# A proposal for realising quantum scars in the tilted 1D Fermi-Hubbard model

Jean-Yves Desaules  School of Physics and Astronomy, University of Leeds,
Leeds LS2 9JT, United Kingdom  Ana Hudomal  School of Physics and Astronomy,
University of Leeds, Leeds LS2 9JT, United Kingdom  Institute of Physics
Belgrade, University of Belgrade, 11080 Belgrade, Serbia  Christopher J.
Turner  School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT,
United Kingdom  Zlatko Papić  School of Physics and Astronomy, University of
Leeds, Leeds LS2 9JT, United Kingdom

(August 27, 2024)

###### Abstract

Motivated by recent observations of ergodicity breaking due to Hilbert space
fragmentation in 1D Fermi-Hubbard chains with a tilted potential [Scherg _et
al._ , arXiv:2010.12965], we show that the same system also hosts quantum
many-body scars in a regime $U\approx\Delta\gg J$ at electronic filling factor
$\nu=1$. We numerically demonstrate that the scarring phenomenology in this
model is similar to other known realisations such as Rydberg atom chains,
including persistent dynamical revivals and ergodicity-breaking many-body
eigenstates. At the same time, we show that the mechanism of scarring in the
Fermi-Hubbard model is different from other examples in the literature: the
scars originate from a subgraph, representing a free spin-1 paramagnet, which
is weakly connected to the rest of the Hamiltonian’s adjacency graph. Our work
demonstrates that correlated fermions in tilted optical lattices provide a
platform for understanding the interplay of many-body scarring and other forms
of ergodicity breaking, such as localisation and Hilbert space fragmentation.

Introduction. Recently, there has been much interest in understanding how
closed many-body quantum systems evolve in time when taken out of their
equilibrium state. While many such systems rapidly return to their equilibrium
state, in accordance with fundamental principles of quantum statistical
mechanics Gogolin and Eisert (2016), much of recent work has focused on
systems that fail to do so as a consequence of ergodicity breaking Nandkishore
and Huse (2015); Abanin  _et al._ (2019), either due to the special
mathematical structure known as integrability or very strong disorder which
leads to (many-body) localisation. Both of these paradigms of behaviour are
actively investigated in experiments on cold atoms, trapped ions and
superconducting qubits Kinoshita  _et al._ (2006); Schreiber  _et al._ (2015);
Smith  _et al._ (2016); Chiaro  _et al._ (2020).

The inability of non-ergodic systems to act as heat reservoirs for their
smaller parts has been traditionally known to affect the entire spectrum of
the system. Recently, however, there has been a flurry of interest in _weak_
ergodicity breaking phenomena Serbyn  _et al._ (2021). The latter refers to
the emergence of a dynamically-decoupled subspace within the many-body Hilbert
space, in general without any underlying symmetry, spanned by ergodicity-
breaking eigenstates. This behaviour was first theoretically established in
the Affleck-Kennedy-Lieb-Tasaki (AKLT) model Arovas (1989); Moudgalya  _et
al._ (2018), followed by the discovery of similar phenomenology in other non-
integrable lattice models Schecter and Iadecola (2019); Iadecola and Schecter
(2020); Bull  _et al._ (2019); Chattopadhyay  _et al._ (2020); Shibata  _et
al._ (2020); Moudgalya  _et al._ (2020a); Surace  _et al._ (2020); Kuno  _et
al._ (2020), models of correlated fermions and bosons Vafek  _et al._ (2017);
Mark  _et al._ (2020); Mark and Motrunich (2020); Moudgalya  _et al._ (2020b,
c); Hudomal  _et al._ (2020); Zhao  _et al._ (2020), frustrated magnets Lee
_et al._ (2020); McClarty  _et al._ (2020), topological phases of matter Ok
_et al._ (2019); Wildeboer  _et al._ (2020), and periodically driven systems
Buca  _et al._ (2019); Tindall  _et al._ (2019); Sugiura  _et al._ (2021);
Mizuta  _et al._ (2020); Mukherjee  _et al._ (2020). In these examples, the
ergodicity-breaking eigenstates are either explicitly embedded into a many-
body spectrum via the mechanism due to Shiraishi and Mori Shiraishi and Mori
(2017), or they form a representation of an algebra Pakrouski  _et al._
(2020); Ren  _et al._ (2020); O’Dea  _et al._ (2020).

A well-known example of weak ergodicity breaking in single-particle systems is
the phenomenon of quantum scars in chaotic stadium billiards Heller (1984). In
this case, the particle’s eigenfuctions exhibit anomalous concentration in the
vicinity of an unstable periodic orbit in the classical limit $\hbar\to 0$
Heller (1991); de Polavieja  _et al._ (1994); Wisniacki  _et al._ (2001),
leading to observable consequences in many physical systems Sridhar (1991);
Wilkinson  _et al._ (1996); Wintgen and Hönig (1989); Arranz  _et al._ (1998).
In recent experiments on interacting Rydberg atom arrays Bernien  _et al._
(2017), weak ergodicity breaking was observed via persistent revivals
following the global quench of the system, prompting the name “quantum many-
body scarring” Turner  _et al._ (2018a); Ho  _et al._ (2019); Turner  _et al._
(2021) by analogy with stadium billiards Choi  _et al._ (2019); Bull  _et al._
(2020). Recently, quantum many-body scarring has been shown to occur in higher
dimensions Michailidis  _et al._ (2020); Lin  _et al._ (2020a); Bluvstein  _et
al._ (2021) and in the presence of certain kinds of perturbations Turner  _et
al._ (2018b); Khemani  _et al._ (2019); Lin  _et al._ (2020b) including
disorder Mondragon-Shem  _et al._ (2020).

On the other hand, it has also been shown that ergodicity breaking can occur
due to a fracturing of the Hilbert space into dynamically disconnected
components Khemani  _et al._ (2020); Pai and Pretko (2019); Moudgalya  _et
al._ (2019); Sala  _et al._ (2020). This typically occurs by the interplay of
local interactions with a higher-moment symmetry such as charge dipole
conservation, which non-trivially intertwines spatial and internal symmetries.
Recent work Scherg  _et al._ (2021) has demonstrated that Hilbert space
fragmentation can be experimentally realised via magnetic field gradient
applied to the Fermi-Hubbard (FH) model in a 1D optical lattice. Apart from
offering a new platform to investigate the link between fragmentation and the
so-called Stark many-body localisation Schulz  _et al._ (2019); van
Nieuwenburg  _et al._ (2019); Yao and Zakrzewski (2020), an immediate question
presents itself: can the tilted FH model realise quantum many-body scars?

