# Two-particle topological Thouless spin pump

J. Mumford  Department of Physics and Astronomy, McMaster University, 1280
Main Street West, Hamilton, Ontario, Canada L8S 4M1

###### Abstract

We show that two particles interacting via spin exchange exhibit topological
features found in one-dimensional single particle lattice models. This is
accomplished by absorbing all of the spatial degrees of freedom of the
lattices into the spin degrees of freedom of the two particles. Comparing the
spin system with the Su-Schrieffer-Heeger model, we show the existence of
topologically protected edge spin states and establish the bulk-edge
correspondence. Modifying the spin system with a chiral symmetry breaking term
results in it resembling the Rice-Mele model and can therefore act as a
Thouless spin pump of one of the particles when periodically and adiabatically
driven. By using the spin states as a synthetic spatial dimension, we show two
particles are enough to simulate well known topological properties in
condensed matter physics.

##  I Introduction

Accurately simulating the degrees of freedom and interactions of large quantum
systems is difficult or impossible for classical computers due to their
complexity. This is because for most quantum systems the amount of memory
required to simulate evolution or even represent a single state of the system
increases exponentially with the number of particles. In recent years, the
motivation to skirt these issues has led to other quantum systems being used
as quantum simulators [1, 2, 3] because they have quantum rules built into
them which do not need to be implemented manually like in classical computers.
Some examples are trapped ions [4, 5, 6] and ultracold atoms in optical
lattices [7, 8, 9, 10] because they are relatively easy to control allowing
for systematic study. The high degree of control has lead to a wide array of
systems which can be simulated in fields as far ranging as statistical
mechanics [11] and astrophysics [12]. However, the main direction in which
quantum simulations are directed is in solid state physics with a focus on
simulating the motion of charged particles in materials [13, 14, 15, 16, 17,
18, 19]. Experiments along these lines vary from simulating well known
phenomena such as the Meissner effect in an optical ladder lattice [20] to
creating new exotic states of matter [21]. They have also made it possible to
achieve magnetic field strengths previously unattainable in real materials
which has led to the observation of the Harper-Hofstadter model [22, 23, 24].

One process which is of particular interest is the Thouless charge pump [25]
which is motion produced from the adiabatic cyclic driving of the lattice the
charges are in. The uniqueness of the Thouless pump comes from the fact that
it does not displace charges continuously like electric or magnetic fields,
but instead displaces them in clumps and is therefore quantized due to the
topology of the pump cycle. Simulations of the Thouless pump have been
successfully realized in, among others systems, ultracold atoms in optical
lattices [26, 27, 28] and in a photonic system with resonator arrays [29].
Thouless spin pumps have also been implemented using ultracold atoms in
optical lattices [30] where particles of opposite spin move in opposite
direction, so there is no charge transport.

In this work, we simulate the Thouless spin pump by completely internalizing
the spatial degrees of freedom of the lattice into spin states. The system
consists of just two particles: one spin-$S$ particle where $S>1$ and one
spin-1/2 particle. We show that two spins interacting via spin exchange is
equivalent to an inhomogeneous version of the Su-Schrieffer-Heeger (SSH) model
[45] which is a one-dimensional (1D) lattice model with a topologically
nontrivial phase. The spin system shares the topological features of the SSH
model which manifest in the form of protected extreme spin states. By
modifying the spin system, we show that it is equivalent to the inhomogeneous
Rice-Mele (RM) model [46] which is another 1D lattice model well known for
being an example of a Thouless charge pump. The spin system also shares the
pumping behavior in the form of a spin pump of the spin-$S$ particle. The
mimicking of the spatial degrees of freedom with spin states is an example of
the use of synthetic dimensions [31, 32, 33]. Synthetic dimensions are a
useful tool in constructing quantum simulators because they provide
versitility in their construction and open up new possibilities of their
applicability. They have been employed in experiments using single particle
states such as spin [34, 35, 36], momentum [37, 38, 39], harmonic oscillator
[40], and rotational [41] states of particles. Recently, many-body cavity
modes of light have also been used to simulate both one-dimensional (1D) and
two-dimensional (2D) lattices [42] and many-body states of matter have been
suggested in constructing synthetic lattices [43, 44]. The goal of this paper
is to use synthetic dimensions to highlight a different way to simulate
quantized motion - one without real-space lattices.

