# Quantized Thouless pumps protected by interactions in dimerized Rydberg
tweezer arrays

Sergi Julià-Farré  0000-0003-4034-5786<EMAIL_ADDRESS>ICFO -
Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
PASQAL SAS, 7 rue Léonard de Vinci - 91300 Massy, Paris, France  Javier
Argüello-Luengo  0000-0001-5627-8907 ICFO - Institut de Ciencies Fotoniques,
The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3,
08860 Castelldefels (Barcelona), Spain  Departament de Física, Universitat
Politècnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain  Loïc
Henriet  PASQAL SAS, 7 rue Léonard de Vinci - 91300 Massy, Paris, France
Alexandre Dauphin  0000-0003-4996-2561 PASQAL SAS, 7 rue Léonard de Vinci -
91300 Massy, Paris, France

###### Abstract

We study Thouless pumps, i.e., adiabatic topological transport, in an
interacting spin chain described by the dimerized XXZ Hamiltonian. In the
noninteracting case, quantized Thouless pumps can only occur when a
topological singularity is encircled adiabatically. In contrast, here we show
that, in the presence of interactions, such topological transport can even
persist for exotic paths in which the system gets arbitrarily close to the
singularity. We illustrate the robustness of these exotic Thouless pumps
through the behavior of the noninteracting singularity, which for sufficiently
strong interactions splits into two singularities separated by a spontaneous
antiferromagnetic insulator. We perform a numerical benchmark of these
phenomena by means of tensor network simulations of ground-state physics and
real-time adiabatic dynamics. Finally, we propose an experimental protocol
with Floquet-driven Rydberg tweezer arrays.

##  I Introduction

In one-dimensional systems, nontrivial topological insulators can only be
defined in the presence of at least one protecting symmetry, leading to the
notion of 1D symmetry-protected phases (SPT) [1]. Interestingly, while at the
ground state level the presence of a symmetry is crucial to distinguish
between trivial and topological ground states, the dynamical response of these
systems to the breaking of such protecting symmetries can also be used to
reveal their topological nature. This is the case of Thouless pumps [2, 3, 4],
in which the breaking of the chiral/inversion symmetry in a periodic parameter
path leads to quantized charge transport. The latter can be understood as a
dimensional reduction of the 2D quantum Hall effects [5, 6], by considering
the time as a second dimension, and provides a direct measure of the
topological invariant of the 1D SPT phase.

In the last decade, we have witnessed the first experimental realizations of
Thouless pumps in photonic platforms [7, 8, 9, 10] as well as in cold atom
quantum simulators [11, 12, 13] with fermions [14, 15] and bosons [16],
modeled by noninteracting Rice-Mele Hamiltonians [17]. In the Rice-Mele model,
the Thouless pump is implemented by modulating a dimerized single-particle
hopping in a closed path that encloses the trivial-to-topological transition
point. The simultaneous periodic modulation of a staggered on-site potential
breaking the protecting chiral/inversion symmetry avoids the singularity at
the transition point and the gap remains open during the cycle, leading to the
quantized pump.

Thouless pumps also provide a natural framework to study the effect of many-
body interactions in topological phases, a topic that has attracted great
interest over the last years [18]. In this direction, several works [19, 20,
21, 22, 23, 24, 25, 26, 27] pointed towards situations where bosonic
interactions do not destroy Thouless pumps, but rather lead to interaction-
induced phenomena. For fermionic systems, the eventual breakdown of quantized
Thouless pumps in the presence of onsite fermionic interactions was
theoretically explained as a direct consequence of emergent many-body Mott
physics [28, 29], and it has been recently observed in the experiment of Ref.
[30]. However, Thouless pumps can also be observed in interacting fermionic
systems due to competing effects [31, 32] in models hosting a spontaneous
bond-order-wave phase [33, 34, 35]. Finally, recent theory [29] and
experimental [36] works have shown that, in the presence of finite-size gaps,
quasi-quantized fermionic Thouless pumps can be observed along displaced paths
that do not encircle the noninteracting topological singularity.

An open question is whether quantized Thouless pumps can be observed in
interacting systems, in the extreme case where the periodic adiabatic path
contains the noninteracting topological singularity. In this work, we show
that such exotic Thouless pumps are indeed possible and that they can be
stabilized even in the absence of finite-size gaps. To present our findings,
we consider the dimerized spin-$1/2$ XXZ chain [37], which we simulate with
matrix-product-states [38], and can be realized in tweezer arrays of Rydberg
atoms [39, 40, 41, 42, 43]. We start by using the density matrix
renormalization group (DMRG) algorithm to revisit the interacting phase
diagram of the system, which for sufficiently large interactions features a
spontaneous antiferromagnetic order between its topological and trivial
dimerized phases [37]. We then introduce a closed path in the Hamiltonian
parameter space, in which both a staggered external field and the dimerized
exchange couplings are varied, while interactions remain fixed, see Fig. 1(d).
Unlike in the paradigmatic Thouless pump, this new path contains the
noninteracting topological singularity, which is split into two singularities
away from the origin in the presence of the intermediate antiferromagnetic
phase. Subsequently, we simulate the real-time dynamics of Thouless pumps
along this newly introduced path with the infinite time-evolving block
decimation (iTEBD) method. This allows us to certify that the system remains
adiabatic, i.e., gapped, in the thermodynamic limit for sufficiently large
interactions, and that a quantized topological transport occurs. As a remark,
note that this exotic quantized transport protocol is in contrast to others
considered [29, 36] in similar split singularity scenarios, in which transport
is quantized only in the presence of finite-size gaps. For completeness, we
also show that nonadiabatic and nonquantized transport is recovered in our
system for vanishing or small interactions, as expected from the gap closing
at the singularity. Due to the adiabatic nature of Thouless pumps, we
furthermore use the time-dependent variational principle (TDVP) in finite
chains, to show that the same phenomena can be studied if, instead of
considering real-time dynamics, one considers the preparation of independent
ground states along the closed path. This is motivated by the experimental
realization of this phenomenon, in which the adiabatic state preparation of
several ground states could be easier to simulate than a dynamical change of
the Hamiltonian parameters. In this line, we also propose an experimental
setup based on Floquet engineering in Rydberg tweezer arrays [43] to realize
the interacting model under consideration and benchmark the adiabatic state
preparation of its ground states, suitable to study the quantization of
Thouless pumps.

