††thanks: These authors contributed equally. Correspondence should be
addressed to K.V. (viebahnk@phys.ethz.ch).††thanks: These authors contributed
equally. Correspondence should be addressed to K.V.
(viebahnk@phys.ethz.ch).††thanks: These authors contributed equally.
Correspondence should be addressed to K.V. (viebahnk@phys.ethz.ch).

# Interaction-induced charge pumping in a topological many-body system

Konrad Viebahn  Institute for Quantum Electronics & Quantum Center, ETH
Zurich, 8093 Zurich, Switzerland  Anne-Sophie Walter  Institute for Quantum
Electronics & Quantum Center, ETH Zurich, 8093 Zurich, Switzerland  Eric
Bertok  Institute for Theoretical Physics, Georg-August-Universität Göttingen,
37077 Göttingen, Germany  Zijie Zhu  Institute for Quantum Electronics &
Quantum Center, ETH Zurich, 8093 Zurich, Switzerland  Marius Gächter
Institute for Quantum Electronics & Quantum Center, ETH Zurich, 8093 Zurich,
Switzerland  Armando A. Aligia  Instituto de Nanociencia y Nanotecnología
CNEA-CONICET, Centro Atómico Bariloche and Instituto Balseiro, 8400 Bariloche,
Argentina  Fabian Heidrich-Meisner  Institute for Theoretical Physics, Georg-
August-Universität Göttingen, 37077 Göttingen, Germany  Tilman Esslinger
Institute for Quantum Electronics & Quantum Center, ETH Zurich, 8093 Zurich,
Switzerland

###### Abstract

A topological ‘Thouless’ pump represents the quantised motion of particles in
response to a slow, cyclic modulation of external control parameters. The
Thouless pump, like the quantum Hall effect, is of fundamental interest in
physics because it links physically measurable quantities, such as particle
currents, to geometric properties of the experimental system, which can be
robust against perturbations and thus technologically useful. So far,
experiments probing the interplay between topology and inter-particle
interactions have remained relatively scarce. Here we observe a Thouless-type
charge pump in which the particle current and its directionality inherently
rely on the presence of strong interactions. Experimentally, we utilise
fermionic atoms in a dynamical superlattice which traces a pump trajectory
that remains trivial in the non-interacting limit. Remarkably, the transferred
charge in the interacting system is half of its usual value in the non-
interacting case, in agreement with matrix-product-state simulations. Our
experiments suggest that Thouless charge pumps are promising platforms to gain
insights into interaction-driven topological transitions and topological
quantum matter.

##  I Introduction

An adiabatic change of an energy landscape represents one of the simplest
strategies to induce controlled particle motion. For example, a sliding
potential minimum can carry a trapped particle from one point to another, both
in classical and quantum mechanics. However, the wave nature of quantum-
mechanical states allows for additional physical effects arising from a
geometric phase change when the Hamiltonian is time-dependent [1]. A geometric
or ‘Berry’ phase is not usually evident from the underlying potential but
requires knowledge of eigenstates and their geometric structure. Thouless
showed that in a periodic system the geometric phase contributions after one
adiabatic cycle sum to integers which correspond to the singularities enclosed
by the pump trajectory [2]. Physically, these integers describe the
quantisation of transported charge. This phenomenon, known as Thouless charge
pumping, is topologically protected against perturbations that are small
compared to the energy gap between ground and excited states [1, 3, 4].
Therefore, topological charge pumps may also become technologically relevant
as sources of quantised current [5]. An alternative (but equivalent)
description of Thouless pumping regards the time-periodic variation as a
Floquet drive [6] which gives rise to a synthetic dimension of photon states.
In this two-dimensional picture, the topological pump represents a quantised
bulk Hall drift.

So far, interactions did not play a major role in pumping experiments [7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17]. Recently, the effects of interactions on
pumping have been explored in two experimental platforms, that is, in the
mean-field regime [18, 19] and in a Fermi-Hubbard system [20]. In both cases,
interactions which exceeded the protecting energy gap caused a suppression of
the quantised pumping motion. In general, theoretical papers have suggested a
variety of interaction-related effects in pumping [21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. However, the question of
whether interactions can cause or even stabilise topological behaviour has
largely remained unanswered on the experimental level [40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51].

Here, we report on the experimental observation of interaction-induced charge
pumping using interacting fermionic atoms in a dynamically modulated optical
lattice. The optical lattice realises a Hubbard model with modulated hopping
matrix elements and onsite potentials. For Hubbard interactions larger than a
nonzero critical value, the atoms exhibit a pumping motion, while they remain
stationary in the non-interacting limit. Our measurements are consistent with
a quantised displacement of the atomic cloud in the first pump cycle, in
quantitative agreement with time-dependent matrix-product-state (MPS)
simulations [52, 53]. Interestingly, the transferred charge per pump cycle is
half of its usual value in a non-interacting system, as predicted in Ref.
[43], and the pumping mechanism does not have a classical counterpart. At very
large interactions and beyond a second critical interaction strength, the
pumped charge vanishes again. Crucially, the region with pumping of one charge
per cycle cannot be adiabatically connected to the non-interacting limit.

Our work establishes an example of topological phase transitions controlled by
the interaction strength. Previously, topological transitions were observed by
tuning external parameters in non-interacting models, such as inversion-
symmetry in the Haldane model [54, 55, 56]. The pump involves the physics of
the ionic Hubbard model [57, 58, 59, 60, 61, 62], which gives rise to the
observed transitions. Due to the inherent SU(2) symmetry in the spin sector,
this model possesses a Mott-insulating region with gapless spin excitations.
Since the pump realised in our experiment traverses through this region,
strictly speaking, there cannot be adiabatic pumping and quantised pumping
must eventually break down [32]. Notably, though, our numerical simulations
for realistic conditions show that the transferred charge in the first pump
cycle remains robust and near-quantised. Moreover, an analysis of correlations
shows that initial spin excitations are converted into defects in the charge
sector with a time delay and therefore do not affect the pumped charge
immediately, thus explaining the experimental observation. Therefore, a
Thouless pump in a two-component Rice-Mele model with Hubbard interactions
allows to systematically study the coupling between the spin and charge
sectors on experimentally and numerically accessible time scales.

