\section{Introduction} 

Important progress is being made in view of realizing strongly-correlated topological phases of ultracold atoms in optical lattices~\cite{Goldman_topology,cooper2019topological}. On the one hand, experimental efforts have been dedicated to the creation of artificial gauge fields~\cite{Dalibard_review,Goldman_review} and topological bands~\cite{cooper2019topological} for neutral atoms, leading to measurements of topological properties~\cite{atala2013direct,Aidelsburger2015,nakajima2016topological,lohse2016thouless,Wu_2016,sun2018uncover,Flaschner2018,Tarnowski2018,sugawa2018second,lohse2018exploring,meier2018observation,de2019observation,genkina2019imaging,asteria2019measuring,rem2019identifying,chalopin2020exploring,wintersperger2020realization}. On the other hand, theoretical studies have identified realistic schemes for preparing small atomic ensembles in fractional Chern insulators (FCIs)~\cite{cooper2013reaching,yao2013realizing,grusdt2014topological,grusdt2014realization,he-PhysRevB.96.201103,motruk-PhysRevB.96.165107,repellin-PhysRevB.96.161111,hudomal2019bosonic}, which are lattice analogues of fractional quantum Hall (FQH) liquids~\cite{fractional_Chern_insulators, Bergholtz_fci_review}; they also proposed methods to probe their characteristic features~\cite{palmer2006high,kjall2012edge,luo2013edge,Grusdt2016,Taddia2017,dong2018edge,Raciunas2018,umucalilar2018time,repellin2019detecting,Yoshida2020,Cian2020}. This progress should soon lead to the realization of FCIs in small atomic ensembles, with $N\lesssim10$ atoms, and to the possibility of observing their properties. However, identifying clear and accessible topological signatures of FCIs in small interacting atomic systems still constitutes a central challenge. In fact, this question concerns a wide range of quantum-engineered platforms, including strongly-interacting photonic systems~\cite{quantumfluids,ozawa2019topological,RoushanSpectroscopicScience2017,RuichaoDissipativelyNature2019,knuppel2019nonlinear}, where FQH-type states of few photons are currently under intense investigation~\cite{Roushan_2016, AndersonEngineeringPRX2016, ClarkObservationNature2020}.

The canonical signature of FQH states is provided by the Hall conductivity, i.e.~the linear-response coefficient relating an induced transverse current to the applied force. In the FQH effect, the Hall conductivity is quantized to a value $\sigma_\text{H}/\sigma_0\!\in\!\mathbb{Q}$ in the thermodynamic limit~\cite{girvin1999quantum}; $\sigma_0^{-1}\!=\!R_{\text{K}}$ is von Klitzing's constant. The Hall response is also accessible in ultracold atoms; it has been measured in weakly-interacting gases through various probes, including center-of-mass (COM) drifts~\cite{choi2013observation,Jotzu2014,Aidelsburger2015,anderson2019conductivity,genkina2019imaging,wintersperger2020realization} and local  currents~\cite{fletcher2019geometric,chalopin2020exploring}, and more indirectly, through collective-mode excitations~\cite{leblanc2012observation} and circular dichroism~\cite{asteria2019measuring}. Whether the Hall response can be extracted and used as a topological marker in few-body interacting systems remain important questions to be addressed.


In this work, we numerically analyze the Hall drift of a small ensemble of strongly-interacting (hardcore) bosons, initially prepared in the ground state of the Harper-Hofstadter-Hubbard model~\cite{hofstadter1976energy,sorensen-PhysRevLett.94.086803}. Building on Refs.~\cite{dauphin2013,Aidelsburger2015}, we monitor the COM of the prepared state upon releasing it into a larger lattice while applying a weak static force [Fig.~\ref{fig1}(a)]. This Hall drift measurement, which provides an estimation of the Hall conductivity in the prepared state, is performed in a wide range of magnetic flux values [Fig.~\ref{fig1}(b)].
From this, we identify an emergent but robust Hall plateau, whose value $\sigma_{\text{H}}/\sigma_0\!\approx0.5$ approaches the many-body Chern number~\cite{NiuThouless,tao1986impurity,hafezi-PhysRevA.76.023613}, a topological marker of the FCI phase. Moreover, the width of the Hall plateau perfectly coincides with the flux window where an FCI state is formed in the initially confined geometry, as we demonstrate based on a static analysis of the ground-state's entanglement spectrum. Our results indicate that Hall signatures of FCI states composed of few bosons ($N\!\ge\!3$) can be identified under realistic experimental conditions. 
We also compare this approach with a direct application of Streda's formula~\cite{Widom,Streda,Streda2,giuliani2005quantum}, which indicates that static density-profile measurements lead to a robust Hall plateau for $N\!\ge\!10$ bosons.