This article is organized as follows. In Section II, we present the
Hamiltonian model and revisit its phase diagram. Section III contains the main
results of this article. We start by introducing the new adiabatic path, and
subsequently study the quantization of Thouless pumps along it, both through
real-time dynamics as well as by means of an adiabatic state preparation
approach. Section IV contains the proposal for the experimental implementation
of the model with Rydberg arrays of optical tweezers. Finally, we conclude in
Section V, where we also provide an outlook of further research directions in
the context of interacting Thouless pumps.

Figure 1: (a) Phase diagram of $H$ in the chiral-symmetric regime
($\Delta=0$). The tricritical point, indicated with a black dot, separates the
three different phases: trivial dimerized (yellow), topologically dimerized
(purple) and spontaneous antiferromagnetic phase (green). (b) Order parameters
across the dashed line depicted in (a) at $\delta=3$. The trivial and
topological chiral-symmetric dimerized phases are characterized by a quantized
Zak phase $\varphi$, whereas the antiferromagnetic phase is signaled by a
spontaneous finite local magnetization $\Delta S_{z}$. (c) Energy gaps along
the same line at $\delta=3$. (d) Closed path used to perform a Thouless pump
at $\delta=3$. Interactions allow one to explore the singularity during the
pump (red point) without closing the bulk gap, due to the presence of a
spontaneous antiferromagnetic phase. (e) Sketch of the instantaneous ground
state during the path of (d).

##  II Model and phase diagram

We consider a one-dimensional chain of $N$ spins, described by a dimerized XXZ
Hamiltonian

$\begin{split}H=&\frac{J}{2}\sum_{i=1}^{N/2}\left(\sigma_{2i-1}^{x}\sigma_{2i}^{x}+\sigma_{2i-1}^{y}\sigma_{2i}^{y}+\delta\sigma_{2i-1}^{z}\sigma_{2i}^{z}\right)\\\
&+\frac{J^{\prime}}{2}\sum_{i=1}^{N/2-1}\left(\sigma_{2i}^{x}\sigma_{2i+1}^{x}+\sigma_{2i}^{y}\sigma_{2i+1}^{y}+\delta\sigma_{2i}^{z}\sigma_{2i+1}^{z}\right)\\\
&+\frac{\Delta}{2}\sum_{i}(-1)^{i}\sigma^{z}_{i},\end{split}$ (1)

where $\sigma^{\mu}$ are the spin-$1/2$ Pauli matrices. Here $J,J^{\prime}>0$
are the dimerized exchange couplings between nearest-neighbor (NN) spins,
which break site-inversion symmetry, $\delta$ is the anisotropy parameter, and
$\Delta$ is a staggered on-site field breaking bond-inversion and chiral
symmetry. First, notice that for $\delta=0$ one recovers the Rice-Mele model,
in which a Thouless pump is implemented by modulating the dimerization
parameter $J^{\prime}/J$ and the chiral symmetry breaking parameter $\Delta$
to encircle the singularity $(\Delta,J^{\prime}/J)=(0,1)$. The motivation to
work with this model is twofold. On the one hand, it accounts for the minimal
modification of the Rice-Mele model needed for the new Thouless pumps proposed
in this work. On the other hand, such a model can be realized with state-of-
the-art programmable arrays of Rydberg atoms [43]. The phase diagram at
$\Delta=0$, obtained with the infinite DMRG method, is shown in Fig. 1(a).

For $\delta<1$, the system is either in a trivial or topological dimerized
phase separated by the transition point at $J^{\prime}/J=1$. Alternatively,
for $\delta>1$ this singular transition point is replaced by an intermediate
antiferromagnetic phase in a finite range of $J^{\prime}/J$ values. In Figs.
1(b)-(c) we study the particular line $\delta/J=3$ in the phase diagram with
the finite DMRG method ($N=64$) and open boundary conditions, to illustrate
the behavior of the relevant order parameters in the different phases. The use
of the finite DMRG method also provides access to the first excited states of
the system. In particular, we compute the two excitation energies
$E_{1},E_{2}$ in the unpolarized ground-state sector ($S_{z,\text{total}}=0$),
and also the first two excitation energies $E_{0}^{s},E_{1}^{s}$ in the spin-
flipped sector ($S_{z,\text{total}}=+1$). The ground-state energy is set to
$E_{0}=0$.

Let us first focus on the trivial dimerized phase that appears at small
$J^{\prime}/J$. As shown in Fig. 1(b), this phase is characterized by a
quantized trivial Zak phase $\varphi=0$, and it is also a fully gapped phase,
as shown in Fig. 1(c). For increasing $J^{\prime}/J$, the system eventually
enters the antiferromagnetic phase, in which there is a spontaneous breaking
of the chiral and bond-inversion symmetries driven by interactions, as
captured by the finite value of $\Delta S_{z}\equiv S^{z}_{i}-S^{z}_{i+1}\neq
0$. At the transition trivial-antiferromagnetic, the spin gap remains open,
but there is an internal gap closing in the sector $S_{z,\text{total}}=0$
associated with the spontaneous antiferromagnetic transition. Inside the
antiferromagnetic phase, there is a ground-state degeneracy ($E_{1}=E_{0}=0$)
associated with the spontaneous symmetry breaking, but this manifold is gapped
from the rest of the spectrum ($E_{0}^{s},E_{2}>0$). Moreover, notice that the
Zak phase is not quantized in the antiferromagnetic region unless one
considers the highly nonlocal symmetric and antisymmetric superpositions
between the two symmetry-broken ground states. For even larger values
$J^{\prime}/J$, there is a transition to the topological dimerized phase. In
this antiferromagnetic-topological phase transition, there is also a bulk gap
closing in the sector $S_{z,\text{total}}=0$ associated with the spontaneous
antiferromagnetic transition. Moreover, the spin gap also closes because the
degenerate spin-polarized topological edge states emerge continuously from the
gapped spin soliton band of the antiferromagnetic phase. In the topological
phase, both $E_{1}$ and $E_{0}^{s}$ coincide with $E_{0}$, due to the
appearance of a degenerate edge-state manifold associated with the open
boundary conditions, while $E_{2}$ and $E_{1}^{s}$ remain finite, showing that
the bulk gap remains open in the topological phase. The topological phase is
characterized by a nonzero quantized Zak phase $\varphi=\pi$, as shown in Fig.
1(b).

