# Zero forcing number of graphs with a power law degree distribution

Alexei Vazquez<EMAIL_ADDRESS>Cancer Research UK Beatson Institute,
Glasgow, United Kingdom  Institute for Cancer Sciences, University of Glasgow,
Glasgow, United Kingdom

###### Abstract

The zero forcing number is the minimum number of black vertices that can turn
a white graph black following a single neighbour colour forcing rule. The zero
forcing number provides topological information about linear algebra on
graphs, with applications to the controllability of quantum dynamical systems.
Here, I investigate the zero forcing number of undirected graphs with a power
law degree distribution $p_{k}\sim k^{-\gamma}$ by means of numerical
simulations. For graphs generated by the preferential attachment model, with a
diameter scaling logarithmically with the graph size, the zero forcing number
approaches the graph size when $\gamma\rightarrow 2$. In contrast, for graphs
generated by the deactivation model, with a diameter scaling linearly with the
graph size, the zero forcing number is smaller than the graph size
independently of $\gamma$. Therefore the scaling of the graph diameter with
the graph size is another factor determining the controllability of dynamical
systems. These results have implications for the controllability of quantum
dynamics on energy landscapes, often characterized by a complex network of
couplings between energy basins.

##  I Introduction

A zero forcing set of an undirected graph, denoted by $S$, is defined as a
subset of vertices that can change a graph coloured white to black according
to the following dynamical rules Group (2008). Start with a set of vertices
coloured black and all other vertices white. At each step, all black vertices
with exactly one white neighbour colour that neighbour black. If the process
ends colouring the whole graph black then the starting set of black vertices
is a zero forcing set. The zero forcing number of a graph $G$, denoted by
$Z(G)$, is defined as the size of the zero forcing set with minimum size.

The zero forcing number has been studied extensively in the context of linear
algebra to calculate bounds on the minimum rank of a graph Group (2008);
Hogben (2010); Trefois and Delvenne (2015). The minimum rank of a graph $G$,
denoted by $mr(G)$, is defined as the minimum rank of all real matrices with
non-zero elements on the graph edges Group (2008). It has been mathematically
proven that Hogben (2010).

$mr(G)\geq N-Z$ (1)

where the equality is warranty when $G$ is an undirected or directed tree.
Equation (1) holds true for graphs without loops (simple), with loops,
undirected and directed, after small changes in the forcing rule Hogben
(2010).

More recently, the zero forcing set has found applications in the
controllability of quantum dynamical systems Burgarth  _et al._ (2013). The
linear control theory of quantum systems is based on time-dependent
Hamiltonians of the form

$H(t)=H_{I}+\sum_{k}u_{k}(t)C^{(k)}$ (2)

operating on a $n$-dimensional Hilbert space ${\cal H}_{n}$. Here $H_{I}$ and
$C^{(k)}$ are $n\times n$ Hermitian matrices encoding the interactions between
internal states and the action of external perturbations, respectively. The
Hamiltonian is fully controllable if for every unitary operation
$U\in\mathrm{SU}(n)$ there is a control profile $\mathbf{u}(t)$ such that
$U={\cal T}\exp[-i\int_{0}^{t}H(\tau)d\tau]$, where ${\cal T}$ is the time
ordering operator Ramakrishna  _et al._ (1995). A sufficient condition for the
controllability of the quantum system (2) is that $H_{I}$, $C^{(k)}$ and their
nested commutators generate the Lie algebra of $\mathrm{SU}(n)$ Ramakrishna
_et al._ (1995); D’Alessandro (2007). Burgarth et al Burgarth  _et al._ (2013)
have identified a class of fully controllable quantum systems where the
$C^{(k)}$ are associated with the dominating set. The quantum system (2) has
an associated backbone graph $G(H_{I})$, where vertices represent the degrees
of freedom and edges represent the couplings between them, the non-zero off-
diagonal elements of $H_{I}$. If $S=\\{v_{1},\ldots,v_{m}\\}$ is a dominating
set of $G(H_{I})$ then the quantum system (2) with

$C^{(k)}_{ij}=\delta_{iv_{k}}\delta_{jv_{k}}$ (3)

is fully controllable Burgarth  _et al._ (2013). Obviously, within this class
of quantum systems, the minimum number of external inputs needed to fully
control the system is the zero forcing number $Z(G)$.

Finding the zero forcing set of undirected graphs is a NP-hard problem Trefois
and Delvenne (2015). Yet, there can be classes of graphs where heuristic
algorithms can find a nearly optimal solution in polynomial time. This is the
case for the minimum vertex cover on graphs with a power law degree
distribution $p_{k}\sim k^{-\gamma}$. A vertex cover is a set of coloured
vertices in a graph such that every edge is incident to a coloured vertex. The
minimum vertex cover is the vertex cover (or covers) of minimum size. Finding
the minimum vertex cover is NP-hard as well. Yet, we can use a leaf-removal
algorithm to find the minimum vertex cover of graphs with a power law degree
distribution Vazquez and Weigt (2003).

Here I investigate the zero forcing number of graphs with a power law degree
distribution $p_{k}\sim k^{-\gamma}$ with $2<\gamma<\infty$. I introduce a
leaf-removal algorithm to identify sub-optimal zero forcing sets. Using graphs
generating by preferential attachment and the leaf-removal algorithm, I
demonstrate numerically that $Z(G)\rightarrow N$ when $\gamma\rightarrow 2$.
In contrast, the size of the minimum vertex cover is of order $N(\gamma-2)$
when $\gamma\rightarrow 2$.

Figure 1: Illustration of the leaf removal rules to construct the zero forcing
set.

##  II Leaf removal algorithm

Leaf-removal algorithms can tackle NP-hard problems on graphs with a power law
degree distribution Vazquez and Weigt (2003); Zhao  _et al._ (2015). A leaf is
defined as a vertex with degree 1. The leaf-removal takes advantage of
scenarios where the optimal choice is evident. Optimal choice will depend on
the specific optimization problem, as shown in Fig. 1 for zero forcing. If a
leaf is at the end of a chain and the leaf at the other end is white (Fig.
1a), one optimal choice is to colour the leaf black and add the leaf to the
forcing set. If a leaf is black and the neighbour is white (Fig. 1b), then the
neighbour is forced to black and the leaf is removed. If the neighbour becomes
a leaf then the procedure can be continue recursively. Yet another scenario is
a vertex with $L$ leafs (Fig. 1c). Here we have no other choice but colouring
black $L-1$ leafs and adding them to the zero forcing set. In some instances
the leaf removal rules do not create new leafs and the algorithm cannot
proceed further. To re-start the process I will remove the vertex with the
largest degree and add it to the zero forcing set. Putting all together the
algorithm proceed as follows.

  1. 1.

Start with all vertices coloured white and an empty zero forcing set.

  2. 2.

Create a list of all leafs in the current graph.

  3. 3.

For each leaf in the list carry on the following rules sequentially.

     * •

If the leaf is white and it is at the end of an isolated chain with a white
leaf at the other end, then colour the leaf black and add it to the zero
forcing set.

     * •

If the leaf is black, remove the leaf and colour the neighbour black. In the
event that the neighbour becomes itself a leaf proceed recursively until no
new leaf is created (Fig. 1a).

     * •

Otherwise, find all leafs at distance 2 from the current leaf, colour them
black, remove them and add them to the zero forcing set. If no leaf is found
at distance 2 no action is taken.

  4. 4.

If the graph is not empty, find a vertex with the current largest degree,
colour the vertex black, remove the vertex, add the vertex to the zero forcing
set, and move all neighbour vertices that become a leaf to the leaf list.

  5. 5.

Remove all isolated vertices, those that are white colour them black and add
them to the zero forcing set.

  6. 6.

If the graph is empty stop, otherwise go back to step 2.

The size of the resulting zero forcing set will be denoted by $Z_{LM}(G)$,
where $LM$ stands for leaf and maximum degree removal. Since the maximum
degree removal, step 4, is not necessarily optimal, this algorithm
overestimates the zero forcing number,

$Z_{LM}\geq Z$ (4)

As a comparison, I will also calculate the minimum vertex covering using an
adaptation of the vertex covering leaf removal algorithm Vazquez and Weigt
(2003). The algorithm proceed as follows. Start with all vertices uncovered
and a list of leafs in the graph. If the leaf list is non-empty, extract a
leaf, remove the leaf, add the leaf neighbour to the vertex covering set and
remove the leaf neighbour. Otherwise, find a vertex with the current largest
degree, add it to the vertex covering set and remove it. Continue until the
graph is empty. The minimum vertex covering will be denoted by $V(G)$. The
vertex cover size estimated by the leaf-maximum degree removal will be denoted
by $V_{LM}(G)$. Once again, because of the maximum degree removal rule, this
algorithm overestimates the minimum vertex covering,

$V_{LM}\geq V$ (5)

Figure 2: Zero forcing fraction $z=Z_{LM}/N$ and minimum vertex cover fraction
$v=V_{L}M/N$ of preferential attachment graphs, with $N=10,000$ and averaged
over 100 graphs.

##  III Preferential-attachment graphs

First, I will consider graphs generated by the preferential attachment model
with initial attractiveness Dorogovtsev  _et al._ (2000) ($G_{PA}$). The graph
is started with a fully connected graph of $m+1$ vertices. Then add new
vertices one by one until the targeted graph size is reached. Each time a new
vertex is added it is connected to $m$ existing and non-overlapping vertices,
each selected with probability

$\pi_{i}=\frac{(a-1)m+k_{i}}{\sum_{j}\left[(a-1)m+k_{j}\right]}$ (6)

where the indexes run over vertices in the current graph and $k_{i}$ denotes
the degree of vertex $i$. The parameter $a>0$ represents a vertex independent
attractiveness named initial attractiveness Dorogovtsev  _et al._ (2000). This
model generates graphs with a power law degree distribution $p_{k}\sim
k^{-\gamma}$ with exponent Dorogovtsev  _et al._ (2000)

$\gamma=2+a$ (7)

By tuning the initial attractiveness $a$ we can obtain exponents in the range
$2<\gamma<\infty$. The case $a=1$ ($\gamma=3$) corresponds to the original
Barabási-Albert model Barabási and Réka (1999).

