4}h_{xs}Z_{YY}-4l^{2}u^{4}h_{xs}Z_{XY}-2l^{2}u^{4}h_{xs}Z_{XX}-$
$\displaystyle
2il^{2}ku^{2}\chi_{y}Z_{Y}-2il^{2}ku^{2}\phi_{y}Z_{X}+2il^{2}ku^{4}\chi_{y}Z_{YY}+2il^{2}ku^{4}\chi_{y}Z_{XY}+2il^{2}ku^{4}\phi_{y}Z_{XY}+$
$\displaystyle
2il^{2}ku^{4}\phi_{y}Z_{XX}+u^{2}f^{\prime}h_{xa}^{\prime}-u^{2}f^{\prime}h_{xs}^{\prime}+u^{2}fh_{xa}^{\prime\prime}-2ufh_{xa}^{\prime}-u^{2}fh_{xs}^{\prime\prime}+2ufh_{xs}^{\prime}+u^{2}h_{tt}^{\prime\prime}-$
$\displaystyle
4uh_{tt}^{\prime}+6h_{tt}-2iku^{2}h_{ty}^{\prime}+4ikuh_{ty}+2iu^{2}\omega
h_{xa}^{\prime}-2m^{2}u^{2}h_{xa}V_{X}-2n^{2}u^{2}h_{xa}W_{Y}-$ $\displaystyle
2iu\omega h_{xa}-2iu^{2}\omega
h_{xs}^{\prime}-2m^{2}u^{4}h_{xs}V_{X}^{\prime}-2n^{2}u^{4}h_{xs}W_{Y}^{\prime}+2iu\omega
h_{xs}-2ikm^{2}u^{2}\phi_{y}V_{X}+$ $\displaystyle
2ikm^{2}u^{4}\phi_{y}V_{X}^{\prime}-2ikn^{2}u^{2}\chi_{y}W_{Y}+2ikn^{2}u^{4}\chi_{y}W_{Y}^{\prime},$
(34) $\displaystyle 0=$ $\displaystyle\
2l^{2}u^{2}h_{xa}Z_{Y}+2l^{2}u^{2}h_{xa}Z_{X}-2l^{2}u^{4}h_{xs}Z_{YY}-4l^{2}u^{4}h_{xs}Z_{XY}-2l^{2}u^{4}h_{xs}Z_{XX}+$
$\displaystyle
2il^{2}ku^{2}\chi_{y}Z_{Y}+2il^{2}ku^{2}\phi_{y}Z_{X}+2il^{2}ku^{4}\chi_{y}Z_{YY}+2il^{2}ku^{4}\chi_{y}Z_{XY}+2il^{2}ku^{4}\phi_{y}Z_{XY}+$
$\displaystyle
2il^{2}ku^{4}\phi_{y}Z_{XX}-u^{2}f^{\prime}h_{xa}^{\prime}-u^{2}f^{\prime}h_{xs}^{\prime}-u^{2}fh_{xa}^{\prime\prime}+2ufh_{xa}^{\prime}-u^{2}fh_{xs}^{\prime\prime}+2ufh_{xs}^{\prime}+u^{2}h_{tt}^{\prime\prime}-$
$\displaystyle 4uh_{tt}^{\prime}+6h_{tt}-2iu^{2}\omega
h_{xa}^{\prime}+2m^{2}u^{2}h_{xa}V_{X}+2n^{2}u^{2}h_{xa}W_{Y}+2iu\omega
h_{xa}-2iu^{2}\omega h_{xs}^{\prime}-$ $\displaystyle
2m^{2}u^{4}h_{xs}V_{X}^{\prime}-2n^{2}u^{4}h_{xs}W_{Y}^{\prime}+2iu\omega
h_{xs}+2ikm^{2}u^{2}\phi_{y}V_{X}+2ikm^{2}u^{4}\phi_{y}V_{X}^{\prime}+2ikn^{2}u^{2}\chi_{y}W_{Y}$
$\displaystyle+2ikn^{2}u^{4}\chi_{y}W_{Y}^{\prime},$ (35) $\displaystyle 0=$
$\displaystyle\
l^{2}uZh_{xs}^{\prime}-2l^{2}u^{2}h_{xs}Z_{Y}-2l^{2}u^{2}h_{xs}Z_{X}+2il^{2}ku^{2}\chi_{y}Z_{Y}+2il^{2}ku^{2}\phi_{y}Z_{X}-ufh_{xs}^{\prime}+2uh_{tt}^{\prime}-$
$\displaystyle
6h_{tt}+iku^{2}h_{ty}^{\prime}-4ikuh_{ty}-k^{2}u^{2}h_{xa}+m^{2}uVh_{xs}^{\prime}+n^{2}uWh_{xs}^{\prime}+2iu^{2}\omega
h_{xs}^{\prime}-3uh_{xs}^{\prime}-$ $\displaystyle
k^{2}u^{2}h_{xs}-2m^{2}u^{2}h_{xs}V_{X}-2n^{2}u^{2}h_{xs}W_{Y}-4iu\omega