##  II Model

Figure 1: Two equivalent nontrivial topological systems. (a) Giant spin-$S$
particle interacting via spin exchange with a spin-1/2 particle. The spin-1/2
particle can make transitions on its own ($v$, purple), however, the spin-$S$
particle can only gain or lose one quantum of spin by exchanging it with the
spin-1/2 particle ($w$, green). (b) Single particle in a one-dimensional (1D)
lattice. The particle can tunnel through the small barrier ($v$, purple) or
the large barrier ($w$, green). There is a constant onsite energy offset
between adjacent sites labeled $\Delta$ and a tunneling offset that changes
along the lattice labeled $\delta$. The energy offset $\Delta$ acts as a
magnetic field applied to the spin-1/2 particle only (not shown) and biases it
toward the up or down state depending on its sign. The tunneling offset
$\delta$ is applied to the lattice to match the spin-$S$ transitions which are
not homogeneous when $S>1$.

The system we will be investigating consists of a giant spin-$S$ particle
interacting via spin exchange with a spin-1/2 particle which can also flip its
spin on its own due to an external field. The Hamiltonian of the system is

$\hat{H}_{1}=-w\left(\hat{S}_{+}\hat{\sigma}_{-}+\hat{S}_{-}\hat{\sigma}_{+}\right)-Sv\hat{\sigma}_{x}$
(1)

where the $S$ operators belong to the spin-$S$ particle and the Pauli spin
matrices belong to the spin-1/2 particle. The factors $w$ and $v$ are the
energies for the spin exchange and the spin-1/2 spin-flip processes,
respectively. For $S\gg 1$ we can perform a mean field approximation of the
spin-$S$ operators by replacing them with their spin-coherent expectation
values $\langle\hat{S}_{\pm}\rangle=\sqrt{S^{2}-n^{2}}e^{\pm i\phi}$ where
$\phi$ is the azimuthal angle on a Bloch sphere of radius $S$ and $n$ is the
projection of the spin onto the $S_{z}$ axis of the Bloch sphere, so
$\hat{S}_{z}|n\rangle=n|n\rangle$. The Hamiltonian becomes

$H_{1}(n,\phi)=-w\sqrt{S^{2}-n^{2}}\left(e^{i\phi}\hat{\sigma}_{-}+e^{-i\phi}\hat{\sigma}_{+}\right)-Sv\hat{\sigma}_{x}\,.$
(2)

The Su-Schrieffer-Heeger (SSH) model Bloch Hamiltonian is

$H(k)=-w\left(e^{ik}\hat{\sigma}_{-}+e^{-ik}\hat{\sigma}_{+}\right)-v\hat{\sigma}_{x}\,,$
(3)

which describes a single particle in a 1D lattice with two sites per unit
cell. Here, the parameters $w$ and $v$ represent the tunneling energies
between unit cells and within unit cells, respectively. Due to the SSH model
having discrete translational invariance, the quasimomentum $k$ is a good
quantum number. Comparing the two Hamiltonians, we see that $\phi$ is similar
to $k$, so it can be thought of as the quasimomentum of a spin-state lattice.
Further comparisons show that the spin exchange process can be thought of as
tunneling between adjacent unit cells and the spin-1/2 spin-flip process can
be thought of as tunneling between the sites within a unit cell. This means
that $n$ is similar to the unit cell number in the SSH model and the two
states of the spin-1/2 particle label the two sites in each unit cell. The
obvious difference between the two systems is the square root factor in Eq.
(2). The source of the square root inhomogeneity comes from the boson
stimulation factors that arise when making transitions between adjacent
spin-$S$ states

$\hat{S}_{\pm}|n\rangle=\sqrt{\left(S\mp n\right)\left(S\pm n+1\right)}\,|n\pm
1\rangle\,.$ (4)

Therefore, the two models are equivalent when the SSH model has broken
translational invariance and the tunneling is unit cell dependent. Figure 1
shows diagrams of the two systems as well as the processes which correspond
between them. Although we cannot use ”unit cell” in the strict sense of the
term due to the inhomogeneity, we will continue to use it for convenience when
making comparisons between the two systems.