In the context of Thouless pumps, the important conclusion from the phase
diagrams of Figs. 1(a)-(c), is that, in the presence of large interactions
$\delta>1$, the singularity $J^{\prime}/J=1$ splits into two singularities
separated by an intermediate antiferromagnetic region. Furthermore, this
antiferromagnetic region is effectively gapped, as any superposition between
the two spontaneous antiferromagnetic Néel states is a thermodynamically
unstable cat state, and they cannot be dynamically coupled with local
Hamiltonians.

##  III Thouless pumps

A Thouless pump can be implemented for the Hamiltonian in Eq. (1), following
the path illustrated in Fig. 1(c), which constitutes the central result of
this work. The path is parametrized by

$\begin{split}J^{\prime}(\theta)/J&=1.4+1.2\cos(\theta)\\\
\Delta(\theta)/J&=\begin{cases}-\sin(a_{1}\theta)/2\,\hskip
14.22636pt(\theta<\pi/2)\\\ -\sin(a_{2}\theta)/2\,\hskip
14.22636pt(\pi/2<\theta<3\pi/2)\\\ +\sin(a_{2}\theta)/2\,\hskip
14.22636pt(3\pi/2<\theta<2\pi)\\\ \end{cases},\end{split}$ (2)

where $\theta\in[0,2\pi)$. Here the coefficients $a_{1},a_{2}$ are fixed by
the condition that the path includes the singularity
$(J^{\prime}/J,\Delta)=(1,0)$. Our results apply to any other closed path
satisfying that (i) the singular line $J^{\prime}/J=1$ is enclosed, (ii) it
avoids a direct phase transition between the trivial and topological phases by
entering the spontaneous antiferromagnetic phase ($\delta>1$), and (iii)
$\Delta(\theta)$ is finite at the dimerized-antiferromagnet transition points,
to avoid a spontaneous symmetry breaking transition. Here $\theta$ can be
interpreted as the time $t$ in real adiabatic time dynamics with $\theta=2\pi
t/T$, where $T$ is the duration of a periodic cycle. Alternatively, $\theta$
can also be understood as a label for independently prepared ground states. In
both cases, as in any Thouless pump encircling a topological singularity in a
finite system with boundaries, one expects the quantization of the spin
transferred from the left edge to the right edge

$\Delta Q^{l}_{\theta}\equiv
Q^{l}_{\theta}-Q^{l}_{0}-\sum_{i\in\text{jumps}}\delta
Q^{l}_{\theta_{i}}=-\Delta Q^{r}_{\theta}.$ (3)

Here $Q^{l}_{\theta}\equiv\frac{1}{2}\sum_{i\in\text{left
edge}}{\sigma}^{z}_{i}$ is the integrated net spin on the left edge of the
system for a given value of $\theta$, and $\delta Q^{l}_{\theta_{i}}$ is the
value of any discontinuity (jump) in $Q^{l}_{\theta}$ during the pump, related
to the crossing of two edge states inside the gap [44]. Importantly, $\delta
Q^{l}_{\theta_{i}}$ always vanishes if $\theta$ is interpreted as the time
variable (i.e., the edge states are never pumped directly in real dynamics).
However, if $\theta$ is interpreted as a label for independently prepared
ground states, $\delta Q^{l}_{\theta_{i}}$ can be finite at values
$\theta_{i}$ for which the edge-state manifold is degenerate. The
correspondence with the topological invariant is given by

$|\Delta Q^{l}_{2\pi}|=|\Delta\varphi|/\pi\in\mathbb{Z}_{2},$ (4)

where $|\Delta\varphi|$ is the change of the topological Zak phase across the
topological singularity.

###  III.1 Real-time dynamics

One option to experimentally realize Thouless pumps for the path illustrated
in Fig. 1(d) is to directly simulate the adiabatic dynamics, which requires
the ability to dynamically modulate $J^{\prime}(t)$ and $\Delta(t)$. To
investigate the quantization of Thouless in this scenario, we simulate the
real-time dynamics of such a protocol using the iTEBD method in an infinite
chain (two-site unit cell). As an initial state, we consider the dimerized
ground state of $H(\theta=0)$, obtained with infinite DMRG. Then, we simulate
the adiabatic dynamics generated by $H(\theta)$, given by Eqs. (2), for a
period of duration $TJ=200$, with a Trotter time-step of $\text{d}tJ=0.01$ in
the iTEBD algorithm, and with a maximum bond dimension
$\chi_{\text{max}}=200$. Note that even in a system without edges (an infinite
or periodic chain), we can still compute the adiabatic spin transfer in the
bulk by integrating the bulk spin current

$\Delta
Q^{l}_{\theta}=\int_{0}^{t(\theta)}\mathcal{J}(t^{\prime})\,\text{dt}^{\prime},$
(5)

where

$\mathcal{J}(t)=\frac{i}{2}\sum_{i=i_{\text{bulk}},i_{\text{bulk}}+1}J_{i}\left\langle{\sigma^{+}_{i}\sigma^{-}_{i+1}-\text{H.c.}}\right\rangle,\
\ J_{i}=J,J^{\prime}.$ (6)

