# Many topological regions on the Bloch sphere of the spin-1/2 double kicked
top

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

###### Abstract

Floquet topological systems have been shown to exhibit features not commonly
found in conventional topological systems such as topological phases
characterized by arbitrarily large winding numbers. This is clearly
highlighted in the quantum double kicked rotor coupled to spin-1/2 degrees of
freedom [Phys. Rev. A 97, 063603 (2018)] where large winding numbers are
achieved by tuning the kick strengths. Here, we extend the results to the
spin-1/2 quantum double kicked top and find not only does the system exhibit
topological regions with large winding numbers, but a large number of them are
needed to fully characterize the topology of the Bloch sphere of the top for
general kick strengths. Due to the geometry of the Bloch sphere it is
partitioned into regions with different topology and the boundaries separating
them are home to $0$ and $\pi$ quasienergy bound states. We characterize the
regions by comparing local versions of the mean field, quantum and mean chiral
displacement winding numbers. We also use a probe state to locate the
boundaries by observing localization as the state evolves when it has a large
initial overlap with bound states. Finally, we briefly discuss the connections
between the spin-1/2 quantum double kicked top and multi-step quantum walks,
putting the system in the context of some current experiments in the
exploration of topological phases.

##  I Introduction

Periodically driven systems, also called Floquet systems, have been a useful
tool in simulating novel phases of matter. Periodic driving allows for control
in the time domain which can lead to exotic phases such as Anderson
localization in time [1, 2] and phase space crystals [3, 4]. Of particular
interest is the creation of effective magnetic fields and spin-orbit couplings
with lasers to simulate topological features found in condensed matter systems
[5, 6, 7, 8]. A notable example along these lines are Floquet topological
insulators which are laser-induced topological states in normal materials
resulting in the creation and control of chiral edge states [9]. In some
cases, the crystal structure of solids is also simulated using optical
lattices [10, 11, 13, 12]. In these systems, the lasers creating the lattice
are periodically driven to induce effective magnetic field strengths
unobtainable in real materials. This has lead to the observation of the
celebrated Harper-Hofstadter model [14, 15] which displays one of the more
striking examples of integer quantum Hall physics.

A special class of Floquet systems involves periodic kicks rather than
continuous driving. Their appeal comes from the fact that only part of the
Hamiltonian is responsible for evolving the system at a given time, so they
are generally easier to conceptualize than their non-kicked counterparts.
Early applications of this method were used to explore connections between
topology and chaos in the kicked Harper model [16]. In recent years, one of
the main focuses has been in controlling topological phases [17], particularly
in generating large topological invariants such as the winding number which
counts the number of symmetry protected states in the system [18, 19]. Double
kicked systems such as the quantum double kicked rotor (QDKR) and the quantum
double kicked top (QDKT) have shown to be promising in this regard [20, 21,
22] where large topological invariants were predicted.

Proposals involving the QDKR coupled to spin-1/2 degrees of freedom have shown
that arbitrarily large winding numbers can be achieved [23, 24]. The
additional spin-1/2 degrees of freedom are significant because they put the
QDKR in the perspective of quantum walks where it plays the role of a coin
which is ’tossed’ each step. The state of the spin determines the direction
the rotor evolves in and is analogous to tossing a coin at regular intervals
and choosing a direction based on whether it is heads or tails in classical
random walks [25]. Of course, the difference in the quantum case is that the
coin can be in a superposition of heads and tails. Quantum walks play an
important part in the building of efficient quantum algorithms [26, 27] and
provide a foundation for quantum computation [28]. Quantum walks have also
been proposed as a method to explore topological phases [29, 30, 31] where
experiments involved the measurement of the response of topological invariants
to disorder [32, 33] and the identification of topologically protected bound
states [34, 35, 36].

In this paper, we perform the natural step of extending the results of the
spin-1/2 QDKR to the spin-1/2 QDKT. This is a generalization which amounts to
extending the Hilbert space of the system from a ring to a sphere [37] \-
commonly referred to as the Bloch sphere. Our main result is that, due to the
change in geometry of the Hilbert space, instead of a single topological
region on the ring, an arbitrarily large number of different topological
regions can be generated on the Bloch sphere depending on the kick strengths.
In general, many winding numbers are needed to characterize the topology of
the entire Bloch sphere and due to the bulk-boundary correspondence, the
boundaries between the regions are home to protected bound states. The number
of bound states at a boundary can also be arbitrarily large, so there can be
regions around the boundaries which are especially dynamically stable.

We also show that the spin-1/2 QDKT dynamics can be framed in terms of a
quantum walk on the Bloch sphere by breaking down the Floquet operator into a
series of coin tosses and coin-dependent rotations. This allows us to make
proper connections between the spin-1/2 QDKT and traditional quantum walk
systems as well as highlight the key difference between them which is the
source of the topological boundaries. For instance, quantum walks are often
implemented with a particle in an optical lattice which has discrete
translational symmetry and the boundary between two different topological
regions is inserted artificially as a local breaking of that symmetry [34, 30,
38]. However, the Bloch sphere, does not have translational symmetry due to
its geometry and we show that it is this inhomogeneity that is responsible for
the topological boundaries. Therefore, the topological boundaries of the
spin-1/2 QDKT can be considered to occur naturally due to the geometry of the
Bloch sphere.

We do not choose a specific model for the top, however, some possibilities for
its physical origin are the angular momentum of a single particle or Fock
states of a collection of two-mode indistinguishable particles [39]. In the
many-body case, the two modes can be external states like the energy levels of
a trapping potential such as the system of a Bose-Einstein condensate (BEC)
occupying the ground states of a double well potential. The two modes can also
be internal states like the hyperfine states of atoms [40, 41] or two
polarizations of light [42]. In either case, the spin-1/2 degrees of freedom
can represent a two-mode particle that is distinguishable from the others and
is either a different species of particle with access to the same two modes
[43, 44] or the same species with a different pair of modes. When the top
operators belong to the angular momentum of a single particle, the spin-1/2
degrees of freedom can represent the internal spin states of that particle.