Figure 2: Energies, wave functions and winding numbers. (a) Spectrum of the
spin system in Eq. (1) as a function of the spin-1/2 spin-flip energy $v$ for
$S=5$. The spectrum features doubly degenerate $E=0$ edge states up until
$v\approx w$ where they begin to significantly diverge. (b) Top and middle-top
panels show the wave functions of the $E=0$, $v=0.4w$ [red dot in (a)] edge
states while the middle-bottom and bottom panels show the edge states have
been destroyed for $v=1.6w$ [blue dots in (a)]. (c) Top panel shows the
localized winding number on either side of the transition point calculated
from Eq. (8): $v=0.5w$ (green) and $v=1.5w$ (orange). The red dashed curve is
the mean field winding number $\nu_{\mathrm{MF}}(n)$ for $v=0.5w$. Bottom
panel shows the average winding number as a function of $v$ for $S=50$ which
has a sudden drop around $v=w$ signifying a topological transition. The
average is taken over the range $-10\leq n\leq 10$.

The SSH model is significant because it can display non-trivial topological
properties in the form of stable zero energy edge states. The topological
phase of the system can be quantified in terms of the winding number

$\displaystyle\nu$ $\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int_{0}^{2\pi}dk\,h(k)^{-1}\partial_{k}h(k)$ (5)
$\displaystyle=$ $\displaystyle\frac{1}{2}\left[1+\mathrm{sgn}(w-v)\right]$

where $\gamma=i\int_{0}^{2\pi}dk\,h(k)^{-1}\partial_{k}h(k)$ is the Berry
phase over the Brillouin zone and $h(k)=v+we^{-ik}$. Here, $w>v$ represents
the topological phase with $\nu=1$ and one edge state at each edge of the
lattice whereas $w<v$ represents the trivial phase with $\nu=0$ and no edge
states. The winding number assumes periodic boundary conditions, so it is a
bulk quantity. The one-to-one correspondence between the winding number and
the number of edge states is a fundamental property of 1D topological systems.
In the following section we will see what topological properties of the SSH
model are shared by the spin system as well as the effects of the square root
factors on those properties.

##  III  Results

###  III.1 Topological transition

We begin by looking at the spectrum of the spin model in Fig. 2(a) for
different values of the spin-1/2 transition energy, $v$. Like the SSH
spectrum, it is symmetric about the $E=0$ axis because both models preserve
chiral symmetry
$\hat{\mathcal{C}}\hat{H}_{1}\hat{\mathcal{C}}^{-1}=-\hat{H}_{1}$, with
$\hat{\mathcal{C}}=\hat{\sigma}_{z}$. This symmetry means that for every
energy eigenstate $|E\rangle$, there is a partner state
$\hat{\sigma}_{z}|E\rangle=|-E\rangle$. Differences between the two models are
most pronounced at $v=0$ where in the SSH model there is $2S$-fold degeneracy
at $E_{\pm}=\pm w$ because the lattice is broken up into $2S$ identical
disconnected sublattices of two sites each. A similar thing happens in the
spin model when $v=0$: for a giant spin-$S$ particle the system breaks-up into
$2S$ pairs of states $|n,\uparrow\rangle$ and $|n+1,\downarrow\rangle$ where
$\hat{\sigma}_{z}|\uparrow(\downarrow)\rangle=+(-)|\uparrow(\downarrow)\rangle$.
Even and odd combinations of these states form the basis and for large $S$
have energies $E_{\pm}(n)\approx\pm w\sqrt{S^{2}-n^{2}}$, so the large
degeneracy found in the SSH model is not present. For $v<w$ the states with
$E\neq 0$ show signs of near double degeneracy which comes from the fact that
$E_{\pm}(n)\approx E_{\pm}(-n)$.