This article is organized as follows: We first analyze the ground-state properties of our model in Section~\ref{section:GS}, setting the focus on the subtle identification of FCI states in confined systems with edges. We then study the center-of-mass Hall drift of this setting, upon release into a larger lattice, and comment on the emergence of quantized Hall plateaus in Section~\ref{section:drift}. We then explore the applicability of the Streda-formula approach in Section~\ref{section:streda}, before concluding with a discussion on the experimental realization of our Hall drift protocol in Section~\ref{section:conclusions}.



\begin{figure}[h!]
\includegraphics[width = \linewidth]{Fig1_v7.pdf}
\caption{(a) Hall drift protocol:~a prepared fractional Chern insulator (FCI) state is released into a larger lattice and a uniform force is applied. The Hall conductivity is extracted from the COM drift transverse to the force [Eq.~\ref{eq:extract}]. (b) Sketch of the Hall conductivity $\sigma_\text{H}(\alpha)$ as a function of the flux density in the Harper-Hofstadter-Hubbard (HHH) model. In the continuum limit $\alpha\!\ll\!1$, the system follows the classical prediction $\sigma_\text{H}\!\sim\!1/\alpha$. In the vicinity of the filling factor $\nu \!=\!\rho_{\text{bulk}}/\alpha\!=\!1/2$, where $\rho_{\text{bulk}}$ is the bulk particle density, an FCI is formed and $\sigma_\text{H}(\alpha)$ depicts a Hall plateau.
}
\label{fig1}
\end{figure}


\section{Fractional Chern insulator in a box: Ground-state properties}\label{section:GS} 

The central scope of this work concerns the emergence of quantized Hall plateaus in the COM dynamics of strongly-interacting bosons moving on a 2D square lattice in the presence of a uniform magnetic flux.
The corresponding Harper-Hofstadter-Hubbard (HHH) Hamiltonian reads~\cite{hofstadter1976energy,sorensen-PhysRevLett.94.086803} 
\begin{align}
\hat{H}_0=&-J\left(\sum_{m, n} \hat{a}^\dagger_{m, n+1} \hat{a}_{m,n} + e^{i 2 \pi \alpha n}\hat{a}^{\dagger}_{m+1, n}\hat{a}_{m,n} + \text{h.c.} \right) \notag \\
&+(U/2) \sum_m \hat{a}^\dagger_{m, n}\hat{a}_{m,n} (\hat{a}^\dagger_{m, n}\hat{a}_{m,n}-1),
\label{eq: time independent H}
\end{align}
where $\hat{a}^\dagger_{m, n}$ creates a boson at lattice site $(m,n)$, $J$ denotes the tunneling amplitude, $U$ is the on-site (Hubbard) interaction strength~\cite{bloch2008many}, and where the Peierls phase factors~\cite{hofstadter1976energy} account for the presence of a flux $\Phi\!=\!2 \pi \alpha$ per plaquette. This model has been experimentally implemented for $N\!=\!2$ strongly-interacting bosonic atoms~\cite{Tai2017}, in a box-type potential~\cite{BauntBosePRL2013, ChomazEmergenceNatComm2015, Tai2017, ChiuQuantumPRL2018, SaintJalmDynamicalPRX2019}. It is the aim of this Section to shed some light on the ground-state properties of this realistic system for $N>2$ bosons.

Numerical simulations using periodic boundary conditions have established that the HHH model hosts a bosonic FCI akin to the Laughlin state~\cite{girvin1999quantum}, for strongly repulsive interactions and filling factor $\nu \!=\!\rho/\alpha\!=\!1/2$, where $\rho$ denotes the particle density; see Refs.~\cite{sorensen-PhysRevLett.94.086803, palmer2006high, hafezi-PhysRevA.76.023613, repellin-PhysRevB.96.161111, he-PhysRevB.96.201103, motruk-PhysRevB.96.165107}. For hardcore bosons, these calculations reveal a stable FCI ground state for $\alpha\!\leq\!0.3$; the bulk gap is maximal around $\alpha\!\approx\!0.2-0.25$~\cite{sorensen-PhysRevLett.94.086803, hafezi-PhysRevA.76.023613, repellin-PhysRevB.96.161111}, and vanishes in the limit $\alpha\!\ll \!1$. This FCI phase is characterized by a fractional many-body Chern number, $\nu_{\text{Ch}}^{\text{MB}}\!=\!1/2$, a topological invariant associated with the ground-state of the many-body system~\cite{NiuThouless,tao1986impurity,hafezi-PhysRevA.76.023613}. In the thermodynamic limit, the Hall conductivity of an incompressible phase approaches the quantized value $\sigma_{\text{H}}/\sigma_0\!=\!\nu_{\text{Ch}}^{\text{MB}}\!\in\!\mathbb{Q}$; see Ref.~\cite{repellin2019detecting} for a numerical analysis of finite-size effects. In the relevant case of the $\nu\!=\!1/2$ FCI phase, a Hall plateau is thus expected at $\sigma_{\text{H}}/\sigma_0\!=\!1/2$; see Fig.~\ref{fig1}(b).