While many aspects of topological band structures were realised with optical
lattices with engineered gauge potentials [54, 63, 55], accessing the
interacting regime still poses a significant challenge. The reason is the
unavoidable heating in a driven many-body system [64, 65] and the problem of
loading the bulk. Therefore, few experiments have explored interacting
topological quantum states [66, 67]. Experiments with Thouless pumps
circumvent these experimental issues and, as we will demonstrate here, enable
the study of interaction effects on the topology of many-body systems.

##  II Interaction-induced charge pump

The specific Hamiltonian studied in this work is the Rice-Mele-Hubbard model

$\displaystyle\hat{H}(\tau)$ $\displaystyle=$
$\displaystyle-\sum_{j,\sigma}\left[t+(-1)^{j}\delta(\tau)\right]\left(\hat{c}_{j\sigma}^{\dagger}\hat{c}_{j+1\sigma}+\text{h.c.}\right)$
$\displaystyle+\,\Delta(\tau)\sum_{j,\sigma}(-1)^{j}\hat{c}_{j\sigma}^{\dagger}\hat{c}_{j\sigma}+U\sum_{j}\hat{n}_{j\uparrow}\hat{n}_{j\downarrow}~{},$

which is parametrised by the time-dependent bond dimerisation $\delta(\tau)$
and the sublattice site offset $\Delta(\tau)$ [Fig. 1(a)]. The fermionic
annihilation and number operators for spin
$\sigma\in\\{\uparrow,\downarrow\\}$ on lattice site $j$ are denoted by
$\hat{c}_{j\sigma}$ and $\hat{n}_{j\sigma}$, respectively. Let us first
consider the non-interacting limit ($U=0$) in which the whole parameter space
is spanned by $\delta$ and $\Delta$ [Fig. 1(b)]. The relevant topological
invariant for charge pumping is the (charge–) Berry phase of the lowest band
which becomes singular at the origin of the $\delta$–$\Delta$ plane. At half-
filling, that is, two fermions per unit cell, a trajectory enclosing the
singularity pumps a total of $\Delta Q=2$ charges to the neighbouring unit
cell per pump cycle (one spin-$\uparrow$, one spin-$\downarrow$). For
trajectories that do not enclose the singularity [Fig. 1(c)] the pumped charge
is zero.

Figure 1: Topological pumping induced by Hubbard interactions. (a) Dynamically
modulated Rice-Mele-Hubbard model. The bond dimerisation ($\delta$) and the
site offset ($\Delta$) are periodically modulated. (b) Pump trajectory centred
at $\Delta=0$, leading to a pumped charge of $\Delta Q=2$ for spinful fermions
at $U=0$. (c-e) Pump trajectories centred at $\Delta>0$. (c) For vanishing
interactions ($U=0$) the pumped charge is zero. (d) A finite Hubbard $U$
causes the splitting of singularities to $\Delta\gtrless 0$, which leads to
interaction-induced topological pumping with $\Delta Q=1$. (e) Once the
Hubbard $U$ is too large, the pump is rendered topologically trivial. (f)
Schematic illustration of the interaction-induced pump on the microscopic
level. The red dot indicates the position of a single singularity at finite
Hubbard $U$.

The Rice-Mele model at finite Hubbard $U$ gives rise to a rich many-body phase
diagram at half-filling [57, 58, 59, 60, 32, 43, 68, 69]. The phases are
governed by the competition and interplay of the parameters $U$, $\delta$, and
$\Delta$. In short, the Su-Schrieffer-Heeger (SSH) lattice with $\Delta=0$
leads to a dimerised Mott insulator for $U\gg t$ [68], whereas the ionic
Hubbard model ($\delta=0$) exhibits band insulating ($\Delta\gtrsim U/2$ for
$U\gg t$) and Mott insulating ($\Delta\lesssim U/2)$ phases, with a small
dimerised intermediate phase around $\Delta\sim U/2$ [58, 59, 33]. Recent
numerical calculations predict a splitting of the non-interacting singularity
at the origin $[(\delta,\Delta)=(0,0)]$ into two, for increasing values of
Hubbard $U$ [43]. The new singularities each exhibit a $2\pi$ winding of the
(charge–)Berry phase and should thus be amenable to topological charge pumping
in the interacting regime.

Figure 2: Experimental observation of interaction-induced charge pumping.
Panel (a) shows a schematic representation of the experimental pump trajectory
(see App. A for details). The singularities are shown as red dots marked (b),
(c), and (d), respectively, for increasing values of Hubbard $U$,
corresponding to the following subplots. (b-d) The scaled centre-of-mass
position (c. o. m.) is plotted as function of time $\tau$ for three different
values of Hubbard $U$ (b): $U=0$, (c): $U/t=9.7(5)$, and (d): $U/t=16.1(8)$.
The dashed line in (c) represents a pumped charge of $\Delta Q=2$, typical for
the usual Rice-Mele pump [9, 10, 20]. Data points and error bars correspond to
mean and standard error of 40 measurements.

The key idea behind interaction-induced pumping is the following: we choose a
pump trajectory centred at a finite site offset $\Delta>0$, which does not
enclose the singularity for $U=0$. This trajectory is topologically trivial in
the non-interacting limit and transfers $\Delta Q=0$ charges per pump cycle as
sketched in Fig. 1(c). Now, we successively tune the value of Hubbard $U$
until one of the two singularities (red dots in Fig. 1) moves along the
$\Delta$-axis into the pump trajectory which, in turn, becomes topologically
non-trivial. Interestingly, the pumped charge in this case is expected to be
$\Delta Q=1$, compared to the usual $\Delta Q=2$ in the ‘traditional’
trajectory centred at the origin.