##  II Model

The system we will be investigating is the spin-1/2 QDKT which is described by
the Hamiltonian

$\displaystyle\hat{H}_{T}$ $\displaystyle=$
$\displaystyle\frac{\Lambda}{j}\hat{J}_{z}^{2}+\alpha_{1}\hat{J}_{x}\hat{\sigma}_{x}\sum_{n}\delta\left[t-nT\right]$
(1)
$\displaystyle+\alpha_{2}\hat{J}_{y}\hat{\sigma}_{y}\sum_{n}\delta\left[t-(n+1/2)T\right]$

where $\hat{J}_{a}$, $a=x,y,z$ are the top operators obeying the usual
commutation relation $[\hat{J}_{i},\hat{J}_{j}]=i\epsilon_{ijk}\hat{J}_{k}$
and the Pauli matrices represent the spin-1/2 degrees of freedom which we
label as $\uparrow$ and $\downarrow$. The parameter $\Lambda$ is the nonlinear
energy, and $\alpha_{1}$ and $\alpha_{2}$ are the first and second kick
strengths, respectively. Each period $T$ there are two kicks where the second
kick is delayed by a time $T/2$.

The dynamics generated after one period is given by the Floquet operator

$\displaystyle\hat{U}_{T}=e^{-i\Lambda\hat{J}_{z}^{2}T/2j}e^{-i\alpha_{2}\hat{J}_{y}\hat{\sigma}_{y}}e^{-i\Lambda\hat{J}_{z}^{2}T/2j}e^{-i\alpha_{1}\hat{J}_{x}\hat{\sigma}_{x}}\,.$

We analyze a simplified version of $\hat{U}_{T}$ by setting the period to
$T=4\pi j/\Lambda$. This is similar to the on-resonance condition found in
variations of the quantum kicked rotor [45, 46, 47] and results in the unitary
operators containing $\hat{J}_{z}^{2}$ becoming unity since the eigenvalues of
$\hat{J}_{z}$, $m$, are integers in the range $-j\leq m\leq j$. The Floquet
operator becomes

$\hat{U}_{T}=e^{-i\frac{\kappa_{2}}{j}\hat{J}_{y}\hat{\sigma}_{y}}e^{-i\frac{\kappa_{1}}{j}\hat{J}_{x}\hat{\sigma}_{x}}$
(3)

where $\kappa_{1}=\alpha_{1}j$ and $\kappa_{2}=\alpha_{2}j$ are the scaled
kick strengths.

If the interactions between the top and the spin-1/2 degrees of freedom in Eq.
(3) are difficult to generate, then a possible solution is to use interactions
of the general form $\hat{H}_{\mathrm{int}}=\hat{J}_{z}\hat{S}_{z}$, where in
our case $\hat{S}_{z}=\hat{\sigma}_{z}$. They can arise in squeezing
experiments involving interactions between matter [48] or between matter and
light [42]. Also, similar interactions are found in a Bose-Fermi mixture in an
optical lattice [49]. Using $\hat{H}_{\mathrm{int}}$ as a starting point, we
can construct $\hat{U}_{T}$ from the rotation operators
$\hat{R}_{x}(a)=e^{-ia\hat{J}_{x}}$, $\hat{R}_{y}(a)=e^{-ia\hat{J}_{y}}$ and
$\hat{R}_{z}(a)=e^{-ia\hat{J}_{z}}$ and the general Pauli rotation operator

$\hat{M}(\alpha,\beta)=\begin{pmatrix}\cos(\alpha/2)&\sin(\alpha/2)e^{-i\beta}\\\
-\sin(\alpha/2)e^{i\beta}&\cos(\alpha/2)\end{pmatrix}\,.$ (4)

The $x$ and $z$ rotation operators can be thought of as a phase accumulation
when only tunneling and only an imbalance between the two modes is switched
on, respectively, and can be used to create the $y$ rotations since
$\hat{R}_{y}(a)=\hat{R}_{z}(\pi/2)\hat{R}_{x}(a)\hat{R}_{z}(-\pi/2)$. The
explicit breakdown of the unitary operators in $\hat{U}_{T}$ is

$\displaystyle e^{-i\kappa_{1}\hat{J}_{x}\hat{\sigma}_{x}/j}$ $\displaystyle=$
$\displaystyle\hat{M}(-\pi/2,0)\hat{R}_{y}(\pi/2)e^{-i\kappa_{1}\hat{J}_{z}\hat{\sigma}_{z}/j}$
$\displaystyle\hskip 20.0pt\times\hat{R}_{y}(-\pi/2)\hat{M}(\pi/2,0)$
$\displaystyle e^{-i\kappa_{2}\hat{J}_{y}\hat{\sigma}_{y}/j}$ $\displaystyle=$
$\displaystyle\hat{M}(-\pi/2,\pi/2)\hat{R}_{x}(-\pi/2)e^{-i\kappa_{2}\hat{J}_{z}\hat{\sigma}_{z}/j}$
(5) $\displaystyle\hskip
20.0pt\times\hat{R}_{x}(\pi/2)\hat{M}(\pi/2,\pi/2)\,.$

A recent proposal of a quantum walk on the Bloch sphere [50] used the Floquet
operator

$\hat{U}_{W}=e^{-i2\kappa\hat{J}_{z}\hat{\sigma}_{z}}\hat{M}(\alpha,\beta)$
(6)

to evolve the system. In this context, the Pauli operators represent a quantum
coin which is tossed at each step via $\hat{M}(\alpha,\beta)$, then a rotation
about the $J_{z}$ axis is performed whose direction depends on the state of
the coin. Therefore, $\hat{U}_{T}$ can be thought of as the Floquet operator
for a quantum walk on the Bloch sphere involving four coin tosses and two
rotations, one about the $J_{y}$ axis and one about the $J_{x}$ axis, at each
step.