In the present work, we consider $N$ hardcore bosons initially confined in a circular box containing $N_s$ lattice sites [Fig.~\ref{fig1}(a)]; such box potentials are indeed available in experiments~\cite{BauntBosePRL2013, ChomazEmergenceNatComm2015, Tai2017, ChiuQuantumPRL2018, SaintJalmDynamicalPRX2019}. In this geometry with edges, we still expect to find the $\nu\!=\!1/2$ FCI phase in the regime $\rho_{\text{bulk}}/\alpha\!\approx\!1/2$, where $\rho_{\text{bulk}}$ denotes the bulk particle density. However, we note that the bulk density $\rho_{\text{bulk}}$, which differs from the total density $\rho\!=\!N/N_s$, has a non-trivial dependence on the flux; see Section~\ref{section:streda}. Hence, we first analyze the ground-state properties of this few-body system in view of determining the values of $\rho$ and $\alpha$ that realize an FCI state. 

We first gain some intuition from the physics of the FQH effect in the (continuum) disk geometry~\footnote{We refer to the infinite plane geometry, where the FQH droplet is confined through total angular momentum conservation, not by a confinement potential.}. For a flux density $\alpha\!\leq\!0.3$, the lowest Bloch band of the single-particle Hofstadter spectrum contains a set of roughly $N_0 (\alpha)$ nearly degenerate states, which are connected to the next band by dispersive edge states; see Appendix~\ref{app:spectrum}. These $N_0 (\alpha)$ states are analogous~\cite{kjall2012edge,luo2013edge} to the orbitals of the lowest Landau level (LLL) in the disk geometry; there, the $\nu\!=\!1/2$ Laughlin state with $N$ bosons occupies $2N\!-\!1$ LLL orbitals. Likewise, we may expect a $\nu\!=\!1/2$ FCI state when $N_0(\alpha)\!\simeq\!2N\!-\!1$. 

We use exact diagonalization to verify the existence of the FCI ground state in our model, and specify its phase boundaries based on (i)~the low-energy spectrum; (ii)~entanglement spectroscopy; (iii)~the occupation of single-particle orbitals.
For concreteness, we analyse $N\!=\!4$ hardcore bosons in $N_s\!=\!60$ sites:

(i) Figure~\ref{fig2}(a) shows the low-energy spectrum of this few-body system; there are three avoided crossings between the ground state and the first excited state  within the range $0\!\leq\!\alpha\!\leq\!0.3$, which we  interpret as the finite-size signatures of three phase transitions.
We focus on the regime $0.15\!<\!\alpha\!<\!0.25$, where the expected FCI bulk gap is the largest~\cite{sorensen-PhysRevLett.94.086803, repellin-PhysRevB.96.161111}, and no phase transition is observed. In this regime, the approximate degeneracy of the lowest band is $N_0(\alpha)\!\simeq\!7$ [see Appendix~\ref{app:spectrum}], which is compatible with an FCI ground state candidate for $N\!=\!4$. We note that the nature of the phases at $\alpha\!<\!0.15$ is likely to be non-universal due to finite-size effects~\footnote{The $\nu\!=\!2/3$ Jain fraction is a possible candidate among these phases~\cite{moller-PhysRevLett.103.105303, kjall2012edge}, but we did not identify any clear topological signature of this phase in our setting.}. 

