# Bose-Hubbard realization of fracton defects

Krzysztof Giergiel  Institute of Theoretical Physics, Jagiellonian University,
30-348 Kraków, Poland  Ruben Lier  Max Planck Institute for the Physics of
Complex Systems, 01187 Dresden, Germany  Piotr Surówka
<EMAIL_ADDRESS>Max Planck Institute for the Physics of Complex
Systems, 01187 Dresden, Germany  Department of Theoretical Physics, Wrocław
University of Science and Technology, 50-370 Wrocław, Poland  Institute for
Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam, The
Netherlands  Dutch Institute for Emergent Phenomena (DIEP), University of
Amsterdam, 1090 GL Amsterdam, The Netherlands  Arkadiusz Kosior
<EMAIL_ADDRESS>Institute for Theoretical Physics, University of
Innsbruck, 6020 Innsbruck, Austria

###### Abstract

Bose-Hubbard models are simple paradigmatic lattice models used to study
dynamics and phases of quantum bosonic matter. We combine the extended Bose-
Hubbard model in the hard-core regime with ring-exchange hoppings. By
investigating the symmetries and low-energy properties of the Hamiltonian we
argue that the model hosts fractonic defect excitations. We back up our claims
with exact numerical simulations of defect dynamics exhibiting mobility
constraints. Moreover, we confirm the robustness of our results against
fracton symmetry breaking perturbations. Finally we argue that this model can
be experimentally realized in recently proposed quantum simulator platforms
with big time crystals, thus paving a way for the controlled study of many-
body dynamics with mobility constraints.

##  I Introduction

Interacting many-body systems exhibit a variety of collective phenomena and
different phases of matter. They are usually accompanied by emergent
quasiparticles, whose properties differ from the elementary excitations. One
example of such a behavior is given by fractons - emergent quasiparticles that
lack the ability to move, either completely or partially. Models with such
excitations fall into two distinct categories: gapless and gapped. Gapless
excitations appear in higher-rank gauge theories that emerge in the
description of spin-liquids Xu (2006); Xu and Hořava (2010); Pretko (2017a,
b); You  _et al._ (2020a), dipole-conserving lattice models Pai  _et al._
(2019); Gromov  _et al._ (2020); Feldmeier  _et al._ (2020); Morningstar  _et
al._ (2020); Iaconis  _et al._ (2021); Moudgalya  _et al._ (2021), elasticity
Pretko and Radzihovsky (2018); Gromov (2019a); Kumar and Potter (2019); Pretko
and Radzihovsky (2018); Pretko  _et al._ (2019); Zhai and Radzihovsky (2019);
Gromov and Surówka (2020); Nguyen  _et al._ (2020); Manoj  _et al._ (2021);
Surówka (2021) and hydrodynamics Doshi and Gromov (2021); Grosvenor  _et al._
(2021); Glorioso  _et al._ (2021). In the gapless theories fractons can be
understood as charges that act as sources to the gauge fields and can be
interpreted as topological defects. Gapped systems hosting fractons have been
identified as certain exactly solvable models Chamon (2005); Haah (2011);
Vijay  _et al._ (2015, 2016); Bravyi  _et al._ (2011); Yoshida (2013) and
Chern-Simons gauge theories You  _et al._ (2020b); Ma  _et al._ (2020); Prem
_et al._ (2018); Pretko (2017c). They display topological ground state
degeneracy accompanied by a unique entanglement structure. Unfortunately all
these models are rather complicated which obstructs either their experimental
realizations or even a detailed numerical study. In order to remedy this we
construct a Bose-Hubbard type Hamiltonian in two dimensions with nearest
neighbor interactions that is feasible to the numerical analysis and has a
potential of being engineered and experimentally studied in recent state-of-
the-art quantum simulator platforms with big time crystals Giergiel  _et al._
(2021) (for reviews see Ref. Sacha and Zakrzewski (2018); Guo and Liang
(2020); Sacha (2020)).

The immobility of fracton excitations can be encapsulated in a generalized set
of global conservation laws, that preserve various multipole moments of the
charge density in addition to the total charge of the system. These
conservation laws arise as a consequence of subsystem symmetry of the
Hamiltonian. The question that we want to address is how to realize extended
symmetries in a realistic Bose system with two-body interactions between the
bosonic constituents taken into account? The standard Bose-Hubbard Hamiltonian
is invariant under a global $U(1)$ symmetry. This means that a constant shift
of the phase leave the theory invariant. As shown in Ref. Gromov (2019b) a
natural generalization of this symmetry to account for higher moment
conservation is to employ the so-called space-dependent polynomial shifts
Pretko (2017a, b). The shift symmetries are broken by the usual hopping terms,
which is natural since the symmetries encode the restricted mobility of
excitations. Hence, the inclusion of constraints on the movement of particles
in the model requires hopping terms that are engineered to account for fracton
mobility constraints. One example of such terms is the ring-exchange term,
which has been extensively studied in a context of exciton Bose liquids
Paramekanti  _et al._ (2002); Tay and Motrunich (2011a); Xu and Fisher (2007);
Balents and Paramekanti (2003) and more recently in the context of fractons
You  _et al._ (2020b, 2021). However, due to the failure of the mean-field
treatment, the ring-exchange models are rather difficult to handle
numerically, especially in the frustrated regime where quantum Monte Carlo
algorithms cannot be applied Tay and Motrunich (2011b); Huerga  _et al._
(2014).

On the other hand, in this work we consider an extended Bose-Hubbard
Hamiltonian at half filling with ring-exchange interactions but on $1\times 2$
and $2\times 1$ plaquettes in a strongly repulsive limit, where the low energy
physics can be efficiently studied due to energetic constraints. This model
provides a complementary realization of gapped fractons. Furthermore, as we
argue below, the defect dynamics in this model resembles the constrained
motion of defects seen for example in elasticity Cvetkovic  _et al._ (2006).
While the ground state of the system realizes a checkerboard charge density
wave (CDW), by means of analytical and numerical analysis we find that the
first excited band consists of fracton excitations (lineons) with the
restricted mobility to one dimensional columns (or rows) of the two
dimensional lattice. At the same time, we show that the second excited band
hosts both completely immobile excitations and two joint excitations which can
move freely on the entire lattice. In the later parts we investigate the
robustness of our findings and concentrate on the experimental realization of
the model.

##  II The model

Figure 1: An illustration of all possible hopping and interaction processes in
the model on a square 2D lattice: standard nearest neighbor hopping $t_{1}$,
ring-exchange terms $t_{2}$ and the nearest neighbor interaction $V$. The
ring-exchange interaction we are considering consists of a simultaneous
tunneling of two particles to the second neighbors along $1\times 2$ and
$2\times 1$ plaquettes. Site labels A and B denote two bipartite sublattices
of a square lattice. Throughout the article we are considering strongly
repulsive regime at half filling where, in the ground state, all the particles
fill completely only one sublattice realizing the checkerboard charge density
wave (CDW) order. The focus of our analysis is put on the lowest lying excited
energy bands consisting of fracton excitations.