In this paper, we show that quantum many-body scars arise in the limit
$U\approx\Delta\gg J$ in the tilted FH model, and that they can be detected
using the quench from a specific initial state at a different filling factor
from the one considered in Ref. Scherg  _et al._ , 2021. We derive an
effective model for this setup, which can be mapped to a spinful
generalisation of the fractional quantum Hall effect on a thin torus Moudgalya
_et al._ (2020c), allowing for a practical experimental realisation. While the
phenomenology of quantum many-body scars is shown to be largely similar to
their realisation in Rydberg atom systems Bernien  _et al._ (2017), including
in particular an extensive set of eigenstates which violate the Eigenstate
Thermalisation Hypothesis (ETH) Deutsch (1991); Srednicki (1994), the origin
of scars is different in the two systems and can be intuitively understood
from a graph-theoretic viewpoint.

Large-tilt limit of the FH model. The 1D FH model is given by the Hamiltonian

$\displaystyle\hat{H}\\!=\\!\sum_{j,\,\sigma=\uparrow,\,\downarrow}\\!-J\hat{c}^{\dagger}_{j,\sigma}\hat{c}_{j+1,\sigma}+{\rm
h.c.}+\Delta
j\hat{n}_{j,\sigma}+U\sum_{j}{\hat{n}_{j,\uparrow}\hat{n}_{j,\downarrow}},\;\;\;\;\;$
(1)

where $\hat{c}_{j,\sigma}^{\dagger}$ denotes the usual electron creation
operator on site $j$ with spin projection $\sigma$,
$\hat{n}_{j,\sigma}\equiv\hat{c}_{j,\sigma}^{\dagger}\hat{c}_{j,\sigma}$ , $J$
and $U$ are the hopping and on-site interaction terms, respectively. Tilt of
the optical lattice is parametrised by $\Delta$, which we take to be spin-
independent Scherg  _et al._ (2021). Note that tilting has the structure of a
dipole term, $\sim j\hat{n}_{j}$. Below we impose open boundary conditions on
the model in Eq. (1), and restrict to the electron filling factor $\nu=1$,
i.e., with $N/2$ fermions with spin $\uparrow$ and $N/2$ fermions with spin
$\downarrow$ on a chain of $N$ sites (assumed to be even). We also set $J=1$
for simplicity. We label the Fock states using $\uparrow$ to denote a fermion
with spin up and $\downarrow$ with spin down, while $0$ stands for an empty
site and $\updownarrow$ denotes a doublon.

We focus on the regime $\Delta\approx U\gg J$. In this case the sum of the
dipole moment and the number of doublons is effectively conserved. The
dominant contribution to the Hamiltonian (using a Schrieffer-Wolff
transformation at first order Bravyi  _et al._ (2011)) is then given by

$\displaystyle\hat{H}_{\rm eff}$ $\displaystyle=$
$\displaystyle-J\sum_{j,\sigma}\hat{c}^{\dagger}_{j,\sigma}\hat{c}_{j+1,\sigma}\hat{n}_{j,\overline{\sigma}}(1-\hat{n}_{j+1,\overline{\sigma}})+\mathrm{h.c.}$
(2) $\displaystyle+$
$\displaystyle(U-\Delta)\sum_{j}{\hat{n}_{j,\uparrow}\hat{n}_{j,\downarrow}}.$

In this effective Hamiltonian, hopping to the left (which decreases the total
dipole moment by 1) is only allowed if it increases the number of doublons by
the same amount ($\bar{\sigma}$ denotes opossite spin from $\sigma$).

The action of the Hamiltonian (2) within the $\nu=1$ sector fragments the
Hilbert space beyond the simple conservation of $U+\Delta$. In this work we
focus on the largest connected component, which is the one containing the
state with alternating $\uparrow$ and $\downarrow$ fermions. In addition to
the symmetries of the full model in Eq. (1), i.e., SU(2) spin symmetry and
spin reversal Essler  _et al._ (2005), the Hamiltonian (2) projected to the
largest sector has an additional symmetry related to spatial inversion and
particle-hole exchange SOM . After resolving these symmetries, we find the
level statistics parameter $\langle r\rangle$ Oganesyan and Huse (2007) to be
close to 0.53 for all symmetry sectors with large numbers of states ($\gtrsim
10^{3}$) SOM . From these values which coincide with the Wigner-Dyson
statistics Mehta (2004), we expect the model in Eq. (2) to be chaotic. We next
outline an intuitive approach for identifying many-body scars in this model.

Figure 1:  Adjacency graph of the effective model in Eq. (2) for $N=6$. Red
vertices denote the states belonging to the hypergrid, with the black vertices
corresponding to $|{-}{+}\rangle$, $|{+}{-}\rangle$ states defined in the
text. Green vertices are the isolated states $|\downarrow 2\uparrow\rangle$,
$|\uparrow 2\downarrow\rangle$ which live on the tails of the graph. For this
graph, hypergrid contains 27 vertices out of 63.

Embedded hypergrid subgraph. A practical diagnostic of quantum many-body scars
is the existence of weakly-correlated states which undergo robust revivals
under quench dynamics, while the majority of other initial states thermalise
fast and do not display revivals. In the Rydberg-blockaded chains Bernien  _et
al._ (2017), the reviving Néel state of atoms is the densest configuration
compatible with the blockade constraint, and it is an extremal vertex of the
Hamiltonian adjacency graph Turner  _et al._ (2018a). In this graph each
vertex corresponds to a basis state, and two vertices are connected by an edge
if the Hamiltonian matrix element between their respective basis states is
non-zero. We next show, by examining the adjacency graph of the model in Eq.
(2), that we can identify a subgraph, weakly coupled to the rest of the
Hilbert space, which contains the reviving initial states and leaves a strong
imprint on the scarred eigenstates. This leads to a transparent manifestation
of scarring in the original Fock basis, in contrast with Rydberg atoms. In the
latter case, the subspace which is weakly coupled to the rest of the Hilbert
space has a much more complicated structure, leading to the wavefunction
spreading across the entire adjacency graph Turner  _et al._ (2018b) before
refocusing onto the Néel state.