For $v=0$, the zero energy states are the edge spin states
$|S,\uparrow\rangle$ and $|-S,\downarrow\rangle$. As $v$ increases, the zero
energy states continue to be localized at the edges because their amplitudes
within the bulk are suppressed due to the large gap above and below $E=0$. The
top and middle-top panels of Fig. 2(b) (red) show the doubly degenerate zero
energy wave functions for $v=0.4$ which are even and odd combinations of
states localized at the edges. For comparison, the bottom and middle-bottom
panels (blue) show the same states for $v=1.6$ where now they mainly occupy
the bulk spin states and have $E\neq 0$. Focusing back on (a), we see that at
$v\approx w$ the zero energy states diverge and their gaps between neighboring
states shrink signaling the delocalization of these states from the edge.

Whether this transition has topological origins like the similar one in the
SSH model requires the establishment of the bulk-edge correspondence. For
that, we need to calculate the winding number and determine if it corresponds
to the number of edge states. To start, the mean field winding number is
calculated by using Eq. (5) by making the substitutions $h(k)\to
h(n,\phi)=Sv+w\sqrt{S^{2}-n^{2}}e^{-i\phi}$ and
$\partial_{k}\to\partial_{\phi}$ and integrating over $\phi$ giving

$\nu_{\mathrm{MF}}(n)=\frac{1}{2}\left[1+\mathrm{sgn}\left(w\sqrt{S^{2}-n^{2}}-Sv\right)\right]\,.$
(6)

The winding number for the SSH model in Eq. (5) which describes the entire
bulk is now spin (site) dependent due to the inhomogeneity. A numerical
calculation from the full quantum model in Eq. (1) is also performed using the
winding number operator [47, 48]

$\hat{\nu}=4\hat{P}_{+-}^{-1}\left[\hat{S}_{z},\hat{P}_{+-}\right]$ (7)

where $\hat{P}_{+-}=\hat{P}_{-+}^{-1}=\hat{\sigma}_{+}\hat{P}\hat{\sigma}_{-}$
is the off-diagonal spin-1/2 component of the projection operator, $\hat{P}$,
of states with $E<0$. Comparing $\hat{\nu}$ with Eq. (5) we have replaced
$h(\phi)$ with $\hat{P}_{\uparrow\downarrow}$ and made use of the fact that
$\phi$ and $n$ are conjugate variables, so $\partial_{\phi}$ is the same as
$-i[\hat{S}_{z},\,]$. The integral over $\phi$ can be replaced with a trace
over the bulk spin states, however, we localize it to a unit cell to compare
it to Eq. (6)

$\nu(n)=\mathrm{Tr}_{\sigma}[\langle n|\hat{\nu}|n\rangle]$ (8)

where $\mathrm{Tr}_{\sigma}$ is the trace over the spin-1/2 degrees of
freedom. Equation (8) is plotted in the top panel of Fig. 2(c) for $v>w$
(orange) and $v<w$ (green). There is a clear difference between the two cases
where the bulk has a winding number of unity when $v=0.5w$ and of zero when
$v=1.5w$. The mean field winding number is also plotted as a red dashed curve
for $v=0.5w$ where both it and the quantum winding number show clear signs of
being bulk quantities as they drop to zero near the edges. The mean field
winding number also provides a good approximation of where the drop takes
place. To get a complete picture of the transition, we plot the bulk average
winding number in the bottom panel of (c). The average is taken over the range
$-10\leq n\leq 10$ where $S=50$, so it is comfortably away from the edges. A
sudden drop from unity to zero is shown at $v\approx w$ agreeing with the
vanishing of edge states shown in Fig. 2(a) and (b). This establishes the
bulk-edge correspondence in the spin system and shows that at $v=w$ there is a
topological transition.