Going forward we will discuss topological regions that emerge in the space of
$\hat{J}_{z}$ eigenstates, $\\{|m\rangle\\}$. To distinguish between the
different topological regions we use terms which are commonly found in
condensed matter physics such as ’bound’ and ’bulk’ to describe the states
located near and away from the boundaries separating the regions,
respectively. We also use the term ’edge’ to refer to the minimum and maximum
angular momentum states of the top, $|m=\pm j\rangle$. Therefore, it is useful
to consider $m$ as the spatial coordinate of a fictitious 1D lattice. An
important property of this lattice is that it does not possess discrete
translational symmetry which can be seen from the raising and lowering
operators of the top

$\hat{J}_{\pm}|m\rangle=\sqrt{(j\mp m)(j\pm m+1)}|m\pm 1\rangle\,.$ (7)

The inhomogeneity comes from the Hilbert space of the top which is a Bloch
sphere of radius $R=\sqrt{j(j+1)}$. In order to see this explicitly one can
imagine a system consisting of a single particle in a 1D lattice, with nearest
neighbor hopping, that has been stretched on a sphere from the north pole to
the south pole. If the tunneling energy is inversely proportional to the
distance between sites the Hamiltonian is

$\hat{H}_{\mathrm{sph}}=-\alpha
j\sum_{m}\frac{1}{d_{m,m+1}}\left(\hat{a}_{m+1}^{\dagger}\hat{a}_{m}+\mathrm{h.c.}\right).$
(8)

Due to the curved surface of the sphere, the distance between adjacent sites
is simply the arclength connecting them $d_{m,m+1}=R|\Delta\theta_{m,m+1}|$
where $\Delta\theta_{m,m+1}=\theta_{m}-\theta_{m+1}$ is the difference between
the polar angle coordinates of the two sites. To make a connection to the
$\hat{J}_{z}$ states of the top, we require that the site label $m$ also be
the polar axis coordinate and that it takes unit increments in the range
$-j\leq m\leq j$. This gives the relation $m=j\cos\theta$ or
$\theta_{m}=\arccos(m/j)$. Finally, we assume a large system size ($j\gg 1$),
so that $R\approx j$ and take the leading order term in a $1/j$ expansion of
the distance $d_{m,m+1}\approx\left[1-(m/j)^{2}\right]^{-1/2}$ which gives

$\hat{H}_{\mathrm{sph}}\approx-\alpha\sum_{m}\sqrt{j^{2}-m^{2}}\left(\hat{a}_{m+1}^{\dagger}\hat{a}_{m}+\mathrm{h.c.}\right)\,.$
(9)

Although Eq. (9) is an approximation, the square root factor is the mean field
version of the ones in Eq. (7), so their difference relative to $j$ vanishes
as $j\to\infty$. Here, we see $\hat{J}_{+}$ has a similar to the single
particle tunneling terms
$\sum_{m}\sqrt{j^{2}-m^{2}}\hat{a}_{m+1}^{\dagger}\hat{a}_{m}$. This quick
analysis is not meant to discuss how $\hat{H}_{\mathrm{sph}}$ can be
implemented, but rather highlight two main points: (1) the eigenvalues of
$\hat{J}_{z}$ are similar to the site label of a 1D lattice stretched over the
semicircle connecting the two poles of a sphere and (2) the square root
factors come from the curvature of the sphere and therefore have a geometric
origin. We stress the second point because, as we will show, it is the square
root factors that are responsible for the breakdown of the state space of the
spin-1/2 QDKT into regions of different topology and therefore the boundaries
between these regions are of geometric origin.

##  III Results

###  III.1 Quasienergy spectrum

When dealing with time periodic systems it is convenient to use Floquet theory
which allows one to write the dynamics over one period in terms of a time
independent effective Hamiltonian

$\hat{U}_{T}=e^{-i\hat{H}_{\mathrm{eff}}T}.$ (10)

The set of eigenvalues of the Floquet operator are $\\{\lambda_{i}\\}$ and
they can be used to calculate the eigenvalues of the effective Hamiltonian
$\\{\varepsilon_{i}\\}=\\{\frac{\mathrm{i}}{T}\mathrm{log}\lambda_{i}\\}$,
however, they are only unique within a range of $2\pi$, so they are referred
to as quasienergies. Going forward we set $T=1$ without loss of generality.
Before we discuss the quasienergy spectrum we will briefly go over some
subtleties in quantifying the topology of Floquet systems.

In static systems, topological phases are characterized by integers such as
the winding number. Through the bulk-boundary correspondence, they count the
number of protected bound states at the boundary of the bulk. In periodically
driven systems it has been shown that, in addition to the usual
$\varepsilon=0$ bound states, there are also $\varepsilon=\pi$ bound states
which come from the fact that the quasienergies are calculated from a unitary
operator and not a Hamiltonian. These states are protected [51] and cannot be
deformed into each other without an energy gap closing, or breaking of some
symmetry, so two winding numbers are required to characterize each phase. Care
must be taken in calculating these numbers for Floquet systems, however, and
it has been shown that the winding numbers of two chiral symmetrized
timeframes which are $w_{1}$ and $w_{2}$, can be used to calculate the winding
numbers that count the number of $\varepsilon=0$ and $\varepsilon=\pi$ bound
states from the relation [52]

$w_{0}=\frac{w_{1}+w_{2}}{2}\hskip 30.0ptw_{\pi}=\frac{w_{1}-w_{2}}{2}\,.$
(11)

In the spin-1/2 QDKT the Floquet operators in the chiral symmetrized
timeframes take the form

$\displaystyle\hat{U}_{T,1}$ $\displaystyle=$ $\displaystyle
e^{-i\frac{\kappa_{1}}{2j}\hat{J}_{x}\hat{\sigma}_{x}}e^{-i\frac{\kappa_{2}}{j}\hat{J}_{y}\hat{\sigma}_{y}}e^{-i\frac{\kappa_{1}}{2j}\hat{J}_{x}\hat{\sigma}_{x}}$
(12) $\displaystyle\hat{U}_{T,2}$ $\displaystyle=$ $\displaystyle
e^{-i\frac{\kappa_{2}}{2j}\hat{J}_{y}\hat{\sigma}_{y}}e^{-i\frac{\kappa_{1}}{j}\hat{J}_{x}\hat{\sigma}_{x}}e^{-i\frac{\kappa_{2}}{2j}\hat{J}_{y}\hat{\sigma}_{y}}\,.$
(13)