Figure 3: Scaling of the zero forcing fraction $z=Z_{LM}/N$) and minimum
vertex cover fraction $v=V_{L}M/N$ of preferential attachment graphs, with
$N=10,000$ and averaged over 100 graphs.

Using the LM algorithms, I have estimated the zero forcing number and minimal
vertex covering of graphs generated by the preferential attachment model (Fig.
2). The zero forcing number and the minimum vertex covering behave quite
differently as $\gamma\rightarrow 2$. The zero forcing number approaches the
graph size while the minimum vertex covering approaches zero. Around
$\gamma=2$ we observe the scalings (Fig. 3)

$N-Z_{LM}(G_{PA})\approx c_{z}N(\gamma-2)$ (8) $V_{LM}(G_{PA})\approx
c_{v}N(\gamma-2)$ (9)

Substituting the scaling (8) into the minimum rank lower bound (1) we obtain

$mr(G_{PA})\geq c_{z}N(\gamma-2)$ (10)

When $\gamma\approx 2$ we cannot exclude that the minimum rank is 0.

Figure 4: Contribution of the maximum degree step to the size of the zero
forcing and vertex covering sets estimated by the LM algorithm, for
preferential attachment graphs with $N=10,000$ and averaged over 100 graphs.

Since the LM algorithm yields upper bounds it is worth asking how tight are
those bounds, specially for $\gamma\approx 2$. To address this question I have
calculated the number of vertices that were forced using the maximum degree
step, denoted by $\Delta Z(G)$ and $\Delta V(G)$ for the zero forcing and
vertex covering algorithms, respectively. We observe that $\Delta
V(G_{PA})/N\approx 0$ for all values of $\gamma$ explored (Fig. 4b). Therefore

$V_{LM}(G_{PA})\approx V(G_{PA})$ (11)

In contrast, $\Delta Z_{LM}(G_{PA})>0$ for most values of $\gamma$ and we
cannot tell how good are the zero forcing number estimates. Nevertheless, for
$\gamma\rightarrow 2$ we have $\Delta Z_{LM}(G_{PA})/N\rightarrow 0$. In the
vicinity of $\gamma=2$ the LM algorithm provides good estimates of the zero
forcing number, i.e.

$Z_{LM}(G_{PA})\approx Z(G_{PA})$ (12)

for $\gamma\approx 2$. This allows us to conclude that the zero forcing number
of preferential-attachment graphs approaches the graph size when
$\gamma\rightarrow 2$.

Figure 5: Star graphs. a) Isolated stars. b) String of stars. Solid circles
are vertices in the zero forcing set.

##  IV Star and deactivation graphs

It is straightforward to show that $\gamma\rightarrow 2$ is not a sufficient
condition to obtain $Z(G)\approx N(G)$. To this end, consider graphs made by
isolated starts with a degree distribution $p_{k}$, $k=1,\ldots$
($G_{0\star}$, Fig. 5a). In this case the degree distribution satisfy the
constraint

$\sum_{k>1}p_{k}k=p_{1}$ (13)

For each isolated star all but 1 leaf need to be included in the zero forcing
set, resulting in

$\frac{Z(G_{0\star})}{N}=\sum_{k>1}p_{k}(k-1)=2p_{1}-1<1$ (14)

A similar result is obtained for graphs made by a string of stars
($G{-\star}$, Fig. 5b). In this case the degree distribution satisfy the
constraint

$\sum_{k>1}p_{k}(k-2)=p_{1}$ (15)

In this case all leafs need to be forced resulting in

$\frac{Z(G_{-\star})}{N}=\sum_{k>1}p_{k}(k-2)=p_{1}<1$ (16)

$Z(G_{0\star})<N$ and $Z(G_{-\star})<N$ regardless of the shape of the degree
distribution. A power low degree distribution $p_{k}\sim k^{-\gamma}$ with
$\gamma\rightarrow 2$ is not a sufficient condition for $Z(G)=N(G)$.

Figure 6: Zero forcing fraction $z=Z_{LM}/N$ and minimum vertex cover fraction
$v=V_{L}M/N$ of deactivation graphs, with $N=10,000$ and averaged over 100
graphs.

The deactivation graphs of Klemm and Eguíluz Klemm and Eguíluz (2002)
($G_{D}$) provide another counter example inspired on natural rules of network
evolution. The basic idea of the deativation graph model is that, with time,
some vertices will no longer participate in the network evolution, becoming
inactive or deactivated. That rule models the retirement of a scientist in the
context of co-authorship networks for example. The deactivation graphs studied
here are generated as follows. Start with a fully connected graph of $m+1$
active vertices. At each graph evolution step, add a new active vertex,
connect the new vertex to the pre-existing $m$ active vertices and set one of
the active vertices ($i\in{\cal A}$) inactive with a probability

$\pi_{i}=\frac{\sum_{s\in{\cal A}}((a-1)m+k_{s})^{-1}}{(a-1)m+k_{i}}$ (17)

The deactivation model generate graphs with a power low degree distribution
$p_{k}\sim k^{-\gamma}$ with exponent Klemm and Eguíluz (2002)

$\gamma=2+a$ (18)

By tuning $a$ we can thus generate power law exponents in the range
$2\leq\gamma<\infty$.

Using the leaf-maximum degree removal algorithms, I have estimated the zero
forcing number and minimal vertex covering of deactivation graphs (Fig. 6).
For the deactivation graphs the zero forcing number does not approach the
graph size when $\gamma\rightarrow 2$ (Fig. 6a). In fact, $Z_{LM}(G_{D})/N<1$
and $V_{LM}(G_{D})/N>0$ for all values of $\gamma$ (Fig. 6a,b).

As shown before, there is a key difference between the preferential-attachment
and deactivation graphs regrading the graph diameter, denoted by $d$. The
preferential-attachment generate small-world graphs for $\gamma>3$ ($d\sim\ln
N$) and ultra-small graphs ($d\sim\ln\ln N$) for $2<\gamma<3$ Cohen and Havlin
(2003). In contrast, deactivation graphs are effectively one-dimensional
($d\sim N$) Vazquez  _et al._ (2003). This, together with the analysis of star
graphs, indicates that the small-world property is a requirement to obtain
$Z(G)\approx N$ when $\gamma\rightarrow 2$.

##  V Conclusions

Using numerical simulations of preferential-attachment graphs, I have
demonstrated that the zero forcing number approaches the graph size
($Z\rightarrow N$) when the exponent of the power law degree distribution
approaches 2 ($\gamma\rightarrow 2)$. This extends a similar result by Liu-
Slotine-Barabási for directed graphs Liu  _et al._ (2011) to undirected
graphs. Through the analysis of some counterexamples, I have shown that the
small-world property of preferential-attachment graphs is a necessary
requirement for this result.

These observations are relevant for the controllability of quantum systems. If
the degrees of freedom are coupled by a small world network with a power law
degree distribution, then we would need to manipulate almost all degree of
freedoms to achieve full controllability. A putative scenario is the
configuration space of energy basins associated with protein folding. The
network of energy basins of a Lennard-Jones gas and a $\beta$-sheet peptide is
characterized by a power law degree distribution, with exponents
$\gamma\approx 2.8$ Doye (2002) and $\gamma\approx 2$ Rao and Caflisch (2004),
respectively. For the Lenard-Jones gas it was further shown that the network
is small world. Controlling the quantum dynamical evolution of these systems
with the zero forcing Hamiltonians (3) would be unfeasible. It would require
the manipulation of almost all energy basins to achieve full control.

## References

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# Characterizing Topological Excitations of a Long-Range Heisenberg Model with
Trapped Ions

Stefan Birnkammer  Department of Physics, Technical University of Munich,
85748 Garching, Germany  Munich Center for Quantum Science and Technology
(MCQST), Schellingstraße 4, 80799 München, Germany  Annabelle Bohrdt
Department of Physics, Technical University of Munich, 85748 Garching, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4,
80799 München, Germany  Fabian Grusdt  Department of Physics and Arnold
Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-
Universität München, Theresienstr. 37, München D-80333, Germany  Department of
Physics, Technical University of Munich, 85748 Garching, Germany  Munich
Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799
München, Germany  Michael Knap  Department of Physics, Technical University of
Munich, 85748 Garching, Germany  Munich Center for Quantum Science and
Technology (MCQST), Schellingstraße 4, 80799 München, Germany

(August 27, 2024)

###### Abstract

Realizing and characterizing interacting topological phases in synthetic
quantum systems is a formidable challenge. Here, we propose a Floquet protocol
to realize the antiferromagnetic Heisenberg model with power-law decaying
interactions. Based on analytical and numerical arguments, we show that this
model features a quantum phase transition from a liquid to a valence bond
solid that spontaneously breaks lattice translational symmetry and is
reminiscent of the Majumdar-Ghosh state. The different phases can be probed
dynamically by measuring the evolution of a fully dimerized state. We moreover
introduce an interferometric protocol to characterize the topological
excitations and the bulk topological invariants of the interacting many-body
system.

Introduction.—Recent progress in realizing synthetic quantum systems has
offered new opportunities for the experimental characterization and control of
topological quantum phases. Topologically nontrivial band structures have been
created by periodic driving Aidelsburger  _et al._ (2013); Miyake  _et al._
(2013); Jotzu  _et al._ (2014); Flaschner  _et al._ (2016); Wintersperger  _et
al._ (2020), interaction-induced chiral propagation of excitations have been
studied in the few-body limit of quantum Hall states Tai  _et al._ (2017);
Roushan  _et al._ (2017), symmetry protected topological (SPT) phases have
been realized Gu and Wen (2009); Pollmann  _et al._ (2012); de Léséleuc  _et
al._ (2019); Sompet  _et al._ (2021), and quantum spin liquids have been
explored with quantum devices Satzinger  _et al._ (2021); Semeghini  _et al._
(2021). While first steps have been laid in realizing interacting topological
phases, several challenges remain, in particular concerning the
characterization and control of individual topological excitations.