In Fig. 1 we plot the adjacency graph of the Hamiltonian in Eq. (2) for a
small system. For the effective model in Eq. (2), it is possible to gauge away
the fermionic minus signs SOM , resulting in an unweighted, undirected graph.
As the Hamiltonian (2) (for $U=\Delta$) has no diagonal elements and the
spectrum is symmetric around zero, all product states are effectively in the
infinite temperature ensemble and are expected to thermalise quickly. As we
confirm numerically below, there are two important exceptions.

First, as highlighted in red colour in Fig. 1, there is a regular subgraph
which has the form of the hypergrid – a Cartesian product of line graphs (in
our case, of length 3), i.e., the hypergrid is isomorphic to an adjacency
graph of a free spin-1 paramagnet. This mapping can be understood by looking
at the state
$|\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow\ldots\rangle$. Each
cell of two sites can take the values ${-}\coloneqq\downarrow\uparrow$,
$2\coloneqq\updownarrow 0$ or ${+}\coloneqq\uparrow\downarrow$, leading to a
three level system. Note that the configuration $0\updownarrow$ is omitted, as
doublons can only be formed by hopping to the left. On the other hand, hopping
between two neighbouring cell will break this mapping and take the system out
of the hypergrid subgraph. Inside the hypergrid, we identify two states for
which the cell alternates between $-$ and $+$. These are the state
$\mbox{$|{-}{+}\rangle$}\coloneqq\mbox{$|{-}{+}{-}{+}\ldots\rangle$}=\mbox{$|\downarrow\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\ldots\rangle$}$
and its spin-inverted partner,
$\mbox{$|{+}{-}\rangle$}=\mbox{$|{+}{-}{+}{-}\ldots\rangle$}\coloneqq\mbox{$|\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow\uparrow\ldots\rangle$}$.
The states $|{-}{+}\rangle$ and $|{+}{-}\rangle$ for $N=6$ are shown in black
colour in Fig. 1. These two states are the only corners of the hypergrid
(state with only $+$ and $-$ cells) with no edges going out of it. As we show
below, either of these states shows persistent oscillations in quench
dynamics, undergoing robust state transfer to their spin-inverted counterpart.
While other corners of the hypergrid also show revivals, they are much smaller
in amplitude and decay faster due to the leakage out of this substructure. The
second example of a reviving state is $\mbox{$|\downarrow
2\uparrow\rangle$}\coloneqq\mbox{$|\downarrow\downarrow\ldots\downarrow\updownarrow
0\uparrow\uparrow\ldots\uparrow\rangle$}$ (and its spin-reversed partner
$|\uparrow 2\downarrow\rangle$), which is situated on a tail-like structure of
length 3 (independent of system size) with minimal connectivity to the rest of
the Hilbert space (green points in Fig 1). Similar tail-like structures occur
in constrained spin models Pancotti  _et al._ (2020).

Many-body scarred dynamics and eigenstates. Having identified candidate states
for revivals, we now scrutinise their quench dynamics using large-scale exact
diagonalisation simulations of the effective model in Eq. (2). Making use of
various symmetries present in the model, we have been able to exactly simulate
dynamics for up to $N=22$ electrons. For convenience, the simulations were
performed in the spin representation of the model SOM .

Figure 2:  Dynamics in the effective model (2) for $N=18$ for
$|{-}{+}\rangle$, $|\downarrow 2\uparrow\rangle$ and randomly chosen initial
states. (a) Entanglement entropy $S_{\mathrm{ent}}$ for an equal bipartition
of the system. Entropy grows linearly in time for all states, consistent with
thermalising dynamics, but it shows oscillations due to many-body scarring.
$\mbox{$|\phi\rangle$}\notin$ HG and $\mbox{$|\phi\rangle$}\in$ HG denote the
average over 10 random product states outside or within the hypergrid,
respectively, and the shading represents standard deviation. (b) Fidelity
dynamics for the same initial states as in (a). Inset shows the finite-size
scaling of the fidelity density
$\frac{1}{N}\ln|\langle\psi_{0}|e^{-i\hat{H}t}|\psi_{0}\rangle|^{2}$ at the
first revival for $|{-}{+}\rangle$ state, demonstrating a value much higher
than $\frac{1}{N}\ln(1/\mathcal{D})$ (with $\mathcal{D}$ the dimension of the
Hilbert space), expected for a random state. Blue dotted line shows the
amplitude of state transfer between $|{-}{+}\rangle$ and $|{+}{-}\rangle$
states. (c) Probability to remain within the hypergrid over time is much
higher for $|{-}{+}\rangle$ than other states.

Fig. 2(a) shows the time dependence of the entanglement entropy
$S_{\mathrm{ent}}(t)$ when the system is quenched from various initial product
states, such as $|-+\rangle$, $|\downarrow 2\uparrow\rangle$ and a few
randomly-chosen product states. $S_{\mathrm{ent}}$ is defined as the von
Neumann entropy of the reduced density matrix for one half of the chain. In
all cases, entropy grows linearly in time, consistent with thermalisation of
the system. However, the coefficient of linear growth is visibly different for
$|-+\rangle$ and $|\downarrow 2\uparrow\rangle$ states, and it is smaller than
that of random states, indicating non-ergodic dynamics. The long-time value of
the entropy is also different for the $|{-}{+}\rangle$ state SOM , hinting
that the wavefunction is still not completely spread into the whole Hilbert
space. The hallmark of many-body scars is the oscillations superposed on top
of the linear growth, as seen in the scarred dynamics in Rydberg atom chains
Turner  _et al._ (2018a). Rapid growth of entropy at short times is a
consequence of the bipartition being located in the middle of an two-site
effective cell.

Entropy oscillations mirror those of the wavefunction return
probability,$|\langle\psi_{0}|e^{-i\hat{H}t}|\psi_{0}\rangle|^{2}$, in Fig.
2(b). For the isolated state $|\downarrow 2\uparrow\rangle$, only a single
revival is clearly visible as the return probability decays rapidly once the
wavefunction leaks out of the tail of the graph. Because of the low
connectivity of the tail, the first revival is still visible on the scale of
Fig. 2(b). The revival time can be accurately estimated by assuming the tail
is completely disconnected, leading to the period $\pi/\sqrt{2}$. In contrast,
the state $|{-}{+}\rangle$ displays several revivals with the sizable weight
of the wavefunction $\sim 40\%$ returning to its initial value. The fidelity
density, $\frac{1}{N}\ln|\langle\psi_{0}|e^{-i\hat{H}t}|\psi_{0}\rangle|^{2}$,
shown in the inset, converges as $1/N$ to a value of $-0.058$. In contrast,
the inverse Hilbert space dimension, $\mathcal{D}^{-1}$, expected for a random
state leads to a fidelity density of $-0.855$ – an order of magnitude higher.
The scarred dynamics in this case can be visualised as the state bouncing
within the hypergrid between $|{-}{+}\rangle$ and its partner
$|{+}{-}\rangle$, illustrated by the dotted line in Fig. 2(b). From the
hypergrid analysis, we expect the revival period to be $\sqrt{2}\pi$, coming
from the $2\pi$ period of free precession and the spin-1 matrix elements
$\sqrt{2}$. This prediction closely matches the revival period observed in
Fig. 2(b).