Figure 2 shows the effect of interactions in Thouless pumps along the path of
Fig. 1(d). Remarkably, for sufficiently strong interactions, in Fig. 2(b) we
observe a perfectly quantized spin transport $|\Delta Q^{l}_{2\pi}|=1$ that,
following Eq. (4), can be used to state that we have crossed a topological
singularity adiabatically (i.e., without closing the bulk gap). The presence
of a finite gap during the whole interacting pump can also be inferred from
Fig. 2(b), which shows the evolution of the antiferromagnetic order.
Importantly, note that such antiferromagnetic insulating order remains robust
except for the points $\theta=0,\pi,2\pi$, where the gap of the system is
dominated by the external dimerization $J^{\prime}(\theta)/J\neq 1$. In
particular, one can see that $\Delta S_{z}$ remains finite around
$\theta=\pi/2,3\pi/2$, where the staggered magnetic field $\Delta$ vanishes,
due to the presence of the spontaneous antiferromagnetic phase. In stark
contrast with the interacting case, in the noninteracting or weakly
interacting cases (orange and purple lines) we observe the expected breakdown
of the Thouless pump, due to the fact that the periodic cycle encounters the
singularity $(\Delta,J^{\prime}/J)=(0,1)$. In this case, $|\Delta
Q^{l}_{2\pi}|$ is not quantized, and we observe strong oscillations in $\Delta
Q^{l}_{\theta}$, as a consequence of the coupling to excited states in the
absence of an energy gap. Figure 2(b) provides further insight into such a
breakdown of the Thouless pump. In the noninteracting case, one can clearly
observe the decrease of the insulating antiferromagnetic order around
$\theta=\pi/2,3\pi/2$, caused by the absence of a staggered magnetic field
$\Delta$. In this region, one also has $J^{\prime}/J\simeq 1$, so that both
the antiferromagnetic and dimerization gaps vanish.

Figure 2: Real-time dynamics (iTEBD) of Thouless pumps for different
interaction strengths $\delta$. (a) Modulation of the staggered detuning
$\Delta$ and dimerization $J^{\prime}$ during the Thouless pump along the
path, given by Eqs. (2) and sketched in Fig. 1(d), with a total duration
$TJ=200$. The values of $\theta$ in which the pump encounters the singularity
$(\Delta,J^{\prime}/J=0,1)$ are indicated with gray vertical lines. (b)
Evolution of the transferred charge $Q^{l}_{\theta}$. (c) Evolution of the
insulating antiferromagnetic order $\Delta S_{z}$.

###  III.2 Independent state-preparation protocol

Another strategy to experimentally observe the quantization given by Eq. (4)
is to adiabatically prepare independent ground states across the path sketched
in Fig. 1(d). That is, as we are interested in slow adiabatic dynamics, the
instantaneous state of the system during the pump corresponds to the ground
state of the instantaneous $H(\theta)$. However, note that when preparing
independent ground states one has to take into account possible degeneracies
in $H(\theta)$, which lead to an ambiguous connection between the
instantaneous state during real-time dynamics and such ground states. On the
one hand, one needs to consider topological degeneracies associated with a
degenerate edge-state manifold. More precisely, while the edge-state
degeneracy does not play a significant role in real-time dynamics, the
instantaneous ground state of $H(\theta)$ in the topological sector is twofold
degenerate at $\Delta=0$ with open boundary conditions, and it exhibits a
discontinuous behavior in its net edge spin when going from $\Delta=0^{-}$ to
$\Delta=0^{+}$ due to the lifting of the edge-state degeneracy. Such jumps in
the edge spin occupation can be easily taken into account following Eq. (3).
On the other hand, in the interacting case $\delta>1$, we have the
abovementioned twofold degeneracy associated with the spontaneous
antiferromagnetic phase at the singularity $(J^{\prime}/J,\Delta)=(1,0)$.
While, as shown in Fig. 2, such a degeneracy does not impact the adiabatic
real-time dynamics. If one aims at preparing the ground state in this regime,
a random arbitrary choice of the initial state would lead to any of the two
degenerate Néel states with equal probability. In this case, the twofold
degeneracy of the ground state at the singularity
$(\Delta,J^{\prime}/J)=(0,1)$ can be effectively avoided with a suitable
state-preparation protocol. We propose to initialize the state-preparation
protocol at $\theta=0$ in one of the two fully polarized antiferromagnets

$\begin{split}\ket{\uparrow\downarrow}^{\otimes
N/2}&\equiv\ket{\uparrow\downarrow\dots\uparrow\downarrow},\\\
\ket{\downarrow\uparrow}^{\otimes
N/2}&\equiv\ket{\downarrow\uparrow\dots\downarrow\uparrow},\end{split}$ (7)

which can be obtained as the ground states of $H$ in Eq. (1) for
$\Delta\to\infty$ and $\Delta\to-\infty$, respectively. From such an initial
state, one then adiabatically varies $\Delta:\pm\infty\to\pm\Delta(\theta)$.
To avoid an arbitrary or random choice between the two possible initial
states, we propose (i) In general, one should choose the initial state as
$\ket{\uparrow\downarrow}^{\otimes N/2}$ or $\ket{\downarrow\uparrow}^{\otimes
N/2}$ for $\Delta(\theta)<0$ or $\Delta(\theta)>0$. (ii) In the special cases
where $\Delta=0$, one chooses the initial state that would mimic the real-time
adiabatic dynamics. At $\theta=0,2\pi$, this is given by the sign of
$\Delta(0^{+})$ and $\Delta(2\pi^{-})$, respectively. At other singular points
$\theta^{*}$ during the pump, the initial state must suddenly change if
$\text{sign}(\Delta(\theta^{-})\neq\text{sign}(\Delta(\theta^{+})$ and $\Delta
S_{z}(\theta^{*})=0$.

Figure 3: Results for independently prepared states across the path sketched
in Fig. 1(d), simulated using an adiabatic TDVP algorithm (see main text for
details). We use a ramping time of $TJ=40$ for the state preparation. (a)-(b)
Evolution of the left-edge net charge $Q^{l}_{\theta}$ for $\delta=3$ (a) and
for the noninteracting case (b). (c)-(d) Evolution of the insulating
antiferromagnetic order $\Delta S_{z}$. In all the panels, the empty circles
show the values after letting the system evolve with the final Hamiltonian
$H(\theta)$ during an extra time $\tau J=10$.