Figure 1: Floquet protocol and phase diagram. (a) Interactions between ions in
a linear Paul trap are of Ising type
$H_{xx}=J/|i-j|^{\alpha}\hat{S}^{x}_{i}\hat{S}^{x}_{j}$ with discrete
$\mathds{Z}_{2}$ symmetry. (b) Periodically applying global $\pi/2$-pulses
around different axes of the Bloch sphere creates interactions along all three
spin directions $H_{\text{xx}}$, $H_{\text{yy}}$, and $H_{\text{zz}}$. (c) The
high-frequency limit of such a protocol realizes a long-ranged Heisenberg
model $H_{\text{LR}}$ with continuous $SU(2)$ symmetry. This model features a
quantum phase transition from a dimerized to a liquid phase. Inset: A Binder
cumulant analysis of the dimerization determines a critical power-law exponent
of $\alpha_{c}\approx 1.66$.

Here, we propose the realization of a dimerized valence bond solid with
topologically nontrivial excitations in a Heisenberg model with power-law
interactions using trapped ions Blatt and Roos (2012). This phase arises due
to frustration from long-range interactions and is adiabatically connected to
the symmetry broken Majumdar-Ghosh phase Majumdar and Ghosh (1969a, b); White
and Affleck (1996). When locally deforming this Hamiltonian to introduce bond
alternating couplings, it realizes a Haldane SPT phase Haldane (1983a). Our
model therefore illustrates the interplay between spontaneous symmetry
breaking and toplogical order. To realize the long-range Heisenberg model in a
trapped-ion setting, we propose a Floquet protocol that consists of periodic
globally-applied $\pi/2$-pulses around different axes of the Bloch sphere, see
Fig. 1 (a, b). We determine the phase diagram and propose an interferometric
protocol to characterize the topological excitations and the bulk topological
invariants of our interacting many-body system.

Model.—We investigate a long-range spin-$1/2$ Heisenberg chain with open
boundaries

${H}_{\text{LR}}(\alpha)\ =\ \ \sum_{i<j}\dfrac{J}{|i-j|^{\alpha}}\
\left[{{\hat{S}}}_{i}^{x}{{\hat{S}}}_{j}^{x}+{{\hat{S}}}_{i}^{y}{{\hat{S}}}_{j}^{y}+{{\hat{S}}}_{i}^{z}{{\hat{S}}}_{j}^{z}\right],$
(1)

where $\alpha$ is the power-law exponent of the long-range interactions and
$J>0$ their typical energy scale (for a spin-$1$ variant of the model see Ref.
Gong  _et al._ (2016)). When considering only nearest and next-to-nearest
neighbor couplings, $H_{\text{LR}}(\alpha)$ reduces to the Majumdar-Ghosh (MG)
model Majumdar and Ghosh (1969a, b), which exhibits a phase transition from a
liquid to a dimerized valence bond solid that breaks the translational
invariance of the lattice White and Affleck (1996).

The long-range Heisenberg model $H_{\text{LR}}$ reduces for $\alpha\to\infty$
to the conventional Heisenberg model with nearest-neighbor couplings, whose
ground state is a gapless spin liquid with power-law decaying
antiferromagnetic correlations Giamarchi (2004). For the opposite limit of
$\alpha\to 0$, each spin interacts equally with all the others and the ground
states correspond to arbitrary singlet pairings (see supplementary materials
sup  and references Cornwell and Buck (1998); Marshall (1955); Haldane (1983b)
therein). For small but finite $\alpha$, it is energetically favorable to form
singlets on neighboring sites as in the Majumdar-Ghosh state

$\left|\text{MG}\right\rangle=\prod_{i}^{L/2}(\left|\uparrow\right\rangle_{2i}\left|\downarrow\right\rangle_{2i+1}-\left|\downarrow\right\rangle_{2i}\left|\uparrow\right\rangle_{2i+1})/\sqrt{2}.$
(2)

As a consequence the ground state breaks translational symmetry. Due to the
different behavior of the ground states in the two limits, at least one
quantum phase transition occurs at some critical $\alpha_{c}$.

To quantitatively determine the phase diagram, we perform Density-Matrix
Renormalization Group (DMRG) simulations Verstraete  _et al._ (2008);
Schollwoeck (2011) and compute the dimerization order parameter

${d}_{i}\ =\ \vec{{S}}_{2i}\cdot(\vec{{S}}_{2i+1}-\vec{{S}}_{2i-1}).$ (3)

as a function of $\alpha$ and for different system sizes $L$, see Fig. 1 (c).
The data for the thermodynamic limit was extrapolated from finite-size results
by a scaling Ansatz $|\alpha-\alpha_{c}|^{\nu}$. Expressions without spatial
indices, indicate in the following averages over the bulk. In our numerical
studies, we represent the Hamiltonian as a matrix product operator, in which
we approximate the power-law coupling by a sum of exponentials Murg  _et al._
(2010); Schollwoeck (2011). An analysis of the Binder cumulant $\left\langle
d^{4}\right\rangle/\left\langle d^{2}\right\rangle^{2}$, that is expected to
be scale-independent at criticality Binder and Heermann (2010); Landau and
Binder (2015), indicates a quantum phase transition from a dimerized phase to
a liquid at $\alpha_{c}\approx 1.66$ (inset). Precisely extracting the
critical point is a formidable challenge, as the transition is in the
Berezinskii-Kosterlitz-Thouless universality class White and Affleck (1996);
Eggert and Affleck (1992); Eggert (1996). We emphasize, however, that for the
following discussions the precise location of the phase transition is not
crucial.

Figure 2: Dynamical phase diagram. (a) Evolution of the dimerization $d$ under
the Floquet dynamics for an initial $\left|\text{MG}\right\rangle$-state of
singlets. The evolution is governed by $H_{\text{LR}}(\alpha)$ for
$\alpha\in\\{0.2,0.5,1.5\\}$ and a chain of 64 sites. Inset: We illustrate the
relative deviation $\sigma\equiv|(d_{\Delta J\rightarrow 0}-d_{\Delta
J})/d_{\Delta J\rightarrow 0}|$ of the Floquet protocol from the exact
evolution for different values of $\Delta J$ and $\alpha=0.5$. (b) Thermal
phase diagram for $H_{\text{LR}}(\alpha)$ including the effective temperature
$T^{*}$ of our system (solid line) and a schematic of the phase boundary
between the dimerized and the translational invariant phases (dashed dome) for
a system of 18 sites.

Floquet Protocol.—Collective vibrations of an ion crystal mediate long-range
Ising interactions $H_{xx}=\sum_{i<j}J/|i-j|^{\alpha}S^{x}_{i}S^{x}_{j}$
Richerme  _et al._ (2014); Jurcevic  _et al._ (2014). Previous works suggested
to use multiple phonon branches Porras and Cirac (2004) and quasi-periodic
driving Bermudez  _et al._ (2017) to realize Heisenberg type interactions, or
have employed digital simulation schemes Lanyon  _et al._ (2011). Here,
instead we suggest periodic driving Choi  _et al._ (2020); Arrazola  _et al._
(2016); Kasper  _et al._ (2020); Agarwal and Martin (2019) to promote the
discrete $\mathds{Z}_{2}$ symmetry of the Ising interactions to the continuous
$SU(2)$ symmetry of the Heisenberg interactions. The protocol consists of
$\pi/2$-pulses around different axes to encircle a surface of the Bloch sphere
Fig. 1 (b); see Ref. Arrazola  _et al._ (2016) for a related protocol and
Refs. Geier  _et al._ (2021); Scholl  _et al._ (2022); Kranzl  _et al._ (2022)
for recent experimental realizations. The duration $\delta$ of the
$\pi/2$-pulses can be chosen to be much shorter than the waiting time
$\Delta$, leading to an effective period of $\approx 3\Delta$. This way the
many-body state rotates periodically from the $x$ over $y$ to $z$ direction.
These unitary transformations can also be interpreted to act on the
Hamiltonian instead of the many-body state, leading to an effective time
evolution with alternating $H_{xx}$, $H_{yy}$, and $H_{zz}$ Ising couplings.
Provided the rate $\Delta^{-1}$ is fast compared to the typical interaction
strength $J$, a high-frequency expansion Bukov  _et al._ (2015) for the
effective periodic drive can be computed, which to leading order yields the
$SU(2)$ invariant Hamiltonian (1).

Singlet evolution.—As a direct application of our Floquet protocol, we compute
the time evolution of the long-range Heisenberg model $H_{\text{LR}}(\alpha)$
for an initial singlet state $\left|\text{MG}\right\rangle$ using the time-
dependent variational principle for MPS Haegeman  _et al._ (2011, 2016);
Žunkovič  _et al._ (2018); Hauschild and Pollmann (2018); Halimeh  _et al._
(2016). For quenches to small $\alpha$, we find that the dimerization remains
finite at long times, whereas it quickly decays to zero for quenches to large
$\alpha$, see Fig. 2 (a). To compare the discrete Floquet evolution with the
exact dynamics of $H_{\text{LR}}(\alpha)$, we introduce the relative deviation
$\sigma\equiv|(d_{\Delta J\rightarrow 0}-d_{\Delta J})/d_{\Delta J\rightarrow
0}|$, shown in the inset of Fig. 2 (a) for $\alpha=0.5$. We find that the
Floquet protocol accurately describes the $SU(2)$ invariant Heisenberg
evolution for $\Delta J=0.1$ (depending on $\alpha$, larger values of $\Delta
J\sim 1$ can be safely reached sup ).