\begin{figure}[h!]
\includegraphics[width = \linewidth]{Fig2_v7.pdf}
\caption{Spectral and topological ground-state properties. We present the ground-state properties for $N\!=\!4$ hardcore bosons in the HHH model, in a circular box of $N_s\!=\!60$ sites. The FCI stability region (shaded) is indicated according to three markers:~(a) Low-energy many-body spectrum (lowest~$10$ energies per discrete rotation-symmetry sector) relative to the ground-state energy $E_{\text{GS}}$. Analyzing absolute energies (not shown) reveals that the three local minima of the gap correspond to level crossings, which are avoided due to finite-size effects (arrows).~(b) Particle entanglement spectrum (PES) for a bipartition with~$2$ particles in each subsystem. In the shaded region, the first~$15$ levels (blue) are well separated from the other levels, revealing that the ground state is topologically equivalent to the Laughlin state.~(c) Occupation of the single-particle orbitals in the ground state (histograms) and in the exact Laughlin state on the disk (line). The orbitals are sorted in increasing energy and angular momentum, respectively.}
\label{fig2}
\end{figure}

(ii) In finite geometries with edges, the topological nature of FCIs can be revealed through the degeneracies of their edge spectrum~\cite{kjall2012edge,luo2013edge}; however, this spectral signature requires a smooth confining potential~\footnote{In the continuum, a hard confinement was shown to suppress the typical Luttinger liquid dispersion relation of edge modes~\cite{fern2017hardconfinement, macaluso2017hardwall}}. Instead, we probe the bulk quasihole excitations of the ground state; their degeneracy is a topological fingerprint of the FCI phase, and it can be extracted from the ground state $\ket{\Psi_{\mathrm{GS}}}$ using the particle entanglement spectrum~\cite{sterdyniak-PhysRevLett.106.100405} (PES). The PES is the spectrum of the reduced density matrix obtained by tracing  $\ket{\Psi_{\mathrm{GS}}} \bra{\Psi_{\mathrm{GS}}}$ over a bipartition containing $N_A$ particles, while keeping the geometry of the system intact. The degeneracy of Laughlin quasiholes is determined through a generalized exclusion principle~\cite{haldane1991fractional}; for $2$ bosons in $7$ orbitals, it is $15$-fold. And indeed, for $0.15\!<\!\alpha\!<\!0.25$ and $N_A\!=\!2$, the PES in Fig.~\ref{fig2}(b) reveals a clear gap above the 15th state, which confirms the FCI nature of $\ket{\Psi_{\mathrm{GS}}}$ in this parameter range.

(iii) To further characterize the ground state, we calculated the occupation of each single-particle orbital. For the Laughlin state on the disk, there is a uniform occupation of all $2N-1$ orbitals in the thermodynamic limit, with moderate deviations from this distribution for small systems. We find a similar distribution in the regime $0.15\!\lesssim\! \alpha\!\lesssim\!0.25$ for the same number of particles [Fig.~\ref{fig2}(c)].

Overall, these probes consistently reveal that the ground state is in the $\nu\!=\!1/2$ FCI phase within the range $0.15\!\lesssim\!\alpha\!\lesssim\!0.25$. 
We have verified that these results are qualitatively robust with respect to changes in the particle number ($N\!=\!3, 4$) and number of sites $N_s$. Quantitatively, we point out that a change in the density $\rho$ leads to a shift of the FCI phase along the flux axis; see Appendix~\ref{section:sizes}.

\section{The center-of-mass Hall drift} \label{section:drift}

We extract the Hall conductivity $\sigma_{\text{H}}$ from the COM drift of an initially prepared state upon applying a weak external force. This COM probe~\cite{dauphin2013,Aidelsburger2015,price2016measurement} is particularly suitable when considering a small ensemble of particles, for which local currents substantially fluctuate. In order to limit boundary effects, we release the initially prepared state into a larger lattice~\cite{dauphin2013,Tran2017} before monitoring the COM drift; see Fig.~\ref{fig1}(a). This protocol is implementable in cold-atom experiments, where tunneling can be dynamically tuned and COM motion measured~\cite{choi2013observation,Jotzu2014,Aidelsburger2015,anderson2019conductivity,genkina2019imaging,wintersperger2020realization}. In our simulations, the time scale associated with the progressive release of the inner system, as well as the duration of the Hall-drift measurement, are adjusted to avoid boundary effects due to the finite simulation box.