Bose-Hubbard models are simple but powerful models to investigate quantum
phases and many-body dynamics of collective excitations Jaksch and Zoller
(2005); Sachdev (2011); Dutta, O. and Gajda, M. and Hauke, P. and Lewenstein,
M. and Lühmann, D.-S. and Malomed, B. A. and Sowiński, T. and Zakrzewski, J.
(2015). Therefore Bose-Hubbard incarnation of fracton quasiparticles is
desirable to understand the dynamics of these excitations. In order to achieve
this let us consider a tight binding hard-core boson Hamiltonian with
repulsive interactions on a two dimensional (2D) square lattice with periodic
boundary conditions,

$\displaystyle\hat{H}=-t_{1}\sum_{\langle
ij\rangle}\hat{b}^{\dagger}_{i}\hat{b}_{j}+\frac{V}{2}\sum_{\langle
ij\rangle}\hat{n}_{i}\hat{n}_{j}-t_{2}\sum_{[ijkl]}\hat{b}^{\dagger}_{i}\hat{b}^{\dagger}_{j}\hat{b}_{k}\hat{b}_{l},$
(1)

with $\hat{b}_{i}$ ($\hat{b}^{\dagger}_{i}$) denoting the standard anihilation
(creation) operator of a bosonic particle localized on the $i$-th site of the
lattice, $\hat{n}_{i}=\hat{b}^{\dagger}_{i}\hat{b}_{i}$ is the particle number
operator. The first two terms of the Hamiltonian (1) describe a familiar hard-
core boson extended Bose-Hubbard model Dutta, O. and Gajda, M. and Hauke, P.
and Lewenstein, M. and Lühmann, D.-S. and Malomed, B. A. and Sowiński, T. and
Zakrzewski, J. (2015) with $t_{1}$ and $V$ being the tunneling amplitude and
the nearest neighbor interaction strength respectively. The symbol $\langle
ij\rangle$ indicate the sum over nearest neighbors on the lattice. The last
term contains simultaneous tunneling processes of two particles, dubbed the
ring-exchange interaction Paramekanti  _et al._ (2002); Sandvik  _et al._
(2002); Rousseau  _et al._ (2004); You  _et al._ (2018), with a hopping
amplitude $t_{2}$. The ring-exchange interaction we are considering is a
combination of two second neighbor hoppings along $1\times 2$ and $2\times 1$
plaquettes which preserves the center of mass of two particles and the number
of particles in bipartite sublattices [cf. Fig. 1 for an illustration]. The
symbol $[ijkl]$ denotes the summation over all possible plaquettes. Throughout
this article we will be considering strongly repulsive limit, i.e. $V\gg
t_{1},t_{2}$, where the ground state realizes the checkerboard charge density
wave (CDW) order. A checkerboard CDW is a single Fock state where all of the
particles occupy one of the two bipartite sublattices. While in the next
section we focus on the symmetries and conservation laws of this model which
are necessary for the appearance of fractons, let us stress that the
underlying CDW order stabilizes the fracton excitations we find in the lowest
lying excited energy bands.

##  III Symmetries of the model

In this section we show that the extended Bose-Hubbard Hamiltonian with the
ring-exchange term (1) possess certain symmetries which enable the model to
host fracton excitations. Our analysis will be phrased in the language of
multipole algebras constructed in Ref. Gromov  _et al._ (2020) (see also Ref.
Griffin  _et al._ (2015)). The basic ingredient of this construction is the
dipole moment conservation, which follows from the so-called polynomial shift
symmetries. These shift symmetries act on a scalar field $\varphi$ that in our
case corresponds to the phase associated with creation and annihilation
operators. Before we proceed let us recall basic facts about this
construction. We consider a low energy dynamics of the phase field given by
the action $S[\varphi]$. We assume that the action is invariant under the
following transformation

$\delta\varphi=\lambda_{\alpha}\mathcal{P}^{\alpha}(\vec{x})\,,$ (2)

where $\lambda_{\alpha}$ is a symmetry parameter and
$\mathcal{P}^{\alpha}(\vec{x})$ is a polynomial. It is enough for our purposes
to consider only homogenous polynomials. Furthermore we assume that our theory
is invariant under global shifts with a corresponding charge density
$\rho(\vec{x})$. It follows from the Noether theorem that charges

$Q^{(\alpha)}=\int d^{2}x\,\mathcal{P}^{\alpha}(\vec{x})\rho(\vec{x})\,.$ (3)

are conserved. Since the polynomials $\mathcal{P}^{\alpha}(\vec{x})$ are
homogenous the symmetry leads to a conservation of proper multipole moments.
The main ingredient of this abstract construction is the conservation of the
monopole and dipole charges.

$\displaystyle\frac{dQ}{dt}=\frac{d}{dt}\int d^{2}x\,\rho=0,$ (4a)
$\displaystyle\frac{dQ_{i}}{dt}=\frac{d}{dt}\int d^{2}x\,x_{i}\rho=0,$ (4b)

where $x_{i}\in\\{x,y\\}$. In addition, if we assume that the system is
translationally invariant, the momentum is also conserved

$\frac{dP_{i}}{dt}=\frac{d}{dt}\int d^{2}x\,p_{i}=0.$ (5)

In order to see the monopole and dipole conservation on a lattice, we first
point out that a 2D theory has to be invariant under the following symmetry
transformation

$\delta\varphi=\lambda_{0}+\lambda_{1}x+\lambda_{2}y,$ (6)

with $\lambda_{\alpha=1,2}$ being symmetry parameters as in Eq. (2).
Therefore, we need to check if the Bose-Hubbard model respects these
symmetries. The interaction terms depend on the densities and the symmetry
transformation in Eq. (6) follows immediately. We therefore only need to check
the symmetry transformations of the hopping term. Of course the usual hopping
implies full mobility and as such breaks the polynomial shift symmetry.
However, the ring-exchange term preserves it. This can be seen explicitly by
making a phase shift of the form

$\displaystyle\varphi\rightarrow\varphi+\zeta(x)+\chi(y),$ (7)

which is a specific local U(1) phase transformation, that changes along rows
or columns Gromov  _et al._ (2020); Xu and Fisher (2007), more general than
the polynomial shift symmetry of Eq. (6). The non-local ring-exchange term is
invariant under the above phase transformation, which can be shown explicitly

$\displaystyle\begin{split}&\hat{b}^{\dagger}_{i}\hat{b}^{\dagger}_{j}\hat{b}_{k}\hat{b}_{l}\rightarrow\hat{b}^{\dagger}_{i}\hat{b}^{\dagger}_{j}\hat{b}_{k}\hat{b}_{l}\cdot
e^{-i\zeta(x_{1})-i\chi(y_{1})}e^{-i\zeta(x_{2})-i\chi(y_{2})}\\\ &\times
e^{i\zeta(x_{2})+i\chi(y_{1})}e^{i\zeta(x_{1})+i\chi(y_{2})}=\hat{b}^{\dagger}_{i}\hat{b}^{\dagger}_{j}\hat{b}_{k}\hat{b}_{l},\end{split}$
(8)

where we have used the fact that the ring-exchange interactions preserve the
center of mass of the tunneling pair, cf. Fig. 1. Note that our symmetry
transformation in Eq. (7) implies a conservation of additional moments.
Similar phenomenon occurs in other models as well. For example, the traceless
scalar theory Pretko (2017b), in addition to the conserved charges $Q,Q^{i}$,
also conserves the moment $Q^{(2)}=\int d^{2}x||\mathbf{x}||^{2}\rho$ and
elasticity conserves one component of the quadrupole moment Pretko and
Radzihovsky (2018). The dipole moment conservation implies that an isolated
charge cannot move. However, if additional moments are preserved it may put
additional constraints on the movement of excitations. Conservation of the
quadrupole moment implies that a dipole can only move in the direction
perpendicular to its vector charge. In two dimensions such a movement is along
a line and thus the quasiparticles giving rise to the dipole charge are called
lineons.