The energy density of the dimerized initial state is
$\left\langle\text{MG}\right|H_{\text{LR}}(\alpha)\left|\text{MG}\right\rangle/L=-0.375J$
independent of $\alpha$, which is larger than the ground-state energy density
of $H_{\text{LR}}(\alpha)$. Hence, the quench deposits an extensive amount of
energy into the system. According to the eigenstate thermalization hypothesis
Deutsch (1991); Srednicki (1994); Rigol  _et al._ (2008), which is expected to
hold for generic interacting systems as this, a subsystem should thermalize to
an effective temperature $T^{*}$, that is consistent with the energy density
deposited in the system. The effective temperature can then be evaluated self-
consistently from the condition
$\left\langle\text{MG}\right|H_{\text{LR}}(\alpha)\left|\text{MG}\right\rangle=\operatorname{\mbox{tr}}[H_{\text{LR}}(\alpha){e}^{-H_{\text{LR}}(\alpha)/T^{*}}/\mathcal{Z}]$
where $\mathcal{Z}$ is the partition sum. We approximate the thermal
expectation value using the typicality approach Bartsch and Gemmer (2009) and
evolve 50 random initial states in imaginary time using exact diagonalization
on 18 spins (see sup  for system size dependence) to extract effective
temperature $T^{*}$ as a function of $\alpha$; Fig. 2 (b). From the dynamical
phase diagram, we find that for $\alpha\lesssim 1$ the effective temperature
is low enough such that a finite dimerization is supported in the steady
state, whereas it decays to zero for $\alpha\gtrsim 1$, consistent with our
observations on the time evolution in Fig. 2 (a). Despite the one-dimensional
character of our system, a sharp finite-temperature phase transition can arise
in the thermodynamic limit because of the long-range interactions Auerbach
(1994).

Measuring the Zak phase.—Starting from the
$\left|\text{MG}\right\rangle$-state, a state close to the ground state of
$H_{\text{LR}}(\alpha)$ can be prepared by adiabatically tuning $\alpha$ as
long as the system remains in the dimerized phase. We will now present a
protocol to measure a topological order parameter of such a state. Following
Refs. Hatsugai (2007, 2006), we introduce the $SU(2)$-transformation

$\Phi:\vec{S}_{i}\cdot\vec{S}_{j}\ \mapsto\
\hat{S}_{i}^{\mathrm{z}}\hat{S}_{j}^{\mathrm{z}}+\dfrac{1}{2}(\mathrm{e}^{\mathrm{i}\varphi}\hat{S}_{i}^{+}\hat{S}_{j}^{-}+\text{h.c.}),$
(4)

where $\varphi$ is a compact variable in the interval $[0,2\pi]$. The
$SU(2)$-transformation is designed such that it affects only couplings
crossing the $\ell$-th bond between sites $\ell$ and $\ell+1$, which separates
our system into a left $S_{L}$ and a right $S_{R}$ part. All interactions
within one subsystem remain unchanged. As a consequence we obtain for every
choice of the bond $\ell$ a new family of Hamiltonians
$H_{\text{LR}}(\alpha;\varphi)$ parametrized by $\varphi\in[0,2\pi]$.

Figure 3: Measuring many-body Zak phases with interferometry. (a) Single
Floquet period and (b) complete pulse sequence for extracting the Zak phase;
see main text for details. (c) The effective field $B^{\text{eff}}$ acting on
the system during the step (iii) of the protocol has a sign change at the bond
where we measure $\gamma^{\text{Zak}}_{\ell}$. (d) Performing measurements for
different $\ell$ yields alternating Zak phases of $0$ or $\pi$. Data is
evaluated for a system of $32$ sites and $\alpha=0.2$.

A key for obtaining a quantized topological order parameter is that the chosen
parametrization retains the time-reversal symmetry of
$H_{\text{LR}}(\alpha;\varphi)$ Hatsugai (2007). Tuning $\varphi$ continuously
through the interval $[0,2\pi]$, describes a closed loop $\mathcal{C}_{\ell}$
within the set of Hamiltonians. This allows us to introduce the Zak phase

$\gamma^{\mathrm{Zak}}_{\ell}=\oint_{\mathcal{C}_{\ell}}\mathrm{d}\varphi\left\langle\psi(\varphi)\right|\mathrm{i}\partial_{\varphi}\left|\psi(\varphi)\right\rangle,$
(5)

where $\left|\psi(\varphi)\right\rangle$ is the ground state of
$H_{\text{LR}}(\alpha;\varphi)$. The Zak phase is well defined, provided the
corresponding path $C_{\ell}$ is followed adiabatically, which can be ensured
because the dimerized phase is gapped, see supplementary material sup . In
order to gain some intuition about the Zak phase, we first apply the
$SU(2)$-transformation to the fully dimerized
$\left|\text{MG}\right\rangle$-state. When the bond $\ell$ lies within a
singlet, the transformation gives
$(\left|\uparrow\downarrow\right\rangle-\mathrm{e}^{\mathrm{i}\varphi}\left|\downarrow\uparrow\right\rangle)/\sqrt{2}$.
Evaluating Eq. (5) for this particular case reveals a Zak-phase of $\pi$
Grusdt  _et al._ (2019, 2014), while it is zero when the bond $\ell$ lies
between two singlets. For the $\left|\text{MG}\right\rangle$-state and thus
also for the adiabatically connected ordered ground states of $H_{\text{LR}}$
we consequently expect to find a Zak phase alternating between values of $0$
or $\pi$ when traversing the bond $\ell$ through the system.

Before we numerically compute the Zak phase of the dimerized state, we
introduce a protocol to experimentally measure it in a trapped ion setting.
Let us first gain some intuition: To realize a transformation similarly to
(4), we can use an effective (in general time-dependent) magnetic field
$B_{i}^{\text{eff}}(t)$ acting on the spins of $H_{\text{LR}}(\alpha)$, that
is proportional to a step function with the step being located at bond $\ell$.
Using the Peierls substitution, the magnetic field can be absorbed into
Hamiltonian as
$H_{\text{LR}}(\alpha;\varphi(t))=\sum_{i<j}J/|i-j|^{\alpha}[S^{\mathrm{z}}_{i}S^{\mathrm{z}}_{j}+\frac{1}{2}(\mathrm{e}^{\mathrm{i}\varphi_{ij}(t)}S^{+}_{i}S^{-}_{j}+\text{h.c.})]$,
where
$\varphi_{ij}(t)\equiv\int_{0}^{t}\mathrm{d}t^{\prime}[B^{\text{eff}}_{j}(t^{\prime})-B^{\text{eff}}_{i}(t^{\prime})]$.
A phase is only picked up, when bond $\ell$ is crossed as $B^{\text{eff}}_{i}$
is assumed to be constant except across bond $\ell$. The time $t$ is chosen
such that the phase $\varphi$ is adiabatically tuned from 0 to $2\pi$. The Zak
phase can be measured using a Ramsey sequence to cancel dynamical phases Atala
_et al._ (2013); Duca  _et al._ (2015); Abanin  _et al._ (2013).

In order to implement this approach in a chain of ions, we identify the left-
most ion as an ancilla qubit $\tau^{z}$ that operates on the same
computational basis and has the same power-law coupling to the other spins of
the chain
$\sum_{i}J/|i|^{\alpha}\tau^{z}\hat{S}_{i}^{z}\equiv\sum_{i}B_{i}^{\text{eff}}\hat{S}_{i}^{z}$.
The protocol then consists of the following steps, see Fig. 3 (a) and (b) for
an illustration: (i) After ground state preparation of the chain initialize
the ancilla qubit in a superposition state by applying a $\pi/2$-rotation.
This leads to an opposite sign in $B^{\text{eff}}$ for the two ancilla states
and in turn allows for a cancellation of the dynamical phase. (ii) Perform
global $\pi/2$-rotations on the system around different axes, as discussed in
Fig. 1 (c), to realize the long-range Heisenberg dynamics. During that time
perform equally spaced $\pi$-rotations on the ancilla qubit to cancel the
phase accumulation from interaction with the remaining chain. (iii) After a
Floquet period, apply a $\pi$-rotation only on the right part of the system to
create a kink in the effective field $B_{i}^{\text{eff}}$ at bond $\ell$, see
Fig. 3 (b). Accumulate phase on the ancilla and apply another $\pi$-rotation
to restore the couplings within the system. Steps (ii) and (iii) are then
repeated until $\varphi$ covers the whole interval $[0,2\pi]$. (iv) Apply a
$\pi-$pulse to the left part of the system enabling an inverse rotation to
compensate the effect of the protocol on the wave function of the system. (v)
Measure the phase of the ancilla, which corresponds to the many-body Zak phase
at bond $\ell$. For a more detailed description see supplementary material sup

We now numerically evaluate the Zak phase Eq. (5). Using DMRG we compute the
ground state for 20 discretized steps along $\mathcal{C}_{\ell}$ for a system
of 32 sites and $\alpha=0.2$, see Fig. 3 (d), which confirms that the Zak
phase is alternating between 0 and $\pi$, a characteristics of a dimerized
state. We also confirm the adiabaticity of the computation by calculating the
product of projectors into the ground state
$\mathcal{P}(\varphi)=\left|\psi(\varphi)\right\rangle\left\langle\psi(\varphi)\right|$
during each step of the protocol, which attains large values between 0.75 to
1.

In the proposed experimental protocol, the effective magnetic field has the
required jump at bond $\ell$ to introduce the local $SU(2)$-transformation,
but also varies slowly across the other bonds. By numerically simulating the
protocol of Fig. 3, we find that the phase accumulated on the ancilla has the
characteristic bond alternating pattern for dimerized states; see
supplementary material sup  for details, where we also analyze the number of
required operations.

Topological excitations.—In order to characterize the topological excitations
of the dimerized phase, we now consider a chain with an odd number of sites.
In this case, singlets cannot fully cover the chain, and hence an unpaired
spin-$1/2$ (spinon) excitation is always present. Due to prominent examples
such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) model Affleck  _et al._ (1987)
we are used to the existence of spin-$1/2$ states in interacting topological
phases. These degenerate modes provide a clear signature for topological order
and are typically localized at the edges of the system. In contrast to the
usual edge modes, a measurement of the magnetization for $H_{\text{LR}}$
indicates that the excitation is delocalized over the entire lattice, see Fig.
4 (a). In order to obtain an analytical understanding, we introduce a
variational state that describes a delocalized spinon with wavevector $q$
separating two $\left|\text{MG}\right\rangle$-states with singlets on even and
odd bonds, respectively sup . The spinon hence represents a defect in the
topological order; inset of Fig. 4 (a). Variationally optimizing the ground
state energy with our Ansatz yields $q=\frac{\pi L}{2(L+1)}$, which is
consistent with the oscillatory magnetization pattern in Fig. 4 (a).