###  III.2 Spin pump

Figure 3: Dynamics of spin displacement and circuit paths in parameter space.
(a) Spin ($S=50$) displacement as a function of time for a circuit in
parameter space that encloses the topological transition point at
$(w-v,\Delta)=(0,0)$ (black) and one that does not (green). For the circuit
that does enclose the transition point, the spin dynamics looks like a
staircase as the average spin increases by one quantum of spin every cycle.
The red line indicates the slope of the ideal spin pump staircase. The
deviation from the ideal case comes from the inhomogeneity of the spin-$S$
transitions and results in a larger energy requirement to make transitions to
higher spin states. (b) Paths taken through parameter space in each cycle. The
numerical values are in units of $w_{0}$.

With the addition of a spin-1/2 offset the Hamiltonian becomes

$\hat{H}_{2}=\hat{H}_{1}-S\Delta\hat{\sigma}_{z}\,.$ (9)

The new term breaks the chiral symmetry and allows for a continuous transition
between the two topological phases. The Hamiltonian resembles the Rice-Mele
(RM) model which is a well known example of a Thouless charge pump. When the
RM model is adiabatically driven in a cycle of period $T$ the charges in the
lattice end up being displaced by a single unit cell if the path in parameter
space encloses the topological transition point at $(w-v,\Delta)=(0,0)$.
Therefore, we should expect the giant spin to change by a single quantum of
spin if an adiabatic circuit in parameter space encompasses the same point.
The time dependent parameters which are used to trace the circuit are

$\displaystyle v(t)$ $\displaystyle=$ $\displaystyle v_{0}\left[1-\sin(2\pi
t/T)\right]/2,$ $\displaystyle w(t)$ $\displaystyle=$ $\displaystyle
w_{0}\left[1+\sin(2\pi t/T)\right]/2,$ $\displaystyle\Delta(t)$
$\displaystyle=$ $\displaystyle\Delta_{0}\cos(2\pi t/T)$ (10)

where $T\gg 1$ is the driving period and is large so the driving is adiabatic.
The other parameter values are $v_{0}=w_{0}$ and $\Delta_{0}=20w_{0}$. The
evolution over one cycle can be imagined with the following major steps: (1)
$t=0$ the system is initialized in the ground state with the spin-1/2 particle
being in the state $|\uparrow\rangle$ due to a large offset from an applied
magnetic field, (2) $0<t\leq T/2$ interactions are ramped up and spin exchange
takes place where the giant spin gains a quantum of spin and
$|\uparrow\rangle\to|\downarrow\rangle$, (3) $T/2<t\leq T$ interactions are
ramped down and the applied magnetic field resets the spin-1/2 particle
$|\downarrow\rangle\to|\uparrow\rangle$. The large offset ensures the state is
nearly completely locked into one of the spin-1/2 states and transitions only
take place at the times $t\approx T/4$ (spin-$S$) and $t\approx 3T/4$
(spin-1/2). The net result is that the giant spin gains one quantum of spin
much the same way a charge is displaced one unit cell in the RM model. To show
the dynamics we calculate the giant spin displacement

$\Delta n(t)=\langle\psi(t)|\hat{S}_{z}|\psi(t)\rangle-n_{0}$ (11)

where $n_{0}=\langle\psi(0)|\hat{S}_{z}|\psi(0)\rangle$ is the initial average
spin. When $S\gg 1$ the continuum approximation can be made and Eq. (11) can
be put into another form (appendix A)

$\displaystyle\Delta s(t)=\int_{0}^{t}dt^{\prime}j(t^{\prime})$ (12)

where $\Delta s(t)=\Delta n(t)/S$ is the continuous spin displacement variable
and $j(t)=\int_{-1}^{1}dsj(s,t)$ is the total probability current. The RHS of
Eq. (12) is also the expression for the transported charge, $Q(t)$, of a
material when $j(t)$ is the electric current averaged over the material, so
$\Delta n(t)$ can be thought of as the scaled transported spin $\Delta
n(t)=SQ(t)$. This is an important connection because in topologically
nontrivial systems, the total charge transported during one adiabatic cycle is
proportional to the change in the winding number $Q(T)\propto\nu(T)-\nu(0)$
[49], so this is where the topological nature of the pump comes from.