The importance of the hypergrid for scarred dynamics is illustrated in Fig.
2(c) which plots the probability to remain in the hypergrid,
$P_{\mathrm{HG}}(t)=\langle\psi_{0}|e^{i\hat{H}t}\hat{P}_{\mathrm{HG}}e^{-i\hat{H}t}|\psi_{0}\rangle$,
where $\hat{P}_{\mathrm{HG}}$ is the projector onto the subspace spanned by
product states belonging to the hypergrid. For the initial state
$|{-}{+}\rangle$, we observe that the wavefunction remains concentrated inside
the hypergrid, even at late times. This is in stark contrast with the PXP
model which describes a chain of Rydberg atoms in the blockade regime Turner
_et al._ (2018b), where the wavefunction spreads across the entire graph by
the time it undergoes the first revival. Furthermore, even at the first
revival peak the fidelity is lower than $P_{\mathrm{HG}}$. This shows that the
wavefunction does not exactly return to itself but gets more spread even
within the hypergrid. Finally, for this initial state after a long time
$P_{\mathrm{HG}}$ converges to a non-zero value which is higher than expected
from the relative size of the hypergrid in the Hilbert space SOM , hinting
that the subgraph could have additional structure that prevents states from
leaking out. The fact that this long-time value is much lower for random
states in the hypergrid than for $|{-}{+}\rangle$ confirms that this is not
simply due to low connectivity between the hypergrid and the rest of the
Hilbert space, but that the special eigenstates indeed play an important role.

Figure 3:  Eigenstate properties of the effective model (2). (a) Overlap of
eigenstates with the $|{-}{+}\rangle$ state as a function of their energy $E$.
(b) Entanglement entropy $S_{\mathrm{ent}}$ of the eigenstates. Data is for
system size $N=16$. Red dots correspond to eigenstates with total spin $S=1$,
while the blue ones ones mark all other spin values. The squares indicate the
eigenstates sitting at the top of each tower of states. These towers have an
energy separation of approximately $\sqrt{2}$, as expected for the spin-1
hypergrid.

Properties of eigenstates of the model (2) are summarised in Fig. 3. The
projection of eigenstates onto $|{-}{+}\rangle$ state, shown in panel (a),
displays prominent tower structures reminiscent of other scarred models Turner
_et al._ (2018a); Bull  _et al._ (2019). The existence of towers implies that
eigenstates tend to concentrate around certain energies in the spectrum,
causing an ETH violation. The separation between the towers is approximately
$\Delta E\approx\sqrt{2}$, as expected from the embedded hypergrid. Note that
the eigenstates have been classified according to the conserved total value of
spin $S$; in contrast, $|{-}{+}\rangle$ state is not an eigenstate of
$\mathbf{S}^{2}$. One can show that for this state,
$\langle\mathbf{S}^{2}\rangle=N/2$, thus $|{-}{+}\rangle$ is predominantly
supported by $S=1$ and $S=2$ eigenstates at the given system size. The $S=1$
eigenstates are indicated by red points in Fig. 3.

Similar violation of the ETH can be seen in the large spread in entanglement
entropy of eigenstates in Fig. 3(b), showing that eigenstates of similar
energy have very different amounts of entanglement. Part of this spreading,
however, can be attributed to the eigenstates belonging to different spin
sectors $S$, giving rise to multiple bands that do not fully overlap at the
system size shown in Fig. 3(b) SOM . The distribution of entropy in $S=1$
sector [red points in Fig. 3(b)] is relatively narrow apart from two
“outliers” shown at energy $E\approx\pm\sqrt{2}$, which sit at the top of the
tower for their sector in Fig. 3(a). The states at the top of each tower are
indicated by squares, but unlike the PXP model Turner  _et al._ (2018b) these
states are not well separated from other states in the same tower.

Figure 4: (a) Occupation imbalance in the full model (in Eq. (1)) with
$N{=}12$ for various values of $U{=}\Delta$ for the initial state
$|{-}{+}\rangle$, and for $U{=}\Delta{=}6$ for the initial states
$\mbox{$|\phi_{0}\rangle$}=\mbox{$|\uparrow\downarrow\uparrow\downarrow\cdots\uparrow\downarrow\rangle$}$
(within the hypergrid) and
$\mbox{$|\phi_{1}\rangle$}=\mbox{$|\downarrow\updownarrow
0\uparrow\updownarrow 0\updownarrow 0\downarrow\updownarrow
0\uparrow\rangle$}$ (outside of it). (b) $U-\Delta$ phase diagram showing the
scarring regime near the diagonal (dashed line). The colour scale represents
the value of the first peak of the imbalance for $N{=}12$.

Experimental implications. The effective model studied above is exact for
$U=\Delta\rightarrow\infty$. For experimental realisations, it is important to
ascertain that the same physics persists for accessible values of $U$,
$\Delta$ and that it can be detected using local measurements. We demonstrate
this in Fig. 4 for the full model in Eq. (1) focusing on the regime
$U,\Delta<10$. Panel (a) shows the dynamics of imbalance on the even/odd
sublattices, $\mathcal{I}=(N_{\rm o}-N_{\rm e})/(N_{\rm o}+N_{\rm e})$, where
$N_{\rm e/o}$ is the total number of fermions on the even or odd sites. The
imbalance is bounded between -1 and 1. We see robust oscillations in
$\mathcal{I}$ with the frequency matching half the wavefunction revival
frequency in Fig. 2(b). The amplitude of the imbalance revival remains close
to the infinite-limit value for $U=\Delta\gtrsim 6$. As further evidence that
the hypergrid is the cause of non-ergodicity, we devised a local perturbation
which effectively disconnects the hypergrid, leading to the improvement of
revivals in the full model SOM .