We extract the Hall velocity $v_{\perp}\!=\!x_{\perp}(t)/t$ from the COM Hall drift $x_{\perp}(t)$ transverse to the applied force $F$, upon reaching a stationary regime within linear response. This Hall velocity is related to the transverse current density through $v_{\perp}\!=\!j_{\perp}/\rho_{\text{bulk}}$, where $\rho_{\text{bulk}}$ denotes the bulk particle density. From the transport equation, $j_{\perp}\!=\!\sigma_{\text{H}} F$, one extracts the Hall conductivity through 
\begin{equation}
\sigma_{\text{H}}/\sigma_0\!=\!(2 \pi \rho_{\text{bulk}}/F)v_{\perp},\label{eq:extract}
\end{equation}
where $\sigma_0\!=\!1/2\pi$ is the conductivity quantum, and we set $\hbar\!=\!1$ except otherwise stated. In the FQH effect~\cite{giuliani2005quantum,mudry2014lecture}, incompressible states exhibit quantized Hall plateaus at fractional values $\sigma_{\text{H}}/\sigma_0\!=\!\nu_{\text{Ch}}^{\text{MB}}\!\in\!\mathbb{Q}$, where $\nu_{\text{Ch}}^{\text{MB}}$ denotes the aforementioned many-body Chern number~\cite{NiuThouless,tao1986impurity}. The main scope of this Section is to analyze the possibility of observing such Hall plateaus, through the Hall drift of few-boson FCI states.

\subsection{Benchmark using non-interacting fermions} 

We first benchmark the Hall-drift measurement by considering non-interacting fermions in the Harper-Hofstadter model, at quarter filling $\rho\!=\!1/4$. In this setting, (integer) Chern insulators are expected around flux densities $\alpha\!=\!1/(4 \mathfrak{n})$, with $\mathfrak{n}\!\in\mathbb{Z}$, where they exhibit quantized Hall plateaus $\sigma_\text{H}/\sigma_0\!=\!\mathfrak{n}$ in the thermodynamic limit. While this result can be directly deduced from a Diophantine equation~\cite{thouless1982quantized,kohmoto1989zero}, the actual size of these plateaus follows a rather complicated law established by the underlying single-particle (Hofstadter-butterfly) spectrum~\cite{hofstadter1976energy}. Furthermore, such Hall plateaus can be altered by finite-size effects. This first numerical study aims to shed some light on these properties.

\begin{figure}[h!]
\includegraphics[width = \linewidth]{Fig3_v9.pdf}
\caption{Benchmark using non-interacting fermions.~Hall conductivity extracted from the Hall drift of $N\!=\!20$ fermions, initially prepared in a small circular box of $N_s\!=\!81$ sites (quarter filling); the release into the larger system ($1005$ sites) and the ramping up of the force ($F\!=\!0.2 J/d$) are performed over a duration $\tau_{\text{ramp}}\!=\!15J^{-1}$, and the Hall drift measurement over $\tau_{\text{hold}}\!=\!13J^{-1}$; the bulk density $\rho_{\text{bulk}}(\alpha)$ in Eq.~\eqref{eq:extract} is evaluated in a central region of 20 sites; 
fit error bars reflect the 95$\%$ confidence interval when extracting the Hall velocity $v_{\perp}$. The gray dashed line indicates the quantized value expected in the thermodynamic limit, $\sigma_\text{H}/\sigma_0\!=\!1$, as dictated by the many-body Chern number of the corresponding insulating state. (Inset) Single-particle spectrum $E(\alpha)$, for the same box ($81$ sites); the Fermi level ($E_F$; grey curve) corresponds to the same number of fermions $N\!=\!20$; filled states are colored in green (bulk states are dark and edge states are light), while empty states are grey; $E_F$ is located in the main bulk gap (Chern insulator) within the blue shaded region.}
\label{fig3}
\end{figure}

We determine the Hall drift by calculating the time-evolution of $N\!=\!20$ non-interacting fermions, initially prepared in the ground-state within a circular box of $N_s\!=\!81$ sites ($\rho\!\approx\!1/4$), which are then released into a larger lattice ($1005$ sites) and subjected to a weak force $F\!=\!0.2 J/d$, where $d$ is the lattice spacing; see Appendix~\ref{methods}. The Hall conductivity extracted from the stationary (linear-response) Hall drift [Eq.~\eqref{eq:extract}] is depicted as a function of the flux density in Fig.~\ref{fig3}. In the low-flux regime ($\alpha\!\lesssim\!0.1$), lattice effects are negligible and the Hall drift follows the classical prediction $\sigma_{\text{H}}\!\sim\! 1 / \alpha$. At $\alpha\!=\!0.25$, the Fermi energy lies within a large bulk gap (see inset of Fig.~\ref{fig3}), which yields an approximately quantized value $\sigma_\text{H}/\sigma_0\!\approx\!0.9$, close to the many-body Chern number $\nu_{\text{Ch}}^{\text{MB}}\!=\!1$ of the corresponding insulating state~\cite{NiuThouless}. In order to explain the deviation from the exact quantized value expected in the thermodynamic limit, we have evaluated the local real-space Chern number~\cite{bianco2011mapping,tran2015topological}, as averaged over 29 bulk sites in the prepared ground state; we have verified that this local real-space Chern number indeed matches the value of the extracted Hall conductivity $\sigma_\text{H}/\sigma_0\!\approx\!0.9$; see Appendix~\ref{app:local} regarding the convergence of the local real-space Chern number as a function of the system size.