Equation (11) is plotted in Fig. 3(a) for the cycle outlined in Eq. (10)
(black) and for the same cycle except $v(t)=v_{0}\left[4-\sin(2\pi
t/T)\right]/2$ (green). A clear difference is displayed where the spin
displacement increases in integer steps each period of the cycle for the
former circuit and does not go beyond unity for the latter circuit. The two
circuits through parameter space are shown in Fig. 3(b) where the black one
encloses the topologically significant point at the origin
$(w-v,\Delta)=(0,0)$ and the green one does not. Other circuits shifted along
the other three axes while not encompassing the origin have also been checked
and no displacement greater than unity was observed in the same time frame.
This behavior matches the quantized displacement found in the RM model and
other systems which are able to act as charge pumps. The red line is a guide
for the ideal case where the black data should intercept it at integer values
of the period $T$. At early times the spin pump does follow the ideal case,
however, at $t\approx 5T$ there starts to be a large discrepancy. The cause of
this is the square root factors in Eq. (4) because they increase the energy
required to transition to larger spin-$S$ states the farther away the state is
from $n=0$. For states around $n=0$, the factors act as a harmonic trap since
to leading order in spin number they are
$-\sqrt{N^{2}/4-n^{2}}\approx-N/2+n^{2}/N$. Similar saturation has been
observed in experiments simulating the RM model with ultracold atoms in an
optical lattice [27] because an overall harmonic trap is required to hold the
atoms in place. The saturation was attributed to the variation of the harmonic
trap becoming comparable to the bandgap. The major difference between the
experiment and the spin system is that the tunneling and the harmonic
confinement in the experiment is controlled separately with different lasers
whereas that is impossible here.

###  III.3 Many-body system

Figure 4: Time dependence of the displacement for different values of the
interaction energy $\Lambda$. Each data point is the displacement at the
beginning of each cycle for the path enclosing the origin in Fig. 3 and the
lines connecting the points are there as a guide. The deviation of the
displacement from the ideal case (straight red line) increases with $\Lambda$.
The number of particles is $N=2S=100$ and the values of $\Lambda$ are in units
of $w_{0}$.

The spin-$S$ operators can also be considered as many-body operators for
$N=2S$ identical spin-1/2 particles
$\hat{S}_{\alpha}=\sum_{i=1}^{N}\hat{\sigma}_{\alpha}^{i}$ where
$\alpha=(x,y,z)$. The Pauli matrices in the sum are distinguished from the
spin-1/2 ones via the superscript. The spin-1/2 particle can represent an
impurity. In this system, the spin number becomes $n=(N_{+}-N_{-})/2$ which is
half the difference between the number of particles in each of the two spin
states labeled $+$ and $-$. Each pump cycle is marked by one of the $N$
particles making a transition from one state to the other depending on the
direction of the pump, so after one period $n\to n\pm 1$. The interaction term
in Eq. (1) can be accomplished if each of the $N$ particles interacts with the
impurity with the same strength which is commonly found in systems with
infinite range interactions (interactions that do not depend on the position
of each particle). We assume the pairwise interactions of the $N$ particles
are infinite range as well and take the same form as their pairwise
interaction with the impurity giving
$-\Lambda\left(\hat{S}_{+}\hat{S}_{-}+\hat{S}_{-}\hat{S}_{+}\right)/N$ which
can also be written as $2\Lambda\hat{S}_{z}^{2}/N$ when $N$ is conserved. Such
interactions can be found in the infinite range isotropic XY model where the
effect for $\Lambda>0$ is harmonic confinement since there is a quadratic
energy cost for states $n\neq 0$. The additional confinement only adds to the
effect of the square root factors causing the pump to substantially deviate
from the ideal case at earlier times which is shown in Fig. 4 for different
values of $\Lambda$ using the circuit in Eq. (10). To reduce clutter in the
graph we only plot the displacement at the start of each cycle.