In fact, the extended Bose-Hubbard model with any number of ring-exchange
interaction terms regardless of their plaquette sizes $n_{1}\times n_{2}$
conserves a component of every $n$-th higher-moment of a given Cartesian
coordinate

$\frac{dQ_{i\ldots i}^{(n)}}{dt}=\frac{d}{dt}\int
d^{2}x(x_{i})^{n}\,\rho(\vec{x})=0,$ (9)

where $n\in\mathbb{N}_{0}$. One can actually choose for $\zeta(x)$ and
$\chi(y)$ a family of functions which form a basis for any integrable
functions of $x$ or $y$ over their respective domains. Completeness of the
basis states allows one to restate this fact as conservation of marginal
distributions of charge:

$\displaystyle f_{1}(x)$ $\displaystyle=\int dy\rho(\vec{x}),$ (10a)
$\displaystyle f_{2}(y)$ $\displaystyle=\int dx\rho(\vec{x}),$ (10b)

where functions $f_{1}$ and $f_{2}$ are conserved under ring-exchange
interactions. It turns out the properties of fracton excitations can be easily
understood by looking at the marginal distributions only, which we will
elaborate on in the next section.

So far we have not used the fact the ring-exchange interactions in our model
take place along rectangular $1\times 2$ and $2\times 1$ plaquettes. Such a
plaquette choice is motivated by the presence of an additional symmetry which
implies the conservation of the number of particles in each bipartite
sublattice

$[N_{A},H]=[N_{B},H]=0.$ (11)

Therefore, as long as the nearest neighbor interaction strength $V$ is a
dominant energy scale, the energy spectrum is divided into isolated bands
composed of Fock states with the same interaction energy. While the ground
state is a simple CDW state, its first two excited bands consists of states
with interaction energy $3V$ and $4V$ that differ from the CDW state by a
single or double dislocations [cf. Sec. IV and Fig. 2].  Let us also stress
that although any system with the ring-exchange and the density-density
interactions obeys subsystem symmetries leading to the multipole moments
conservation, not all of such processes will lead to the mobility restrictions
of localized excitations. In our case, due to the conservation of the number
of particles in each bipartite sublattice, the ring-exchange term connects
configurations with the same nearest neighbor interaction energy. As such, it
constitutes a minimal Bose-Hubbard model with the ring-exchange interactions
hosting single fractonic excitations.  Due to the conservation laws discussed
in this section, which are exact for $t_{1}=0$ and approximate for $t_{1}\neq
0$,  the Hilbert space of dislocations is fragmented into disjoint sectors. As
we show in the next section the latter enforces the mobility restrictions of
the excitations. We note that this fragmentation is distinct from the
dynamical Hilbert space shattering Khemani  _et al._ (2020); Sala  _et al._
(2020); Iaconis  _et al._ (2021); Li  _et al._ (2021) where fragmentation
happens for fixed symmetry sectors due to dynamical constraints, which are
responsible, among others, for the subdiffusion in the tilted Fermi-Hubbard
models Guardado-Sanchez  _et al._ (2020); Scherg  _et al._ (2021).

Figure 2: Schematic view of the CDW dislocation movement due to ring-exchange
interactions. White circles show particles missing from the checkerboard
pattern and red circles show particles present in a sublattice, which is empty
in the ground state. A red cross indicates a prohibited hopping. When the
system is in the ground state (GS) there are no dislocations and no hopping
can take place. For non-local dislocations, hopping is still not possible, and
hence the motion of such excitations is zero-dimensional (0D). For a single
local dislocation one has one-dimensional (1D) motion and for a double local
dislocation one has two-dimensional (2D) motion.

##  IV Fracton excitations

Before we turn to the description of fracton excitations and their dynamics we
start this section by looking at lowest energy excitations of the model in a
strongly interacting regime $V\gg t_{1},t_{2}$ we consider in this work. These
lowest energy excitations can be constructed by looking at single particle
dislocations in the CDW ground state. A single dislocation state is created
when one creates a particle on an empty site in the CDW state and also
annihilates a particle on the occupied sublattice to preserve half filling.
The space of lowest energy excitations consists of single local particle
dislocations

$\displaystyle\left(D^{x}_{ij}\right)^{\dagger}|\textrm{CDW}\rangle$
$\displaystyle=\left(\hat{b}_{i+1,j}^{\dagger}\hat{b}_{i,j}+\hat{b}_{i,j}^{\dagger}\hat{b}_{i+1,j}\right)|\textrm{CDW}\rangle,$
(12a) $\displaystyle\left(D^{y}_{ij}\right)^{\dagger}|\textrm{CDW}\rangle$
$\displaystyle=\left(\hat{b}_{i,j+1}^{\dagger}\hat{b}_{i,j}+\hat{b}_{i,j}^{\dagger}\hat{b}_{i,j+1}\right)|\textrm{CDW}\rangle,$
(12b)

whose interaction energy is $3V$. Note that the above definition is
indifferent to the choice of a filled sublattice in the charge density wave.
If the dislocation is not local i.e. the two sites that deviate from the CDW
state are not nearest neighbors, then such a state belongs to the second
excited energy band with an eigenenergy of $4V$. The second excited energy
band is completed with states that compose of two neighboring local
dislocations

$\displaystyle|D^{(2)}_{ij}\rangle$
$\displaystyle=\left(D^{x}_{ij}\right)^{\dagger}\left(D^{x}_{i,j+1}\right)^{\dagger}|\textrm{CDW}\rangle$
$\displaystyle=\left(D^{y}_{ij}\right)^{\dagger}\left(D^{y}_{i+1,j}\right)^{\dagger}|\textrm{CDW}\rangle.$
(13)

Other states, such as non-neighboring double dislocations, belong to higher
energy bands and are beyond the scope of interest of this article. In the next
parts we argue the single dislocation states correspond to either completelely
immobile zero dimensional fractons or partially mobile lineons (one
dimensional fractons), whereas the two dislocation state
$|D^{(2)}_{ij}\rangle$ can be interpreted as two bound lineon states which is
allowed to move in both directions, see Fig. 2 for an illustration. First, we
support our arguments by constructing approximate eigenstates of the model in
the $t_{1}\ll t_{2}$ limit and then we numerically study the robustness of
fracton excitations using exact time propagation.

Figure 3: Exact time propagation of elementary dislocations. While panels
(a-c) show a propagation of a single local dislocation, panels (d-f)
illustrate a motion of a bound double dislocation. Each row is associated with
time point $\tau=0$, $\tau=3.5/t_{2}$ and $\tau=7/t_{2}$. The panels confirm
the analytical arguments of the main text that a single local dislocation can
propagate only in one dimension but two bound dislocation state propagates
freely in two dimension. The movement of the later is twice slower compared to
a single dislocation state. The lattice size is 48x48 sites, each site is
represented by a single pixel. Positive probability (blue) shows particle
density in empty lattice, as compared to the CDW ground state. Negative
probability (red) is used to show hole density, as compared to the CDW ground
state. The color scale is cut at low values to be consistent along all images
in a column. Model parameters are $t_{1}=0$, $t_{2}=0.01V$.  Figure 4:
Stability of the fractonic behavior of a single initial dislocation as seen in
the top-left panel of Fig. 3. The color scale represents root sum of squared
differences of initial and final marginal distributions. The plotted
probability is the value achieved during time evolution on $20\times 20$
lattice for time $\tau=2/t_{2}$. $V=1$ is assumed. Figure 5: Exact time
propagation for a single dislocation at time $\tau=4/t_{2}$ for $t_{2}=1V$ and
(a) $t_{1}=0$, for (b) $t_{1}=0.2V$ and for (c) $t_{2}=0.1V$ and $t_{1}=0.2V$.
The lattice size is 20x20 sites, each site is represented by a single pixel.