h_{xs}+2ikm^{2}u^{2}\phi_{y}V_{X}+2ikn^{2}u^{2}\chi_{y}W_{Y},$ (36)
$\displaystyle 0=$ $\displaystyle\
l^{2}uZ_{X}f^{\prime}\phi_{y}^{\prime}+l^{2}ufZ_{X}\phi_{y}^{\prime\prime}+2l^{2}u^{2}fZ_{XY}\phi_{y}^{\prime}+2l^{2}u^{2}fZ_{XX}\phi_{y}^{\prime}-2l^{2}fZ_{X}\phi_{y}^{\prime}+l^{2}uZ_{X}h_{ty}^{\prime}+$
$\displaystyle
2l^{2}u^{2}h_{ty}Z_{XY}+2l^{2}u^{2}h_{ty}Z_{XX}-2l^{2}h_{ty}Z_{X}+il^{2}kuh_{xa}Z_{X}-il^{2}ku^{3}h_{xs}Z_{XY}-il^{2}ku^{3}h_{xs}Z_{XX}$
$\displaystyle-l^{2}k^{2}u\phi_{y}Z_{X}-l^{2}k^{2}u^{3}\chi_{y}Z_{XY}-l^{2}k^{2}u^{3}\phi_{y}Z_{XX}+2il^{2}u\omega
Z_{X}\phi_{y}^{\prime}+2il^{2}u^{2}\omega\phi_{y}Z_{XY}+$ $\displaystyle
2il^{2}u^{2}\omega\phi_{y}Z_{XX}-2il^{2}\omega\phi_{y}Z_{X}+m^{2}uf^{\prime}V_{X}\phi_{y}^{\prime}+2m^{2}u^{2}fV_{X}^{\prime}\phi_{y}^{\prime}+m^{2}ufV_{X}\phi_{y}^{\prime\prime}-2m^{2}fV_{X}\phi_{y}^{\prime}+$
$\displaystyle
m^{2}uh_{ty}^{\prime}V_{X}+2m^{2}u^{2}h_{ty}V_{X}^{\prime}-2m^{2}h_{ty}V_{X}+ikm^{2}uh_{xa}V_{X}-ikm^{2}u^{3}h_{xs}V_{X}^{\prime}-k^{2}m^{2}u\phi_{y}V_{X}-$
$\displaystyle
k^{2}m^{2}u^{3}\phi_{y}V_{X}^{\prime}+2im^{2}u^{2}\omega\phi_{y}V_{X}^{\prime}+2im^{2}u\omega
V_{X}\phi_{y}^{\prime}-2im^{2}\omega\phi_{y}V_{X},$ (37) $\displaystyle 0=$
$\displaystyle\
l^{2}uZ_{Y}f^{\prime}\chi_{y}^{\prime}+l^{2}ufZ_{Y}\chi_{y}^{\prime\prime}+2l^{2}u^{2}fZ_{YY}\chi_{y}^{\prime}+2l^{2}u^{2}fZ_{XY}\chi_{y}^{\prime}-2l^{2}fZ_{Y}\chi_{y}^{\prime}+l^{2}uZ_{Y}h_{ty}^{\prime}+$
$\displaystyle
2l^{2}u^{2}h_{ty}Z_{YY}+2l^{2}u^{2}h_{ty}Z_{XY}-2l^{2}h_{ty}Z_{Y}+il^{2}kuh_{xa}Z_{Y}-il^{2}ku^{3}h_{xs}Z_{YY}-il^{2}ku^{3}h_{xs}Z_{XY}$
$\displaystyle-l^{2}k^{2}u\chi_{y}Z_{Y}-l^{2}k^{2}u^{3}\chi_{y}Z_{YY}-l^{2}k^{2}u^{3}\phi_{y}Z_{XY}+2il^{2}u\omega
Z_{Y}\chi_{y}^{\prime}+2il^{2}u^{2}\omega\chi_{y}Z_{YY}+$ $\displaystyle
2il^{2}u^{2}\omega\chi_{y}Z_{XY}-2il^{2}\omega\chi_{y}Z_{Y}+n^{2}uf^{\prime}W_{Y}\chi_{y}^{\prime}+2n^{2}u^{2}fW_{Y}^{\prime}\chi_{y}^{\prime}+n^{2}ufW_{Y}\chi_{y}^{\prime\prime}-2n^{2}fW_{Y}\chi_{y}^{\prime}+$
$\displaystyle
n^{2}uh_{ty}^{\prime}W_{Y}+2n^{2}u^{2}h_{ty}W_{Y}^{\prime}-2n^{2}h_{ty}W_{Y}+ikn^{2}uh_{xa}W_{Y}-ikn^{2}u^{3}h_{xs}W_{Y}^{\prime}-k^{2}n^{2}u\chi_{y}W_{Y}-$
$\displaystyle
k^{2}n^{2}u^{3}\chi_{y}W_{Y}^{\prime}+2in^{2}u^{2}\omega\chi_{y}W_{Y}^{\prime}+2in^{2}u\omega
W_{Y}\chi_{y}^{\prime}-2in^{2}\omega\chi_{y}W_{Y}.$ (38)