We develop an intuition for the interaction-induced pump by adiabatically
following the ground states in the interacting Rice-Mele model. Let us start
on the left of the $(\delta,\Delta)$–plane [Fig. 1(f)] which is characterised
by predominant singlet-correlations along the strong bonds. Following a
hypothetical singlet-pair (marked as (i) and highlighted in orange) out of the
full many-body ground state, the two atoms are converted into a double
occupancy for large sublattice offsets (ii, $\Delta>U/2$). Subsequently, the
double occupancy is converted back to a singlet ($\delta>0$), albeit shifted
by half a unit cell (iii). In the last section of the pump trajectory, the
pair of atoms remains in place, leading to an overall shift of half a unit
cell per pump cycle (pumped charge $\Delta Q=1$ per unit cell) while the local
spin and charge correlations adiabatically adjust, ensuring return to the
initial state (iv – v). In comparison, a non-interacting atom would only
oscillate to and fro during this process (i – v), leading to zero net current.
Note that the sketch describes an idealised adiabatic situation in which the
system remains in its instantaneous ground state at all times. Specific
aspects of the experiment and of the concrete model can affect the
adiabaticity, which we will address in Sec. V below.

Crucially, the lattice potential [schematics in Fig. 1(f)] does not exhibit an
underlying ‘sliding lattice’, which is used in the usual Rice-Mele pump [24,
9, 10]. Instead, a ‘long’ and a ‘short’ lattice are slowly oscillating with
respect to one another. Consequently, this interaction-induced pump does not
have a classical, non-interacting counterpart.

##  III Experimental realisation

We use fermionic 40K atoms in a dynamically controlled optical lattice to
realise the Hamiltonian in Eq. (II). Here, atoms take the roles of pumped
charges. The value of average tunnelling is
$t/h=$371(19)\text{\,}\mathrm{H}\mathrm{z}$$, where $h$ is Planck’s constant;
the Hubbard $U$ is widely tuneable via a magnetic Feshbach resonance. The
lattice laser setup, derived from a single laser source at
$\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$, is described in detail in Ref.
[20]. In short, a combination of standing waves in $x$–, $y$–, and
$z$–directions and a running wave component along the pumping ($x$–)direction
superimpose to form effectively one-dimensional tubes of superlattices (size
of one unit cell $\equiv d=\lambda$). The relative phase $\varphi$ between
interfering beams along $x$ and $z$, as well as the lattice depth
$V_{\text{Xint}}$ of the interfering $x$-lattice give independent control over
$\delta$ and $\Delta$ (see App. A). Prior to pumping, we maximise the
proportion of doubly occupied unit cells, as described previously [20], and
calibrate this value to be 0.415(8), independent of the value of Hubbard $U$,
where the number in brackets denotes the standard deviation. Subsequently,
pumping is initiated by sinusoidally oscillating $\varphi$ and
$V_{\text{Xint}}$ out of phase with respect to one another, causing the ‘long’
lattice (lattice spacing $=d$) to periodically move back and forth over the
‘short’ lattice (lattice spacing $=d/2$). The trajectory starts at $\Delta=0$
and $\delta<0$, then crosses the vertical axis at the maximal $\Delta$ above
the singularity, passes below the singularity, and finally returns to its
initial position, as shown in Fig. 2(a). In order to invert the pumping
direction, denoted by negative time $\tau$ in Figs. 2-4, we start on the
opposite side of the vertical axis at $\delta>0$ and again move upwards to
larger values of $\Delta$. The ‘boomerang’ shape of the experimental pump
trajectory is a consequence of having only two modulation parameters (similar
to Ref. [10]). In addition to varying $\delta$ and $\Delta$, the change in
lattice potential leads to a variation in average tunnelling $t$ by as much as
$60\%$, but this variation does not affect any conclusions, based on our
observations.

In a first experiment, we measure the in-situ centre-of-mass (c. o. m.)
position of the atomic cloud in units of unit cells ($d$) as function of time
$\tau$. In order to quantify the transferred charge per cycle for doubly-
occupied unit cells, the centre-of-mass displacement is divided by the
independently calibrated filling fraction 0.415(8) described above. The pump
period is fixed to $T=23\,\hbar/t=$10\text{\,}\mathrm{m}\mathrm{s}$$, chosen
to be much longer than the maximal inverse single-particle band gap
$1/($1.4\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$)=$0.7\text{\,}\mathrm{m}\mathrm{s}$$.
In the non-interacting limit, we find no significant displacement and a linear
fit yields a slope of $0.01(1)\,d/T$. This reflects the topologically trivial
nature of the pump trajectory for $U=0$ [Fig. 2(b)]. The situation changes
when performing the same experiment at a Hubbard interaction of $U/t=9.7(5)$
[Fig. 2(c)]. Here, we measure a slope of $0.51(4)\,d/T$, consistent with the
expected value of $\Delta Q=1$ pumped charge per cycle and unit cell in a
quantised, interaction-induced Thouless pump. Compared to a usual ‘Rice-Mele’
pump with $\Delta Q=2$ and a measured slope of $1\,d/T$, as observed in
previous experiments [9, 10, 20] and plotted as a dashed line, the
interaction-induced pump transfers only half the amount of charge. A third
experiment, this time at $U/t=16.1(8)$ yields no significant displacement
[slope $=-0.03(7)\,d/T$], since the singularity has moved out of the pump
trajectory [Fig. 2(d)]. These findings constitute the main qualitative result
of our experiments.