###  IV.1 Fracton eigenstates

Let us first look at the eigenstates of the Hamiltonian when normal hopping is
set to zero, i.e. $t_{1}=0$. First of all, since the ring-exchange interaction
is a combination of two simultaneous local particle hoppings, the non-local
dislocation state is a completely immobile eigenstate of the model with the
eigenenergy equal to the interaction energy $4V$ and corresponds to two
isolated fractons, with one being a particle, and the other one a hole [cf.
Fig. 2]. The first mobile excitation one can have is a fracton dipole which
can be constructed from single local dislocation states [cf. Eq. (12a)-(12b)].
The fracton dipoles can only move orthogonally to the dipole moment You  _et
al._ (2020a, 2018); Pretko (2017b). In two dimensions that means the state is
only allowed to move along a line and thus these states are also called
lineons [cf. Fig. 2]. The momentum state of fracton dipole with momentum
orthogonal to the dipole moment pointing in the $y$-direction reads Sous and
Pretko (2020)

$\displaystyle|k_{x},m\rangle=\frac{1}{\sqrt{L}}\sum_{n}e^{2iank_{x}}\left(D^{y}_{2n,m}\right)^{\dagger}|\textrm{CDW}\rangle,$
(14)

where $L=L_{x}=L_{y}$ is the linear lattice size, $a$ is the lattice constant.
The state $|k_{x},m\rangle$ is an eigenstate of the Hamiltonian, which we can
check explicitly, i.e.,

$\displaystyle\frac{1}{2}\sum_{\langle
kl\rangle}\hat{n}_{k}\hat{n}_{l}|k_{x},m\rangle$
$\displaystyle=3\,|k_{x},m\rangle,$ (15)
$\displaystyle\sum_{[klmn]}\hat{b}^{\dagger}_{k}\hat{b}^{\dagger}_{l}\hat{b}_{m}\hat{b}_{n}|k_{x},m\rangle$
$\displaystyle=2\cos(2k_{x}a)|k_{x},m\rangle.$ (16)

It is easy to find out that the same results will be given by an analogous
momentum state $|k_{y},n\rangle$ for dipoles pointing in the $x$-direction. We
thus conclude that the energy dispersion of the first excited band is 2L-fold
degenerate and given by

$\displaystyle E^{(1)}(k_{i},j)=3V-2t_{2}\cos(2k_{i}a).$ (17)

This restricted mobility of lineons can be also understood from the marginal
distributions. The charge density state has constant (featureless) marginal
distributions $f_{1}(x)$ and $f_{2}(y)$, Eq. (10a)-(10b). The $x$ dipole is
generated from CDW by moving particle in direction $x$, which leaves a hole
and creates a peak in $f_{1}(x)$ distribution and leaves $f_{2}(y)$
featureless. This means that there can be no movement in direction $x$ due to
ring-exchange interaction.

In a similar way we can construct the second kind of mobile excitations in the
lowest energy sector as a momentum state in a two bound dislocations subspace
[cf. Eq (13)],

$|k_{x},k_{y}\rangle=\frac{1}{L}\sum_{m,n}e^{i(k_{x}m+k_{y}n)a}|D^{(2)}_{ij}\rangle,$
(18)

which is an approximate eigenstate of the model with the eigenenergy

$\displaystyle E^{(2)}(k_{x},k_{y})\approx
4V-2t_{2}\left[\cos(k_{x}a)+\cos(k_{y}a)\right].$ (19)

The full mobility of this state can be explained by the fact that its marginal
distributions are featureless and look the same as for ground state. The
latter is understandable since, due to the indistinguishability of bosons, a
state $|D^{(2)}_{ij}\rangle$ can be viewed as two $x$ or two $y$ dipoles [cf.
Eq (13)].

If the two dislocations were far apart, then its interaction energy would be
equal to $6V$, twice the energy of a single lineon. However, two dislocations
close to each other form a bound state with an interaction energy $4V$. The
momentum state Eq. (18) is only approximately the eigenstate because in
principle two dislocations can dissociate when either moves in the opposite
direction, but this would require a surplus energy to break a bound state.
Only processes that keep the bound state together do not change its
interaction energy. Each ring-exchange interaction moves the center of mass of
a bound state by one lattice site, an hence the state propagates with twice
smaller group velocity $v=\mathrm{max}[\partial_{k_{i}}E^{(2)}]$.

###  IV.2 Time evolution of localized dislocations

In this section we first confirm numerically the considerations of the
previous section and then investigate the robustness of our findings by
looking at the time propagation of dislocations with non-zero normal hopping
on smaller lattices. The numerical time propagation is feasible when we are in
the strongly interacting regime of parameters and can cut the Hilbert space
using the energy criterion.

Figure 6: Exact time evolution of few fractons for $t_{1}=0$. Columns present
snapshot of the evolution for times $\tau=0,1,7$. Panels (a-c) show the
evolution of a double defect for $t_{2}=0.5V$ Here the value of $t_{2}$ is
much larger than in panels (d-f) in Fig 6, and because of this, one-
dimensional defects, which have a larger interaction energy, appear out of the
squared shape because of their larger group velocity. Panels (d-f) show the
evolution of two orthogonal separated single defects for $t_{2}=0.5V$. We see
that these defects move straight through eachother and no two-dimensional
mobility arises , but due to their interaction the time evolution is
asymmetric as the propagation is delayed after the collision. Panels (g-i)
show the scattering of a lineon on a zero-dimensional immobile fracton.

The Hilbert space dimension for $20\times 20$ lattice is then about $2\cdot
10^{6}$ for repulsive energies $E_{R}\leq 5V$, which is perfectly fine for
vector exponentiation using Krylov techniques Moler and Loan (2003). We first
discuss the ideal case at $t_{1}=0$ where the symmetries are perfectly
satisfied. In Fig. 3 a single and double defect is initialized and time-
evolution is studied. Panels (a-c) show that a localised local dislocation can
propagate only in a direction orthogonal to its dipole moment. On the other
hand, as shown in panels (d-f), two close dislocations form a bound state
which does not have any mobility restrictions.

Now we consider the case of a finite $t_{1}$. For $t_{1}\neq 0$, the defects
will eventually dissolve, because the normal hopping operator allows to
transition to states that have a different number of defects. However, our
considerations concern states that only locally deviate from the ground state.
Although the local dislocations will in principle decay for a finite
$V/t_{1}$, due the Lieb-Robinson Lieb and Robinson (1972) bound the speed of
the information propagation will be always finite ($\sim 2|t_{1}|$), and
therefore a state deviating locally from a ground state will never quantum
thermalize for an infinite system. For this reason, the stability of fractonic
excitations for $t_{2}\neq 0$ is based on the stability of the charge density.
A zero dimensional (0D) immobile excitation is an exact eigenstate due the
hardcore boson constraint and therefore it does not thermalize at all. Of
course, the lack of quantum thermalization does not imply that the 1D
fractonic behavior will be always present in all parameter regimes. To
guarantee the latter, the lineon group velocity $(4|t_{2}|)$ must an order of
magnitude greater than the Lieb-Robinson velocity, i.e. $|t_{1}|\ll 2|t_{2}|$.
To show the strength of this effect, in Fig. 4 we consider the degree to which
a single defect signature disappears from the marginal distribution after some
time $\tau=2/t_{1}$. We see that for larger values of $t_{1}$, the
destablizing effect of the normal hopping operator $t_{1}$ starts to become
noticeable past lines of $t_{1}\approx\pm 0.2t_{2}$, which agrees with our
theoretical preditions. Note that Fig. 4 is symmetric for $t_{1}\rightarrow-
t_{1}$, which is because the normal hopping operator plays the same role of
creating and annihilating defects regardless of the sign of $t_{1}$. There is
no such symmetry for $t_{2}\rightarrow-t_{2}$, because for finite $t_{1}$ and
negative $t_{2}$ there is frustration which is not the case for positive
$t_{2}$. Moreover, in Fig. 5, we show the evolution of single dislocations
after a time $\tau$ for some specific cases values of $t_{1}$ and $t_{2}$. It
is clear from comparing the ideal case of panel (a) to panel (b) that for a
small value of $t_{1}$ the mobility restrictions effectively remain, whereas
for large $t_{1}$ compared to $t_{2}$, as shown in (c), the behavior starts to
become two-dimensional, but for a finite evolution time it is still possible
to distinguish a one dimensional density profile.