Figure 4: Topological excitations. (a) Local magnetization for a spinon in a
lattice of 33 sites and $\alpha=0.2$. We compare the DMRG result with a
variational Ansatz delocalizing a single spinon. (b) This bulk excitation
swaps the Zak phase of even and odd bonds as the bond $\ell$ traverses the
system.

We also characterize the Zak phase for this state. For odd numbers of sites,
the dimerized ground state of the system is two-fold degenerate due to the
$SU(2)$ symmetry of our model and hence we cannot construct an adiabatic path
$\mathcal{C}_{\ell}$. To lift the degeneracy, we apply a weak local magnetic
field in the center of the system, which introduces a small gap. This magnetic
field breaks time reversal symmetry. From the arguments of Hatsugai Hatsugai
(2007, 2006) it then follows that the Zak phase is not quantized in general.
As a consequence, we expect a monotonous change in the Zak phase as we
traverse the system which can be interpreted as a mobile domain wall
separating two distinct topological phases. This is consistent with our
results in Fig. 4 (b), which also show that the Zak phase of even and odd
bonds differ by $\pi$ as advocated by the domain wall picture.

Outlook.—The Heisenberg model with long-range antiferromagnetic interactions
exhibits a phase transition from a liquid to a dimerized valence bond solid
that spontaneously breaks the lattice translational invariance. We propose
Floquet protocols for trapped ions to realize this model and to characterize
the nature of the delocalized topological excitations as well as the bulk
topological invariants.

For future studies, it would be interesting to introduce an easy-axis
anisotropy by adjusting the Floquet periods between the $\pi/2$-pulses to
realize a deconfined quantum critical point between a dimerized and a Néel
ordered phase in our one-dimensional model Mudry  _et al._ (2019); Roberts
_et al._ (2019) or to explicitly break the translational symmetry by
introducing bond-alternating couplings to realize a Haldane symmetry protected
topological (SPT) phase with localized edge states Haldane (1983a). A future
challenge is to create interacting higher-dimensional topologically ordered
many-body states with synthetic quantum matter, that are characterized by a
topological entanglement entropy and fractional excitations Kitaev and
Preskill (2006). With our protocols, interactions in two dimensional
triangular lattices of trapped ions Bohnet  _et al._ (2016), could be promoted
from $\mathbb{Z}_{2}$ to $SU(2)$ symmetry, with the prospect of realizing
exotic frustrated mangetic states or even quantum spin liquids.

Acknowledgements.—We thank R. Blatt, E. Demler, Ch. Maier, F. Pollmann, and R.
Verresen for insightful discussions. Our tensor-network simulations were
performed using the TeNPy Library Hauschild and Pollmann (2018). We
acknowledge support from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germanys Excellence Strategy–EXC–2111–390814868,
TRR80 and DFG grant No. KN1254/2-1, No. KN1254/1-2, Research Unit FOR 2414
under project number 277974659, and from the European Research Council (ERC)
under the European Unions Horizon 2020 research and innovation programme
(grant agreement No. 851161), as well as the Munich Quantum Valley, which is
supported by the Bavarian state government with funds from the Hightech Agenda
Bayern Plus.

Supplemental Materials

All-to-all Interactions.—Here, we discuss the limiting case $\alpha=0$ of
$H_{\text{LR}}$, in which our one-dimensional model describes a cluster of
spins forming one self-interacting spin of extensive magnitude
$\vec{{\mathcal{S}}}$:

$\displaystyle\hat{H}_{\text{LR}}(\alpha=0)$
$\displaystyle=J\sum_{i<j}\vec{{S}}_{i}\cdot\vec{{S}}_{j}$
$\displaystyle=\dfrac{J}{2}\sum_{i,j}\vec{{S}}_{i}\cdot\vec{{S}}_{j}-\dfrac{J}{2}\sum_{i}\vec{{S}}_{i}^{2}$
(6)
$\displaystyle=\dfrac{J}{2}\vec{{\mathcal{S}}}\cdot\vec{{\mathcal{S}}}-\dfrac{3J}{8}L.$

To arrive at the final expression we introduced
$\vec{{\mathcal{S}}}\equiv\sum_{i}\vec{{S}}_{i}$ and made use of the identity
for amplitudes of spin operators $\vec{{S}}\cdot\vec{{S}}=s(s+1)$.

The energy spectrum of the model can now be completely determined by
representation theory of the group $SU(2)$. To this end, we look at all
possible irreducible representations arising from a product of $L$ spin-$1/2$
degrees of freedom. We will first assume an even number of spins $L$. The
arising representations are given by

$\bigotimes_{i=1}^{L}\leavevmode\nobreak\ {\bf
D}^{1/2}=\bigoplus_{j=0}^{L/2}\leavevmode\nobreak\
\bigoplus_{n=1}^{N_{j}}\leavevmode\nobreak\ {\bf D}^{j},$ (7)

where we denoted an irreducible representation of spin $j$ as ${\bf D}^{j}$
following the notation of Cornwell and Buck (1998). Expression (7) indicates
that we have $N_{j}$ distinct irreducible $SU(2)$-representations ${\bf
D}^{j}$, where j ranges up to total spin $L/2$. The energy $E_{j}$
corresponding to each spin representation ${\bf D}^{j}$ is according to
Hamiltonian (Characterizing Topological Excitations of a Long-Range Heisenberg
Model with Trapped Ions) given as

$E_{j}=\dfrac{J}{2}j(j+1)-\text{const}.$ (8)

The ground states of the theory are hence given by all singlet representations
arising from the product of $L$ spins, which is also conjectured by the
theorem of Marshall for quantum antiferromagnetic Heisenberg models Marshall
(1955); Auerbach (1994). Next we determine the degeneracies of energy levels,
which are directly related to the set $\\{N_{j}\\}$, because we find precisely
$N_{j}(2j+1)$ states of energy $E_{j}$. First, we note that these degeneracies
carry a dependence on the system size $L$. This is required to ensure an
exponential growth of the Hilbert space as we extend our system with
additional spins. Although also higher spin representations will be made
accessible by additional spins, latter only cause a linear increase in the
dimension of the Hilbert space.

Figure 5: Energy Gaps. Energy gap $\Delta$E above the ground state (a) as a
function of the lattice size $L$ and (b) as a function of $\alpha$.

We can now use the $L$-dependence to determine recursive relations for every
$N_{j}(L)$ respectively. For this purpose we will first look at a small system
and then generalize our observations to arbitrary system sizes. We start by
adding an additional spin doublet to a system of only $L=2$ spins. The
corresponding decomposition into irreducible representations ${\bf D}^{j}$ is
given by

$\displaystyle\big{(}{\bf D}^{1/2}\otimes{\bf D}^{1/2}\big{)}$
$\displaystyle\otimes\big{(}{\bf D}^{1/2}\otimes{\bf D}^{1/2}\big{)}$ (9)
$\displaystyle=\big{(}{\bf D}^{0}\oplus{\bf D}^{1}\big{)}\otimes\big{(}{\bf
D}^{0}\oplus{\bf D}^{1}\big{)}$ $\displaystyle=\big{(}{\bf D}^{0}\oplus{\bf
D}^{1}\big{)}\oplus\big{(}{\bf D}^{1}\oplus{\bf D}^{0}\oplus{\bf
D}^{1}\oplus{\bf D}^{2}\big{)}$ $\displaystyle=\leavevmode\nobreak\
\leavevmode\nobreak\ {\bf D}^{0}\oplus{\bf D}^{0}\oplus{\bf D}^{1}\oplus{\bf
D}^{1}\oplus{\bf D}^{1}\oplus{\bf D}^{2}.$

Here, we used distributivity as well as the Clebsch-Gordan series for products
of two irreducible $SU(2)$-representations.

We notice that the added singlet representation reproduces all given
representations of the previous system. The triplet contained in the added
spin doublet will produce three new representations for every previous one.
For the example of (9) this concretely increases $N_{0}$ by $1$, $N_{1}$ by
$2$ and $N_{2}$ by $1$. If we look at the situation, given in (9), on more
general grounds we recognize that ${\bf D}^{0}$ can only result from a product
of two singlets or two triplets. All other spin representations can, however,
be generated from previous ones in four different ways. The spin can on the
one hand be kept constant by an product with ${\bf D}^{0}$ or ${\bf D}^{1}$.
On the other hand it can be increased or decreased by $1$ using the product
with ${\bf D}^{1}$. This behavior is summed up to the following recursive
sequence for the set of $\\{N_{j}(L)\\}$.

$\displaystyle N_{0}(L+2)$ $\displaystyle=N_{0}(L)+N_{1}(L)$ $\displaystyle
N_{1}(L+2)$ $\displaystyle=N_{0}(L)+2N_{1}(L)+N_{2}(L)$ $\displaystyle
N_{2}(L+2)$ $\displaystyle=N_{1}(L)+2N_{2}(L)+N_{3}(L)$ (10)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \vdots$ $\displaystyle
N_{n}(L+2)$ $\displaystyle=N_{n-1}(L)+2N_{n}(L)+N_{n+1}(L)$

This set of equations can be implemented and solved symbolically, determining
the entire spectrum of $H_{\text{LR}}(\alpha=0)$.

After investigating the exact limit $\alpha=0$ let us now discuss the
implications for small but positive $\alpha$. Recalling Marshall’s theorem
Marshall (1955) we still expect to find an overall singlet as the ground state
of our system. On the contrary to the previous discussion we do, however, not
expect to find a degeneracy in the ground state. High degeneracies are
expected for the case $\alpha=0$ as interactions do not introduce a definite
order in the system and we can reorder the spins in all possible ways. This is
no longer possible for $\alpha>0$, where the decaying interactions favor
nearby singlets. For this reason we expect the ground state to exhibit a large
overlap with $\left|\mathrm{MG}\right\rangle$, defined in the main text and
possesses a finite dimerization $d$.