##  Appendix C Some results for the double-axion model with interaction terms

Let us turn on the interaction term with a non-zero $l$ and check how it
changes the shear viscoelasticity and the shear hydrodynamic modes by varying
its value. It is found that such an interaction term does not change the whole
story but just modify the behavior of these quantities quantitatively.

Figure 13: The shear elastic modulus $G$ as the function of $m/T$ for
$l/m=\\{0,1/2,1,2\\}$ (from black to green). We fix $M=5,N=3$ and $P=2,Q=7$.
Left: $n/m=1$. Right: $n/m=2$.

Figure 14: The shear viscosity $\eta$ as the function of $m/T$ for
$l/m=\\{0,1/2,1,2\\}$ (from black to green). We fix $M=5,N=3$ and $P=2,Q=7$.
Left: $n/m=1$. Right: $n/m=2$.

Figure 15: The transverse sound speed $v_{T}$ as the function of $m/T$ for
$l/m=\\{0,1/2,1,2\\}$ (from black to green). We fix $M=5,N=3$ and $P=2,Q=7$.
Left: $n/m=1$. Right: $n/m=2$.

Figure 16: The transverse sound attenuation $\Gamma_{T}$ as the function of
$m/T$ for $l/m=\\{0,1/2,1,2\\}$ (from black to green). We fix $M=5,N=3$ and
$P=2,Q=7$. Left: $n/m=1$. Right: $n/m=2$.

Figure 17: The transverse diffusivity $D_{T}$ as the function of $m/T$ for
$l/m=\\{1/2,1,2,3\\}$ (from red to orange). We fix $M=5,N=3$ and $P=2,Q=7$.
Left: $n/m=1$. Right: $n/m=2$.

##  Appendix D A check about the diffusion bound

In this section, we verify the bounds for diffusive modes in both transverse
and longitudinal channels. According to Baggioli and Li (2020), the butterfly
velocity $v_{B}$ is expressed as

$v_{B}^{2}=\frac{\pi T}{u_{h}}$ (39)

Figure 18: The transverse diffusivity $D_{T}$ as the function of $m/T$ for
$n/m=\\{1/2,1,2,3\\}$ (red, blue, green, orange). Left: $M=5,N=7$. Right:
$M=5,N=3$.

Figure 19: The ratios $D_{\phi}T/v_{B}^{2}$ and $D_{\chi}T/v_{B}^{2}$ as
functions of $m/T$ for the double-axion model with $M=5$ and $N=7$. Here, we
set $n/m=\\{0,1/2,1,2,3\\}$ (black, red, blue, green, orange). Dashed line:
$\text{AdS}_{2}$ bound.

Figure 20: The ratios $D_{\phi}T/v_{B}^{2}$ and $D_{\chi}T/v_{B}^{2}$ as
functions of $m/T$ for the double-axion model with $M=5$ and $N=3$. Here, we
set $n/m=\\{0,1/2,1,2,3\\}$ (black,red, blue, green, orange). Dashed line:
$\text{AdS}_{2}$ bound.


Oliver<EMAIL_ADDRESS>James Charleshttp://www.jjcvision.com1 Roberto
<EMAIL_ADDRESS>Machine Intelligence Lab
Department of Engineering
University of Cambridge
Cambridge, U.K. \- 3D Model of Articulated Human Feet

#  : An Unsupervised Implicit 3D Model of Articulated Human Feet

###### Abstract

In this paper we present a high fidelity and articulated 3D human foot model.
The model is parameterised by a disentangled latent code in terms of shape,
texture and articulated pose. While high fidelity models are typically created
with strong supervision such as 3D keypoint correspondences or pre-
registration, we focus on the difficult case of little to no annotation. To
this end, we make the following contributions: (i) we develop a Foot Implicit
Neural Deformation field model, named FIND, capable of tailoring explicit
meshes at any resolution i.e. for low or high powered devices; (ii) an
approach for training our model in various modes of weak supervision with
progressively better disentanglement as more labels, such as pose categories,
are provided; (iii) a novel unsupervised part-based loss for fitting our model
to 2D images which is better than traditional photometric or silhouette
losses; (iv) finally, we release a new dataset of high resolution 3D human
foot scans, Foot3D. On this dataset, we show our model outperforms a strong
PCA implementation trained on the same data in terms of shape quality and part
correspondences, and that our novel unsupervised part-based loss improves
inference on images.