The chiral symmetry that Eqns. (12) and (13) possess is defined in terms of
the relations $\hat{\Gamma}\hat{U}_{T,1}\hat{\Gamma}=\hat{U}_{T,1}^{\dagger}$
and $\hat{\Gamma}\hat{U}_{T,2}\hat{\Gamma}=\hat{U}_{T,2}^{\dagger}$ for the
operator $\hat{\Gamma}=\hat{\sigma}_{z}$. This means that for any state with
quasienergy $\varepsilon$ there is a partner state with quasienergy
$-\varepsilon$ through the relation
$\hat{\sigma}_{z}|\varepsilon\rangle=|-\varepsilon\rangle$. The Floquet
operators in Eqns. (3), (12) and (13) are separated by unitary
transformations, so they have the same spectrum. Figure 1 shows the spectrum
for a fixed value of $\kappa_{2}=0.5\pi$ and variable $\kappa_{1}$ for $j=50$.
Increasing $\kappa_{1}$ creates new $\varepsilon=0$ and $\varepsilon=\pi$
states at even and odd integer multiples of $\pi$, respectively. These states
are the bound states mentioned earlier and are protected in the sense that one
can perturb the Floquet operators with a general term in the exponentials that
respects the chiral symmetry (terms proportional to $\hat{\sigma}_{x}$ or
$\hat{\sigma}_{y}$) without changing the number of bound states.

We note that the bound states do not have quasienergies $\varepsilon=0$ and
$\varepsilon=\pi$ exactly due to finite size effects. Evidence of this can be
seen around $\kappa_{1}=4\pi$ where there is a slight wiggle in the energy
around $\varepsilon=0$. Nevertheless, for the range of values and system size
shown, the bound states are quite stable. Additionally, we note that the form
of the spectrum depends on the value of $\kappa_{2}$ and for general values
the number of bound states does not always increase monotonically as
$\kappa_{1}$ increases. This is highlighted in the phase diagram of the
spin-1/2 QDKR [23] which quickly becomes complicated as $\kappa_{1}$ and
$\kappa_{2}$ increase. With the intent of keeping our analysis simple, we set
$\kappa_{2}=0.5\pi$ for the remainder of the paper.

###  III.2 Locations of bound states

Figure 1: Quasienergies as a function of kick strength. Eigenvalues of
$\hat{U}_{T}$ in Eq. (3) as a function of $\kappa_{1}$ for a fixed value of
$\kappa_{2}=0.5\pi$ and $j=50$. When $\kappa_{1}/\pi$ is even or odd a new
pair of $\varepsilon=0$ or $\varepsilon=\pi$ states are formed, respectively.

To gain a better insight into the bound states as well as the topology of the
system it would be useful to calculate $\hat{H}_{\mathrm{eff}}$ for the chiral
symmetrized Floquet operators in Eqns. (12) and (13), however, their forms are
not obvious. Some progress can be made by performing a mean field
approximation of the top operators by transforming them into their coherent
state expectation values

$\langle\hat{\bm{J}}\rangle=\langle(\hat{J}_{x},\hat{J}_{y},\hat{J}_{z})\rangle=j(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\,.$
(14)

The angles $\theta$ and $\phi$ are the polar and azimuthal angles,
respectively, of the Bloch sphere of the top with radius $R=\sqrt{j(j+1)}$.
Instead of the polar angle, we find it useful to use the eigenvalue label of
$\hat{J}_{z}$, $m$, as a variable through the previously mentioned relation
$m=j\cos\theta$. Defining the new parameters

$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\kappa_{1}\sqrt{1-(m/j)^{2}}\cos\phi$ $\displaystyle K_{2}$
$\displaystyle=$ $\displaystyle\kappa_{2}\sqrt{1-(m/j)^{2}}\sin\phi\,,$ (15)

the mean field versions of Eqns. (12) and (13) become

$\displaystyle\hat{U}_{T,1}^{\mathrm{MF}}$ $\displaystyle=$ $\displaystyle
e^{-i\frac{K_{1}}{2}\hat{\sigma}_{x}}e^{-iK_{2}\hat{\sigma}_{y}}e^{-i\frac{K_{1}}{2}\hat{\sigma}_{x}}$
(16) $\displaystyle\hat{U}_{T,2}^{\mathrm{MF}}$ $\displaystyle=$
$\displaystyle
e^{-i\frac{K_{2}}{2}\hat{\sigma}_{y}}e^{-iK_{1}\hat{\sigma}_{x}}e^{-i\frac{K_{2}}{2}\hat{\sigma}_{y}}\,.$
(17)

In this form, the effective Hamiltonians can be calculated exactly giving
$\hat{H}_{\mathrm{eff},i}^{\mathrm{MF}}=\varepsilon\bm{n_{i}}\cdot\hat{\bm{\sigma}}$
(Appendix A), where $i=1,2$ for the two symmetrized timeframes. Both the
quasienergy $\varepsilon$ and the vector $\bm{n_{i}}=(n_{ix},n_{iy})$, which
has unit length, depend on the state of the top. This means $\bm{n_{i}}$ maps
points on the Bloch sphere of the top to points on the equator of the spin-1/2
Bloch sphere.

The quasienergy is the same for both timeframes in Eqns. (16) and (17) and is

$\varepsilon=\arccos\left[\cos(K_{1})\cos(K_{2})\right]\,.$ (18)

Topological transitions in Floquet systems are marked by bound states which
close the quasienergy gaps at $\varepsilon=0$ and $\varepsilon=\pi$. Looking
at Eq. (18), we see that these energies occur when $\cos(K_{1})\cos(K_{2})=\pm
1$ which in turn occurs when $\kappa_{1}\sqrt{1-(m/j)^{2}}\cos\phi=\mu\pi$ and
$\kappa_{2}\sqrt{1-(m/j)^{2}}\sin\phi=\nu\pi$, where $\mu$ and $\nu$ are
integers. Combining these two results gives

$\frac{\pi^{2}}{1-(m/j)^{2}}\left[\frac{\mu^{2}}{\kappa_{1}^{2}}+\frac{\nu^{2}}{\kappa_{2}^{2}}\right]=1\,.$
(19)

which we use to find the mean field values of $m$ that correspond to the
quasienergies $\varepsilon=0$ and $\varepsilon=\pi$

$m_{\mu,\nu}=\pm
j\sqrt{1-\pi^{2}\left(\frac{\mu^{2}}{\kappa_{1}^{2}}+\frac{\nu^{2}}{\kappa_{2}^{2}}\right)}\,.$
(20)

Equation (20) is the mean field prediction of the locations of localized bound
states and consequently gives the locations of the boundaries separating
different topological regions. The source of these boundaries comes from the
mean field square root factor, $\sqrt{1-(m/j)^{2}}$, which is due to the
curvature of the Bloch sphere. The geometric origins of these boundaries is
reminiscent of geometry induced domain walls which appear in a 2D lattice of
dipoles on the surface of a torus [53].