Figure 3: Scan of interaction-induced charge pumping as a function of Hubbard
$U$. Intermediate interactions lead to an extended region of interaction-
induced pumping with slopes around $0.5\,d/T$. The Thouless pump is rendered
trivial for very strong and very weak interactions, in agreement with theory.
Data points denote fitted slopes to time traces such as those shown in Figs.
2(b)-(d), error bars correspond to the standard error of the fitted slopes.
The ‘gap’ in the data points around $U/t\simeq 6$ is due to the use of
different hyperfine mixtures of 40K (App. A). The solid line is a MPS
calculation taking into account the experimental trajectory, as well as the
trapping potential (App. B).

##  IV Stability of the pump

A key aspect of our work is the ability to tune interactions and other
external parameters in a controlled fashion. In a second set of experiments we
map out the parameter regions in which interaction-induced pumping occurs.
Here, we vary the Hubbard $U$ from $0$ to $17\,t$, fit lines to the data such
as Fig. 2, and plot the resulting slopes in Fig. 3. From this data we can
identify three distinct regimes. Firstly, we find vanishing displacements for
interaction strengths up to $U/t\lesssim 6$ which matches the topologically
trivial ‘pump’ with zero transferred charge. Secondly, we observe
displacements of roughly $0.5\,d/T$ for intermediate interactions ($6\lesssim
U/t\lesssim 12$), reflecting a large range of Hubbard $U$ for which
interaction-induced pumping occurs with $\Delta Q\simeq 1$. Thirdly, a re-
entrant phase appears for $U/t\gtrsim 12$ when the interactions dominate and
the singularity exits the pump trajectory for strong interactions. This
interaction-induced transition to a topologically trivial situation with
$\Delta Q=0$ (or even into a nonadiabatic regime) also occurs for other types
of pumps [20], as well as interacting topological insulators [70, 46, 71]. The
observed transitions at $U/t\simeq 6$ and $U/t\simeq 12$ roughly coincide with
the extremal $\Delta$–values within the pump trajectory,
$2\Delta_{\text{min}}/t=5.4$ and $2\Delta_{\text{max}}/t=12$. The scatter in
the experimental data is due to drifts and shot-to-shot fluctuations of the
atomic cloud, as well as our finite imaging resolution. We measure
sub-$\text{\,}\mu\mathrm{m}$ movements on an atomic cloud using an imaging
system with a resolution of only $5\text{\,}\mu\mathrm{m}$ while the total
cloud diameter is on the order of $25\text{\,}\mu\mathrm{m}$.

Figure 4: Stability of interaction-induced pumping for different pump periods
$T$. For pump periods longer than 20 tunnelling times, the transferred charge
per cycle becomes largely independent of $T$, supporting the existence of
stable interaction-induced pumping. For shorter pump periods, the measured
slopes are reduced due to non-adiabatic effects. Data points and error bars
are analogous to Fig. 3.

We compare the experimental data with numerical simulations using matrix-
product states in Fig. 3. These theory calculations were performed with the
experimental parameters of the time-dependent Hamiltonian [Eq. (II)] including
the trapping potential. The starting state was approximated by disjoint
singlet states on each dimer with no long-range correlations, in order to take
our loading protocol and finite temperature into account (see App. B for
details). The results of the $U$-scan are plotted as a line in Fig. 3 and we
find good agreement between theory and experiment, within the error bars of
the data points. Smoothening of the topological transitions is present both in
theory and experiment, which is a result of nonadiabatic effects. The full-
width-half-maximum of interaction-induced pumping plateau is located at
$U/t=5.8(3)$ and $U/t=11.5(6)$, respectively, which is consistent with the
experimental observations.

An important control parameter for adiabatic pumping is the duration of one
pump cycle $T$, which is investigated experimentally in the following. Keeping
the interaction strength [$U/t=8.9(4)$] and all other experimental parameters
fixed, we vary the pump period $T$ over two orders of magnitude from 2 to 100
tunnelling times ($\hbar/t$). The results are plotted in Fig. 4 and the data
suggests that for slow-enough pump periods the measured slope converges
towards the quantised value of $0.5\,d/T$ (dashed line) and becomes largely
independent of $T$. This observation supports the conclusion that stable
interaction-induced pumping occurs for a large range of parameters. The
experimental data roughly agree with MPS calculations (solid line in Fig. 4).
For fast pump periods below 20 tunnelling times, the interaction-induced
displacement clearly breaks down due to nonadiabatic effects.

##  V Dynamics of spin- and charge-correlations during pumping

So far, both the experimental data and the MPS simulations demonstrate near-
quantised pumping in the first pump cycle for a range of finite interactions
strengths, neighboured by regions of no pumping at small and large values of
$U/t$, respectively. Since there are spin-gapless excitations in the region of
a nonzero pumped charge, in principle, adiabaticity is not guaranteed. We now
utilise numerical simulations to explain, on the one hand, why the pumping is
initially still robust, consistent with the experimental data, and on the
other hand, to develop a microscopic picture of the evolution of correlations
in the pumping process.

The presence of a sublattice site offset ($\Delta$) in the model gives rise to
a coupling of spin- and charge-degrees of freedom, which can result in complex
many-body dynamics. In our case, the ionic Hubbard model ($\delta=0$) [57, 58,
59, 60] is of particular interest due to the presence of two subsequent gap
closings in its many-body spectrum. For concreteness, let us fix the value of
$U/t$ and vary $\Delta$. A gap related to reordering of the charges vanishes
only at the single critical points $\Delta=\pm\Delta_{c}$ which determine the
topology of the charge sector. The spin gap vanishes for
$-\Delta_{s}\leq\Delta\leq\Delta_{s}$, where $\Delta_{c}-\Delta_{s}\lesssim t$
[57, 58, 59, 60], and we will call it ‘spin-gapless line’ in the following,
denoted by an orange line in Fig. 5(a). The parameter trajectory for the
interaction-induced pump crosses this line and spin excitations can occur, in
principle, at zero energy cost.