##  1 Introduction

Shape reconstruction from single-view or few-view images is a vital but
difficult computer vision task. Current approaches are dependent on strong
priors to help constrain this ill-posed problem, usually in the form of well
constructed and parameterised 3D models [Loper et al.(2015)Loper, Mahmood,
Romero, Pons-Moll, and Black, Zuffi et al.(2017)Zuffi, Kanazawa, Jacobs, and
Black]. The process of model creation is also often time consuming, requiring
a lot of supervision e.g. registration and correspondence annotation. For the
case of human feet, the current state-of-the-art in reconstruction involves
the use of expensive scanning equipment [Vol(), yet(), Art(a)], unavailable to
the average consumer at home. Thus, there is growing interest for easier
methods of foot reconstruction, particularly for home health monitoring,
custom orthotics and the growing online shoe retail industry. As such, these
models must be capable of running on low powered devices such as mobile
phones, but also displayable at high resolution e.g. for a doctor’s
inspection. To this end, we develop FIND, a Foot Implicit Neural Deformation
field model, which is easy to train as it requires little to no labels and can
produce explicit meshes at tailored resolutions, useful for targeting specific
hardware or needs. We can improve the model further by introducing minimal
extra supervision: (i) we add the weak label of foot identities to disentangle
shape and pose; and (ii) pose descriptions to produce a more interpretable
pose latent space.

We outline our key contributions as follows: (1) a novel implicit foot model,
FIND, capable of producing high fidelity human feet and fully paramaterised by
a disentangled latent space of shape, texture and pose. The model can also be
tuned to specific hardware considerations or needs; (2) an approach to train
the model under various levels of weak supervision, with stronger supervision
producing better disentanglement; (3) a novel, unsupervised part-based loss
for fitting FIND to images using unsupervised feature learning; and (4) we
release Foot3D, a dataset of high resolution foot scans, providing shape and
texture information for training of 3D foot models.

Figure 1: We produce (1) a coordinate based multilayer perceptron (MLP)
architecture, FIND, which receives a 3D vertex position on a template mesh,
and a shape, pose and texture embedding, and predicts a corresponding vertex
deformation and colour. (2) We train our model with 3D based losses against
ground truth scans, and learn a sensible pose space using a contrastive loss
(3) We use differentiable rendering at inference to enforce a silhouette based
loss, in addition to our novel unsupervised part-based loss.

##  2 Related work

#### Generative shape models.

These models endeavour to capture the distribution of an object category’s
shape in a set of controllable parameters e.g. height, width, pose, etc. Much
of the work in generative shape modelling looks at human bodies where a number
of parameterised models have been built [Loper et al.(2015)Loper, Mahmood,
Romero, Pons-Moll, and Black, Zuffi and Black(2015)]. For construction, the
models are typically trained with strong supervision via 2D annotation
[Andriluka et al.(2014)Andriluka, Pishchulin, Gehler, and Schiele, Lin et
al.(2014)Lin, Maire, Belongie, Hays, Perona, Ramanan, Dollár, and Zitnick] of
rich 3D scans [Ionescu et al.(2014)Ionescu, Papava, Olaru, and Sminchisescu,
von Marcard et al.(2018)von Marcard, Henschel, Black, Rosenhahn, and Pons-
Moll]. The building of generative models of more niche object categories often
uses similar principles, but usually on much smaller, custom datasets, often
due to the time and expense in collecting the data. For examples of the
difficulties involved, analogous generative quadruped models have been
constructed from scans of toys of real animals [Zuffi et al.(2017)Zuffi,
Kanazawa, Jacobs, and Black] (as real animals don’t keep still long enough),
and parametric hand models have been constructed from 3D scans [Romero et
al.(2022)Romero, Tzionas, and Black] and even at larger expense from MRI scans
[Li et al.(2021)Li, Wu, Zhang, Xu, and Yu]. These models typically use
Principal Component Analysis (PCA) to compose a small set of parameters which
control linear offsets from a template mesh. We show here, that our implicit
surface deformation field performs better against this type of strong PCA
baseline on feet.

#### Implicit shape representations.

Recent interest has shifted towards implicit representations of shape -
functions, often deep networks, that take in coordinates corresponding to a
point in space, and return spacial information relating to an object - such as
whether the point in space is inside the object [Chen and Zhang(2019)], or how
near that point is to the object’s surface [Park et al.(2019)Park, Florence,
Straub, Newcombe, and Lovegrove]. These methods are naturally differentiable
and can represent scenes at arbitrary scales, so lend themselves to many
reconstruction tasks - for example, reconstructing meshes from point clouds
[Hanocka et al.(2020)Hanocka, Metzer, Giryes, and Cohen-Or] or single view
images [Wang et al.(2018)Wang, Zhang, Li, Fu, Liu, and Jiang]. One particular
method, Neural Radiance Fields (NeRFs) [Mildenhall et al.(2020)Mildenhall,
Srinivasan, Tancik, Barron, Ramamoorthi, and Ng], has become increasingly
popular due to the very high fidelity in which it can synthesis novel views of
a scene. Many extensions to NeRF have investigated how to parameterise shape
and texture for generic object classes [Jang and Agapito(2021)], decompose
scenes into individual, controllable components [Niemeyer and Geiger(2021)],
and directly make edits to geometry and texture [Liu et al.(2021)Liu, Zhang,
Zhang, Zhang, Zhu, and Russell]. This has also been extended to posed humans,
with several works [Su et al.(2021)Su, Yu, Zollhoefer, and Rhodin, Xu et
al.(2021)Xu, Alldieck, and Sminchisescu] showing promise in modelling dynamic
human models using NeRF architectures. We take inspiration from this approach
and, similar to [Michel et al.(2022)Michel, Bar-On, Liu, Benaim, and Hanocka],
build our implicit foot model FIND by sampling points on the surface of a
manifold, which in our case are vertices of a high resolution template foot
mesh, and learn a parameterised deformation field for deforming vertices to
different types of foot shapes and poses.