Finally, we stress that our approach allows for the study of more complex
phenomena related to the evolution of fractonic defects, such as few and many
fracton interaction dynamics with and without the presence of disorder.
Although a detailed analysis of interaction phenomena goes beyond the scope of
this paper, in Fig 6 we present some basic results: panels (a-c) show the
decays of a two dislocation state into two orthogonal lineons, panels (d-f)
show lineon collision and panels (g-i) show a scattering of lineon on an
immobile (0D) fracton.

##  V Experimental proposal

A pleasant feature of the model (1) in a strongly repulsive regime at half
filling is that its low-energy physics can be studied analytically (for
$t_{1}=0$) and efficiently simulated numerically (for $t_{1}\neq 0$) which is
due to the energetic cut-off of the basis states. Nevertheless other parameter
regimes still present a challenge for a numerical analysis. On the other hand,
for a few decades now, quantum simulators have provided a fertile ground for
the study of many body physics. Recently it was also realized that condensed
matter phenomena can appear in time crystals. Although most theoretical
Wilczek (2012); Sacha (2015); Khemani  _et al._ (2016); Else  _et al._ (2016);
Yao  _et al._ (2017); Lazarides and Moessner (2017); Russomanno  _et al._
(2017); Ho  _et al._ (2017); Huang  _et al._ (2018); Iemini  _et al._ (2018);
Wang  _et al._ (2018); Zeng and Sheng (2017); Surace  _et al._ (2019); Mizuta
_et al._ (2018); Giergiel  _et al._ (2018); Kosior and Sacha (2018); Kosior
_et al._ (2018); Pizzi  _et al._ (2019); Liang  _et al._ (2018); Bomantara and
Gong (2018); Fan  _et al._ (2020); Kozin and Kyriienko (2019); Matus and Sacha
(2019); Pizzi  _et al._ (2021); Syrwid  _et al._ (2020a, b); Russomanno  _et
al._ (2020); Giergiel  _et al._ (2020); Wang  _et al._ (2021); Kuroś  _et al._
(2020) and experimental Zhang  _et al._ (2017); Choi  _et al._ (2017); Pal
_et al._ (2018); Rovny  _et al._ (2018a); Autti  _et al._ (2018); Kreil  _et
al._ (2019); Rovny  _et al._ (2018b); Smits  _et al._ (2018); Liao  _et al._
(2019); Autti  _et al._ (2021) works on time crystals have focused on the
period-doubled discrete time crystals, the upcoming gravitational bouncer
experiment Giergiel  _et al._ (2020) is expected to demonstrate big time
crystals, where discrete time translation symmetry broken states
$\phi_{n}(\tau)$, $n=1,\ldots,s$ evolve with a period $sT$, up to $s=100$
times longer than the periodicity of the drive $T$. The symmetry broken states
$\phi_{n}(\tau)=\phi_{n+1}(\tau+T)$ are localized wavepackets which evolve
periodically in the laboratory frame of reference along a classical trajectory
of the particle in the gravitational field. As such, they can be interpreted
as temporal analogs of the familiar Wannier states localized on temporally
equidistant points on the closed classical trajectory. Although the temporal
Wannier states $\phi_{n}(\tau)$ break time translation symmetry of the
periodic potential, they constitute a convenient basis to study many-body
temporal lattice models. Once the field operator $\hat{\psi}$ describing
particles in a periodically driven system is restricted to the states
$\phi_{n}(\tau)$, the effective Floquet Hamiltonian
$H_{F}=H(\tau)-i\partial_{\tau}$ describing the quasi-energy structure of the
system takes the form of a Bose-Hubbard Hamiltonian

$\displaystyle H_{F}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}d\tau\int
dxdy\;\hat{\psi}^{\dagger}\left(H_{0}(\tau)+\frac{g_{0}}{2}\hat{\psi}^{\dagger}\hat{\psi}-i\partial_{\tau}\right)\hat{\psi}$
(20) $\displaystyle\approx$ $\displaystyle
t_{1}\sum_{<i,j>}\hat{b}_{i}^{\dagger}b_{j}+\sum_{ijkl}U_{ijkl}\hat{b}^{\dagger}_{i}\hat{b}_{j}^{\dagger}\hat{b}_{k}\hat{b}_{l},$
(21)

where $H_{0}(\tau)=H_{0}(\tau+T)$ is a single particle’s time $T$-periodic
Hamiltonian, $\hat{\psi}\approx\sum_{n}\phi_{j}(\tau)\hat{b}_{j}$ is a field
operator with bosonic operators $\hat{b}_{j}$ annihilating a particle
occupying the $\phi_{j}$ state. The parameter $t_{1}$ is the tunneling
amplitude between neighboring sites and $U_{ijkl}$ is the interaction strength
of a two particle scattering process between sites $k,l$ and $i,j$. Note that
the quasi-energies of a time periodic problem are unbounded and defined up to
a shift by $\omega=2\pi/T$. Here we assume that $\omega$ is large compared to
other energy scales in a system so that an infinite Floquet matrix can be
reduced to a single diagonal block as in the effective Hamiltonian (20)
Eckardt (2017).

The above description can be generalised to all motional degrees of freedom of
a particle, which would result in multidimensional time crystals Sacha (2020).
In particular, a particle moving along closed periodic trajectories in two
orthogonal spatial directions in a gravitational bouncer model is described by
$\phi_{n,m}(x,y,\tau)=\psi^{(x)}_{n}(x,\tau)\psi_{m}^{(y)}(y,\tau)$, where
$\psi^{(x_{i})}_{n}(x_{i},\tau)=\psi_{n+1}^{(x_{i})}(x_{i},\tau+T)$. Another
relevant feature of the gravitational bouncer model is that the microscopic
parameters of $H_{F}$ can be in principle manipulated experimentally by means
of the Feshbach resonance Chin  _et al._ (2010) which modifies atomic
interactions and hence the coefficients $U_{ijkl}$. This gives unique
possibilities to investigate the full phase diagrams of various many body
lattice models experimentally.

In a recent Letter Giergiel  _et al._ (2021) it was shown that the
gravitational bouncer model with two periodically oscillating mirrors has
properties of a two-dimensional time crystal where the geometry of a temporal
lattice can be arbitrarily shaped. In particular, one example shows how to
design a Lieb lattice with a well separated middle flat energy band where the
dynamics is governed by a many-body Hamiltonian with long range density-
density and long-distance ring-exchange interactions fulfilling fracton
symmetries discusses in Section III. This physical system should host
fractons, but possesses additional complications, because the Bose-Hubbard
model analyzed in Ref. Giergiel  _et al._ (2021) exists on a Möbius strip and
the ring-exchange term does not exhibit the full lattice translation symmetry.

Here we overcome these difficulties by showing how to realize a minimally
extended Bose-Hubbard model hosting fractons, as in Eq. (1). The experimental
proposal relies on two basic steps:

  1. 1.