Conclusions and discussion. We have proposed an experimental realisation of
quantum many-body scars in the regime $U=\Delta$ of the tilted FH model. We
have identified product states $|{-}{+}\rangle$, $|{+}{-}\rangle$ at filling
factor $\nu=1$ which give rise to scarred dynamics and reveal towers of
ergodicity-breaking many-body eigenstates, allowing to investigate the
interplay of many-body scarring with other facets of weak ergodicity breaking
such as localisation and Hilbert space fragmentation. In addition to the
filling factor $\nu=1$, we have also studied the filling $\nu=1/2$ used in
Ref. Scherg  _et al._ , 2021. In the latter case, taking the large-tilt limit
$\Delta\gg U,J$ and using a Schrieffer-Wolff transformation up to third order,
we found analogous signatures of scars SOM , provided we neglect the diagonal
terms in the effective Hamiltonian. Under these assumptions, the resulting
model can be viewed as a spinful generalisation of the fractional quantum Hall
effect on a thin torus Moudgalya  _et al._ (2020c). By contrast, the approach
presented here for $\nu=1$ is considerably simpler as it allows to
conveniently eliminate the undesirable diagonal terms.

Acknowledgements. We acknowledge support by the Leverhulme Trust Research
Leadership Award RL-2019-015 (ZP, AH), and by EPSRC grants EP/R020612/1 (ZP),
EP/R513258/1 (JYD), and EP/M50807X/1 (CJT). Statement of compliance with EPSRC
policy framework on research data: This publication is theoretical work that
does not require supporting research data. A.H. acknowledges funding provided
by the Institute of Physics Belgrade, through the grant by the Ministry of
Education, Science, and Technological Development of the Republic of Serbia.
Part of the numerical simulations were performed on the PARADOX-IV
supercomputing facility at the Scientific Computing Laboratory, National
Center of Excellence for the Study of Complex Systems, Institute of Physics
Belgrade.

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Supplementary online material for “A proposal for realising quantum scars in
the tilted 1D Fermi-Hubbard model”

Jean-Yves Desaules1, Ana Hudomal1,2, Christopher J. Turner1, and Zlatko Papić1

1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United
Kingdom
2Institute of Physics Belgrade, University of Belgrade, 11080 Belgrade, Serbia

In this Supplementary Material, we discuss details of the numerical
implementation of the tilted 1D Fermi-Hubbard model using the Jordan-Wigner
mapping to spins, the symmetries of the model, and we present more details on
the level statistics and eigenstate properties of the model. We provide
further evidence for the importance of the hypergrid subgraph for many-body
scarring, including a perturbation which improves the revivals. Finally, we
also explain how our results relate to a different regime of the model at high
tilt ($\Delta\gg U,J$) and electronic filling $\nu=1/2$.

##  I Jordan-Wigner transformation and symmetries

###  S1 Jordan-Wigner transformation

In order to simplify the computations, we perform a Jordan-Wigner
transformation to express the Hamiltonian in terms of spin operators. As a
convention, we will set that the spin-down fermions are located “on the right”
of the up-spin ones. For open boundary conditions the resulting Hamiltonian
for the effective model is :

$\displaystyle\hat{H}_{\rm eff,JW}=$
$\displaystyle-J\sum_{j}\hat{S}^{+}_{j,\downarrow}\hat{S}^{-}_{j+1,\downarrow}\hat{P}_{j,\uparrow}\left(1-\hat{P}_{j+1,\uparrow}\right)$
(S1)
$\displaystyle+J\sum_{j}\hat{S}^{+}_{j,\uparrow}\hat{S}^{-}_{j+1,\uparrow}\hat{P}_{j,\downarrow}\left(1-\hat{P}_{j+1,\downarrow}\right)$

where

$\hat{S}^{z}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad\hat{S}^{+}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\quad{\rm and}\quad\hat{S}^{-}=\begin{pmatrix}0&0\\\
1&0\end{pmatrix}$

are the usual Pauli spin operators and
$\hat{P}_{j,\sigma}=\frac{1+\hat{S}_{j,\sigma}^{z}}{2}$ is the projector on
the excited state of the Jordan-Wigner spin. In order to minimise the
confusion between the spin of the original fermions and the Jordan-Wigner
spins we will use “spin” to only refer to the fermion spin. For the Jordan-
Wigner spins we will describe the physics in term of “excitations”, e.g., the
action of $\hat{S}^{+}_{j,\downarrow}$ is to create an excitation on site $j$
with spin down.

###  S2 Symmetries

In the component of the Hilbert space we are interested in (i.e., the one
containing the state with alternating $\uparrow$ and $\downarrow$ fermions, as
explained in the main text), the Hamiltonian in the Jordan-Wigner spin
representation has two $\mathbb{Z}_{2}$ symmetries. The first one, denoted as
$\hat{z}_{1}$, is given by the joint action of the doublon parity operator and
the spin inversion operator:

$\hat{z}_{1}=\prod_{j=1}^{N}(-1)^{\hat{P}_{j,\downarrow}\hat{P}_{j,\uparrow}}\left(\hat{S}_{j,\uparrow}^{+}\hat{S}_{j,\downarrow}^{-}{+}\hat{S}_{j,\downarrow}^{+}\hat{S}_{j,\uparrow}^{-}{+}\frac{1{+}\hat{S}^{z}_{j,\uparrow}\hat{S}^{z}_{j,\downarrow}}{2}\right)$
(S2)

The doublon parity is diagonal in the product basis and gives +1 if there is
an even number of doublons and -1 otherwise. The spin inversion simply changes
the excitations between up spin and down spin.

The second symmetry, denoted as $\hat{z}_{2}$, corresponds to the joint
application of the spatial inversion operator and of the particle-hole
operator:

$\displaystyle\hat{z}_{2}=\prod_{j=1}^{N/2}\prod_{\sigma}\Big{(}$
$\displaystyle\hat{S}_{j,\sigma}^{+}\hat{S}_{N-j-1,\sigma}^{+}+\hat{S}_{j,\sigma}^{-}\hat{S}_{N-j-1,\sigma}^{-}$
(S3)
$\displaystyle+\frac{1-\hat{S}^{z}_{j,\sigma}\hat{S}^{z}_{N-j-1,\sigma}}{2}\Big{)}.$

The spatial inversion swaps sites $j$ and $N-j$, while the particle-hole
conjugation puts an excitation in every empty site and vice-versa. Performing
a Jordan-Wigner transformation on the full Fermi-Hubbard model with the same
fermion ordering convention produces a spin Hamiltonian which has the symmetry
generated by $\hat{z}_{1}$ but not by $\hat{z}_{2}$.