To test the mean field predictions we calculate the probability for a given
$\hat{J}_{z}$ state to be in an eigenstate of the Floquet operator in Eq. (3),
$P(m,\varepsilon_{i})=|\langle m|\varepsilon_{i}\rangle|^{2}$. Figure 2 shows
a density plot of the probability with the $m$ state label and the quasienergy
on the $x$ and $y$ axes, respectively, along with the $m$ dependence of the
mean field quasienergy for the azimuthal angle $\phi=0$ (dashed, red). The
kinks in the mean field result at $\varepsilon=0$ and $\varepsilon=\pi$ are
the locations of the localized bound states predicted from Eq. (20) and they
agree quite well with the quantum probability which shows localized states
around these points. The creation (destruction) of bound states takes place at
the equator of the Bloch sphere at $m=0$. This means that as $\kappa_{1}$
decreases in Fig. 2, the $\varepsilon=0$ bound states at $m\approx 67$ will
move toward $m=0$ until they reach that point when $\kappa_{1}=4\pi$. Further
decreasing $\kappa_{1}$ results in the bound states’ destruction as they are
absorbed into the bulk.

Figure 2: Density plot of the probability for a given eigenstate of the top to
have a given quasienergy. Darker shades of black show higher values of the
probability $|\langle m|\varepsilon_{i}\rangle|^{2}$ where the $\hat{J}_{z}$
eigenstate label, $m$, and the quasienergy, $\varepsilon_{i}$, vary along the
$x$ and $y$ axes, respectively. The red dashed curve is the mean field
quasienergy from Eq. (18) for $\phi=0$. The points of interest are the kinks
in the mean field result at $\varepsilon=0,\pi$ where protected edge states
are shown in the density plot as localized states around these points. The
parameter values are $\kappa_{1}=4.25\pi$, $\kappa_{2}=0.5\pi$ and $j=200$.

###  III.3 Local winding numbers

The boundaries partition the Bloch sphere into regions of different topology
which are quantified in terms of winding numbers. Under the mean field
approximation the winding number has a simple geometric interpretation as the
number of times the vector $\bm{n_{i}}$ winds around the origin as $\phi$ goes
from $-\pi$ to $\pi$. However, due to the square root factors, the winding
number is $m$ dependent, so we calculate local versions of them. The mean
field local winding number is calculated with the equation [32]

$w_{\mathrm{MF},i}(m)=\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\bm{n_{i}}\times\partial_{\phi}\bm{n_{i}}\,.$
(21)

The quantum winding number calculation must also be performed locally due to
$\phi$ not being good quantum number. The calculation relies on the flat-band
transformation of the effective Floquet Hamiltonian,
$\hat{Q}=\hat{P}_{+}-\hat{P}_{-}$, where $\hat{P}_{+}$ is the projector onto
eigenstates with $\varepsilon>0$ and $\hat{P}_{-}$ is the projector onto
eigenstates with $\varepsilon<0$. The chiral symmetry of
$\hat{H}_{\mathrm{eff}}$ allows us to write the flat-band projector as
$\hat{Q}=\hat{Q}_{\uparrow\downarrow}+\hat{Q}_{\downarrow\uparrow}$ where
$\hat{Q}_{\uparrow\downarrow}=\hat{\Gamma}_{\uparrow}\hat{Q}\hat{\Gamma}_{\downarrow}$
and $\hat{\Gamma}_{\uparrow}$, $\hat{\Gamma}_{\downarrow}$ are the projectors
onto the spin-1/2 $\uparrow$ and $\downarrow$ states, respectively. The
spin-1/2 projectors form the chiral symmetry operator
$\hat{\Gamma}=\hat{\Gamma}_{\uparrow}-\hat{\Gamma}_{\downarrow}$ and the
symmetrized winding number operator is then [55, 56, 54]

$\hat{w}_{i}=\left(\hat{Q}_{\downarrow\uparrow,i}[\hat{J}_{z},\hat{Q}_{\uparrow\downarrow,i}]+\hat{Q}_{\uparrow\downarrow,i}[\hat{J}_{z},\hat{Q}_{\downarrow\uparrow,i}]\right)/2$
(22)

where the index $i=1,2$ is for the two chiral timeframes from Eqns. (12) and
(13). Therefore, the winding number localized to a single $\hat{J}_{z}$ state
is $w_{i}(m)=\mathrm{Tr}_{\sigma}\langle m|\hat{w}_{i}|m\rangle$ where the
trace is over the spin-1/2 degrees of freedom. It is expected that for large
system sizes and for states comfortably within a bulk (away from the
topological boundaries), the quantum and mean field calculations will agree.

In Fig. 3 (a) and (b), we plot the $m$ dependence of the mean field (black)
and quantum (green) values of $w_{0}$ and $w_{\pi}$, respectively, for
$\kappa_{1}=4.25\pi$ and $\kappa_{2}=0.5\pi$. The different topological
regions are shown as plateaus while the boundaries are shown as a steps in the
mean field case and jumps/dips in the quantum case. In each image the step
locations are calculated from Eq. (20) $m_{\mu,\nu}=\pm
j\sqrt{1-16\mu^{2}/289-4\nu^{2}}$ where we find five pairs of integers which
satisfy the equation: $0\leq\mu\leq 4$ and $\nu=0$. The quantum local winding
number agrees with the mean field result strongly in the central bulk and the
agreement falls off moving closer to the edge at $m=\pm 200$. This is expected
since finite size effects become pronounced at the edges of the system.
Another obvious feature of the quantum local winding number is the sudden
jumps/dips at the boundaries between different topological regions. These are
attributed to the fact that the boundaries are home to exponentially localized
bound states, so they are close to being eigenstates of $\hat{J}_{z}$.