#### Unsupervised representation learning for 3D objects.

Generative Adversarial Networks (GANs) [Goodfellow et al.(2014)Goodfellow,
Pouget-Abadie, Mirza, Xu, Warde-Farley, Ozair, Courville, and Bengio] have
shown promise in learning unsupervised representations of 2D data, even
learning information about the underlying 3D geometry [Pan et al.(2020)Pan,
Dai, Liu, Loy, and Luo]. The intermediate feature representations are
sufficiently powerful that Zhang  _et al_ [Zhang et al.(2021)Zhang, Ling, Gao,
Yin, Lafleche, Barriuso, Torralba, and Fidler] showed they can be trained to
solve downstream tasks, such as semantic segmentations, with very simple MLP
classifiers and an incredibly small amount of ground truth labels. We leverage
this for our novel unsupervised part-based loss, allowing us to learn foot
part classes in 3D (from 2D images) to aid fitting to 2D images.

#### Foot reconstruction.

Solutions exist for high resolution, accurate 3D scanners [Vol(), yet(),
Art(a)]. These are expensive and difficult to use outside of specific
environments, so do not lend themselves to the average consumer. While PCA
parameterised foot models do exist [Amstutz et al.(2008)Amstutz, Teshima,
Kimura, Mochimaru, and Saito], many existing foot reconstruction solutions
instead directly predict an unconstrained mesh from pointclouds or depth maps
[Lunscher and Zelek(2017)]. Large scale, proprietary datasets of feet, scanned
at low resolution [Jurca et al.(2019)Jurca, Žabkar, and Džeroski, Jurca et
al.(2011)Jurca, Kolsek, and Vidic] only release the population measurement
statistics to the public. For our work, we build a new foot scan dataset,
Foot3D, for model building and evaluation, which we release to the community.

##  3 Method

###  3.1 Foot3D Dataset

|  |  |  |
---|---|---|---|---
T-Pose |  Toe flexion |  Medial rotation, Dorsiflexion |  Toe Abduction |  Eversion
Figure 2: 5 feet from our dataset and corresponding pose descriptions. The
pose types are based on foot articulation described in foot anatomy literature
[Houglum and Bertoti(2011)] \- further details in the supplementary

We produce Foot3D, a dataset of high resolution, textured 3D human feet. For
acquisition, we use an Artec Leo 3D scanner [Art(a)], which has a 3D point
accuracy of up to 0.1mm. A total of 61 scans of the left feet on 34 subjects
in a variety of poses was collect. To capture the entire surface of the foot,
subjects would sit with their leg on a table for stability, and their foot
suspended over the edge, this allowed the scanner to view the entire foot
surface. Subjects then hold a static pose for approximately 2 minutes while
the scan takes place. Details of the nature of the articulation requested can
be found in the supplementary.

####

The raw data is then processed in Artec Studio 16 [Art(b)] to produce 100K
polygon meshes. These meshes were cut-off at an approximate position on the
shin, by a plane approximately perpendicular to the leg. Next, we leverage
this slice-plane as a basis for loosely registering all of the meshes in the
dataset, such that these planes are parallel to the XY-plane and a vector from
the slice plane to the foot’s centroid lies in the X direction. We then slice
each foot at a uniform height above the heel to provide a consistent ankle
length. Figure 2 shows dataset samples, and further details about this process
are provided in the supplementary.

###  3.2 The FIND model

Figure 3: Our implicit surface field is an MLP which takes as input a 3D point
queried on a template mesh’s surface, and texture, shape and pose embeddings,
and provides as output a vertex colour value and displacement.

The FIND model is controlled using a shape, pose and texture embedding to
produce a surface deformation and colour field over a template mesh. This
function is implemented as an implicit coordinate based neural network. The
architecture is now explained in detail.

#### Positional Encoding.

Texture and shape have both high frequency and low frequency information
across a foot’s surface. As a result, we leverage positional encoding to allow
for the network to capture the higher frequency signals [Mildenhall et
al.(2020)Mildenhall, Srinivasan, Tancik, Barron, Ramamoorthi, and Ng, Rahaman
et al.(2019)Rahaman, Baratin, Arpit, Draxler, Lin, Hamprecht, Bengio, and
Courville],

$\gamma_{N}(x)=\\{x,\sin(2\pi x),\cos(2\pi x),\dots,\sin(2N\pi x),\cos(2N\pi
x)\\}$ (1)

#### Network.

After we encode our position with $N=10$, we learn a network $F$ with weights
$\theta$, which takes shape, pose and texture encodings $z_{s}$, $z_{p}$ and
$z_{t}$, and predicts vertex offsets $\Delta x$ relative to a template mesh
and vertex colours $c$,

$F(\theta,\gamma(x),z_{s},z_{p},z_{t})\rightarrow(\Delta x,c)$ (2)

The main body of the network is 4 $256\times 256$ linear layers, encoding the
positional encoding $\gamma(x)$ into a feature, $g$.