Figure 6: Convergence of the Floquet Protocol. The convergence of the Floquet
Hamiltonian towards its infinite-frequency limit $H_{\text{LR}}$ is examined
by comparing time-evolution of $\left|\mathrm{MG}\right\rangle$-states in
different regions of the phase diagram: (a) $\alpha=0.1$, (b) $\alpha=1.0$,
and (c) $\alpha=2.0$ for different values of the Floquet period $3\Delta$.

Energy Gaps of Ordered Phase.—After investigating the ground state of the
system, we can use our previous results to discuss also excited states and
especially the existence of finite energy gaps in the dimerized phase. The
latter are especially important for an adiabatic implementation of our
protocol to measure geometric Zak phases.

For the case of even $L$ and very small $\alpha$ we expect to find an overall
spin-singlet even for the first excited states, since all spin-$0$
configurations only receive small corrections increasing with $\alpha$
compared to the degenerate case of $\alpha=0$. The gaps of all higher spin
representations are, however, still determined up to a factor by (8) to be
finite and $\mathcal{O}(J)$. Even though an analytical calculation of the gap
is challenging, intuition can be gained from mapping our long-ranged spin
model $H_{\text{LR}}$ to a Non-Linear Sigma model (NL$\sigma$M) with a
topological term Bermudez  _et al._ (2017); Haldane (1983b, a). A
renormalization analysis indicates that such a system either flows to a
gapless fixed point of a $SU(2)_{1}$ Wess-Zumino-Witten conformal field theory
or approaches a gapped fixed point. These limits are related to the disordered
gapless and the gapped ordered phase of our model. We test this conjecture
numerically in Fig. 5. For small $\alpha$ our finite size analysis supports a
converged energy gap above the ground state. As the critical point
$\alpha_{c}\approx 1.66$ is approached, the required system sizes to observe
an onset of saturation increases.

A short remark should be addressed to the case of odd-numbered system sizes.
Whereas finite energy gaps above the ground state were shown to exist
naturally in the ordered phase for systems with an even number of sites, the
situation is different for odd $L$. Although the previous arguments for a gap
between the various spin representations apply in the same way for odd $L$,
they also conserve the degeneracy within each representation. This fact
directly results from the $SU(2)$ symmetry of $H_{\text{LR}}$. Contrary to the
spin singlet for even lattice sizes transforming trivially under application
of group generators from $su(2)$, the ground state spin-$1/2$-representation
for odd $L$ allows a mixing of both polarization directions of the spinon
excitation. The ground state degeneracy will hence remain two-fold unless we
break the underlying $SU(2)$ symmetry. This can for instance be achieved with
a small but finite local magnetic field, causing an energy splitting within
each spin representation and creating a finite gap above the ground state.

Convergence of the Floquet Protocol.—We numerically study the convergence of
the driven Hamiltonian to its high-frequency limit $H_{\text{LR}}$. To this
end, we investigate the time-evolution of a maximally dimerized
$\left|\mathrm{MG}\right\rangle$-state for both cases. The numerical results
are shown in Fig. 6 for Floquet step durations of $\Delta\in\\{0.1,1.0\\}$ and
$\alpha\in\\{0.1,1,2\\}$. Note that for the following investigations we
explicitly set the overall coupling strength to $J=1$, causing time intervals
like $\Delta$ to be dimensionless quantities. We find fast convergence to the
evolution governed by $H_{\text{LR}}$ even for comparatively low driving
frequencies ($\Delta=1.0$) both deep in the dimerized phase ($\alpha=0.1$) and
in the fluid phase ($\alpha=2.0$), whereas closer to the critical regime
shorter periods are required. All in all, the effective Floquet Hamiltonian
approximates $H_{\text{LR}}$ reasonably well already for not-too-high driving
frequencies which simplifies an experimental realization.

Figure 7: Short-time evolution. (a) We provide numerical results for the
double commutator
$\left\langle\mathrm{MG}\right|[H_{\mathrm{LR}},[H_{\mathrm{LR}},d]]\left|\mathrm{MG}\right\rangle$
characterizing the decay of the order parameter obtained from a short-time
expansion. (b)-(d) Comparison of the perturbative to the exact the time-
evolution of the order parameter for values of the scaling parameter
$\alpha\in\\{0.1,0.5,1.0\\}$. Figure 8: Size-dependence of effective
temperature. Effective temperature $T^{*}$, at which the thermal energy of the
system reaches the energy of a fully dimerized initial state, displayed for
values of the scaling parameter $\alpha\in\\{0.2,0.5,1.0,1.5\\}$. The
estimates were gathered using the typicality approach with 40 random states
and system sizes from 12 to 20 sites.

Short-time expansion.—Besides questions on the convergence of our protocol, a
point of experimental interest should be in timescales. To estimate the
typical relaxation time scale of the dimerization, we perform a short-time
expansion of the quantum evolution

Figure 9: Simulation of the Zak protocol. We show the phase accumulation of
the ancilla qubit $\left\langle\tau^{x}_{A}\right\rangle$ during the Zak
protocol illustrated in Fig. 3 (a) and (b) in the main text. We consider an
effective evolution with $H_{\mathrm{LR}}(\alpha)$ for a time span of
$T_{1}=2.0$ and $\alpha=0.2$. The phase $\varphi$ is introduced during the
evolution via application of $H_{\mathrm{Zak}}$ for durations of
$T_{2}\in\\{0.1,0.2,0.5\\}$. Interferometric measurements should be performed
at $\varphi=\pi$ to extract the relevant phase accumulation (orange dashed
lines). The results are compared with an ideal step function coupling the
ancilla and the system with $T_{2}/T_{1}=0.05$ (red dashed line).

$\displaystyle\left\langle\mathrm{MG}\right|d(t)\left|\mathrm{MG}\right\rangle=\left\langle\mathrm{MG}\right|U^{\dagger}(t,0)\leavevmode\nobreak\
d\leavevmode\nobreak\ U(t,0)\left|\mathrm{MG}\right\rangle$ (11)
$\displaystyle\quad\quad\mathrm{and}\quad\quad
U(t,0)=\mathcal{T}\exp\Big{(}-\mathrm{i}\int_{0}^{t}\mathrm{d}t^{\prime}H_{\mathrm{LR}}(t^{\prime};\alpha)\Big{)},$

where $U(t,0)$ denotes the usual time evolution operator from time $0$ to $t$
involving time ordering $\mathcal{T}$ in the Dyson sense. To simplify our
notation we will suppress the powerlaw exponent $\alpha$ of the Hamiltonian
from now on. Expanding expression (11) up to second order we get

$\displaystyle d(t)$
$\displaystyle=\left\langle\mathrm{MG}\right|d\left|\mathrm{MG}\right\rangle+\mathrm{i}\int_{0}^{t}\mathrm{d}t_{1}\left\langle\mathrm{MG}\right|[H(t_{1}),d]\left|\mathrm{MG}\right\rangle$
$\displaystyle-\dfrac{1}{2}\int_{0}^{t}\mathrm{d}t_{1}\mathrm{d}t_{2}\left\langle\mathrm{MG}\right|\mathcal{T}\Big{\\{}[H(t_{1}),[H(t_{2}),d]]\Big{\\}}\left|\mathrm{MG}\right\rangle.$
(12)

Next we would like to comment on the terms of Eq. (Characterizing Topological
Excitations of a Long-Range Heisenberg Model with Trapped Ions) individually.
The first term describes just a static offset given by the maximal
dimerization value $d_{0}=0.75$. The linear contribution in $t$ [second term
in Eq. (Characterizing Topological Excitations of a Long-Range Heisenberg
Model with Trapped Ions)] will vanish since the dimer order parameter only is
summed up from two projectors on the singlet configuration
$\left|\mathrm{MG}\right\rangle$ translated with respect to each other by one
lattice site. This implies that
$d\left|\mathrm{MG}\right\rangle\propto\left|\mathrm{MG}\right\rangle+\left|\phi\right\rangle$,
where $\left|\phi\right\rangle$ is related to $\left|\mathrm{MG}\right\rangle$
by a projection onto the singlet configuration between two consecutive
singlets of $\left|\mathrm{MG}\right\rangle$. As a consequence the linear part
takes the form
$\propto\operatorname{Im}[\left\langle\mathrm{MG}\right|H(t_{1})\left|\phi\right\rangle]$,
which in fact vanishes.
To simplify the last term, we choose $t$ to be commensurate with full Floquet
periods

$\displaystyle
d(t)=d_{0}-\dfrac{t^{2}}{2}\left\langle\mathrm{MG}\right|[H_{\mathrm{LR}},[H_{\mathrm{LR}},d]]\left|\mathrm{MG}\right\rangle$
(13)

The last expression describes the short time evolution of the order parameter
up to quadratic order and can be further simplified by evaluating the double
commutator, see Fig. 7a, which determines (the square of) the typical decay
rate. The short-time expansion up to quadratic order is also compared to an
exact numerical time evolution in Fig. 7 (b - d).

System size dependence of thermal expectation values.—We evaluate the system
size dependence of the thermal phase diagram of Fig. 2 (b) of the main text.
In the main text, we already mentioned that computations for the finite
temperature expectations of dimer order parameter and energy were performed
based on the concept of typicality. Concretely we generate a number of $N$
random states, which will be evolved in imaginary time to $-\beta/2$. The
entire set of $N$ evolved states can now be used as an approximation of the
density matrix of the system. Here, we evaluate the effective temperature
after the quench for different system sizes ranging from 12 to 20 sites and
different values of the long-range exponent $\alpha\in\\{0.2,0.5,1.0,1.5\\}$.
The results are illustrated in Fig. 8 for a summation over $N=40$ random
states. We find good agreement with the data of Fig. 2 (b) of the main text
for all system sizes $L$.