We simulate such a protocol across the path of Fig. 1(d) in a finite open
chain of $N=64$ spins and consider a total duration of the ramping protocol of
$TJ=40$, followed by a free evolution under $H(\theta)$ during an extra time
$\tau J=10$, to ensure that the protocol leads to a stationary eigenstate of
$H(\theta)$. The results for the left net spin $Q^{l}_{\theta}$ and insulating
antiferromagnetic order $\Delta S_{z}$ are shown in Fig. 3. A first
observation is that by comparing these results with the real-time dynamics of
Figs. 2(b)-(c), there is a perfect agreement in the interacting case
$\delta=3$ (blue points). The latter confirms that preparing independent
ground states is a useful tool to mimic adiabatic dynamics in the system. Note
that the main difference in this approach is that within such an independent
state-preparation protocol, one, in general, would observe a quantized
discontinuity in $Q^{l}_{\theta}$, which in Fig. 3 indeed occurs from
$\theta=2\pi\rightarrow 0$. In the noninteracting case, the charge transfer is
not quantized, as expected from the real-time dynamics results, but we observe
important differences with respect to the iTEBD results. This is expected, as
in the noninteracting case the population of excited states around the
singularity $(J^{\prime}/J,\Delta)=(1,0)$ leads to a breaking of the adiabatic
nature of the dynamics, which cannot be captured by independently prepared
ground states. Nevertheless, in Fig. 3(b) we can also clearly see the breaking
of adiabaticity in the noninteracting case, which shows a nonstationary value
of $Q^{l}_{\theta}$ after the extra time evolution of $\tau J=10$. This
highlights that one cannot adiabatically prepare the ground state when the gap
of $H(\theta)$ vanishes.

##  IV Implementation with programmable Rydberg arrays

Figure 4: Pulse sequence associated to one Floquet cycle in the implementation
of the XXZ model in Eq. (1). Inside each pulse, we indicate its effective area
(see text). Global pulses are shown in green, and those only applied on even
sites, in red.

The dimerized XXZ Hamiltonian in Eq. (1) can be implemented with dipolar
interactions. As a particular example, here we introduce a scheme that can be
directly implemented in current setups based on Rydberg arrays and optical
tweezers [45, 46, 47, 48]. The internal structure of these atoms can be
simplified as a two-level system with Rydberg states of opposite parity
$\ket{nS}=\ket{\downarrow}$, and $\ket{nP}=\ket{\uparrow}$ coupled to each
other through resonant dipole interactions of the form,

$H_{\textrm{XX}}=\sum_{i\neq
j}\frac{J_{ij}}{2}(\sigma_{i}^{x}\sigma_{j}^{x}+\sigma_{i}^{y}\sigma_{j}^{y})\,.$
(8)

The coupling strength, $J_{ij}=C_{3}[1-3\cos^{2}(\theta_{ij})]/(2r_{ij}^{3})$
depends on both the interatomic separation, $r_{ij}$, and the relative angle
with the quantization axis, $\theta_{ij}$, which can be tuned by appropriately
choosing the position of the optical tweezers.

These internal states can also be externally driven with a resonant microwave
field of the form,

$H_{\text{driven}}=\hbar\Omega\sum_{i}(\cos(\phi)\sigma_{i}^{x}+\sin(\phi)\sigma_{i}^{y}),$
(9)

where $\Omega$ and $\phi$ are the amplitude and phase of the drive, which can
be modified over time. An appropriate sequence of these pulses allows one to
engineer the interactions of the system [49, 50].

For example a sequence of four global $\pi/2$ pulses of duration $\tau_{p}$
and constant phase $\phi=(0,-\pi/2,\pi/2,\pi)$ rotate the effective plane of
the dipole interactions. This leads to a time-independent Hamiltonian of the
form,

$H_{\text{av}}=\sum_{ij}\frac{J_{ij}}{t_{c}}[(\tau_{1}+\tau_{2})\sigma_{i}^{x}\sigma_{j}^{x}+(\tau_{1}+\tau_{3})\sigma_{i}^{y}\sigma_{j}^{y}+(\tau_{2}+\tau_{3})\sigma_{i}^{z}\sigma_{j}^{z}]\,,$
(10)

where $\tau_{1,2,3}$ is the delay between pulses, and
$t_{c}=2\sum_{i=1}^{3}\tau_{i}$ is the duration of the sequence, as it was
experimentally implemented in Ref. [43]. Using this procedure, one can access
XXZ models with $\delta\in[0,2]$.

In the following, we introduce in more detail how the independent state
preparation approach described in Sec. III.2 applies to the simulation of the
effective Hamiltonian in Eq. (1), which is engineered with the combination of
the natural XX dipole interaction and global/dimerized rotations along the $x$
and $y$-axis.

  * •

The system is formed by a dimerized chain of atoms placed in positions where
the XX interactions satisfy the desired ratio $J^{\prime}/J$. For
$J^{\prime}\sim J$, this can correspond to a bipartite 1D chain with
$r_{2}/r_{1}=(J^{\prime}/J)^{1/3}$. Configurations where $J^{\prime}/J\sim 0$
can also benefit from double ladders with neighbor atoms oriented along the
magic angle $\theta_{ij}\sim 54.7\,^{\circ}$, where the dipole interactions
vanish, as experimentally realized in Ref. [42].

  * •

To prepare an antiferromagnetic state, one initializes all atoms in state
$\ket{\downarrow}$ using a stimulated Raman adiabatic passage. Next, a spatial
light modulator (SLM) induces an Autler-Townes splitting of the atoms placed
in odd positions, bringing them out-of-resonance from a last $\pi$ microwave
rotation along the $y$-axis. Even if applied globally, the rotation only acts
on the atoms that are masked by the SLM, providing experimental control of
which atoms are flipped. This allows one to induce the fully polarized
antiferromagnets of Eq. (7),

$\begin{split}\ket{\uparrow\downarrow}^{\otimes
N/2}&\equiv\ket{\uparrow\downarrow\dots\uparrow\downarrow}\,,\\\
\ket{\downarrow\uparrow}^{\otimes
N/2}&\equiv\ket{\downarrow\uparrow\dots\downarrow\uparrow}\,.\end{split}$ (11)

  * •

To induce the XXZ terms of the Hamiltonian, we perform a Floquet engineering
of the effective dynamics following the procedure discussed above [49, 50]. In
particular, we choose $\tau_{1}=\tau_{2}$, and
$\delta=(\tau_{1}+\tau_{2})/(\tau_{1}+\tau_{3})$.