Figure 5: Dynamics of pumped charge, charge-, and spin-correlations computed
with time-dependent Lanczos. (a) Schematic of pump trajectory. The charge- and
spin-gap closings along the ionic Hubbard axis ($\delta=0$) are shown in red
and orange, respectively. The spin-gapless line extends from $+\Delta_{s}$ to
$-\Delta_{s}$. (b) Exemplary spin excitation (not the lowest), which can
prevent further pumping after one period. (c,d,e) Dynamics of pumped charge,
spin correlator, and charge correlator, respectively, calculated with Lanczos
for $L=10$, $T=100\hbar/t$, $\delta\tau=0.1\hbar/t$ and antiperiodic boundary
conditions (see App. B). The blue line corresponds to counter-clockwise
pumping (blue arrow), whereas the grey lines corresponds to clockwise pumping
(grey arrow, plotted in reverse time $-\tau$ for comparison). The crosses in
(c) indicate quantised displacement. The orange arrows in (d) denote the
crossing of the spin-gapless line. The black arrows in (e) denote the crossing
of the $\Delta=\Delta_{c}$ line. All three grey curves have been offset on the
vertical axis for clarity.

The schematics in Fig. 5(b) illustrate how spin- and charge-dynamics can
become intertwined during the course of a pump cycle. While not the exact
ground state, it is instructive to start from an anti-ferromagnetically
ordered state in (i) and incur a spin excitation when crossing the spin-
gapless line for $\Delta<\Delta_{s}$ [see (ii)]. The spin excitation can take
the form of a spin triplet or more complex patterns; here a spin-flip from
$\uparrow$ to $\downarrow$ is shown for simplicity. In the subsequent half-
cycle, two neighbouring spins have to form a double occupancy
($\Delta>\Delta_{c}$) in order to ensure pumping. However, the spin-excited
pair precludes the formation of a double occupancy, thereby preventing pumping
(iii).

In the following we investigate the mechanism described above using numerical
simulations of pumped charge per unit cell $Q(\tau)$, spin-correlations, as
well as charge-correlations as functions of time, shown in Figs. 5(c)-(e).
Here, we choose an idealised system consisting of ten lattice sites,
antiperiodic boundary conditions, and an elliptical pump trajectory in order
to draw generic, qualitative conclusions, leaving out some of the specifics of
our experimental system (see App. B). We compare two distinct trajectories,
both starting on the left of the ($\delta,\Delta$)-plane, as shown in Fig.
5(a). The ‘blue’ trajectory crosses the spin-gapless line at $\tau=3T/4$ while
the ‘grey’ one crosses the spin-gapless line at $\tau=T/4$. In Figs. 5(c)-(e),
the blue trajectory is always plotted forwards in time (starting from
$\tau=0$) while the grey trajectory is plotted backwards in time (starting
from $\tau/T=1$ and offset on the vertical axis) in order to simplify the
comparison. The blue trajectory clearly shows a quantised response (black
crosses) for one period, while the pumped charge is visibly reduced during the
second period. The grey trajectory follows the blue trajectory for half a
period but departs afterwards.

In order to understand the charge dynamics described previously we now
consider the dynamics of spin- and charge-correlations in Figs. 5(d) and (e).
The spin-correlations visibly start to oscillate when crossing the spin-
gapless line after $3/4$ (blue) and $1/4$ (grey) of the pump period (orange
arrows). Indeed, the charge dynamics remains smooth up until
$\Delta\sim\Delta_{c}$ which happens after $1$ (blue) and $1/2$ (grey) period,
respectively (black arrows). We can conclude that the spin excitations are
converted to charge excitations with a delay of a quarter period.
Interestingly, this effect is largely independent of system size, as shown in
Fig. A3. To summarise, crossing the spin-gapless line does not necessarily
lead to an immediate breakdown of pumping, but excitations first have to
spread to the charge sector in order to influence the pumped charge.

The spin-gapless line plays an important role in determining the long-time
behaviour of the interaction-induced charge pump [32]. In the experiment, we
observe a reduction of the pumped charge after the first cycle (Fig. A2),
which could indicate a breakdown of pumping after one cycle. Indeed, the
interplay between spin- and charge-degrees of freedom implies that the
dynamics of the second and subsequent pump cycles will become increasingly
dependent on the precise system parameters, such as system size and pump
trajectory. We investigate these effects using numerical calculations in Fig.
A4. The precise values of the excitation gaps matter crucially, which can be
tuned by modifying the pump cycle and the driving protocol. Altering the pump
trajectory in the experiment corresponds to including higher harmonics in the
driving waveform which we plan to investigate in the future.

Similarly, the addition of an Ising-type interaction term leads to an explicit
opening of the spin gap in the Mott-insulating regime of the ionic Hubbard
model, which can improve the stability of the pump even further (Fig. A6).
Ising-anisotropies have been realised with two-component bosons [72].

A third approach involves the intermediate (third) phase in the ionic Hubbard
model, called the spontaneously dimerised insulator (SDI). The three phases of
the ionic Hubbard model for repulsive interactions are, in ascending order of
$\Delta$, Mott insulator, SDI, and band insulator. The transition from a Mott
insulator to the SDI is accompanied by a spin-gap opening. The transition from
the SDI to the band insulator is due to a crossing of two ground states with
different charge distributions. Thus, adiabatic pumping could potentially be
stabilised by enlarging the SDI phase and avoiding the Mott insulator
altogether. It has been suggested that including a density-dependent hopping
term into the ionic Hubbard model can enlarge the SDI phase [73]. A density-
dependent hopping can be engineered by near-resonant Floquet modulation [74,
75, 76, 77].