Attractive interactions ($\Lambda<0$) energetically promote the spreading of
states away from $n=0$. For weak attractive interactions, the ground state at
$t=0$ spreads, but still remains centered near $n=0$ due to the square root
factors dominating. However, there is a critical interaction strength where a
quantum phase transition (QPT) occurs resulting in the shift of the center of
the ground state to $n\neq 0$. The critical value is (Appendix B)

$\Lambda_{c}=-\frac{w(w+v)}{\sqrt{\Delta^{2}+(w+v)^{2}}}\,.$ (13)

Care must be taken when $\Delta\gg w,v$, which is what we have used, because
the system is sensitive to any attractive interactions since the critical
value $\Lambda_{c}\approx w(w+v)/\Delta$ will be close to zero. Preparing the
initial state under these conditions could result in a large shift of the
center of the state toward the edges at $n=\pm N/2$ where the winding number
drops rapidly, as seen in the bottom panel of Fig. 2(c), resulting in the
destruction of the pump. This issue can be resolved by simply preparing the
initial state in another segment of the cycle around $\Delta\approx 0$.
However, if $\Lambda<\Lambda_{c}^{\mathrm{min}}$, where
$|\Lambda_{c}^{\mathrm{min}}|$ is the minimum interaction strength over one
cycle, then the system will ramp through the critical point twice per cycle
resulting in unwanted excitations.

##  IV Conclusion

We have shown that a system consisting of spinful particles interacting via
spin exchange can simulate inhomogeneous versions of the SSH and RM models.
The spin system shares all of the topological features of both models which
includes topologically protected edge states and a quantized spin pump in the
form of a Thouless pump. We also showed that when the spin system is
interpreted as a many-body system, repulsive interactions lead to decrease
efficiency of the pump. The majority of simulations of condensed matter
systems depend on spatial degrees of freedom such as optical lattices and
resonator arrays to simulate the crystal structure of solids, however, all
that is needed are discrete sets of states which are abundant in quantum
systems. Here, we have presented an extreme case of this by absorbing all of
the spatial dependence into the internal spin states of two particles.

The large values of $S$ throughout this paper are used to highlight the
topological features of the model. In reality, the spins of individual
particles are much smaller, so the topological transition point at
$(w-v,\Delta)=(0,0)$ becomes a transition region around this point due to
finite size effects. This means the smaller $S$ is, the smaller $v$ must be to
observe the edge states and the larger the circuit must be to observe the spin
pump. Of course, a bulk must exist which requires $S>1$ since there should be
two bulk unit cells to pump between. Therefore, the minimum case has $n=\pm
3/2$ and $n=\pm 1/2$ as the edge and bulk states, respectively. To observe
larger $S$ effects, the infinite range XY model can be used where the spin-1/2
particle plays the role of an impurity. We found the best spin pump result is
achieved when there are no interactions between the $N$ identical particles in
the XY chain, but there are equal interactions between the particles in the
chain and the impurity. This can happen in a variation of the central spin
model [50] where the chain makes a horseshoe shape with the impurity at the
center, so the impurity is roughly equal distance from each particle along the
chain and the edges are maintained.

##  Appendix A Derivation of $\Delta s(t)$

We start with Eq. (11) and expand the $|\psi\rangle$ states in terms of the
$\hat{S}_{z}$ eigenstates, $|\psi(t)\rangle=\sum_{n}c_{n}(t)|n\rangle$ where
$c_{n}(t)=\langle n|\psi(t)\rangle$, giving

$\Delta n(t)=\sum_{n}n\left[|c_{n}(t)|^{2}-|c_{n}(0)|^{2}\right]\,.$ (14)

The continuum approximation is used by assuming $S\gg 1$, so that $s=n/S$ is a
continuous variable. We also identify $\rho(s,t)=|c_{n}(t)|^{2}$ as the
probability density where $\int_{-1}^{1}ds\rho(s,t)=1$, so Eq. (14) becomes