  * •

The final step consists of inducing a staggered driving of the form
$H_{z}=\Delta/2\sum_{i}(-1)^{i}\sigma_{i}^{z}$. After the first $\pi/2$
$x$-rotation of the Floquet pulse, one can observe that, the operator
$\sigma^{z}$ is mapped to $\sigma^{y}$. One can then induce $H_{z}$ by
appropriately modifying the subsequent $(-Y)$-rotation of the Floquet
sequence, which should be different on even and odd sites. In particular, one
can apply a corrected effective area, $\pi/2-\varphi_{\Delta}$ on the global
$(-Y)$-rotation, where $\varphi_{\Delta}=t_{c}\,\Delta/(1+\delta/2)$ (see the
scheme in Fig. 4). Immediately after, the SLM is used to continue the
rotation, on the even sites of the array, inducing a site-dependent short
Gaussian $Y$-pulse of duration $\tau_{p}$ and area $2\varphi_{\Delta}$ that
only affects those positions. We highlight that the pattern of the SLM remains
constant throughout the implementation, preventing any limitation coming from
its reconfiguration time. Furthermore, the strength of $\Delta$ can be tuned
over time by appropriately choosing its value on each cycle of the sequence.

  * •

For the measurement stage of the protocol, a combination of fluorescence
imaging and microwave rotations is conventionally used to measure the internal
state of each atom [48].

This protocol results in the implementation of the effective Hamiltonian in
Eq. (1) provided that the duration of each pulse is much shorter than $t_{c}$,
and that this is shorter than the average interactions timescale,
$N/\sum_{ij}J_{ij}$. Let us now numerically benchmark this implementation for
relevant parameters corresponding to a 1D tweezer array of 87Rb atoms with
pseudospin states $\ket{\downarrow}=\ket{75S_{1/2},m_{J}=1/2}$ and
$\ket{\uparrow}=\ket{75P_{3/2},m_{J}=3/2}$, which are separated by $2\pi\times
8.5$ GHz [43]. These atoms are alternatively spaced with distances
$r_{12}=19\,\mu$m, and $r_{23}=23.9\,\mu$m, so that the associated interaction
is $J_{12}\approx 2\pi\times 270$ kHz, and $J^{\prime}/J=2$. Site-dependent
microwave pulses can be applied by using the SLM to induce a light shift on
the $6P-75S$ transition on the undesired atoms, where laser beams of waist
$\sim 2\,\mu$m are achievable.

In Fig. 5, we calculate the infidelity due to the state preparation
($J^{\prime}=2J$, $\delta$=2, $\Delta=J$) when an initial AF state in the
$x$-basis is evolved along the effective Hamiltonian in Eq. (1) (blue line),
or the Floquet protocol described above (orange). There, we consider a cycle
duration of $\tau_{c}=210$ ns, Gaussian pulses with a $1/e^{2}$ width of $16$
ns and $J=1.7$ MHz. Following the procedure described above, we monotonously
modify the effective area of the intermediate site-dependent $X$-pulses
according to $\varphi_{\Delta_{\text{F}}(t)}$, with
$\Delta_{\text{F}}(t)=\Delta+9\Delta\left[(T-t)/T\right]^{1/4}$. For a
conservative sequence of 10 cycles, we obtain a preparation infidelity of
order $0.6\%$.

Figure 5: Infidelity in the preparation of state ($J^{\prime}=2J$,
$\delta=2J$, $\Delta=J$),
$\mathcal{F}=\sqrt{\innerproduct{\psi_{\Delta}}{\psi(t)}}$ when an initial AF
state in the $\sigma^{x}$-basis is evolved along the effective Hamiltonian in
Eq. (1) (blue line), or after each cycle of the Floquet protocol shown in Fig.
4 (orange). Here we consider a 10-cycle preparation sequence of a chain with
12 atoms, and a cycle duration of $\tau_{c}=210$ ns.

##  V Conclusions and outlook

In this work, we analyzed a new class of Thouless pumps in interacting spin
systems. In particular, we showed that, for sufficiently large interactions,
the dimerized XXZ chain exhibits quantized topological transport along a newly
introduced path that contains the topological singularity, and that such an
effect is completely absent for vanishing or small interactions. We could
understand this phenomenon by the fact that interactions replace the original
singularity by an intermediate antiferromagnetic phase that, despite being
topologically trivial, has a finite bulk gap that can be used to preserve the
adiabaticity of the cycle. Consequently, we expect that similar phenomena
could also be observed in other models in which a direct topological phase
transition is replaced by an intermediate spontaneous symmetry-breaking phase
induced by interactions [51]. There, we stress that one should always ensure
that the adiabatic path is such that it avoids bulk gap closings associated
with spontaneous symmetry-breaking transitions. Finally, our experimental
proposal paves the way for the experimental realization of the phenomena
studied in this work using Rydberg tweezer arrays [40, 41], as well as other
emergent platforms also providing extended dipolar interactions [52].

###### Acknowledgements.

The tensor-network calculations were performed using the TeNPy Library [53].
The ICFO group acknowledges support from ERC AdG NOQIA; MCIN /
AEI(PGC20180910.13039 / 501100011033, CEX2019-000910S / 10.13039 /
501100011033, Plan National FIDEUA PID2019-106901GB-I00, Plan National
STAMEENA PID2022-139099NB-I00 project funded by MCIN/AEI/10.13039/501100011033
and by the “European Union NextGenerationEU/PRTR” (PRTR-C17.I1), FPI);
QUANTERA MAQS PCI2019-1118282); QUANTERA DYNAMITE PCI2022-132919 (QuantERA II
Programme co-funded by European Union’s Horizon 2020 program under Grant
Agreement No 101017733), Ministry of Economic Affairs and Digital
Transformation of the Spanish Government through the QUANTUM ENIA project call
– Quantum Spain project, and by the European Union through the Recovery,
Transformation, and Resilience Plan – NextGenerationEU within the framework of
the Digital Spain 2026 Agenda; Fundació Cellex; Fundació Mir-Puig; Generalitat
de Catalunya (European Social Fund FEDER and CERCA program, AGAUR Grant No.
2021 SGR 01452, QuantumCAT U16-011424, co-funded by ERDF Operational Program
of Catalonia 2014-2020); Barcelona Supercomputing Center MareNostrum
(FI-2023-1-0013); EU Quantum Flagship (PASQuanS2.1, 101113690); EU Horizon
2020 FET-OPEN OPTOlogic (Grant No 899794); EU Horizon Europe Program (Grant
Agreement 101080086 — NeQST), ICFO Internal “QuantumGaudi” project; European
Union’s Horizon 2020 program under the Marie Sklodowska-Curie grant agreement
No 847648; “La Caixa” Junior Leaders fellowships, La Caixa” Foundation (ID
100010434): CF/BQ/PR23/11980043. Views and opinions expressed are, however,
those of the author(s) only and do not necessarily reflect those of the
European Union, European Commission, European Climate, Infrastructure and
Environment Executive Agency (CINEA), or any other granting authority. Neither
the European Union nor any granting authority can be held responsible for
them. AD acknowledges funding from the European Union under Grant Agreement
101080142 and the project EQUALITY.