##  VI Outlook

In conclusion, we have experimentally demonstrated an interaction-induced
charge pump using ultracold fermions in a dynamical superlattice. The observed
transport is consistent with quantised pumping within a range of repulsive
interactions, while it has no classical counterpart and the pumped charge is
zero below a critical Hubbard $U$. Pumping of one charge per cycle is not
possible for $U=0$, therefore, the pumping protocol is not adiabatically
connected to the non-interacting limit. Our numerical simulations unveil the
mechanism for the breakdown of adiabatic pumping, in which spin excitations
are transferred to the charge sector only later in the pump cycle.

These results open up multiple avenues for future research into topological
many-body systems. For example, the presence of trapping potentials in the
experiment could lead to interaction-induced edge physics [78], possibly
giving rise to novel topological boundary modes [79]. Novel cooling mechanisms
in optical lattices may be enabled via density redistribution [80], making use
of the inherent backaction between density and pump-induced currents in the
interaction-induced charge pump. In addition, the coupling mechanism between
spin and charge degrees of freedom could be harnessed to realise singlet
pumping, in view to realise measurement-based quantum computation [81]. In
general, our work establishes Thouless charge pumping as a promising system to
investigate interaction-driven physics in topological systems and topological
quantum matter. Having cross-validated theory and experiment in the limit of
one-dimensional dynamics, the experimental platform can be extended to two
[82, 83] and even three dimensions, eventually addressing questions beyond the
reach of numerical simulations.

## Acknowledgements

We would like to thank Kaden Hazzard and Philipp Preiss for comments on a
previous version of the manuscript. K.V., A.-S.W., Z.Z., M.G., and T.E.
acknowledge funding by the Swiss National Science Foundation (Grants No.
182650, 212168, NCCR-QSIT, as well as advanced grant AdiaPump 209376) and
European Research Council advanced grant TransQ (Grant No. 742579). E.B. and
F.H.-M. acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – 277974659, 436382789, 493420525 via DFG Research
Unit FOR 2414 and large-equipment grants (GOEGrid cluster). A.A.A.
acknowledges financial support provided by PICT 2017-2726 and PICT 2018-01546
of the ANPCyT, Argentina. This research was supported in part by the National
Science Foundation under Grant No. NSF PHY-1748958.

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##  Appendix A Experimental details

The time-dependent optical lattice potential is given by

$\begin{split}&V(x,y,z,\tau)=\\\ &\quad-
V_{\text{X}}I_{\text{self}}\cos^{2}(kx+\vartheta/2)\\\ &\quad-
V_{\text{Xint}}(\tau)I_{\text{self}}\cos^{2}(kx)\\\ &\quad-
V_{\text{Y}}\cos^{2}(ky)\\\ &\quad-V_{\text{Z}}\cos^{2}(kz)\\\
&\quad-\sqrt{V_{\text{Xint}}(\tau)V_{\text{Z}}}\cos(kz)\cos[kx+\varphi(\tau)]\\\
&\quad-
I_{\text{XZ}}\sqrt{V_{\text{Xint}}(\tau)V_{\text{Z}}}\cos(kz)\cos[kx-\varphi(\tau)]~{},\end{split}$
(A1)

where $k=2\pi/\lambda$ and $\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$. The
constant lattice depths $[V_{\text{X}},V_{\text{Y}},V_{\text{Z}}]$ used in
this paper are given by $[7.95(7),20.08(4),17.0(1)]E_{R}$, measured in units
of recoil energy $E_{R}=h^{2}/2m\lambda^{2}$, where $m$ the mass of the atoms.
The values in brackets denote the standard deviations of the lattice depths
over 5700 individual measurements. Contrary to our previous work [20], the
value of $V_{\text{Xint}}(\tau)$ is time-dependent:

$V_{\text{Xint}}(\tau)=V_{0}[1+A\times\sin(2\pi\tau/T)]\,.$ (A2)

The average lattice depth is $V_{0}=0.30(2)E_{R}$ and the amplitude is
$A=0.67(7)$. Likewise, the superlattice phase is time-dependent:

$\varphi(\tau)=\pi/2\times[1+\cos(2\pi\tau/T)]\,.$ (A3)

The imbalance factors are $I_{\text{self}}=0.98(2)$ and
$I_{\text{XZ}}=0.81(2)$.

Figure A1: Experimental pump trajectory. (a) Rice-Mele parameters as function
of time. (b) The ‘boomerang’ pump trajectory. (c) The single-particle band gap
varies during the course of a pump cycle. Its minimum is
$1.4\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. All energies are given in kHz
($\times h$).

The time-dependent lattice parameters $V_{\text{Xint}}(\tau)$ and
$\varphi(\tau)$ lead to a periodic variation of the Rice-Mele parameters
$\delta$ (dimerisation), $\Delta$ (sublattice site offset), and $t$ (average
tunnelling), as plotted in Fig. A1(a). The resulting pump trajectory is
‘boomerang’-shaped [Fig. A1(b)]. In addition, we plot the minimal single-
particle band gap in Fig. A1(c). The minimum band gap of
$1.4\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ can be used to estimate an
adiabatic timescale in the non-interacting limit. The time-dependence of the
average hopping matrix element $t$ causes a slight shift of the critical
$\Delta_{c}$ at a fixed value of $U$ by an amount which is smaller than $t$.

A large range of Hubbard $U$ can be accessed by utilising the Feshbach
resonances in the hyperfine ground-state manifold $F=9/2$ of 40K. For small
and intermediate values of $U<6t$, we use the mixture
$m_{F}=\\{-\nicefrac{{9}}{{2}},-\nicefrac{{7}}{{2}}\\}$. For stronger
interactions ($U>7t$), we use the mixture
$m_{F}=\\{-\nicefrac{{9}}{{2}},-\nicefrac{{5}}{{2}}\\}$. In between the two
ranges there is a small gap in the measured data points (Fig. 3). The
bracketed errors of the values of $U/t$ are dominated by the statistical
uncertainty of $5\%$ in the value of $t$.