Effectively switching off the nearest neighbor tunnelings $t_{1}$ by building
a deep temporal lattice with well localized states on a 2D torus, where
$t_{1}$ are negligible.

  2. 2.

Selecting the relevant interactions by the Feshbach resonance.

In order to achieve this we choose a symmetric drive of two orthogonal mirrors
giving rise to a square $n\times n$ temporal lattice [cf Fig. 7(a)]. In the
first step only a subset of ring-exchange interactions are realised. However,
one can restore the full translation symmetry by adiabatically moving lattice
in the $x$ or $y$ direction [cf Fig. 7(b-c)]. After $n$ such steps, all of the
ring-exchange tunnelings are realized and the time averaging reproduces
exactly the Hamiltonian (1), which we analyze in the main part of this paper.
A schematic of the protocol is presented in Fig. 7(d). Note that the limit on
the speed of the adiabatic shift of the lattice is given by the energy
separation of the lowest and first excited quasi-energy bands, which is the
higher, the deeper is the lattice. This works in a favour of our experimental
proposal, as the fractons are expected in parameter range $t_{1}\approx 0$,
where the driving change can be the fastest.

Figure 7:  Schematic of the experimental protocol. Panels (a-c) illustrate
subsequent building blocks of the protocol, which is summarized on panel (d).
Note that for illustrative purposes each panel presents a relatively small
$5\times 5$ lattice, but experimentally relevant sizes can go up to $100\times
100$ sites [cf. Ref. Giergiel  _et al._ (2020) for accessible lattice sizes in
1D experiments]. Panel (a) shows a 2D time crystal with $2\times 1$ and
$1\times 2$ ring-exchange interactions. The axes $\theta_{x}$ and $\theta_{y}$
represent directions of the flat torus shown as black rectangle, with parts of
it repeated to illustrate the boundary conditions. Such a lattice is achieved
using drives:
$\text{drive}_{x}(\tau)=\text{drive}_{y}(\tau)=\cos(5\omega\tau)$. Note that
in the first step only a subset of the ring-exchange interactions is realized
and only a few of them are shown for illustrative purposes. Panel (b) shows
the adiabatic movement step, which is introduced to restore the full
translational symmetry of the effective Hamiltonian. This is achieved by
introducing a slow adiabatic shift in one of the drives, e.g.
$\text{drive}_{x}(\tau)=\cos[5\omega\tau+\theta_{0}(\tau)]$. This drive change
is stopped after the lattice is moved by a single position to the right
$\theta_{0}(\tau_{end})=2\pi$, which also means that the $x$ drive made one
more oscillation then the $y$ drive. After this step new set of ring-exchange
interactions is realized [panel (c)]. After $5$ such steps, we return to
original positions. Time averaged Bose-Hubbard Hamiltonian realizes exactly
the model (1) described in this article.

##  VI Conclusions and perspectives

We have studied an extended Bose-Hubbard model with ring-exchange interactions
in a strongly interacting limit at half filling. We have argued that this
model has a ground state that corresponds to a charge density wave and hosts
fracton excitations in the excited states, which appear due to mobility
constraints arising from symmetries and conservation laws of the model. We
have studied the dynamics of these excitations and concluded that a single
fractonic defect is either completely immobile or can propagate only in one
dimension (lineon). On the other hand, a bound state of two lineons is free to
propagate in all directions. We have argued that these properties can be also
understood in terms of conserved marginal charge distributions. We have also
addressed the stability issue, showing that these states are stable against
small symmetry breaking perturbations such as the usual nearest neighbor
hopping term. Finally we have devised an experimental proposal to realize the
fractonic extended Bose-Hubbard model in the upcoming platforms hosting big
time crystals. As a result our construction provides a clear path towards
understanding of many body dynamics with restricted mobility.

The most interesting future extension of this work involves the investigation
of the full phase diagram of our model. Although the analysis of this paper
has been restricted to the excited states of the charge density wave, it is
known that both a simple and extended Bose-Hubbard models can enter a
superfluid regime Jaksch and Zoller (2005); Sachdev (2011); Dutta, O. and
Gajda, M. and Hauke, P. and Lewenstein, M. and Lühmann, D.-S. and Malomed, B.
A. and Sowiński, T. and Zakrzewski, J. (2015) characterized by a spontaneously
broken $U(1)$ symmetry and associated Goldstone modes. The notion of a
fractonic analogues of superfluids was recently introduced in Yuan  _et al._
(2020); Chen  _et al._ (2021) (see also Argurio  _et al._ (2021)), using low-
energy Hamiltonians defined in continuum. It is therefore desirable to have a
microscopic understanding of such phases and a detailed study of phases and
phase transitions. Although arguments presented in Ref. Yuan  _et al._ (2020);
Chen  _et al._ (2021) based on the Mermin-Wagner theorem indicate that $d=3$
is the lowest stable dimension at zero temperature, a phase with a quasi long-
range order is possible in two spatial dimensions. This is similar to the
ordinary Bose-Hubbard model, which at zero temperature can have a quasi long-
range order for $d=1$ and true long-range order only for $d=2$ Lewenstein  _et
al._ (2017). However, the phase with a quasi long-range order will effectively
be a superfluid phase for small system sizes. A detailed numerical study can
shed light onto exotic superfluids which go beyond the mean field treatment.

## Acknowledgments

We acknowledge discussions with Arkadiusz Kuroś, Roderich Moessner, Krzysztof
Sacha, and Yizhi You. KG was supported by the Narodowe Centrum Nauki (NCN)
grant 2019/32/T/ST2/00413 and the Fundacja na rzecz Nauki Polskiej (FNP) START
grant. RL was supported, in part, by the cluster of excellence ct.qmat (EXC
2147, project-id 39085490). PS acknowledges the support of the NCN Sonata Bis
grant 2019/34/E/ST3/00405 and the Nederlandse Organisatie voor
Wetenschappelijk Onderzoek (NWO) Klein grant via NWA route 2. AK acknowledges
the support of Österreichische Akademie der Wissenschaften (ÖAW) ESQ-Discovery
Grant.

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# An Application to Generate Style Guided Compatible Outfit

∗Debopriyo Banerjee1,2, ∗Harsh Maheshwari1, ∗Lucky Dhakad1, Arnab
Bhattacharya1, Niloy Ganguly2, Muthusamy Chelliah1, Suyash Agarwal1
<EMAIL_ADDRESS>1Flipkart Internet Pvt. Ltd., 2Indian Institute of
Technology KharagpurIndia∗equal contributions

(2022)

###### Abstract.

Fashion recommendation has witnessed a phenomenal growth of research,
particularly in the domains of shop-the-look, context-aware outfit creation,
personalizing outfit creation etc. Majority of the work in this area focuses
on better understanding of the notion of complimentary relationship between
lifestyle items. Quite recently, some works have realised that _style_ plays a
vital role in fashion, especially in the understanding of compatibility
learning and outfit creation. In this paper, we would like to present the end-
to-end design of a methodology in which we aim to generate outfits guided by
styles or themes using a novel style encoder network. We present an extensive
analysis of different aspects of our method through various experiments. We
also provide a demonstration api to showcase the ability of our work in
generating outfits based on an anchor item and styles.

complete the look, neural networks, outfit compatibility, style

††journalyear: 2022††copyright: rightsretained††conference: 5th Joint
International Conference on Data Science & Management of Data (9th ACM IKDD
CODS and 27th COMAD); January 8–10, 2022; Bangalore, India††booktitle: 5th
Joint International Conference on Data Science & Management of Data (9th ACM
IKDD CODS and 27th COMAD) (CODS-COMAD 2022), January 8–10, 2022, Bangalore,
India††doi: 10.1145/3493700.3493737††isbn: 978-1-4503-8582-4/22/01††ccs:
Computing methodologies Learning latent representations††ccs: Computing
methodologies Neural networks

##  1\. Introduction

Outfit recommendation is a relatively well studied area in which researchers
aim to recommend outfits based on the notion of _compatibility_ between
lifestyle items, see (Vasileva et al., 2018; Song et al., 2019; Li et al.,
2020; Wang et al., 2021) for more details. A substantial volume of work has
also been done on the specific area of personalised recommendations (Zhan et
al., 2021; Lu et al., 2021). However, none of them specifically take style
into account while learning compatibility within outfits. We realise that
style is an essential component in modelling outfit compatibility as an outfit
may look compatible under one style construct but not in another.