The Hall response remains approximately constant for a wide range of flux, $\alpha\!\in\![0.21 , 0.32]$, in agreement with the Fermi level's location within the main bulk gap of the butterfly spectrum [Fig.~\ref{fig3}]; we verified that the flatness of this emergent Hall plateau, as well as the approached quantized value, improve as the system size $N_s$ increases. Besides, we also observe the emergence of additional plateaus in Fig.~\ref{fig3}, as the Fermi level visits other bulk gaps in the spectrum. These results illustrate how the Hofstadter-butterfly spectrum dictates the size of emergent Hall plateaus in realistic finite-size (non-interacting) settings.


\subsection{Hall drift of interacting bosons} 
A system of $N\!=\!4$ hardcore bosons is initially prepared in the ground state of the HHH model, using a circular box  of $N_s\!=\!60$ sites. At $t\!=\!0$, this bosonic cloud is slowly released into a larger circle ($124$ sites), while the force is ramped up to the value $F\!=\!0.01 J/d$; see Appendix~\ref{methods} for details on the ramps used to reach a stationary regime. 
Our numerics show that a stationary COM motion takes place after the ramp, over a few tunneling times [inset of Fig.~\ref{fig4}], from which we extract the (constant) Hall velocity $v_{\perp}$. For longer times, the COM motion is affected by the edge of the finite simulation box, which sets the end of the stationary regime; we point out that this numerical constraint is not an experimental one, since the prepared state can be released into a much larger lattice in realistic setups. 

We extract the Hall conductivity $\sigma_{\text{H}}$, from the stationary Hall velocity $v_{\perp}$ and bulk density $\rho_{\text{bulk}}$ [Eq.~\eqref{eq:extract}], for a large range of flux values; see Fig.~\ref{fig4}. First, we find that the classical behavior $\sigma_{\text{H}}\!\sim\! 1 / \alpha$ is recovered~\cite{lindner2009vortex} in the low-flux (continuum) limit. The correlated behavior of our interacting system then appears for $\alpha\!\geq\!0.1$. Most strikingly, $\sigma_{\text{H}}(\alpha)$ depicts an emergent Hall plateau, whose width matches the flux window associated with the $\nu\!=\!1/2$ FCI phase [Fig.~\ref{fig2}]. In this region, the extracted Hall conductivity approaches the quantized value $\sigma_{\text{H}}/\sigma_0\!\approx\!0.5$, which is expected in the thermodynamic limit. Calculations performed on larger systems (up to $N\!=\!10$ bosons in $N_s\!=\!120$ sites) indeed demonstrate convergence to a quantized Hall plateau; see Appendix~\ref{section:sizes}. Interestingly, a second plateau appears in the range $\alpha\!\in\![0.1, 0.15]$, which may signal the onset of the $\nu\!=\!2/3$ FCI phase~\cite{moller-PhysRevLett.103.105303, kjall2012edge}.

In the prospect of experimental implementation, we sought for minimal configurations that exhibit Hall-plateau signatures of the FCI state, by simulating our Hall drift protocol for various system sizes $N_s$, considering $N\!=\!3, 4$ bosons; see Appendix~\ref{section:sizes}. Our results show that emergent Hall plateaus, compatible with the $\nu\!=\!1/2$ FCI state, can be detected by measuring the COM Hall drift of systems as small as $3$ bosons in $N_s\!=\!40$ sites and $4$ bosons in $N_s\!=\!49$ sites. 

\begin{figure}[h!]
\includegraphics[width = \linewidth]{Fig4_v10.pdf}
\caption{Hall drift of hardcore bosons in the HHH model. Initially, $N\!=\!4$ bosons are prepared in the ground state within a circular box of $N_s\!=\!60$ sites. The bosons are then progressively released into a larger box containing $124$ sites and subjected to a uniform force $F\!=\!0.01 J/d$. At the end of this ramp, the COM motion is stationary for a few tunneling times (inset), permitting the extraction of the Hall velocity $v_{\perp}$; the bulk density $\rho_{\text{bulk}}(\alpha)$ is evaluated in a central region of 16 sites; fit error bars reflect the 95$\%$ confidence interval when extracting the Hall velocity $v_{\perp}$. The FCI phase (shaded region from Fig.~\ref{fig2}) coincides with the emergent Hall plateau. The gray dashed line indicates the quantized value expected in the thermodynamic limit, $\sigma_\text{H}/\sigma_0\!=\!1/2$, as dictated by the many-body Chern number.}
\label{fig4}
\end{figure}