#### Heads.

We concatenate $g$ with the relevant shape codes and pass to a head specific
for each task. The colour head receives $g$ and the texture code $z_{t}$, and
the displacement head receives $g$ and the shape and pose codes, $z_{s}$ and
$z_{p}$. Both heads are composed of 3 $256\times 256$ linear layers. We choose
$\tanh$ as our activation function, and normalize outputs such that colour
values are in the range $[0,1]$, and displacements in the range $[-0.1,0.1]$.

#### Defining the surface.

We define the surface over which the network is trained by using a template
mesh, which we take as a foot in our dataset outside of our training and
validation sets. The vertex positions $x$ are found by sampling across the
surface of a template mesh. At inference time, all of these vertices can be
used to output a full mesh.

#### Latent vectors.

The shape, texture and pose encodings, all of size 100, are jointly learned
during training. We place the constraint that texture and shape codes must be
shared between different scans of the same foot. We also learn a per-foot
registration to the ground truth: an Euler rotation $r\in\mathbb{R}^{3}$,
global translation $t\in\mathbb{R}^{3}$, and global XYZ scaling
$s\in\mathbb{R}^{3}$.

Figure 4: We show the effect of slicing and masking on our dataset and our
differentiable renderer.

###  3.3 Learning part-segmentation

Here we aim to develop an inference-time supervisory loss that learns part-
awareness in an unsupervised manner.

#### Differentiable rendering.

In order to optimise the model to fit to real images, it is necessary to be
able to differentiably render the model. We use PyTorch3D’s [Ravi et
al.(2020)Ravi, Reizenstein, Novotny, Gordon, Lo, Johnson, and Gkioxari]
differentiable renderer. As feet are not complete, watertight objects on their
own, we choose to mask out the slice plane applied to the feet models when
rendering, as we argue this better reflects how real feet appear in images
(post-segmentation). To achieve this, all faces on this slice plane are marked
when formulating the dataset, and these faces are masked out of the renders
produced by the differentiable renderer. Figure 4 shows the effect of this
process.

#### Extracting features.

Using the knowledge that 2D GANs can learn 3D geometrical information [Pan et
al.(2020)Pan, Dai, Liu, Loy, and Luo], we train a StyleGAN2 [Karras et
al.(2020)Karras, Laine, Aittala, Hellsten, Lehtinen, and Aila] network to
generate rendered feet from StyleGAN2 latent codes. We train on 100k 128x128
rendered images from models within Foot3D, with camera viewpoints in a
hemisphere above the foot. To use this network as an inference loss, invert
StyleGAN to obtain the codes using the Restyle [Alaluf et al.(2021)Alaluf,
Patashnik, and Cohen-Or] encoder. When passing a rendered image from our
differentiable renderer through the encoder-decoder, similar to [Zhang et
al.(2021)Zhang, Ling, Gao, Yin, Lafleche, Barriuso, Torralba, and Fidler], we
can use the features maps after the AdaIN layers of StyleGAN2 as a form of
high level information about the foot in the image.

#### Learning unsupervised 2D parts.

To learn foot parts from these feature maps, we observe that clustering them
with k-means produces meaningful part segmentations, as can be seen in Figure
5, which correspond between different feet. As a result, we train a simple
linear classifier to produce these per-pixel part segmentations from a subset
of the StyleGAN feature maps. The whole classification process is fully
differentiable.

#### Moving to 3D.

We then use this classifier to learn a per-vertex part probability on the
template mesh i.e. a $C$ length vector $c\in[0,1]$ where $C$ is the total
number of classes. These vectors can be directly ‘rendered’ using our
differentiable renderer to any new $C$ channel image producing a per pixel
foot part probability map.

#### Unsupervised part-based loss.

Using this part classification pipeline, we enforce a cross-entropy loss on
our $C$ cluster classes when optimising to images. Given our rendered part
probability map $P\in\mathbb{R}^{H\times W\times C}$, and our ‘ground-truth’
labels from our classifier $L\in\mathbb{R}^{H\times W}$, we define our loss,

$Loss_{\textrm{CE}}=-\frac{1}{H\times
W}\sum^{H}_{y=1}{\sum^{W}_{x=1}{\sum^{C}_{c=1}\ln{\frac{\exp{P_{y,x,L_{y,x}}}}{\sum^{C}_{i=1}\exp{P_{y,x,i}}}}}}$
(3)

Figure 5: 20 clusters on the 8th feature map of our Restyle encoder.