Working in the fermionic language, the corresponding set of symmetry operators
are

$\hat{y}_{1}=\prod_{j}e^{-i\pi\frac{1}{2}(i\hat{c}^{\dagger}_{j,\downarrow}\hat{c}_{j,\uparrow}-i\hat{c}^{\dagger}_{j,\uparrow}\hat{c}_{j,\downarrow})}$
(S4)

and

$\hat{y}_{2}=\prod_{j}(-1)^{\hat{n}_{j,\uparrow}\hat{n}_{j,\downarrow}}\mathcal{P}\mathcal{R}\text{,}$
(S5)

using the particle-hole operator $\mathcal{P}$ and spatial-reflection operator
$\mathcal{R}$ defined by the adjoint actions,

$\displaystyle\mathcal{P}c_{j,\sigma}^{\dagger}$
$\displaystyle=c_{j,\sigma}\mathcal{P}\text{,}$
$\displaystyle\mathcal{R}c_{j,\sigma}^{\dagger}$
$\displaystyle=c_{N-j-1,\sigma}^{\dagger}\mathcal{R}\text{.}$ (S6)

The particle-hole operator also has a non-trivial action on the vacuum state

$\mathcal{P}\mbox{$|0\rangle$}=\prod_{j}c_{j,\uparrow}^{\dagger}c_{j,\downarrow}^{\dagger}\mbox{$|0\rangle$}\text{.}$
(S7)

These are equivalent to $\hat{z}_{1}$ and $\hat{z}_{2}$ respectively, up to an
overall spin rotation and some charge-dependent phase factors. The first one
is similar to the spin-reversal symmetry from Eq. (2.57) in Ref. Essler  _et
al._ (2005).

###  S3 Negative matrix elements

The spin formulation of the model, after the Jordan-Wigner transformation,
offers a convenient framework for numerics. However, it is clear that the
Hamiltonian in Eq. (S1), expressed in the product state basis, has both
negative as well as positive matrix elements. In order to treat this model as
an unweighted graph, we show that a simple basis transformation is enough to
get rid of all minus signs in the Hamiltonian. These occur only when moving a
spin down excitation. Here we propose the following convention for fixing the
signs: we assign to each product a sign equal to the parity of the number of
spin down excitations on the even sites. This corresponds to a change of basis
using the diagonal matrix
$\hat{T}=\sum_{j=1}^{N/2}(-1)^{\hat{P}_{j,\downarrow}}$. If a spin up
excitation is moved, this does not change this quantity and it follows that
both states connected by this move have the same sign. Whether it is +1 or -1
is irrelevant as the matrix element will get a factor equivalent to their
product, which is always +1. On the other hand, if a spin down excitation is
moved, this parity number will always change by 1, and the states connected by
the move will have opposite signs. From this, it follows that all the matrix
elements of $\hat{T}\hat{H}_{\rm eff,JW}\hat{T}$ are equal to +1, thus we can
view the model in terms of an undirected, unweighted graph.

Alternatively, the same result can be obtained by choosing a different
convention for the Jordan-Wigner transformation. Instead of interleaving the
fermions with a different spin-projection, one can first place all up-spin
fermions and then all down-spin ones with a simple linear order for each
species. This has the effect of ensuring that each of the ‘hopping’ terms
never takes a fermion past another. Then, all these matrix elements are equal
and can be brought to $1$ by the choice of the coupling constant $J$.

Figure S1: Overlap of the eigenstates with $|{-}{+}\rangle$ state in the
effective model for $N=16$ in each total spin sector. The red crosses indicate
the states at the top of each tower in the relevant sector.  Figure S2:
Entropy of the eigenstates with $|{-}{+}\rangle$ state in the effective model
for $N=16$. The red crosses indicate the states at the top of each tower in
the relevant sector and are the same as in Figure S1.

##  II Individual symmetry sectors

Because of SU(2) symmetry, the number of total spin sectors is extensive in
system size. In Fig. S1 we show results for the overlap of eigenstates with
$|{-}{+}\rangle$ state, while Fig. S2 shows the entropy of eigenstates in
different total spin sectors. We focus on system size $N=16$ and neglect the
sector with maximum total spin ($S=8$), since this only has a single state.
All sectors display towers of states with the same approximate spacing of
energy equal to $2\sqrt{2}$ (Figure S1). However as the total spin increases
the number of towers decreases as $N/2+1-S$. Because of the change in parity
of that quantity, the location in energy of the towers is also dependent on
the parity of the total spin. Sectors with an odd number of towers will have
them at $E\approx\pm n2\sqrt{2}$, with $n$ being a positive integer or zero.
On the other hand, sectors with an even number of towers will have them at
$E\approx\pm(1+2n)\sqrt{2}$.

For the largest sectors ($S\leq 4$) the entropy band is relatively narrow,
with only a few outliers corresponding to the top state of some of the towers
(Figure S2). Interestingly, the states with a very low entropy are only
present for odd total spin for $N=16$. This is linked with the dependence of
the location on the towers on the total spin. Indeed, the most atypical states
have a fixed energy $E\approx\pm\sqrt{2}$ for all system sizes attainable by
exact diagonalisation. Hence they will only be present in sectors with an even
number of towers, which means an odd total spin for $N=16$.

After all the symmetries have been resolved, one can study energy level
statistics in each symmetry sector separately (Figure S3).

Figure S3: Number of states and level statistics in each symmetry sector
$(z1,z2)$ in the effective model for $N=16$.

All sectors with a large number of states ($\mathcal{D}>10^{3}$) have mean
level spacing $\langle r\rangle$ very close to 0.53, compatible with the
Wigner-Dyson distribution. We also show a histogram of level spacing after
unfolding Mehta (2004). This is done for the sector $z_{1}=1$, $z_{2}=1$,
$S=2$ in Figure S4. This sector was chosen because it has the largest overlap
with $|{-}{+}\rangle$ state at the system size studied and also has a large
number of states. The histogram of the level spacing confirms that this sector
indeed corresponds to a chaotic quantum system.

Figure S4: Distribution of the energy level spacing after unfolding Mehta
(2004) in the sector $z_{1}=1$, $z_{2}=1$, $S=2$ for $N=16$. The result for
this sector is very close to a Wigner-Dyson distribution.