An example of style guided outfit creation is provided in Figure 1. The same
top item (highlighted by a dotted rectangle) is used to create two outfits
under two different styles, namely _Formal_ and _Casual_. This is useful in
the situation where we consider that a user likes the top item but is doubtful
about making the final purchase. A style-guided algorithm will have two
advantages: (a) it will reject an outfit which may be otherwise compatible but
not in accordance with the desired style, and (b) it will pick compatible
outfits from different styles, hence expanding the choice to the user. A
style-independent algorithm, on the other hand, gets biased towards the
dominant style (assuming _formal_ style is dominant).

Figure 1. Illustration of the effectiveness of style-guided outfit generation
over a style-independent variant. If _formal_ is the dominant style, the
latter will only accept outfits from _formal_ style while rejecting from
others. Style-guided methods, however, will accept outfits from multiple
styles.

Style guided recommender system requires special attention from the research
community as most of the work are unsupervised in nature. Some example
research in this area are listed below: (a). Kuhn et al. (Kuhn et al., 2019)
refer an outfit as a style fit and do not explicitly use style information for
modeling compatibility; (b). Jeon et al. (Jeon et al., 2021) extract fashion
attributes from full-body outfit images for classifying outfit style; (c). Li
et al. (Li et al., 2019) models outfit level style from item descriptions;
(d). Singhal et al. (Singhal et al., 2020) models context and type jointly
using Graph Neural Network (GNN), and style between item pairs are modeled
using autoencoder without any explicit style information. Each of these works
lack in one way or the other the ability to generate style guided outfits.

Lai et al. proposed the Theme Matters paper (Lai et al., 2020) (archived work)
which comes closest to our work. It projects a supervised approach that
applies theme-aware attention to item pairs with fine-grained category tags
(e.g., long-skirt, mini-skirt, t-shirt, etc.). There are two specific
drawbacks, the first of which is that such fine-grained category information
is not always available and can be ambiguous if done manually. Secondly, the
size of the model increases exponentially with the number of fine-grained
categories.

We propose a Style-Attention-based Compatible Outfit Generation (SATCOGen)
framework that uses high-level categories (e.g., topwear, bottomwear,
footwear, accessory, etc.) and outfit-level style information. It consists of
a Style-Compatibility-Attention Network (SCA Net) (Lin et al., 2020) and a
novel style encoder network called Variational Style Encoder Network (VSEN)
which encodes the style of an outfit into a latent space. This encoding is
used to provide style-specific subspace attention, along with category
information during the computation of embedding. Multiple loss functions
ensure style encoding, general as well as style-specific compatibility. For
the generation task, given an anchor item, beam search is used to generate
style-specific outfit. We have provided a demonstration api to showcase the
kind of outfits that are generated for an anchor item given various styles.

##  2\. Methodology

The fundamental philosophy guiding the work in this paper is that in practical
circumstances compatibility between lifestyle items present within an outfit
is contingent on the style to which the outfit belongs. In a nutshell, we make
use of Style-Compatibility-Attention Network (SCA Net), a compatibility
learning framework that makes use of features extracted from the image of an
item based on category information as previously done by (Lin et al., 2020)
and then add style component to it. The methodology of SATCOGen is explained
in greater detail below.

A smooth latent vector representation for outfit style is learnt using
Variational inference in a novel style encoder named _Variational Style
Encoder Network_ (VSEN). There are two main trends in denoting an outfit, as
an ordered sequence of items (Han et al., 2017; Nakamura and Goto, 2018) or as
_set_ (Bettaney et al., 2019; Chen et al., 2019). We choose the latter
representation, which brings in two important properties, namely permutation
invariance and allowance for varying length. This assumption enables us to
select the _set transformer_ approach proposed in (Lee et al., 2019) for our
style encoder job. Keeping in mind that our work is not restricted only to
compatibility learning and also involves outfit generation, we ensure that
every outfit style is represented by the first two moments of a Gaussian
distribution which is proximal to the unit Gaussian
$\mathcal{N}(0,\mathbbm{1})$, a mechanism we borrowed from Variational
inference (Blei et al., 2017). This further ensures smooth representation of
the latent style space. The advantage of this step will be clear during the
outfit generation stage.

A vector, sampled from the Gaussian distribution representing the style of the
outfit is used to classify the style of the outfit. This ensures that VSEN is
able to capture specific information about the style of the outfit as well.
Thus, given the styles and their corresponding style vectors, this module
solves a multi-class classification problem using an MLP with $n$ layers.

We modify the subspace attention network proposed in (Lin et al., 2020) to
learn compatibility between items in an outfit. In the previous network, the
image of an item within an outfit is passed through a ResNet18, and the
embedding vector output is multiplied by learnt masks that help to learn the
subspaces. The item and target categories are then consumed to estimate the
subspace attention weights which subsequently leads to a weighted average of
the masked embeddings to be denoted as the final embedding of the item in the
tuple <item, item category, target category>. We tweak this and estimate the
subspace attention weights by providing the outfit specific sample style
vector from VSEN as an additional input. This helps to learn compatibility
conditional on the style of the outfit.

There are four loss functions used in our method for learning style specific
compatibility. We have the KL divergence loss from the VSEN network and the
classification loss from the downstream job. We also have the compatibility
loss from the SCA Net which is based on the popular triplet loss. And finally,
we introduce one more loss function to account for penalisation when the wrong
style is specified for an outfit. The overall loss for our method is given as
the weighted sum of these four individual losses.

Figure 2. Beam search: Given a top-wear chosen by a user, and a template, the
algorithm would go about generating outfits by sequentially adding items from
each category in the template.

###  2.1. Outfit generation

A globally optimal outfit generation task is a non-trivial task since it is
infeasible to look into all possible combinations. An approximate solution is
provided in this case. First, embeddings are created for different target
categories for an item and an associated style. Note that we know from the
previous section that embedding computation requires us to provide a style
vector for every item. If there is a reference outfit from the same style
which we want to emulate, it is trivial to generate a style vector from VSEN
using that outfit. However, in the absence of a reference outfit, there is no
specific distribution to sample from. For this we pool the mean and variance
of all outfits belonging to that specific style, and use a Gaussian with
pooled moments. This distribution can be assumed to be representative of the
style in question and enables us to generate a style vector from it.

Once we have estimated embeddings for each item, one can generate outfits
based on the well known _beam search method_ (Zhang et al., 2021), as is shown
in Figure 2.

Table 1. Distribution of compatible outfits across different styles.

Style | Work | Casual | Party | Relax | Travel | Athleisure | Sporty
---|---|---|---|---|---|---|---
Train | 841 | 13062 | 1215 | 473 | 2128 | 1160 | 534
Val | 108 | 1679 | 156 | 61 | 272 | 149 | 68
Test | 251 | 3917 | 362 | 140 | 631 | 348 | 160

Figure 3. Annotation interface for curating the style of Zalando data. A
complete outfit representing a _look_ from the website is provided along with
individual items from the outfit. The annotator has to assign one or more
appropriate style tags to the outfit, along with selecting the suitable gender
for it. One can also discard the item if there is any form of discrepancy.