\subsection{Stability of the Hall plateau and bulk density evaluation} \label{subsec: stability}

Extracting the Hall conductivity $\sigma_{\text{H}}$ from the center-of-mass (COM) Hall drift requires the evaluation of the bulk density $\rho_{\text{bulk}}$, according to Eq.~\eqref{eq:extract}. For large enough systems, one can evaluate $\rho_{\text{bulk}}$ by averaging  the density over a central circular region, whose radius is large compared to the lattice spacing but small compared to the radius of the atomic cloud; see Fig.~\ref{figStreda} and Section~\ref{section:streda}. In few-boson systems, however, such scale separation may not exist. For instance, in the configuration studied in Fig.~\ref{fig4}, i.e.~$N\!=\!4$ bosons initially confined in $N_s\!=\!60$ sites, one notices substantial spatial fluctuations of the particle density. 

Here, we analyze the impact of such density fluctuations on the extracted Hall conductivity shown in Fig.~\ref{fig4}. To do so, we evaluate $\rho_{\text{bulk}}$ as the average particle density within a small and central circular region of radius $r_{\text{bulk}}$, considering values in the range $r_{\text{bulk}}\!\in\![0.4r_0 , 0.6r_0 ]$, where $r_0$ is the radius of the (initial) circular box containing $N_s$ sites. We illustrate the impact of this radius choice in Fig.~\ref{figsuppImpactBulkDensity}, which shows $\sigma_{\text{H}}$ as extracted from the COM Hall drift [Eq.~\eqref{eq:extract}] for three different choices. Interestingly, the Hall plateau displayed in Fig.~\ref{fig4} is shown to be very robust:~neither its existence, nor its position and range on the flux axis, depend on the radius $r_{\text{bulk}}$ used to evaluate the bulk density. While the exact value of $\sigma_{\text{H}}/\sigma_0$ depends on $r_{\text{bulk}}$, one notices that $\sigma_{\text{H}}/\sigma_0\!\approx\!0.5$ on this plateau for all choices.\\


\begin{figure}[h!]
\includegraphics[width = \linewidth]{FigS5_v5.pdf}
\caption{Impact of the bulk density evaluation on the determination of $\sigma_{\text{H}}$ for the few-boson system in Fig.~4. Here, we determined $\rho_{\text{bulk}}$ as the average particle density within a central circular region of radius $r_{\text{bulk}}$, with $r_{\text{bulk}}\!=\!0.4 r_0$, $r_{\text{bulk}}\!=\!0.5 r_0$ and $r_{\text{bulk}}\!=\!0.6 r_0$, where $r_0$ is the radius of the initial circular box ($N_s\!=\!60$); these three radii correspond respectively to $12$, $16$ and $24$ sites. In this small system, the radius choice affects the value of $\sigma_{\text{H}}$, as extracted from the COM Hall drift [Eq.~\eqref{eq:extract}]; however, the existence of the Hall plateau, as well as its position and range along the flux axis, are robust.
}
\label{figsuppImpactBulkDensity}
\end{figure}




\section{Extracting the Hall plateau from the particle density:~Streda's formula}\label{section:streda}

In this Section, we consider an alternative approach based on density-profile measurements. For an incompressible phase, the variation of the bulk density $\rho_{\text{bulk}}$ with respect to the flux density $\alpha$ is directly related to the quantized Hall conductivity. This relation, which is known as Streda's formula~\cite{Widom,Streda,Streda2,giuliani2005quantum,price2016measurement,Oktel,Fukui_Streda}, reads in our units
\begin{equation}
\sigma_{\text{H}}/\sigma_0\!=\!\frac{\partial \rho_{\text{bulk}}}{\partial \alpha}.\label{streda} 
\end{equation}
Applying this approach to the $\nu\!=\!1/2$ FCI phase, one thus expects the existence of a plateau $\partial \rho_{\text{bulk}}/\partial \alpha \!\approx\!0.5$ within the corresponding flux range. 