Numerical investigations of the dynamical Zak protocol.—The protocol depicted
in Fig. 3 (a) and (b) of the main text can be used to measure topological
phases in trapped ion experiments. The connection between this dynamical
protocol and the conventional transformation of the Hamiltonian in Eq. (5) of
the main text is corroborated in this section. To this end, we simulate the
protocol of Fig. 3 (a) and (b) using exact diagonalization methods for systems
of $L=15$ ions including the ancilla qubit.

We start by implementing the initial state of our evolution protocol. We
consider a system with an even number of sites, determine the ground state
$\left|\psi_{0}\right\rangle$ of this system, and append an ancilla spin in
state $\left|\uparrow\right\rangle_{A}$ to its left. The adiabatic ground
state preparation is discussed in the next section. After performing a
$\pi/2-$flip to the ancilla the system is described by the product state
$\left|\psi\right\rangle\equiv(\left|\uparrow\right\rangle_{A}+\left|\downarrow\right\rangle_{A})/\sqrt{2}\otimes\left|\psi_{0}\right\rangle$.
This state can now be evolved according to the dynamical protocol of Fig. 3
(a) and (b).

For sake of simplicity we will at first neglect the discretization of the
original Floquet protocol within the time interval $[0,T_{1}]$ of a single
period in our simulation. Instead we will directly use the effective
description obtained in the high-frequency limit in terms of the long-ranged
Heisenberg model $H_{\mathrm{LR}}$. Below, we will systematically study the
full Floquet evolution. In this first part of the protocol, depicted in Fig. 3
(a), the ancilla qubit gets effectively decoupled from the remaining systems
by $\pi-$pulses frequently applied to it. For the second part of the protocol
within the time interval $[T_{1},T_{1}+T_{2}]$ the decoupling of the ancilla
no longer takes place. More precisely, the ancilla interacts with the
remaining system via a long-ranged Ising Hamiltonian, whose orientation can be
chosen to point along the $z$-direction. Interactions will furthermore exhibit
a characteristic sign change in their prefactors as they cross the critical
bond $i_{c}$. The evolution of the system during the second part of the
protocol is hence described by the Hamiltonian

$\displaystyle H_{\mathrm{Zak}}(\alpha;i_{c})=\sum_{0\leq
i<j}J_{ij}(\alpha;i_{c})S^{z}_{i}S^{z}_{j}$ (14)
$\displaystyle\quad\text{with}\quad
J_{ij}(\alpha;i_{c})=\frac{3\operatorname{{sign}}(i_{c}-i)\operatorname{{sign}}(i_{c}-j)}{|j-i|^{\alpha}}.$
(15)

In this expression we identify the left most spin with the ancilla qubit
$S^{z}_{0}\equiv\tau^{z}_{A}$. The critical bond $i_{c}$ in this notation
takes half-integer values within $[1/2,L-3/2]$. In every period of our Floquet
protocol the evolution governed by $H_{\mathrm{Zak}}$ will induce an
additional phase $\Delta\varphi(\alpha;i_{c})$ to the Hamiltonian. Following
the notation of the main text, the Hamiltonian will transform as

$\displaystyle H_{\mathrm{LR}}(\alpha,\varphi)\longmapsto$
$\displaystyle\begin{cases}H_{\mathrm{LR}}(\alpha,\varphi+\Delta\varphi)&\text{for
$\left|\uparrow\right\rangle_{A}$}\\\
H_{\mathrm{LR}}(\alpha,\varphi-\Delta\varphi)&\text{for
$\left|\downarrow\right\rangle_{A}$}\end{cases}$ (16)
$\displaystyle\text{with}\quad\Delta\varphi(\alpha;i_{c})$
$\displaystyle\equiv\Big{(}J_{0,i_{c}-\frac{1}{2}}(\alpha)-J_{0,i_{c}+\frac{1}{2}}(\alpha)\Big{)}\frac{T_{2}}{2}$
(17) $\displaystyle\equiv\Delta J(\alpha;i_{c})\frac{T_{2}}{2}.$ (18)

Figure 10: Spatial dependence of Zak phases from the simulated protocol. The
phase accumulated at the ancilla $\left\langle\tau^{x}_{A}\right\rangle$ shows
characteristic alternations between odd and even bonds. Imperfections of a
power-law implementation of $H_{\mathrm{Zak}}$ become relevant at bonds
further away from the ancilla qubit.

The factor of $\textstyle{\frac{1}{2}}$ is resulting from the fact that we use
the spin operator of the ancilla $\tau^{z}_{A}$ in $H_{\mathrm{Zak}}$ instead
of the related Pauli matrix.

The absolute phase $\Delta\varphi(\alpha;i_{c})$ acquired by the Hamiltonian
thereby strongly depends on the value of the power-law exponent $\alpha$ as
well as on the chosen critical bond $i_{c}$. Our implementation will describe
an overall closed protocol if the relative phase between both ancilla states
$\left|\uparrow\right\rangle_{A}$ and $\left|\downarrow\right\rangle_{A}$
equals $2\pi$. When we start from the initial experimentally accessible
Hamiltonian $H_{\mathrm{LR}}(\alpha,\varphi=0)$ we will find after $n$ periods
Hamiltonians $H_{\mathrm{LR}}(\alpha,\varphi=\pm n\Delta\varphi)$ associated
to the different $z-$basis states of the ancilla. This determines the number
of Floquet periods $N$ needed for measuring the Zak phase as a function of
$T_{2},\alpha$ and $i_{c}$

$N=\frac{2\pi}{\Delta J(\alpha;i_{c})T_{2}}.$ (19)

For large values of $\alpha$ and critical bond $i_{c}$, i.e. far away from the
ancilla, $\Delta J(\alpha;i_{c})$ can be very small, which requires either
large application times $T_{2}$ or a high number of Floquet periods $N$ to
successfully describe a closed protocol. We, however, find numerically that
the number of Floquet periods needed for the considered systems stays
reasonable small, as discussed below.

Figure 11: Zak phase measurements for Floquet protocol. (a) Change of the
ancilla magnetization $\left\langle\tau^{x}_{A}\right\rangle$ during the
evolution with a discretized version of the dynamical Zak protocol for
$\alpha=0.2$ and application times for the sequence of Ising evolutions of
$T_{1}=0.9$. The discretized protocol shows good agreement with the continuous
evolution using $H_{\mathrm{LR}}$ (red dashed lines) for $T_{2}\lesssim 0.2$.
(b) The results for interferometric measurements at $\varphi=\pi$ show the
expected alternating behavior for different choices of $i_{c}$. For large
$T_{2}$ associated to a non-adiabatic implementation the protocol does not
allow to reach
$\left\langle\tau^{z}_{A}\right\rangle=-\textstyle{\frac{1}{2}}$ at odd bonds.
However, the alternating nature of the pattern can be resolved even away from
the adiabatic limit $T_{2}\ll T_{1}$.

Before investigating detailed results of numerical simulations, we want to
address another aspect important for measurements of topological phases at the
ancilla qubit. Besides the desired effect of transferring the topological Zak
phase to the ancilla spin, we also need to study the consequences of our
protocol of Fig. 3 (a) on the physical spin system. To this end, we interpret
the evolution governed by $H_{\mathrm{Zak}}$ as a rotation for each spin with
specific angle $\delta\varphi$ around the $z$-direction,
$R_{\delta\varphi}\equiv\exp(-\mathrm{i}T_{2}H_{\mathrm{Zak}})$. To simplify
our argument, we will for the moment consider a coupling in the form of step
function ($\alpha\to 0$ case). Spins located left/right to the critical bond
$i_{c}$ will thereby acquire rotations with opposite chirality. Acting with
these rotations on our Hamiltonian, i.e., computing
$R_{\delta\varphi}^{\dagger}H_{\mathrm{LR}}(\alpha,\varphi)R_{\delta\varphi}^{\phantom{\dagger}}$,
exactly reproduces the transformation of Eq. (16). The rotation
$R_{\delta\varphi}$ hence allows us to describe the evolution with all
Hamiltonians within the family
$\\{H_{\mathrm{LR}}(\alpha,\varphi_{n})\\}_{n}$, although not directly
accessible in experiments. In total, we find for the evolution in our closed
protocol of Fig. 3 (b),

$\displaystyle\prod_{n=1}^{N}\Big{(}\mathrm{e}^{-\mathrm{i}T_{1}H_{\mathrm{LR}}(\alpha,\varphi_{n})}\Big{)}$
$\displaystyle=\prod_{n=1}^{N}\Big{(}\big{(}R^{\dagger}_{\delta\varphi}\big{)}^{n}\mathrm{e}^{-\mathrm{i}T_{1}H_{\mathrm{LR}}(\alpha,0)}\big{(}R^{\phantom{\dagger}}_{\delta\varphi}\big{)}^{n}\Big{)}$
$\displaystyle=\big{(}R^{\dagger}_{\delta\varphi}\big{)}^{N}\prod_{n=1}^{N}\Big{(}\mathrm{e}^{-\mathrm{i}T_{1}H_{\mathrm{LR}}(\alpha,0)}R^{\phantom{\dagger}}_{\delta\varphi}\Big{)},$
(20)

where $\\{\varphi_{n}\\}_{n}$ denotes a suitable discretization of the
interval $[0,2\pi]$. The second factor of the last equality is precisely the
evolution described by steps (ii) and (iii) in the protocol of Fig. 3 (b) in
the main text. The first factor, moreover, indicates an inverse rotation
around an angle $N\delta\varphi$, which also has to be applied in our protocol
before measuring. This inverse rotation takes care of an additional unwanted
phase acquired during the protocol evolution in the physical spin system.
Neglecting this contribution would result in interferometric measurements not
only detecting the desired topological phase at the ancilla, but also an
unknown phase acquired in the system. The required inverse rotation of Eq.
(Characterizing Topological Excitations of a Long-Range Heisenberg Model with
Trapped Ions) is obtained by applying $-H_{\mathrm{Zak}}$ for the same
duration as the overall application time of $H_{\mathrm{Zak}}$ during the
protocol. The Hamiltonian $-H_{\mathrm{Zak}}$ can thereby be generated
analogously to its positive counterpart $H_{\mathrm{Zak}}$. Specifically,
after the last Floquet period of the protocol, a single global $\pi$-pulse to
the left part of the system (in contrast $H_{\mathrm{Zak}}$ was generated by a
$\pi$-pulse acting on the right part) implements $-H_{\mathrm{Zak}}$ and
subsequent time evolution finally allows to unwind the unwanted phase
accumulation of the physical spin system. This subtlety is taken into account
in step (iv) of the protocol of Fig. 3 (b). After this procedure the Zak phase
of the ancilla can be measured in the conventional way on the ancilla qubit.