As described in the main text, we extract the scaled centre-of-mass position
to measure the pumped charges. In Fig. A2 below we plot the measured dynamics
beyond one pump cycle. The outlier in Fig. A2(a) can be attributed to slow
drifts as well as statistical noise. Similar outliers are responsible for the
scatter in Fig. 3 in the main text. Interaction-induced pumping is evident
from Fig. A2(b), but the slope does not persist beyond one pump cycle.

Figure A2: Measured centre-of-mass displacements for two pump cycles. The
panels show different values of Hubbard $U/t$: (a) $0$, (b) $9.7$, and (c)
$16$.

##  Appendix B Numerical simulations

###  B.1 Real-time simulations

For the theory curves in Figs. 2, 3 and 4, we use a variational MPS method for
ground-state calculations [52, 53] and time-evolving-block-decimation (TEBD)
[84, 85, 86] for the time-evolution. The pumping is always started slowly via
a quadratic ramp-up of the driving to minimise non-adiabatic effects [87]. In
order to make the initial state resemble experimentally realistic conditions,
we start with decoupled dimer states. Concretely, we prepare each pair of two
sites (a dimer) in their ground state and then form a product of such dimers.

Technically, we start from the ground state for the decoupled Hamiltonian

$\displaystyle\hat{H}_{{\mathcal{D}_{N}}}=\sum_{(i,j)\in\mathcal{D}_{N}}\hat{H}_{i,j}.$
(A4)

$\mathcal{D}_{N}=\\{\dots,(c-2,c-1),(c,c+1),(c+2,c+3),\dots\\}$ is a set of
uncoupled dimer sites centred around site $c$, such that we fill $N$ particles
in total. Each initial dimer contains 2 particles (one spin-up and one spin-
down fermion). $c$ is chosen to be compatible with the dimerised ground state
of the whole system. We use an open system with size $L=49$, a trap strength
of $V/t=0.034$ and $N=24$ particles. For the ground-state search, we converge
the local density to a tolerance of $10^{-4}$. The time-evolution is carried
out using TEBD with a time step of $d\tau=0.01/t$ (where $t$ is the average
hopping rate) and an adaptive bond dimension with a cutoff of $10^{-12}$. The
pump cycle is identical to the experimental one. The pumped charge is computed
from the centre-of-mass displacement of the cloud.

For Fig. 5, we study a finite system with open-shell boundary conditions
(periodic boundary conditions for a number of sites $L$ multiple of four,
antiperiodic for even $L$ not a multiple of four) to allow for the resolving
of gap closings. We consider an elliptical pump cycle

$\displaystyle(\Delta/t,\delta)=\left[\Delta_{c}+R_{\Delta}\sin(\theta),R_{\delta}\cos(\theta)\right]~{},$
(A5)

with $R_{\Delta}/t=2.10$ and $R_{\delta}=0.88$. The time-evolution is carried
out with a time-dependent Lanczos method [88] with a tolerance of $10^{-12}$
and a time-step of $d\tau=0.01/t$. The initial state is chosen as the many-
body ground state at $\theta=0$. The pumped charge is computed as the integral
over the local particle current over one period:

$\displaystyle Q(\tau)$
$\displaystyle=\int_{0}^{T}d\tau^{\prime}\left\langle\hat{J}\left(\tau^{\prime}\right)\right\rangle,$
(A6) $\displaystyle\hat{J}$
$\displaystyle=\frac{i}{2}\sum_{j=1,2;\alpha=\uparrow,\downarrow}\left(t_{j}\hat{c}_{j,\alpha}^{\dagger}\hat{c}_{j+1,\alpha}-\text{
H.c. }\right),$ (A7)

with $t_{j}(\tau)=t+(-1)^{j}\delta(\tau)$.

In order to explore the breakdown of quantised charge pumping due to low-lying
spin excitations, we calculate the charge and spin nearest-neighbour
correlators:

$\displaystyle C_{\hat{n}}(t)$
$\displaystyle=\frac{1}{L}\langle\Psi(t)|\sum_{j=1}^{L}\hat{n}_{j}\hat{n}_{j+1}|\Psi(t)\rangle$
(A8) $\displaystyle C_{\hat{S}}(t)$
$\displaystyle=\frac{4}{L}\langle\Psi(t)|\sum_{j=1}^{L}\hat{S}_{j}^{z}\hat{S}_{j+1}^{z}|\Psi(t)\rangle,$
(A9)

where $\hat{n}_{j}=\hat{n}_{\uparrow,j}+\hat{n}_{\downarrow,j}$ is the total
particle number operator and
$\hat{S}_{j}^{z}=(\hat{n}_{\uparrow,j}-\hat{n}_{\downarrow,j})/2$ is the total
spin projection. $|\Psi(t)\rangle$ are the time-propagated states.

In the calculations for Fig. 5, we simplify the setup and discard the trap,
the variation of the hopping matrix element $t$, and the shape of the pump
cycle. As the initial state, we choose the ground state at the start of the
cycle. We verified that this simplification does not affect the main
qualitative conclusions from Fig. 5.

We next discuss the individual effect of some of the potential sources for
nonadiabatic and nonquantised pumping.

###  B.2 Effects of trap, particle number and state preparation

As the experiment works with a harmonic trap, there are limitations concerning
the maximum number of cycles that can be carried out. At the latest, when the
increase of the onsite potential due to the trap overcomes the required
variation of potentials during the pump cycle, quantised and unidirectional
pumping will break down. However, this happens much later than the first pump
cycle [78].

Another possible source of imperfect pumping could be the inhomogeneous
distribution of particle numbers in the various one-dimensional systems probed
simultaneously in our experiment. While it is impossible to simulate the
dynamics of the full distribution for an interacting system with realistic
particle numbers, we studied the behavior when increasing $N$ at fixed trap
strength for the initial state used in the experiment. All other parameters
are kept fixed and correspond to the situation discussed in the main text in
the context of Fig. 3.