###  3.4 Training Losses

Our training loop and inference process uses several different losses:

#### Chamfer loss.

We sample 5000 vertices uniformly from the surface of each of the ground truth
and predicted meshes, to get sampled vertices $v_{\textrm{pred}}$ and
$v_{\textrm{gt}}$, and evaluate,

$L_{\textrm{chamf}}=\sum_{j}{\min_{i}{\left\lVert
v_{\textrm{gt},i}-v_{\textrm{pred},j}\right\rVert_{2}}}+\sum_{i}{\min_{j}{\left\lVert
v_{\textrm{gt},i}-v_{\textrm{pred},j}\right\rVert_{2}}}$ (4)

#### Smoothness loss.

We enforce a loss on the smoothness of the predicted mesh, which is a
combination of a Laplacian smoothness [Nealen et al.(2006)Nealen, Igarashi,
Sorkine, and Alexa] metric, and an average edge length regularizer.

#### Texture loss.

We sample 1000 vertices $v_{\textrm{gt}}$ uniformly over the ground truth mesh
surface, collect their colours $c_{\textrm{gt}}$, and enforce a loss
encouraging the network to map $v_{\textrm{gt}}$ to $c_{\textrm{gt}}$,

$L_{\textrm{tex}}=\left\lVert
c_{\textrm{gt}}-F(v_{\textrm{gt}},z_{s},z_{p},z_{t})_{\textrm{col}}\right\rVert_{2}$
(5)

#### Contrastive loss.

To encourage the latent pose space to meaningfully partition the data, we
enforce a contrastive loss between selected pairs of pose latent codes
$z_{p}$. Given the latent vectors $z_{p,1}$ and $z_{p,2}$, and corresponding
‘label’ vectors $l_{1},l_{2}$, we enforce the loss,

$L_{\textrm{contr}}=yd^{2}+(1-y)\max(M-d,0)^{2},\quad\textrm{where}\,d=\left\lVert
z_{p,1}-z_{p,2}\right\rVert_{2},y=l_{1}\cdot l_{2}$ (6)

where $M=0.5$ is a selected margin distance to encourage between dissimilar
pose vectors. Details of the formation of the labels $l$ can be found in the
supplementary.

###  3.5 Training

Our full training loop consists of three stages: (1) registration, in which we
optimise per-foot registration parameters to align our template mesh to the
$Foot3D$ dataset; (2) network training, in which we optimise network weights
$\theta$ and latent codes $z_{s}$, $z_{p}$, and $z_{t}$ in accordance with our
training losses; and (3) latent refinement, where we fix network weights
$\theta$ and fully optimise the latent codes to best match the data. In this
loop, we minimise:

$\begin{matrix}E_{\textrm{registration}}(r,t,s)=\lambda_{\textrm{chamf}}L_{\textrm{chamf}}\\\\[5.0pt]
E_{\textrm{network
training}}(\theta,z_{s},z_{p},z_{t})=\lambda_{\textrm{chamf}}L_{\textrm{chamf}}+\lambda_{\textrm{smooth}}L_{\textrm{smooth}}+\lambda_{\textrm{tex}}L_{\textrm{tex}}\\\\[5.0pt]
E_{\textrm{latent
refinement}}(z_{s},z_{p},z_{t})=\lambda_{\textrm{chamf}}L_{\textrm{chamf}}+\lambda_{\textrm{smooth}}L_{\textrm{smooth}}+\lambda_{\textrm{tex}}L_{\textrm{tex}}\end{matrix}$
(7)

We select
$\lambda_{\textrm{chamf}}=10^{4},\lambda_{\textrm{smooth}}=10^{3},\lambda_{\textrm{tex}}=1$,
and train our network and latent parameters with an Adam optimizer [Kingma and
Ba(2014)], with a learning rate of $5\times 10^{-5}$.

#### Levels of supervision.

We are able to train our model with three levels of supervision, to provide
better pose disentanglement and control: (i) Fully unsupervised, training only
from mesh data; (ii) Using foot identity as a weak label, fixing shape between
feet of the same identity for disentanglement of shape and pose; (iii) Adding
pose descriptions as an additional weak label to allow for a more
interpretable pose space via our contrastive loss.

##  4 Experiments

#### Foot3D dataset.

Our training set contains 27 feet in which the users were instructed to hold a
neutral, ‘T-pose’ position, and 51 feet total, including articulated feet. We
use a further 8 feet for validation, of which 4 are ‘T-pose’.

#### Evaluation metrics.

We evaluate the shape accuracy of our models using three metrics: a mean
chamfer loss between a predicted model and the ground truth (as in Equation
4); a mean euclidean distance between 6 hand-labelled keypoints on the model’s
template mesh and the ground truth meshes; and an Intersection over Union
(IoU) metric on silhouttes produced from 50 rendered viewpoints sampled
uniformly in an arc across the foot.

#### Baseline.

In order to compare our method with typical PCA implementations, we generate a
PCA model. To do this without dense correspondences, we first use a FoldingNet
[Yang et al.(2018)Yang, Feng, Shen, and Tian] implementation fitted to all of
our data to produce a set of 1600-vertex point clouds with learned weak
correspondences, and fit a PCA model to this data.