We have validated this prediction by analyzing the density profiles of $N\!=\!10$ hardcore bosons confined in $N_s\!=\!120$ sites, for various values of the flux $\alpha$. We display three representative profiles in Fig.~\ref{figStreda} (a), which illustrate the existence of a density plateau in the bulk. One verifies that the bulk density $\rho_{\text{bulk}}$, as extracted from the density plateau, increases linearly as a function of the flux $\alpha$ within a well-defined flux window. This result is represented in Fig.~\ref{figStreda} (b), which depicts the derivative of the extracted bulk density [Eq.~\eqref{streda}]. The clear plateau at $\sigma_{\text{H}}/\sigma_0\!=\!\frac{\partial \rho_{\text{bulk}}}{\partial \alpha}\approx1/2$ offers a striking signature of the FCI phase within this flux window. One verifies that this ``Streda-Hall" plateau matches the plateau obtained from the Hall drift protocol using the same system parameters; see Fig.~\ref{figsuppDMRG} in Appendix~\ref{section:sizes}.

\begin{figure}[h!]
\includegraphics[width = \linewidth]{FigS4_v6_a.pdf}
\caption{Streda analysis of the HHH ground-state, for $N\!=\!10$ hardcore bosons in a circular box of $N_s=120$ sites, using DMRG: (a) Particle density profiles; (b) Derivative of the bulk particle density $\rho_{\text{bulk}}$ with respect to the flux density; in the FCI phase, it satisfies Streda's formula in Eq.~\eqref{streda}. The bulk density $\rho_{\text{bulk}}$ was evaluated as the average density in a circle comprising $16$ sites. The gray dashed line indicates the quantized value expected in the thermodynamic limit, $\sigma_{\text{H}}/\sigma_0\!=\!\partial \rho_{\text{bulk}}/\partial \alpha=1/2$, as dictated by the many-body Chern number.
}
\label{figStreda}
\end{figure}

For smaller systems, the irregular density profile complicates the extraction of $\rho_{\text{bulk}}$, as already explained in Section~\ref{subsec: stability}. Nevertheless, evaluating $\rho_{\text{bulk}}$ in a small circular region of radius $r_{\text{bulk}}$, we still observe emergent plateaus of $\partial \rho_{\text{bulk}}/\partial \alpha$ within the FCI phase of smaller systems ($N\!\sim\!4$). In those configurations, the value on emergent plateaus displays a strong dependence on $r_{\text{bulk}}$, which complicates a quantitative application of Streda's formula for such small systems.



\section{Experimental considerations}\label{section:conclusions}

For a possible implementation of the proposed Hall-drift protocol, we consider the following experimental scheme. First, the FCI is prepared through adiabatic quantum state engineering, starting from a topologically trivial state and inducing a topological phase transition to an FCI by slowly tuning the Hamiltonian parameters~\cite{grusdt2014topological,grusdt2014realization,he-PhysRevB.96.201103,motruk-PhysRevB.96.165107,repellin-PhysRevB.96.161111,hudomal2019bosonic}. The adiabaticity of this preparation relies on the small number of particles and size of the system, which prevents the many-body gap from vanishing at the transition point. In our scheme, the FCI would be embedded in a lattice with more sites, which are initially uncoupled either by switching off the tunneling to those sites (as assumed in our numerical calculations), or by increasing their energy with a repulsive potential. The drift protocol is initiated by restoring the coupling to outer sites and simultaneously ramping up a force induced by, for instance, an optical or magnetic potential gradient~\cite{Jotzu2014,Aidelsburger2015}. Finally, the Hall drift is detected by measuring the COM position after variable drift times. Detecting COM displacements smaller than one lattice site, as depicted in Fig.~\ref{fig4}, is within the current capabilities of cold-atom experiments~\cite{Tai2017}. Yet, we expect that even stronger signatures are possible because experiments allow access to total system sizes and drift times beyond the reach of exact numerics. In addition, the ability to choose finite interactions and to tune them dynamically opens up the possibility for advanced transport studies. The simplicity of this realistic experimental scheme paves the way to the exploration of quantum transport in strongly-correlated ultracold topological matter.\\

During completion of our manuscript, we became aware of
a recent work ~\cite{Motruk_prep}, which also analyses the Hall response of an FCI in the HHH model. \\

\paragraph*{Acknowledgments} 
We thank M.~Greiner for insightful discussions, and I.~Carusotto and M.~Hafezi for comments.
We acknowledge J. Motruk and I. Na for sharing their manuscript~\cite{Motruk_prep} before submission, and for their comments on our work. DMRG calculations were performed using the TeNPy Library (version 0.4.0)~\cite{package}. C.R. is supported by the Marie Sklodowska-Curie program under EC Grant agreement 751859. J. L. acknowledges support from the Swiss National Science Foundation. N.G. is supported by the FRS-FNRS (Belgium) and the ERC Starting Grant TopoCold.