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# Evaluating Uncertainty Calibration for Open-Set Recognition

Zongyao Lyu, Nolan B. Gutierrez, and William J. Beksi The authors are with the
Department of Computer Science and Engineering, University of Texas at
Arlington, Arlington, TX, USA. Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Despite achieving enormous success in predictive accuracy for visual
classification problems, deep neural networks (DNNs) suffer from providing
overconfident probabilities on out-of-distribution (OOD) data. Yet, accurate
uncertainty estimation is crucial for safe and reliable robot autonomy. In
this paper, we evaluate popular calibration techniques for open-set conditions
in a way that is distinctly different from the conventional evaluation of
calibration methods on OOD data. Our results show that closed-set DNN
calibration approaches are much less effective for open-set recognition, which
highlights the need to develop new DNN calibration methods to address this
problem.

##  I Introduction

Deep neural networks (DNNs) have been very successful at vision-based tasks,
which has led to their widespread deployment for real-world applications
(e.g., object recognition, image segmentation, etc.). Nevertheless, a
challenge of DNN-based vision systems lies in the network’s inclination to
produce overconfident predictions during inference, especially when facing
categories not seen during training (i.e., out-of-distribution (OOD) data)
[1]. Within the realm of robotics, this motivates research questions such as
how much trust can we put into the predictions of a DNN when
misclassifications may have catastrophic consequences [2]? Methods have been
proposed to mitigate the overconfidence problem by calibrating the predictive
probabilities [3], or estimating the predictive uncertainties [4, 5], for
various vision-based objectives (e.g., image classification [6], semantic
segmentation [7], and object detection [8, 9]).

When deployed in real-world environments, we desire a DNN to have the
discriminative ability to separate query inputs into either known (i.e., seen
during training) or unknown (i.e., not yet seen during training) classes. This
problem has previously been formalized and studied under the context of open-
set recognition (OSR) [10]. OSR is the extension of object recognition from
closed-set to open-set conditions, where classes outside of the training data
can appear at inference time. Specifically, OSR trains a model on $K$ classes
of an $N$ class training dataset. Then, at test time, the model is faced with
$N$ different classes of which $N-K$ were not seen during training. An
explicit none-of-the-above (or unknown) class placeholder (i.e., the $K+1$
class) is added during inference. OSR aims to assign correct labels from $K$
known classes to seen test-time samples, while detecting and placing unseen
samples to the unknown class. Robotic applications of OSR include the division
of semantically known places from semantically unknown [11], failure
identification for self-driving cars [12], and entity detection [13].

Calibration refers to the issue of obtaining a model that has a predictive
probability that reflects the true correctness likelihood. More formally, a
model is well calibrated if outcomes predicted to occur with probability $p$
occur approximately $p$ fraction of the time [14]. In robotics, there is a
clear need for uncertainty calibration of policies (e.g., autonomous driving
[15]), especially for OSR. Previous work has evaluated calibration methods on
both in distribution (ID) cases and under distributional shifts [16, 17]. The
experimental settings in [16, 17] follow those in standard OOD detection works
where a model trained on the ID set is evaluated for its reliability in
identifying test images as either ID or OOD [18]. Although dealing with
similar problems, OOD detection and OSR have been studied separately with
different evaluation protocols. To the best of our knowledge, the evaluation
of calibration methods for the OSR problem (i.e., open-set calibration) has
not been explored. Hence, our work serves as a step toward this direction.

##  II Open-Set Recognition Evaluation

The first step of our evaluation is to enhance a model to perform OSR. To do
this, we create a baseline by thresholding on the maximum probability which
indicates the mostly likely predicted class. We also apply OpenMax [19], a
well-known OSR method, for comparison. Following the standard experimental
setup in OSR works, we use ResNet [20] as the backbone classification network
and evaluate on the CIFAR10 benchmark dataset [21]. To test under OSR
conditions, we follow the most common data partitioning protocol [22, 23, 24,
25, 26, 27]. Specifically, we split each dataset at random such that 6 classes
are chosen to be known and the remaining 4 classes are unknown. We repeat the
experiment over 5 runs and report the average score. Thus, the model is
trained with 6 known classes and tested on all 10 classes. We map the randomly
selected 6 known classes using indices 0-5 and set the other 4 unknown classes
to start at index 6. A perfect OSR model should be able to assign correct
labels to images belonging to any of the 6 known classes and identify images
from all other classes as unknowns.

In the second step of our evaluation, we calibrate the trained model on the
validation set in order to improve calibration during inference. Calibration
is performed using temperature scaling [3], a simple and effective technique
that makes adjustments to a model’s logits by a single temperature parameter
$T$. Logits are probability vectors produced by a network before the softmax
layer for a given input image. Additionally, $T$ is optimized on a held-out
validation set and it can rescale a model’s confidence without affecting
accuracy. After acquiring an OSR model, we apply temperature scaling on the
model’s predictive confidence and evaluate the calibration performance in this
novel problem setting. Concretely, to apply temperature scaling we reserve 10%
of the training data as a validation set for optimizing $T$ and use the
remaining 90% for training the model. To evaluate model calibration, we use
the expected calibration error (ECE) [14] to measure the absolute difference
between predictive confidence and accuracy. As a result of its simplicity and
effectiveness, ECE is the most commonly used metric for model calibration. We
also evaluate the Brier score within the framework of OSR as follows. If a
trained classifier’s prediction $\hat{\bm{Y}}\in\mathbb{R}^{N\times K},$ where
$N$ is the number of samples and $K$ is the number of known classes, then we
can obtain a new prediction
$\hat{\bm{Y}}_{\textrm{OSR}}\in\mathbb{R}^{N\times(K+1)}$ that takes into
account the unknown class. For each prediction $\hat{\bm{y}}\in\mathbb{R}^{K}$
we estimate the probability of being unknown, i.e.,

$\hat{y}_{\textrm{unknown}}=\prod\limits_{i=1}^{K}(1-\hat{y}_{i}).$ (1)

Next, we calculate the Brier score by

$\frac{1}{N}\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{K}(\hat{Y}^{ij}_{\textrm{OSR}}-Y^{ij})^{2},$
(2)

where the predicted probability and ground truth of the $j$th class are
$\hat{Y}^{ij}_{\textrm{OSR}}$ and $Y^{ij}$, respectively.