Looking at Fig. 1 we see that the parameter values of $\kappa_{1}=4.25\pi$ and
$\kappa_{2}=0.5\pi$ mean there are five $\varepsilon=0$ states and four
$\varepsilon=\pi$ states. In Fig. 3, the quantum local winding numbers for the
central region are $|w_{0}|\approx 5$ and $|w_{\pi}|\approx 4$ which matches
the number of bound states and therefore supports the bulk-boundary
correspondence. To determine how the outer regions support the bulk-boundary
correspondence we once again turn to the mean field results, specifically the
parameters in Eq. (15). There we see that moving away from the equator of the
Bloch sphere at $m=0$ has the same effect on $K_{1}$ and $K_{2}$ as decreasing
the scaled kick strengths $\kappa_{1}$ and $\kappa_{2}$. Figure 1 shows that
as $\kappa_{1}$ decreases, bound states are destroyed which is why we see the
mean field local winding number decrease in steps in Fig. 3 as one moves
toward the edges at $j=\pm 200$. We see the same decrease in the quantum local
winding numbers, however, the fluctuations increase as the edge is approached
due to finite size effects.

Figure 3: Local winding numbers. (a) Comparison of the mean field (solid
black), quantum (solid green) and mean chiral displacement (dashed red)
predictions of the $\varepsilon=0$ local winding number over the space of
$\hat{J}_{z}$ states for $\kappa_{1}=4.25\pi$, $\kappa_{2}=0.5\pi$ and
$j=200$. The locations of the steps in the mean field prediction can be
calculated from Eq. (20). (b) Same as (a) except the calculations are
performed for the $\varepsilon=\pi$ local winding number.

The winding number is not always directly measurable in experiments and it is
often easier to extract information from the dynamics of the system. To this
end, the chiral displacement [32, 57]

$\displaystyle C_{i}(m,n)$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{\sigma}\langle
m|\hat{U}_{T,i}^{-n}\hat{J}_{z}\hat{\sigma}_{z}\hat{U}_{T,i}^{n}|m\rangle$
$\displaystyle=$
$\displaystyle\left(\sum_{a=\uparrow,\downarrow}\sum_{j,k}c_{j,a}^{m*}c_{k,a}^{m}e^{-i(\varepsilon_{j}-\varepsilon_{k})n}\langle\varepsilon_{k}|\hat{J}_{z}\hat{\sigma}_{z}|\varepsilon_{j}\rangle\right)_{i}$

has been shown to be related to the quantum winding number. In the second line
of Eq. (LABEL:eq:CD), we take the spectral decomposition of the Floquet
operators, $c_{j,a}^{m}=\langle a,m|\varepsilon_{j}\rangle$ and the subscript
$i=1,2$ indicates which Floquet operator we are using from Eqns. (12) and
(13). The relation to the winding number comes from the fact that the local
winding number at $m=0$ can be approximated as [54]

$\displaystyle w_{i}(0)$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{\sigma}\langle 0|\hat{w}_{i}|0\rangle$
$\displaystyle=$
$\displaystyle\left(\sum_{a=\uparrow,\downarrow}\sum_{j,k}c_{j,a}^{0*}c_{k,a}^{0}\langle\varepsilon_{k}|\hat{J}_{z}\hat{\sigma}_{z}|\varepsilon_{j}\rangle\right)_{i}\,.$

Comparing this equation with Eq. (LABEL:eq:CD), we see that a long time
average of the chiral displacement at $m=0$

$\overline{C_{i}(0)}=\lim_{\mathcal{N}\to\infty}\frac{1}{\mathcal{N}}\sum_{n}^{\mathcal{N}}C_{i}(0,n)$
(25)

will be equal to the winding number if the off-diagonal terms of
$\langle\varepsilon_{j}|\hat{J}_{z}\hat{\sigma}_{z}|\varepsilon_{k}\rangle$
can be neglected since they get ’washed-out’ in the averaging process. This
was found to be the case for the spin-1/2 QDKR [23] and in a synthesized 1D
topological wire based on the momentum states of a BEC [54]. Although these
terms are small for our system, they are non-negligible, so they need to be
included in order for the dynamics to display an accurate winding number.
Therefore, short time averages are better in order for the off-diagonal terms
to not vanish completely. The dashed red curves in Fig. 3 (a) and (b) show the
chiral displacement averaged over $\mathcal{N}=20$ steps for all states (not
just the $m=0$ state). We see it maintains a similar shape to the local
winding number calculated from Eq. (22) (green) including the sudden
jumps/dips at the boundaries, however, it does fall short of the mean field
and quantum local winding numbers over the majority of the states due to the
averaging process. A comparison between short and long time averages of the
chiral displacement can be found in Appendix B.

###  III.4 Using a probe state to locate topological boundaries

Figure 4: Evolution of probability distributions for different initial states.
(a) Evolution generated by $\hat{U}_{T}$ in Eq. (3) of the probability
distribution of an initial Gaussian state centered on the boundary between two
different topological regions. The boundary is located near $\lfloor
m_{4,0}\rfloor=67$ and corresponds to the right closest step to the center in
Fig. 3 (a), so it is the boundary between the $(w_{0},w_{\pi})=(5,-4)$ and the
$(w_{0},w_{\pi})=(3,-4)$ topological regions. (b) Same initial Gaussian state
as in (a) except it is centered at $\lfloor m_{4,0}+m_{3,0}\rfloor/2=104$
which is halfway between the boundary separating the $(w_{0},w_{\pi})=(5,-4)$
and the $(3,-4)$ regions and the boundary separating the
$(w_{0},w_{\pi})=(3,-4)$ and the $(3,-2)$ regions. (c) Same initial Gaussian
state as in (a) except it is centered at $\lfloor m_{3,0}\rfloor=141$ which is
the boundary separating the $(w_{0},w_{\pi})=(3,-4)$ and the $(3,-2)$ regions
which is the right closest step to the center in Fig. 3 (b). The parameters
are $\kappa_{1}=4.25\pi$, $\kappa_{2}=0.5\pi$ and $j=200$. Figure 5: Inverse
participation ratio of Gaussian states in the Floquet basis. The IPR given in
Eq. (26) of the state $|\psi_{0}\rangle=\sqrt{\frac{1}{\Delta
m\sqrt{\pi}}}\sum_{m}e^{-\frac{(m-m_{0})^{2}}{2\Delta
m^{2}}}|m,\uparrow\rangle$ in the basis states of Eq. (3) is plotted as a
function of $m_{0}$. The red vertical lines are the predicted locations of the
topological boundaries from Eq. (20) which line up with peaks in the IPR where
the Gaussian strongly overlaps the bound states located at the boundaries. The
peaks match the locations of the steps in the mean field winding number and
the jumps/dips in the quantum winding number in Fig. 3. The parameter values
are $\Delta m=10$, $\kappa_{1}=4.25\pi$, $\kappa_{2}=0.5\pi$ and $j=200$.