Our results are shown in Fig. A3. Clearly, regardless of how many particles
are placed into the trap (as long as we do not reach the region of a steep
potential increase), during the first three quarters of the pump cycle, the
behavior is largely independent thereof and the particle number affects the
pumped charge only mildly. After crossing the gapless line, however, the pump
efficiency depends on $N$, with smaller $N$ being detrimental to efficient
pumping. Notably, the pumped charge quickly becomes independent of particle
number for $N\gtrsim 12$ and the pumped charge converges to a quantised value
in the first pump cycle. Since particle numbers are much higher in most of the
one-dimensional systems realised in the experiment, we conclude that
variations in $N$ can be excluded as a dominant source of imperfect pumping on
the time scales investigated.

Another potential source of imperfect pumping could be initial-state-
preparation defects, e.g., hole and doublon defects as well as empty unit
cells in the string of dimers. Exemplary simulations show that every string of
dimers pumps independently on the relevant time scales and hence these defects
are not expected to play a crucial role.

Figure A3: Pumping for different particle numbers. TEBD calculation for a
harmonically trapped system with initial strings of dimer states with
different particle numbers $N=2,6,12$, and $24$. The pump charge is calculated
via the centre-of-mass displacement. Calculated for the experimental pump
cycle with $L=49$, $U/t=8$, $T=23.3\hbar/t$, $d\tau=0.1\hbar/t$ and
$V_{0}/t=0.034$.

###  B.3 Role of the pump cycle

For a strictly adiabatic situation, the choice of the pump cycle does not
matter. In our situation, where a spin-gapless region inevitably exists in the
thermodynamic limit, a dependence on details of the pump cycle is, in
principle, expected. Moreover, the experiment operates at a finite entropy
density.

For a finite-size system, there is, strictly speaking, no gapless continuum of
spin excitations yet, hence the minimum finite-size gap along the pump cycle
should matter. To address this point, we compare the boomerang-shape pump
cycle used in the experiment (including the actual variation of the average
hopping matrix element) to an elliptical cycle as used for Fig. 5 where we
keep the average hopping matrix element constant, both with $N=10$ particles.
These simulations are carried out for a closed system using the time-dependent
Lanczos technique for a slow pumping protocol. We remove the trap to single
out the effect of the minimum gap.

The results presented in Fig. A4 illustrate that during the first pump cycle,
there is no significant difference between the two pump paths. Beyond that,
the behavior is strongly path-dependent, but not quantised in either case, yet
somewhat larger for the elliptical cycle. This is reflected in the minimum
gaps along the two cycles, which are $\Delta E_{\mathrm{exp}}=0.04t$, $\Delta
E_{\mathrm{ellip}}=0.18t$.

One may further wonder about the effect of total particle number at fixed
filling. For the elliptical cycle with $N=L=6$ and $10$, the respective
minimum gaps are $\Delta E_{L=6}=0.28t$, $\Delta E_{L=10}=0.18t$,
respectively. Consistently, the pump efficiency for $L=10$ is much worse than
for $L=6$ (results not shown).

Figure A4: Pumping with different trajectories and system sizes. Time-
dependent Lanczos calculation of the pumped charge $Q(\tau)$ in an open-shell
antiperiodic system, started from the ground state. Experimentally realised
pump-cycle (‘exp’) and elliptical pump cycle (‘ellip’) $L=10$. $U/t=10$,
$d\tau=0.01\hbar/t$, $T=100\hbar/t$.

###  B.4 Effect of dimer initial state instead of ground state.

Figure A5: Starting from the ground state versus dimer state. Difference of
the pumped charge between a pump starting from the ground state ($Q_{gs}$) and
one starting from the dimer state ($Q_{dim}$) for the experimental and the
elliptical pump cycle, $L=6$, $U/t=10$, $d\tau=0.01\hbar/t$ and
$T=100\hbar/t$.

In the experiment, the loading scheme leads to an initial state that can be
approximated by a product of dimers, each with two particles and in their
ground state. We compare pumping starting from either this state or the ground
state, from the same point in the pump cycle. The results of time-dependent
Lanczos calculations are shown in Fig. A5. The difference in the pumped charge
is initially small yet increases significantly once the minimum gap is
crossed. The comparison between the two different pump cycles – experimental
versus elliptical one – shows that the former leads to smaller differences in
the first pump cycle because the starting point has a larger value of $\delta$
and therefore a stronger dimerization.

Figure A6: Pumping with an additional interaction term. Time-dependent Lanczos
calculation of the pumped charge $Q(\tau)$ from the ground state in an open-
shell antiperiodic system for the experimentally realised pump-cycle, $L=10$,
$U/t=10$, $d\tau=0.01\hbar/t$, $T=100\hbar/t$ and two different Ising-coupling
strengths ($J=0$ and $J=2t$).

###  B.5 Improving the robustness of the interaction-induced pump via opening
the spin gap

As already discussed in Ref. [43], the SU(2) symmetry of the ionic Hubbard
model can be lifted by various perturbations which at the same time gap out
the spin sector. This leads to robust quantised pumping over many cycles.

We here demonstrate that this prediction remains valid also for the
experimental pump cycle, by adding an Ising interaction to the Hamiltonian
(other examples were studied in Ref. [43]):

$\displaystyle\hat{H}_{\mathrm{Z}}=J\sum_{j=1}^{L}\hat{S}_{j}^{z}\hat{S}_{j+1}^{z}.$
(A10)

The comparison between a simulation with and without this Ising term is shown
in Fig. A6. The results establish that opening the spin gap significantly
stabilises the pump, leading to robust pumping for many pump cycles compared
to the bare Rice-Mele-Hubbard model.