##  3\. Dataset and Metrics

We created a female outfit dataset with style annotations.

Zalando Dataset (Zal): This dataset consists of items and outfits from the
Zalando website111https://www.zalando.co.uk/. In the website, _outfit looks_
are displayed with the option of shopping items from them. We merged
semantically similar fine-grained item categories, which resulted in nine
higher level categories. We scraped the looks and the corresponding set of
items present in each of the looks. Since style tags are unavailable for
majority of the looks, we recruited two human annotators to perform the task
of annotating style tags in outfits. We present the annotation interface in
figure 3, where we displayed an outfit look image along with the set
individual items in the look on the left and center and a list of style
options on the right. The annotator can assign one or more appropriate style
tags to each outfit. We also provided a dropdown list to select the gender for
which the outfit is primarily suitable. In case there is a mismatch in gender
of the outfit look and the associated items or if there are any other
discrepancies, the annotator has the option to select _Discard_. Finally,
there is an option to edit the immediately previous outfit and the number of
completed annotations are shown below as a summary. It is ensured that each
outfit is shown only once to each annotator.

Even though an outfit can have multiple style tags, for this work we ensure
that each outfit has a single style. In the situation that an outfit has
different style tags from the two annotations, we choose the randomly select a
style. After the annotation task, it was found that some of the styles were
heavily under-represented. To mitigate this issue, we merged certain styles
that are semantically similar and discarded some as well. For example, party
and wedding were merged into party while all outfits from classic and trendy
were discarded. At the end we had ~28K outfits.

Metrics: Two well known metrics are used to evaluate the performance of an
outfit compatibility prediction model (Han et al., 2017; McAuley et al.,
2015).
Fill-in-the-blank Accuracy (FITB Acc.): Given a set of items of an outfit with
one missing item as query, the task is to predict the correct missing item
from a list of four option items (where one is correct and three are
incorrect) based on compatibility of each option item with the query set.
Compatibility AUROC (Compat. AUC): Given a set of positive and negative outfit
samples, this metric helps in measuring the quality of predictions for
compatible and incompatible outfits.

We constructed separate test sets for FITB and compatibility tasks similar to
(Vasileva et al., 2018) by creating soft negative (SN) and hard negative (HN)
samples corresponding to each positive outfit sample. In case of SN, we sample
random items from the matching higher level categories (e.g., topwear,
bottomwear, footwear, etc.), whereas for HN, we do the sampling from matching
fine-grained categories222We make use of fine-grained category information
only during evaluation or testing phase. (e.g., t-shirts, heels, shoes, etc.).
It is to be noted that _HN_ samples are relatively harder to differentiate
from positives than SN samples. This gradation helps in evaluating the
performance of SATCOGen at various difficulty levels. We consider five FITB
and Compatibility test datasets and report the mean performance.

##  4\. Implementation Details

The two modules of SATCOGen, namely VSEN and SCA Net make use of RestNet18 as
the backbone CNN architecture. We freeze all the layers of ResNet18 except the
last convolutional block, which is connected to a new fully connected layer
that outputs a 64 dimensional embedding vector similar to the state-of-the-art
compatibility learning methods (Lin et al., 2020; Vasileva et al., 2018).

VSEN aggregates the CNN features of all the items in an outfit using the SAB
Set Transformer (Lee et al., 2019) with a hidden dimension of 32 and two heads
for mean and variance. The output of style vector of VSEN has a dimension of
64 which is followed by two fully connected MLP layers for the style
classification task. Here we consider batch size as 128 and employ the Adam
optimizer (Kingma and Ba, 2015) with an initial learning rate of $5\times
10^{-6}$. When combining the KL-Divergence loss with the cross entropy loss of
classification, we consider 0.05 as the weight coefficient corresponding to
KL-Divergence loss. After training the VSEN module, we freeze all of its
parameters and use it in evaluation mode to train the rest of the SATCOGen
framework. For learning the parameters of the SCA Net (five subspaces), we use
the Adam optimizer with batch size of 32 triplets, and initial learning rate
of $1\times 10^{-6}$. The Attention Network of SCA Net transforms the
concatenated one-hot-encoded category vectors and the extracted style vectors
from VSEN into 32 dimensions, respectively (using a single fully connected
layer), which are then forwarded to two fully connected layers (after
concatenation). Finally, the output is the five subspace attention weights.

For outfit generation, we have optimized the inference code using native spark
implementation to get item embedding given style and category information and
to run beam search at scale on a large scale of volume of anchor and candidate
items, resulting in 20X run time reduction compared to single box
implementation. We compared the run time on 60K anchor items with 30K average
child candidates per category, 5 average categories in beam search template
and beam width of 3.

Figure 4. Demonstration of how SATCOGen is able to choose diverse style
relevant bottomwears for a given parent topwear.  Table 2. Comparison of
compatibility learning for different methods on the Zalando dataset. We
compute FITB and Compatibility ROC AUC with both hard and soft negatives.

Method | Type | FITB Acc. | Compat. AUC
---|---|---|---
TM | SN | $47.79\pm 0.07$ | $76.73\pm 0.06$
HN | $43.78\pm 0.25$ | $75.97\pm 0.08$
SATCOGen | SN | $59.10\pm 0.34$ | $88.58\pm 0.08$
HN | $55.90\pm 0.31$ | $86.96\pm 0.06$

##  5\. Results

We show the efficacy of SATCOGen in generating superior outfits for online
shopping portals by comparing its performance against Theme Matters (TM) (Lai
et al., 2020) based on FITB Acc. and Compat. AUC scores. Table 2 presents the
FITB Acc. and Compat AUC results on SN and HN samples of Zal dataset for TM
and SATCOGen (our proposed model). SATCOGen outperforms the TM results, mainly
because of the requirement of fine-grained category information to have
improved performance. Fig. 4 shows an example of the quality of parent-child
category combinations with style pre-conditioning.

##  6\. Demonstration Interface

We have created a demonstration api to showcase the outfits generated by using
SATCOGen framework given an anchor item, a style and candidate items from
other categories. The outfit templates have been identified after discussion
with fashion experts. An example of such template would be, (’dress’, ’heels’,
’bag’, ’jewellery’). In the api, we provide users to select a category and
subsequently an anchor item from the category for which the user wants to view
outfits under different legitimate styles. The top-5 outfits per style are
displayed for the selected anchor item. In figure-5, we are showcasing the
top-3 outfits for an anchor t-shirt given style _sporty_ and template
’t-shirt’, ’leggings’, ’trainer-shoes’, ’bra’. The demo-api along with details
and screenshots can be found here 333https://github.com/Lucky-Dhakad/SATCOGen-
Demo-api.

Figure 5. Screenshot of the Web Interface used for Demonstration.

##  7\. Conclusion

In this paper, we presented SATCOGen - a novel outfit generation framework
based on styles and evaluated its performance on Zal - a newly introduced
outfit dataset with style tags associated with each outfit. In general, the
outfit generation process using beam search algorithm is time consuming and
not scalable for datasets having thousands of items for each category. We
ensured scalability, by optimizing the code and making it suitable for
execution on Hadoop Clusters, which reduced the execution time drastically.
Finally, we presented a web-interface that demonstrates outfit generation
starting from a chosen anchor item, a pre-defined style, and a template. In
the future, we are planning to extend the outfit dataset with ethnic outfits
(specific to Indian context) and productionize the SATCOGen framework to
provide an enhanced shopping experience to the customers.

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