The boundaries separating different topological regions can also be located
using another dynamical method which takes advantage of the fact that the
boundaries are home to the $\varepsilon=0$, $\pi$ bound states. If an initial
state has a strong overlap with the bound states at a boundary, then it will
remain localized there for a long period of time. In contrast, an initial
state comfortably away from a boundary will explore more of the Hilbert space
as it evolves because of the larger overlap with the bulk states. We choose
the initial probe state to be Gaussian in shape
$|\psi_{0}\rangle=\sqrt{\frac{1}{\Delta
m\sqrt{\pi}}}\sum_{m}e^{-\frac{(m-m_{0})^{2}}{2\Delta
m^{2}}}|m,\uparrow\rangle$ with $\Delta m=10$. One of the benefits of using
this method is that the details of the initial state are not important as long
as it can pick out one boundary over another. In Fig. 4 (a), we show the
evolution of the probability distribution in the $\uparrow$ subspace of the
initial Gaussian state centered on the boundary located at $m_{0}=\lfloor
m_{4,0}\rfloor=67$. This boundary is shown in Fig. 3 (a) as the first step
away from the $m=0$ state. The strong overlap between the initial state and
the bound states keeps the distribution localized over the period shown. In
Fig. 4 (b), the initial state is centered at $m_{0}=104$ which is halfway
between two boundaries. Clearly the probability distribution does not remain
localized and has a checkered pattern which is due to parts of the wave
function occupying the $\downarrow$ subspace which is not shown. The reason
why the checkered pattern appears in (b) and not (a) is because the initial
state has a larger overlap with $\varepsilon\neq 0,\pi$ states in (b) and
these states have equal support on both subspaces due to chiral symmetry
whereas the $\varepsilon=0,\pi$ bound states have support on a single
subspace. In Fig. 4 (c), the initial Gaussian is centered on another boundary
at $m_{0}=\lfloor m_{3,0}\rfloor=141$ which is the first step from the center
in Fig. 3 (b). Again, the probability distribution is fairly localized with
small parts of it propagating away due the Gaussian having a weaker overlap
with the bound states compared to the ones in (a).

In order to get a sense of the overall quality of the specific initial state
we chose, we take a look at the inverse participation ratio (IPR) which is a
good measure of the localization of a state in a given basis

$\mathrm{IPR}=\sum_{i}|\langle\varepsilon_{i}|\psi_{0}\rangle|^{4}\,.$ (26)

Here, $|\varepsilon_{i}\rangle$ are the eigenstates of $\hat{U}_{T}$ in Eq.
(3) and $|\psi_{0}\rangle$ is the initial Gaussian state. The two extreme
values are $\mathrm{IPR}=1$ and $\mathrm{IPR}=\frac{1}{2(2j+1)}$ when
$|\psi_{0}\rangle$ is completely localized and completely spread in the basis,
respectively. In Fig. 5 the IPR is plotted as a function of the center of the
Gaussian $m_{0}$ where the red vertical lines are the locations of the
boundaries predicted from Eq. (20). The peaks correspond to the initial state
being localized at the boundaries where it strongly overlaps with the bound
states. The relative height of the peaks, including the peak at $m=0$, are
state dependent, so Gaussians with different widths will produce different
qualitative results, however, the peak locations will remain the same.
Nevertheless, the chosen Gaussian does an excellent job at picking out the
boundary locations with the only discrepancy being near the edge of the system
at $m=200$. There the distance between boundaries is similar to $\Delta m=10$,
so the initial state lacks the resolution required to locate those boundaries
completely.

##  IV Summary and Discussions

We have shown that, like the spin-1/2 QDKR, the spin-1/2 QDKT has topological
regions with large winding numbers. However, the two models differ in the
number of topological regions for a given pair of kick strengths. Whereas the
spin-1/2 QDKR has a single region, different regions proliferate in the Bloch
sphere of the spin-1/2 QDKT as the kick strengths increase. We quantify the
topology of each region by comparing local versions of the mean field, quantum
and mean chiral displacement winding numbers. We find that away from the edge
of the system they agree with the number of bound states at the boundaries
separating each region which supports the bulk-boundary correspondence.
Finally, we used a simple dynamical method to locate the boundaries by
preparing a Gaussian initial state and evolving it. When the initial state is
centered on a boundary the state remained localized due to the large overlap
with the bound states exponentially localized there.

The spin-1/2 QDKT is a rich topological system with many possible avenues to
explore. As previously mentioned, it is related to quantum walks and can be
used to investigate the effects of multiple topological boundaries in the walk
space. A notable departure from former quantum walk studies is the source of
the boundaries. Usually they are implemented via insertion of an inhomogeneity
at a specific site, whereas here, they come from the geometry of the Bloch
sphere in which the walk is taking place and can therefore be considered as
being built-in by nature. The positions of the boundaries are controlled by
the kick strengths which leads to some interesting possibilities for using the
topology of the spin-1/2 QDKT as a tool to control the state of the system.
One example of this is in the creation of cat states. One can imagine
initially setting each kick strength close to some multiple of $\pi$, so that
a pair of boundaries just forms at the equator of the Bloch sphere of the top
($m=0$), then preparing the initial product state
$|\psi_{0}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)|\theta_{0}=\pi/2\rangle$,
where $|\theta_{0}=\pi/2\rangle$ is a coherent state centered on the equator.
Provided the coherent state has a strong overlap with the bound states at the
boundary it will remain clamped there as we showed in Fig. 4. If one of the
the kick strengths is made to increase slowly in time, then the coherent state
is effectively pulled apart with parts of it moving toward opposite poles of
the Bloch sphere as the boundaries move away from the equator resulting in the
final state
$|\psi_{f}\rangle=\frac{1}{\sqrt{2}}(|\uparrow,\theta_{0}+\theta\rangle+|\downarrow,\theta_{0}-\theta\rangle)$.
The final state is often referred to as a Bell-cat state and is used to test
entanglement generation and efficient information extraction beyond the
original two-qubit Bell state [58]. The process is possible due to the chiral
symmetry of the system which forces the bound states at the two newly formed
boundaries to have support in opposite spin-1/2 subspaces.