##  III Influence of the hypergrid

The influence of the hypergrid subgraph is visible in the dynamics, i.e., in
the revivals from a few product states as demonstrated in the main text. In
this section we illustrate the impact of the hypergrid on eigenstates and the
long-time behaviour of the system.

In a fully ergodic system, one expects the support of an eigenstate on a
subset of states to be approximately equal to the ratio of the number of
states belonging to this subset and the total dimension of the Hilbert space.
Performing this computation for the effective model and choosing the hypergrid
subgraph as the subset shows anomalous concentration of some eigenstates – see
Figure S5.

Figure S5: Expectation value of the projector onto the hypergrid for the
eigenstates at $N=16$. The colours indicate the value of total spin for each
eigenstate and the dashed line is the average value of $P_{\rm HG}$ for the
system. Crosses are those states with highest overlap around their energy
irrespective of spin.

As the overlap with the $|{-}{+}\rangle$ state forms a lower bound to the
support on the hypergrid, we find the same kind of towers of states, located
at the same energies, in this plot. However, where the maximum overlap with
$|{-}{+}\rangle$ was approximately 0.011 for $N=16$, the maximum support on
the hypergrid is instead close to 0.76 for the same size. This confirms that
the hypergrid subgraph still leaves a large imprint on the eigenstates, even
when it only comprises less than $2.5\%$ of the Hilbert space for $N=16$.

Another signature of non-ergodicity caused by the hypergrid subgraph can be
seen in the long-time expectation value of the projector onto the hypergrid,
$P_{\mathrm{HG}}(t)$, when starting from the $|{-}{+}\rangle$ state. It
converges to a non-zero value which is higher than what would be expected from
the relative size of the hypergrid in the Hilbert space (Figure S6). This
hints that the embedding of the hypergrid in the effective model is non-
trivial and that the long-lived oscillations in the dynamics cannot simply be
explained by the relative size of this substructure. This can also be seen in
the long-time value of the entanglement entropy (Figure S7). For a large
enough system, most states saturate to the same entropy but for
$|{-}{+}\rangle$ this long-time value is lower, showing that the wavefunction
is not entirely spread into the whole Hilbert space. As is demonstrated in the
inset of Figure S7, this lower saturation entropy is not visible for generic
states within the hypergrid. It is also interesting to note that the long-time
entropy value of $|{-}{+}\rangle$ is still higher than what would be expected
from a system with the same dimension as the hypergrid.

Figure S6: Expectation value of the projector on the hypergrid states over
time for various system sizes with the initial state $|{-}{+}\rangle$. The
dotted black lines show the average of this projector at long times. The inset
shows these long-time value compared to the expected value from the relative
size of the hypergrid for various system sizes. Figure S7: Entanglement
entropy as a function of time for different initial states and system size
$N=16$. $\mbox{$|\phi\rangle$}\notin$ HG and $\mbox{$|\phi\rangle$}\in$ HG
denote the average over 10 random product states, respectively outside of and
within the hypergrid, that have been projected into the two symmetry sectors
in which the $|{-}{+}\rangle$ state has support. This was done to remove the
influence of these symmetries in the entanglement entropy. The shaded area
corresponds to the standard deviation. The inset shows the long-time value for
different initial states as a function of system size. For the system sizes
investigated, all states except $|{-}{+}\rangle$ converge towards the same
entropy value.

##  IV Full model

In this section we provide further results showing that the scarred dynamics
persists in the full model in Eq. (1) of the main text at finite values of
$\Delta$ and $U$. In our analysis of the effective model, we looked at the
fidelity of wavefunction revivals; here we show that the wavefunction revivals
are also visible in the full model for a relatively broad range of parameters
$U$ and $\Delta$ (Figure S8).

Figure S8: Maximum revival fidelity in the full model between $t=1$ and $t=10$
when starting from the $|{-}{+}\rangle$ state. The revivals show an
increasingly better fidelity on the $U=\Delta$ diagonal as the value of these
parameters get larger. The high fidelity in the top left corner corresponds to
the high-tilt regime investigated in Scherg  _et al._ (2021), where a large
number of states show revivals due to the conservation of the dipole moment
and of the number of doublons _separately_. Figure S9: Overlap of
$|{-}{+}\rangle$ state with eigenstates of the full model for $N=10$,
$U=\Delta=5$, and $J=1$. The colour indicates the expectation value of the
number of doublons plus the dipole moment. We see that already at this value
of the parameters the eigenstates start to form distinct towers.

We also plot the overlap of $|{-}{+}\rangle$ state with the eigenstates of the
full model in Figure S9. Colouring each eigenstate with the expectation value
of
$U\sum_{j}\langle\hat{n}_{j,\uparrow}\hat{n}_{j,\downarrow}\rangle+\Delta\sum_{j,\sigma}j\langle\hat{n}_{j,\sigma}\rangle$
highlights the fact that the eigenstates start to form distinct towers at
these parameter values. Each of these towers correspond to an expectation
value of this operator with a value close to an integer multiplied by
$U=\Delta$. As this parameter is increased the mixing between the towers is
reduced until they are completely separated in the $U=\Delta=\infty$ limit. In
this case only eigenstates in the central tower have a non-zero overlap with
the $|{-}{+}\rangle$ state.

##  V Improving the revival by perturbation

As the revivals emerge due to an embedded subgraph, we can devise a
perturbation that enhances the revivals by decreasing the leakage out of the
hypergrid. For the effective model, this can be achieved by a dimerised
perturbation given by

$H_{\rm Pert,eff}{=}\lambda
J\sum_{j=1}^{N/2-1}\sum_{\sigma}\hat{c}^{\dagger}_{2j,\sigma}\hat{c}_{2j+1,\sigma}\hat{n}_{2j,\overline{\sigma}}(1-\hat{n}_{2j+1,\overline{\sigma}}){+}\mathrm{h.c.},$
(S8)

i.e., the perturbation only affects interaction between the unit cells,
leaving the hypergrid subgraph untouched. For $\lambda=1$, the hopping term
between cells in the Hamiltonian and in the perturbation cancel each other. As
a result, the hypergrid becomes completely isolated from the rest of the
Hilbert space, resulting in the perfect wavefunction revival. This can be seen
in Figure S10 for $N=14$.