With the basics of our simulations set, let us have a look at concrete
results. In Fig. 9 we provide results for dynamical evolution of the initial
state $\left|\psi\right\rangle$ for different choices of $T_{2}/T_{1}$ and the
critical bond $i_{c}$ in a system with power-law scaling $\alpha=0.2$. We show
the magnetization of the ancilla qubit in $x-$basis
$\left\langle\tau^{x}_{A}\right\rangle$ against the absolute phase $\varphi$
acquired by the Hamiltonian $H_{\mathrm{LR}}(\alpha,\varphi)$. The
interferometric measurement to extract the value of the Zak phase should be
performed at $\varphi=\pi$ (orange dashed lines), where both absolute phases
add up to a relative phase difference of $2\pi$. We can clearly identify odd
bonds (top row: bond 1, bond 3, …) and even bonds (bottom row: bond 2, bond 4,
…) in Fig. 9. Whereas odd bonds approach the value
$\left\langle\tau^{x}_{A}\right\rangle=-\textstyle{\frac{1}{2}}$ for
$\varphi\to\pi$, even bonds will almost return to the starting value
$\left\langle\tau^{x}_{A}\right\rangle=+\textstyle{\frac{1}{2}}$. This refers
to values of the Zak phase of $\gamma_{\mathrm{Zak}}=\pi$ and
$\gamma_{\mathrm{Zak}}=0$, respectively, considering an ancilla state of

$\left|\psi(\varphi=\pi)\right\rangle_{A}=\frac{1}{\sqrt{2}}\big{(}\left|\uparrow\right\rangle+\mathrm{e}^{\mathrm{i}\gamma_{\mathrm{Zak}}}\left|\downarrow\right\rangle\big{)}$
(21)

at absolute implemented phase of $\varphi=\pi$. To investigate the impact of a
power-law decay in the couplings of $H_{\mathrm{Zak}}$ we furthermore show in
Fig. 9 results obtained using a perfect step function ($\alpha\to 0$) in
$H_{\mathrm{Zak}}$ (red dashed lines). We find that deviations between power-
law and step function are most pronounced for even bonds: Whereas the step
function implementation returns to the starting value at absolute phases of
$\varphi=\pi$, the power-law results yield values of
$\left\langle\tau^{x}_{A}\right\rangle<\frac{1}{2}$. This imperfection is,
however, expected as a power-law entails small but finite values for the
implemented phases also away from the critical bond $i_{c}$. As a consequence
our protocol will apart from the chosen critical bond $i_{c}$ also effect
couplings across other bonds; most dominantly the first one. These phases
implemented at other bonds will in general prohibit recurrence to the initial
configuration at $\varphi=\pi$. This also applies in similar form for a phase
accumulation of $\varphi=2\pi$, shown in Fig. 9. At $\varphi=2\pi$, where we
expect our protocol to reproduce the initial state independent of the choice
for $i_{c}$, only the step function implementation yields fully accurate
results. Despite the power-law decay of interactions causing some deviations
from the ideal behavior, we find that the protocol of Fig. 3 (a) and (b)
allows clear discrimination of even and odd bonds, respectively. This becomes
clear when extracting the values of $\left\langle\tau^{z}_{A}\right\rangle$
interferometrically measured at the ancilla for $\varphi=\pi$ (orange dashed
lines) as a function of the chosen bonds $i_{c}$, see Fig. 10.

Now, we can repeat the analysis for the evolution under the discretized
Floquet protocol of consecutive rotated long-ranged Ising Hamiltonians. We
proceed in exactly the same manner as before and expect the results to be
qualitatively equivalent if we consider the adiabatic limit $T_{2}\ll T_{1}$
with the drive frequency of the Floquet protocol $\propto 1/T_{1}$ still being
sufficiently fast. We show the results for the discretized version of the
dynamical protocol in Fig. 11. Indeed we find that for example for a choice of
$(\alpha,T_{1})=(0.2,0.9)$ values of $T_{2}\lesssim 0.2$ are already
sufficient to obtain the correct behavior we already encountered for evolution
governed by the high-frequency Hamiltonian $H_{\mathrm{LR}}$ (red dashed
lines). This is illustrated in Fig. 11 (a) for evolution associated to $i_{c}$
located at the first two bonds in the system (continuous data obtained for
$T_{1}=0.9$ and $T_{2}=0.1$). Extracting the results for the interferometric
measurement for all bonds indicates that also larger $T_{2}\approx 0.4$
already yield the characteristic alternating pattern of the Zak phase in the
topological phase, as illustrated in Fig. 11 (b).

Given all this input, we can estimate the number of Floquet periods needed to
obtain the results of Fig. 9, Fig. 10 and Fig. 11. According to Eq. (19) the
number of Floquet periods $N$ strongly depends on $\alpha$ and the chosen
$i_{c}$. For our simulations this requires for example $N\leq 9$ Floquet
periods to reach the desired absolute phase of $\varphi=\pi$ for
$(\alpha,T_{2})=(0.2,0.2)$ and $(i_{c}-\textstyle{\frac{1}{2}})\leq 13$. For
larger values of $\alpha$ (e.g. $\alpha=0.8$), the required number of periods
can vary more drastically with $i_{c}$ ($N=7$ for $i_{c}=1$ or $N=40$ for
$i_{c}=12$). Overall, the requirements are, however, not too severe
highlighting the experimental relevance of the proposed protocol.

Figure 12: Adiabatic ground state preparation. (a) We show the $L-$th root of
the absolute overlap between an in $M-$steps adiabatically prepared state
$\left|\phi_{M}\right\rangle$ and the ground state
$\left|\psi_{0}\right\rangle$ of $H_{\mathrm{LR}}(\alpha)$ for different
values of the power-law scaling $\alpha$ and application times of
$T_{0}\in\\{2.0,6.0\\}$ for each evolution step. We find that already a
comparable small number of evolution steps $M$ is sufficient to reach large
overlap with the actual ground state for small $\alpha\lesssim 0.8$. (b) Using
the adiabatically prepared states as initial states for our dynamical protocol
we find qualitatively similar results as for the previous simulations.

Adiabatic ground state preparation.— In the previous section we assumed that
the system can initially be prepared in its ground state
$\left|\psi_{0}\right\rangle$. This can for example be achieved by adiabatic
evolution. To this end, we start out with a fully dimerized state
$\left|\mathrm{MG}\right\rangle$. To approximate the ground state of
$H_{\mathrm{LR}}(\alpha)$ we sequentially evolve
$\left|\mathrm{MG}\right\rangle$ with $H_{\mathrm{LR}}(\alpha_{m})$ for a
given time interval $T_{0}$. The values of $\alpha_{m}$ thereby approach the
final value of $\alpha$ in $M$ steps, i.e. $\alpha_{m}\equiv m\;\delta\alpha$
for $m\in\\{1,2,,\dots,M\\}$ with $\delta\alpha=\textstyle{\frac{\alpha}{M}}$.
If the resulting state $\left|\phi_{M}\right\rangle$ is sufficiently close to
the ground state of the system $\left|\psi_{0}\right\rangle$ it should allow
for measurements of Zak phases with results similar to the ones displayed in
Fig. 10 and 11.

We show the $L-$th root of the absolute overlap
$|\left\langle\psi_{0}|\phi_{M}\right\rangle|^{1/L}$ between the adiabatically
prepared state and the true ground state in Fig. 12 (a) for various choices of
$\alpha$ as a function of the number of evolution steps $M$. During the
evolution every Hamiltonian $H_{\mathrm{LR}}(\alpha_{m})$ is applied for a
time interval of $T_{0}=2.0$ respectively $T_{0}=6.0$. We notice that already
a comparable small number of evolution steps $M$ provides a good approximation
of the actual ground state for small values of $\alpha\lesssim 0.8$. We
moreover repeat the simulations for the interferometric protocol starting from
these states. Our results, shown in Fig. 12 (b), reveal that adiabatically
prepared initial states yield similarly to previous results an alternating
pattern in the interferometrically measured magnetization
$\left\langle\tau^{z}_{A}\right\rangle$ at the ancilla, as expected from the
good fidelity of the adiabatically prepared ground state.

Combining all insight we gained within the last paragraphs we can estimate the
experimental requirements for interferometric measurements of topological Zak
phases. Once more concrete statements highly depend on the chosen parameters.
We will in the following hence consider exemplary choices reasonable close to
experimental possibilities. Recalling the $T_{2}-$dependence of Eq. (19) we
want to choose the application time of $H_{\mathrm{Zak}}$ as large as
possible, while still remaining close enough to both the adiabatic and the
high-frequency limit. Our simulations of the discretized protocol indicate
that this criterion is e.g. satisfied for $(T_{1},T_{2})=(1.2,0.4)$. Assuming
initial ground state preparation within $M=8$ evolution steps we expect
$\gamma_{\mathrm{Zak}}$ to be measurable with $50$ global and $32$ local
pulses at the first two bonds of a system characterized by a power-law
exponent $\alpha=0.8$. This estimate includes $1$ local and $3M$ global pulses
for preparation of the initial state, $6N$ local respectively $5N$ global
pulses for evolution as well as $1$ additional global and local pulse for
unrotating the system and performing the interferometric measurement.

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