$\Delta s(t)=\int_{-1}^{1}dss\left[\rho(s,t)-\rho(s,0)\right]\,.$ (15)

The probability density difference can be put into integral form
$\rho(s,t)-\rho(s,0)=\int_{0}^{t}dt^{\prime}\partial\rho(s,t^{\prime})/\partial
t^{\prime}$ and the probability continuity equation,
$\partial\rho(s,t)/\partial t=-\nabla j(s,t)$, is used to give

$\Delta s(t)=-\int_{0}^{t}dt^{\prime}\int_{-1}^{1}dss\nabla
j(s,t^{\prime})\,.$ (16)

Next, we use integration by parts and set the boundary terms to zero by making
the assumption that the probability current falls off at the edges of the
system which leads to the final result

$\Delta s(t)=\int_{0}^{t}dt^{\prime}\int_{-1}^{1}dsj(s,t^{\prime})\,.$ (17)

Defining $j(t)=\int_{-1}^{1}dsj(s,t)$ as the total probability current gives
Eq. (11) in the main text.

##  Appendix B Derivation of $\Lambda_{c}$

Figure 5: Quantum and mean field ground state energies. The ground state
energy shows a shift around the critical interaction energy of $\Lambda_{c}$
signaling a QPT. The parameters used in the calculations are $v=1$, $\Delta=4$
(both in units of $w$) and $N=400$.

We are interested in the ground state QPT, so we start by finding an
expression for the ground band energy. The Hamiltonian including interactions
is

$\hat{H}_{3}=\frac{2\Lambda}{N}\hat{S}_{z}^{2}-w\left(\hat{S}_{+}\hat{\sigma}_{-}+\hat{S}_{-}\hat{\sigma}_{+}\right)-Nv\hat{\sigma}_{x}/2-N\Delta\hat{\sigma}_{z}/2$
(18)

and making a mean field approximation by replacing the spin-$S$ operators with
their spin coherent expectation values gives

$\frac{2H_{3}}{N}=\frac{\Lambda}{2}\cos\theta^{2}-w\sin\theta\left(e^{i\phi}\hat{\sigma}_{-}+e^{-i\phi}\hat{\sigma}_{+}\right)-v\hat{\sigma}_{x}-\Delta\hat{\sigma}_{z}$
(19)

where we have written the spin number in terms of the polar angle on the Bloch
sphere $n=(N/2)\cos\theta$. Setting $\phi=0$ minimizes the Hamiltonian further
and diagonalizing it gives an expression for the ground band energy

$\frac{2E_{0}(\theta)}{N}=\frac{\Lambda}{2}\cos\theta^{2}-\sqrt{\Delta^{2}+(v+w\sin\theta)^{2}}\,.$
(20)

It is expected that the phase transition occurs when there is a new ground
state. Due to the symmetry around the ground state at $\theta=\pi/2$ ($n=0$),
we expect a second order phase transition in the ground band where the initial
ground state transforms into a local maximum and two new degenerate
$\theta\neq\pi/2$ ground states appear when $\Lambda<\Lambda_{c}$. We expand
$E_{0}(\theta)$ around $\theta=\pi/2$

$\frac{2E_{0}(\theta)}{N}\approx A+B\delta\theta^{2}+C\delta\theta^{4}$ (21)

where $\delta\theta=\theta-\pi/2$ and note that new minima occur when $B<0$
while $C$ is positive. The constant $B$ is

$B=\Lambda+\frac{w(w+v)}{\sqrt{\Delta^{2}+(w+v)^{2}}}\,,$ (22)

so $\Lambda_{c}$ is Eq. (13) in the main text. A phase transition is marked by
a shift in the ground state energy which we plot in Fig. 5 where the black
solid curve is calculated by diagonalizing Eq. (18) and the dashed red curve
is calculated by finding the minimum of Eq. (20). The image shows excellent
agreement between the quantum and mean field predictions and that there is a
shift in the ground state energy at $\Lambda\approx\Lambda_{c}$.

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