##  III Evaluation Results

We visualize model calibration via reliability diagrams [28, 29], where the
diagonal represents perfect calibration and any deviation from a perfect
diagonal indicates miscalibration. The top rows of Fig. 1 and Table I present
the reliability diagrams and scores for closed-set classification before and
after calibration by temperature scaling. We see that temperature scaling
significantly improves the model’s ability to match the true correctness
likelihood in calibrating the predictions for this conventional 10-class
closed-set classification problem. In the middle rows of Fig. 1 and Table I,
the reliability diagrams and scores of a classifier operating in open-set
conditions by thresholding on its predictive probabilities before and after
calibration by temperature scaling are shown. In this scenario, temperature
scaling still provides a clear improvement over uncalibrated models.
Nonetheless, there exist clear gaps between temperature scaling calibrated
confidence and perfectly calibrated confidence thus indicating that the
calibration problem for open-set conditions is much more challenging. The
bottom rows of Fig. 1 and Table I demonstrate the reliability diagrams and
scores of a classifier operating in open-set conditions by applying OpenMax
before and after calibration. Although OpenMax improves classification
accuracy, it exhibits no obvious improvement in terms of calibration. In
addition, we also conducted these experiments with disparate classification
networks such as DenseNet [30] and EfficientNet [31] on the SVHN dataset [32]
(see appendix). In all of these experiments, we observed the same findings as
above.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: The reliability diagrams of the closed-set (top row), open-set
(middle row), and open-set + OpenMax (bottom row) methods before (left column)
and after (right column) performing temperature scaling calibration.

##  IV Conclusion

In this paper, we assessed the efficacy of calibration methods for OSR. From
the evaluation results, we observe that the calibration problem is much more
challenging under open-set conditions in comparison to closed-set conditions.
Previous techniques that calibrate confidence really well for closed-set
classification only provide limited calibration performance for OSR. Moreover,
traditional OSR approaches improve accuracy, yet they do not take calibration
into consideration. Given these observations, we want to bring the community’s
attention not only to the pursuit of more accurate OSR methods, but also well-
calibrated OSR systems that can perform reliably and safely in open-set
settings.

Method | Measure |  Brier $\downarrow$ |  ECE $\downarrow$ | Accuracy
---|---|---|---|---
Closed-set | Before calibration | 0.255 | 0.069 | 0.832
After calibration | 0.244 | 0.011
Open-set | Before calibration | 0.814 | 0.346 | 0.603
After calibration | 0.647 | 0.230
Open-set + OpenMax | Before calibration | 0.841 | 0.335 | 0.632
After calibration | 0.703 | 0.197
TABLE I: Brier and ECE scores for closed-set and open-set calibration on
CIFAR10 using ResNet.

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Method | Measure |  Brier $\downarrow$ |  ECE $\downarrow$ | Accuracy
---|---|---|---|---
Closed-set | Before calibration | 0.085 | 0.009 | 0.943
After calibration | 0.084 | 0.006
Open-set | Before calibration | 0.434 | 0.220 | 0.793
After calibration | 0.406 | 0.204
Open-set + OpenMax | Before calibration | 0.496 | 0.224 | 0.811
After calibration | 0.489 | 0.218
TABLE II: Brier and ECE scores for closed-set and open-set calibration on SVHN using ResNet. Method | Measure |  Brier $\downarrow$ |  ECE $\downarrow$ | Accuracy
---|---|---|---|---
Closed-set | Before calibration | 0.246 | 0.070 | 0.835
After calibration | 0.234 | 0.012
Open-set | Before calibration | 0.788 | 0.357 | 0.645
After calibration | 0.643 | 0.269
Open-set + OpenMax | Before calibration | 0.861 | 0.369 | 0.666
After calibration | 0.755 | 0.268
TABLE III: Brier and ECE scores for closed-set and open-set calibration on CIFAR10 using DenseNet. Method | Measure |  Brier $\downarrow$ |  ECE $\downarrow$ | Accuracy
---|---|---|---|---
Closed-set | Before calibration | 0.091 | 0.008 | 0.939
After calibration | 0.090 | 0.006
Open-set | Before calibration | 0.447 | 0.229 | 0.816
After calibration | 0.376 | 0.178
Open-set + OpenMax | Before calibration | 0.507 | 0.231 | 0.822
After calibration | 0.483 | 0.209
TABLE IV: Brier and ECE scores for closed-set and open-set calibration on SVHN using DenseNet. Method | Measure |  Brier $\downarrow$ |  ECE $\downarrow$ | Accuracy
---|---|---|---|---
Closed-set | Before calibration | 0.233 | 0.032 | 0.839
After calibration | 0.230 | 0.012
Open-set | Before calibration | 0.767 | 0.373 | 0.633
After calibration | 0.665 | 0.275
Open-set + OpenMax | Before calibration | 0.841 | 0.334 | 0.636
After calibration | 0.719 | 0.151
TABLE V: Brier and ECE scores for closed-set and open-set calibration on CIFAR10 using EfficientNet. Method | Measure |  Brier $\downarrow$ |  ECE $\downarrow$ | Accuracy
---|---|---|---|---
Closed-set | Before calibration | 0.101 | 0.010 | 0.932
After calibration | 0.100 | 0.004
Open-set | Before calibration | 0.394 | 0.207 | 0.845
After calibration | 0.377 | 0.197
Open-set + OpenMax | Before calibration | 0.466 | 0.209 | 0.857
After calibration | 0.450 | 0.193
TABLE VI: Brier and ECE scores for closed-set and open-set calibration on SVHN
using EfficientNet.