##  5 Results

Figure 6: Our network can be sampled at different resolutions for lower-
resolution applications (_e.g_ mobile phones), or higher-resolution
applications. Query times are shown on a Macbook M1 CPU. Figure 7: Here we
show 5 samples within the latent spaces of shape, pose and texture. Shape
variations visible are toe lengths, separation and arch of the foot; pose
variations visible are toe flexion and foot plantarflexion.

#### Multi-resolution.

As a coordinate based model, our neural surface deformation field can be
sampled at as many 3D vertices as desired to produce an output mesh. As
visualised in Figure 6, we leverage this to sample at high resolutions for
precise applications, or at lower resolutions for memory and CPU-critical
applications, such as mobile phones. This requires template mesh to be
reproduced at different resolutions, possible through off-the-shelf mesh
simplification and subdivision tools [Cignoni et al.(2008)Cignoni, Callieri,
Corsini, Dellepiane, Ganovelli, and Ranzuglia].

Figure 8: We visualise the FIND pose space on all training data in its two
principal directions. We apply k-means clustering with 4 clusters to show
partitioning of the space. We identify feet with poses in one of four pose
categories to showcase the effective pose space partitioning due to our
contrastive loss.

#### Latent spaces.

Visualised learned shape, pose, and texture spaces of FIND are shown in Figure
7. Observe that our model has learned to disentangle shape and pose with
minimal supervision using the simple constraint of fixing shape and texture
codes between scans of feet belonging to the same person. Furthermore, we show
in Figure 8 that our contrastive loss ensures pose space is suitably segmented
according to pose type.

Top Side Zoom |  Top Side Zoom |  Top Side Zoom
---|---|---
(a) GT |  (b) PCA |  (c) Ours
Figure 9: Qualitative results of our 3D fits to 8 validation feet, rendered from 3 views each. Model | Trained on |  Chamfer, µm $\downarrow$ |  Keypoint, mm $\downarrow$ |  IoU $\uparrow$
---|---|---|---|---
PCA | T-Pose only | 12.9 | 15.0 | 0.892
PCA | All | 11.7 | 15.7 | 0.892
Ours | T-pose only | 9.0 | 6.4 | 0.923
Ours | All | 7.3 | 5.9 | 0.931
Table 1: Comparison of FIND and a PCA baseline on our 8 validation feet.

#### 3D evaluation.

We compare our model to the PCA baseline. For both, we train 2 models: one on
the 27 ‘T-pose’ feet in the training dataset, and one on all 51 feet. Our
‘T-pose‘ trained model is restricted from learning a pose space. Table 1 shows
a significant quantitative improvement over the baseline, and Figure 9 shows
qualitative results. Notice our model does a much better job of toe
separation.

| 2 view | 5 view
---|---|---
Optimisation loss |  Chamfer, µm $\downarrow$ |  Keypoint, mm $\downarrow$ |  Chamfer, µm $\downarrow$ |  Keypoint, mm $\downarrow$
Sil | 81.8 | 14.4 | 16.8 | 7.7
Sil + VGG | 78.7 | 13.1 | 15.9 | 7.3
Sil + CE Loss | 45.8 | 10.3 | 15.7 | 6.4
Table 2: Quantitative results on inference pipeline on our validation feet. Model |  Chamfer, µm $\downarrow$ |  Keypoint, mm $\downarrow$ |  IoU $\uparrow$
---|---|---|---
Full model | 7.3 | 5.9 | 0.931
$-$ Pose vector | 8.0 | 6.2 | 0.926
$-$ Texture loss | 7.6 | 6.8 | 0.928
Table 3: Ablation study.

#### Inference evaluation.

We show in Table 2 our results of optimising FIND to renderings of validation
feet, using only 2D losses to fit the registration and latent parameters. In
our experiments, we sample 2 and 5 viewpoints in an arc around the foot. We
compare three training losses: silhouette only; silhouette + an off-the-shelf
VGG-16 [Simonyan and Zisserman(2014)] perceptual loss; and silhouette + our
novel Cross Entropy (CE) unsupervised part-based loss. We show that our loss
improves the quality of these reconstructions. We find that our loss has a
more substantial improvement in the recovered shape when fewer views are
available for optimisation - we see this as likely being due to the extra 3D
information this loss has available about unseen viewpoints.

###  5.1 Ablation study

We investigate in Table 3 the effect on our full model of several features. We
observe that the shape and pose disentanglement improves the shape
reconstruction quality. Additionally, the texture loss, which is only able to
learn from vertex colours, is also responsible for some of the reconstruction
quality. We predict that this might be due to the surface colours providing
supervision for vertex correspondences.

##  6 Conclusion

We demonstrate that our neural implicit foot model, FIND, outperforms typical
PCA implementations of statistical shape learning, providing a higher fidelity
shape space, whilst also able to learn texture and pose with minimal
supervision. Furthermore, we have shown that simple foot identity and pose
labels can be used to disentangle pose and shape, and produce a meaningful,
interpretable pose space. Finally, we have shown that our unsupervised part-
based loss improves upon typical shape model inference methods by enforcing
part consistency.

##  7 Acknowledgements

The authors acknowledge the collaboration and financial support of Trya Srl.

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