Figure S10: Fidelity revivals for the perturbed effective model for $N=14$. At
$\lambda=1$ the hypergrid subgraph is completely isolated from the rest of the
Hilbert space and the revivals from the $|{-}{+}\rangle$ state (and from any
other of the hypergrid corners) become perfect.

These results show clearly that the perturbation has a very weak effect on the
period of the revivals, but only modulates their amplitude. Furthermore, as
expected the fidelity peaks at $\lambda=1$, when the system revives perfectly.

The same perturbation can also be applied to the full model, in which case it
takes the form

$\hat{H}=\lambda
J\sum_{j=1}^{N/2-1}\sum_{\sigma}\hat{c}^{\dagger}_{2j,\sigma}\hat{c}_{2j+1,\sigma}+{\rm
h.c.}.$ (S9)

The optimal value of $\lambda$ for the first peak of the fidelity revivals is
also present at $\lambda=1$, but the revivals are no longer perfect (the first
peak reaches $\sim 70\%$). Furthermore, unlike in the effective model, looking
at the second revival peak gives an optimal value of lambda closer to 0.3.

Figure S11: Fidelity revivals for the full model with perturbation for $N=12$
starting from the $|{-}{+}\rangle$ state. As for the effective model
$\lambda=1$ gives the best revivals for the this state.

The fact that this perturbation still significantly improves the revivals in
the full model strongly suggests that the oscillatory is also due to the
imprint of the hypergrid in this case.

##  VI Filling $\nu=1/2$

While most of our work has focused on the sector with filling $\nu=1$, we find
similar dynamics in the $\nu=1/2$ sector which was explored in Ref. Scherg
_et al._ (2021). In the high tilt regime $\Delta\gg U,J$, the effective
Hamiltonian is given by

$\displaystyle\hat{H}_{\textrm{eff}}^{\mathrm{dip}}=$ $\displaystyle
J^{(3)}\hat{T}_{3}+U\Big{(}1-\frac{4J^{2}}{\Delta^{2}}\Big{)}\sum_{i}\hat{n}_{j,\uparrow}\hat{n}_{j,\downarrow}$
$\displaystyle+2J^{(3)}\hat{T}_{XY}+2J^{(3)}\sum_{j,\sigma}\hat{n}_{j,\sigma}\hat{n}_{j+1,\bar{\sigma}},$
(S10)

up to some constant terms, where

$\displaystyle J^{(3)}=\frac{J^{2}U}{\Delta^{2}}$ (S11)
$\displaystyle\hat{T}_{3}=\sum_{i,\sigma}{\hat{c}_{i,\sigma}\hat{c}_{i+1,\sigma}^{\dagger}\hat{c}_{i+1,\bar{\sigma}}^{\dagger}\hat{c}_{i+2,\bar{\sigma}}}+\text{h.c.},$
(S12)
$\displaystyle\hat{T}_{XY}=\sum_{i,\sigma}\hat{c}_{i,\bar{\sigma}}^{\dagger}\hat{c}_{i+1,\bar{\sigma}}\hat{c}_{i+1,\sigma}^{\dagger}\hat{c}_{i,{\sigma}}$
(S13)

The action of this Hamiltonian also produces a fragmentation of the Hilbert
space, and we focus on the sector with $N/4$ fermions with spin down and $N/4$
with spin up, and containing the charge density wave (CDW) state studied in
experiment with fermions only on even sites.

The effective Hamiltonian projected to the sector of the above CDW state can
be further simplified by noticing that some terms exactly vanish. The
$\hat{T}_{3}$ operator only allows to squeeze two fermions with opposite spin
to create a doublon between them. As all fermions are originally separated by
an empty site it is impossible for this operator to reach a state with
neighbouring fermions: they will always be either on the same site or with an
empty site between them. Hence, the operator $\hat{T}_{XY}$ and the potential
terms $\hat{n}_{j,\sigma}\hat{n}_{j+1,\bar{\sigma}}$ are always zero. As the
doublon-counting term is diagonal, the only term that allows for hopping
between product states is $\hat{T}_{3}$. By studying the effect of this
operator in a 4-site cell we will show that it leads to the same constrained
Hilbert space structure as the sector studied at filling $\nu=1$.

Indeed, for $\nu=1$ with two sites we have the possible states
$\mbox{$|\uparrow\downarrow\rangle$}\leftrightarrow\mbox{$|\updownarrow
0\rangle$}\leftrightarrow\mbox{$|\downarrow\uparrow\rangle$}$ forming a three
level system. For $\nu=1/2$, the operator $\hat{T}_{3}$ also only allows 3
states but now for four sites as $\mbox{$|\uparrow 0\downarrow
0\rangle$}\leftrightarrow\mbox{$|0\updownarrow
00\rangle$}\leftrightarrow\mbox{$|\downarrow{}0\uparrow{}0\rangle$}$. As all
actions of the Hamiltonian can be understood in terms of these cells, it
ensues that both models have _exactly_ the same off-diagonal matrix elements
if $N_{\nu=1/2}=2N_{\nu=1}$. The only difference is the diagonal term counting
the number of doublons for $\nu=1/2$ filling.

To better grasp the consequences of this potential term, it is useful to
rewrite the effective Hamiltonian in Eq. (VI) without the redundant terms and
factorising the common prefactors to obtain

$\displaystyle\hat{H}_{\textrm{CDW}}$
$\displaystyle=J^{(3)}\hat{T}_{3}+U\Big{(}1-\frac{4J^{2}}{\Delta^{2}}\Big{)}\sum_{i}\hat{n}_{j,\uparrow}\hat{n}_{j,\downarrow}$
(S14)
$\displaystyle=J^{(3)}\left[\hat{T}_{3}+\Big{(}\frac{\Delta^{2}}{J^{2}}-4\Big{)}\sum_{i}\hat{n}_{j,\uparrow}\hat{n}_{j,\downarrow}\right].$

As the Hamiltonian in Eq. (VI) is only valid for $\Delta\gg U,J$, this is also
the case for the one in Eq. (S14). However, it is easy to see that as
$\Delta/J$ increases, the potential term will dominate the dynamics. So there
is no way to obtain the Hamiltonian $\hat{T}_{3}$ on its own. Set $\Delta$ too
low and the higher order terms in the Schrieffer-Wolff transformation will not
be negligible, set it too high and the potential term will dominate the
dynamics. On the other hand, this issue does not arise in the $\nu=1$ sector
as only the equivalent of $\hat{T}_{3}$ is present in the effective
Hamiltonian at leading order.