Another topic to explore is the connection between topology and chaos. Some
initial numerical results of the level spacings of the quasienergies indicate
that the spin-1/2 QDKT displays chaotic behavior as the kick strengths
increase. It has been shown that nonlinear effects can lead to chaotic
behavior in the bulk of a system while the boundaries have topological order
[59]. However, the spin-1/2 QDKT is unique in that the number of boundaries
also increases as the kick strengths increase. This raises the question as to
how a bulk can be chaotic when boundaries proliferate in the system.

##  Appendix A Derivation of winding vector

We can find $\bm{n_{i}}$ by expanding the unitary operators in Eqns. (16) and
(17) in terms of trigonometric functions using the identity

$e^{-ia\bm{b}\cdot\hat{\bm{\sigma}}}=\cos(a)-i\bm{b}\cdot\hat{\bm{\sigma}}\sin(a)$
(27)

which gives

$\displaystyle\hat{U}_{T,1}^{\mathrm{MF}}$ $\displaystyle=$ $\displaystyle\cos
K_{1}\cos K_{2}-i\left(\sin K_{1}\cos K_{2}\hat{\sigma}_{x}+\sin
K_{2}\hat{\sigma}_{y}\right)$ $\displaystyle\hat{U}_{T,2}^{\mathrm{MF}}$
$\displaystyle=$ $\displaystyle\cos K_{1}\cos K_{2}-i\left(\sin
K_{1}\hat{\sigma}_{x}+\sin K_{2}\cos K_{1}\hat{\sigma}_{y}\right)\,.$

For $\hat{U}_{T,1}^{\mathrm{MF}}$ we define

$\displaystyle\cos\varepsilon$ $\displaystyle=$ $\displaystyle\cos K_{1}\cos
K_{2}$ $\displaystyle\sin\varepsilon$ $\displaystyle=$
$\displaystyle\sqrt{\sin^{2}K_{1}\cos^{2}K_{2}+\sin^{2}K_{2}}$

and similarly for $\hat{U}_{T,2}^{\mathrm{MF}}$ we define

$\displaystyle\cos\varepsilon$ $\displaystyle=$ $\displaystyle\cos K_{1}\cos
K_{2}$ $\displaystyle\sin\varepsilon$ $\displaystyle=$
$\displaystyle\sqrt{\sin^{2}K_{2}\cos^{2}K_{1}+\sin^{2}K_{1}}$

which allows us to write the Floquet operators as

$\hat{U}_{T,i}^{\mathrm{MF}}=\cos\varepsilon-i\sin\varepsilon\left(n_{ix}\hat{\sigma}_{x}+n_{iy}\hat{\sigma}_{y}\right)$
(30)

where the vector components are

$\displaystyle n_{1x}$ $\displaystyle=$ $\displaystyle\frac{\sin K_{1}\cos
K_{2}}{\sin\varepsilon},\hskip 20.0ptn_{1y}=\frac{\sin
K_{2}}{\sin\varepsilon}$ (31) $\displaystyle n_{2x}$ $\displaystyle=$
$\displaystyle\frac{\sin K_{1}}{\sin\varepsilon},\hskip
50.0ptn_{2y}=\frac{\sin K_{2}\cos K_{1}}{\sin\varepsilon}\,.$ (32)

It is clear from Eq. (30) that the effective Hamiltonian mentioned in the main
text takes the form
$\hat{H}_{\mathrm{eff},i}^{\mathrm{MF}}=\varepsilon\bm{n_{i}}\cdot\hat{\bm{\sigma}}$.
We note that the derivation of the winding vector components is the same as
the one for the spin-1/2 QDKR in Ref. [23] with the only difference being that
here we have $K_{1}=\kappa_{1}\sqrt{1-(m/j)^{2}}\cos\phi$ and
$K_{2}=\kappa_{2}\sqrt{1-(m/j)^{2}}\sin\phi$ and they have $m=0$.

##  Appendix B Short and long time averages of the chiral displacement

Figure 6: Comparison of short time and long time averages of the chiral
displacement. The solid orange and dashed blue curves show time averages of
the chiral displacement using Eq. (25) for $\mathcal{N}=10$ and
$\mathcal{N}=1000$, respectively. The long time average has less agreement
with the mean field prediction (solid black) than the short time average
because important terms get ’washed-out’ in the averaging process. The
parameter values are $\kappa_{1}=7.5\pi$, $\kappa_{2}=0.5\pi$ and $j=200$.

We find that the short time average rather than the long time average of the
chiral displacement in Eq. (LABEL:eq:CD) captures the winding number. Figure 6
explicitly shows the difference between short and long time average
calculations of $w_{\pi}$ for $\kappa_{1}=7.5\pi$ and $\kappa_{2}=0.5\pi$. The
solid orange and dashed blue data is calculated with $\mathcal{N}=10$ and
$\mathcal{N}=1000$, respectively, and the solid black curve is the mean field
prediction. The long time average result has less fluctuations than the short
time average, but has a larger disagreement with the mean field result over
the range of states. This is because the long time average does not keep terms
like
$\langle\varepsilon_{k}|\hat{J}_{z}\hat{\sigma}_{z}|\varepsilon_{j}\rangle$,
where $j\neq k$, which are necessary to accurately predict the winding number
from the chiral displacement. However, the long time average still exhibits
the sudden jumps at the boundaries, so it can still be used to locate the
boundaries.

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