#  Wave packet dynamics and edge transport in anomalous Floquet topological
phases

Miguel F. Martínez  Department of Physics, KTH Royal Institute of Technology,
106 91, Stockholm, Sweden  TCM Group, Cavendish Laboratory, University of
Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom  F. Nur Ünal
<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University of Cambridge, JJ
Thomson Avenue, Cambridge CB3 0HE, United Kingdom

###### Abstract

The possibility of attaining chiral edge modes under periodic driving has
spurred tremendous attention both theoretically and experimentally, especially
in light of anomalous Floquet topological phases that feature vanishing Chern
numbers unlike any static counterpart. We here consider a periodically
modulated honeycomb lattice and experimentally relevant driving protocols,
which allows us to obtain edge modes of various character in a simple model.
We calculate the phase diagram over a wide range of parameters and recover an
anomalous topological phase with quasienergy gaps harbouring edge states with
opposite chirality. Motivated by the advances in single-site control in
optical lattices, we investigate wave packet dynamics localized at the edges
in distinct Floquet topological regimes that cannot be achieved in
equilibrium. We analyse transport properties in edge modes which originate
from the same bands, but with support at different quasienergies and
sublattices as well as possessing different chiralities. We find that an
anomalous Floquet topological phase can in general generate more robust chiral
edge motion than a Haldane phase, allowing for more effective loading of the
wave packet into edge channels. Our results demonstrate that the rich
interplay of wave packet dynamics and topological edge states can serve as a
versatile tool in ultracold quantum gases in optical lattices.

##  I Introduction

Topologically protected phenomena entail a prominent research direction in
condensed matter physics Hasan and Kane (2010); Qi and Zhang (2011). A wide
range of novel phases arising from the interplay of topology and symmetries
have been theorised and observed, with intriguing features being unearthed
regularly especially in highly nontrivial many-body or out-of-equilibrium
settings Thouless  _et al._ (1982); Bernevig  _et al._ (2006); Fang  _et al._
(2012); Kane and Mele (2005); Slager  _et al._ (2013); Kruthoff  _et al._
(2017); Po  _et al._ (2017); Bradlyn  _et al._ (2017); Jotzu  _et al._ (2014);
Aidelsburger  _et al._ (2015); Tran  _et al._ (2017); Asteria  _et al._
(2019); Kemp  _et al._ (2022); Tan  _et al._ (2019); Xu  _et al._ (2022);
Jangjan  _et al._ (2022); Jangjan and Hosseini (2020). Regarding the latter,
rapid developments have extended topological characterisations to periodically
driven Floquet systems Roy and Harper (2017); Oka and Aoki (2009); Kitagawa
_et al._ (2010); Rudner  _et al._ (2013); Ünal  _et al._ (2019a);
Wintersperger  _et al._ (2020) and dynamic quench settings with new invariants
Wang  _et al._ (2017); Tarnowski  _et al._ (2019); Ünal  _et al._ (2019b); Hu
and Zhao (2020), even reaching to exotic multi-gap topologies with non-Abelian
braiding properties Ünal  _et al._ (2020); Slager  _et al._ (2022); Ahn  _et
al._ (2019); Bouhon  _et al._ (2020a, b); Jiang  _et al._ (2021a, b). From an
experimental point of view, Floquet engineering Goldman and Dalibard (2014);
Eckardt (2017); Bukov  _et al._ (2015) has been established as a powerful tool
to realise paradigmatic models in periodically driven non-equilibrium quantum
matter in platforms such as ultracold atoms Cooper  _et al._ (2019); Wang  _et
al._ (2018); Račiūnas  _et al._ (2018); Wintersperger  _et al._ (2020); Reichl
and Mueller (2014) and photonic lattices Maczewsky  _et al._ (2017); Mukherjee
_et al._ (2017), allowing for not only high degree of control and efficient
quantum simulations but also exploring new regimes unattainable in
equilibrium.

In a Floquet system, where energy is not a conserved quantity due to broken
continuous time-translation invariance, one can adopt a description in terms
of a periodic quasienergy since discrete time translations are still present.
An effective quasienergy spectrum and the topological information are encoded
by the time evolution over a period, $T$. Upon evaluating stroboscopically
Eckardt (2017), a quasienergy can be defined as phase eigenvalues of the time
evolution operator, namely as $\varepsilon_{n}T\in[-\pi,\pi)$ for $n$ number
of bands, modulo $2\pi$ in a Floquet Brillouin Zone (FBZ). The fact that
quasienergy bands are phases forming a circle induces one additional,
anomalous, $\pi$-gap connecting the bands through the FBZ edge. The
periodicity of the Floquet spectrum has paved the way for novel phases that
truly arise from this out-of-equilibrium nature such as helical edge states
crossing across the FBZ, anomalous Floquet Anderson insulators and anomalous
Dirac string phases Budich  _et al._ (2017); Titum  _et al._ (2016); Slager
_et al._ (2022).

Most interestingly, the possibility to obtain anomalous edge states in the
FBZ-edge gap renders the equilibrium topological classification in terms of
the Chern number, $C_{n}$, in two dimensions inept to characterize driven
systems Kitagawa  _et al._ (2010); Rudner  _et al._ (2013). Rather than
invariants of individual bands, one needs to consider winding numbers,
$W_{g}$, associated with gaps centered around, e.g. $g=0$ and $g=\pi$ for two
levels. Consequently, the anomalous Floquet topological phase has attracted
great attention, characterised by the winding number combination
$[W_{0},W_{\pi}]=[1,1]$ harbouring edge states in both gaps despite a
vanishing Chern number Wintersperger  _et al._ (2020); Rudner  _et al._
(2013). Experimentally, individual edge modes have been probed in photonic
lattices Maczewsky  _et al._ (2017); Mukherjee  _et al._ (2017). Advances in
optical lattices have allowed for directly measuring the topological
invariants Wintersperger  _et al._ (2020); Tarnowski  _et al._ (2019); Jotzu
_et al._ (2014) and in particular distinguishing the quasienergy gaps to
unambiguously assign the observed winding numbers to individual gaps. However,
coherent edge dynamics and transport properties in different quasienergy gaps,
particularly with respect to each other and equilibrium phases, remain an open
question. Recent advances in single-site accessibility Gross and Bakr (2021)
in optical lattices now offer new possibilities for the creation of sharp
edges and probing topological edge modes by using localised wave packets to
investigate unique Floquet topological features as well as the effect of
different quasienergy gaps associated to different branch cuts.

In this work, we consider a periodically driven two-band model in two
dimensions (2D) that is also experimentally relevant and analyse transport
properties in different quasienergy gaps focusing on the distinct Floquet
nature Ünal  _et al._ (2019a); Wintersperger  _et al._ (2020). We calculate
the phase diagram over a wide range of parameters and contrast different
driving protocols. Going beyond the anomalous $[1,1]$-phase that has been
originally introduced, this allows us to reach an unexplored anomalous Floquet
topological phase where a pair of edge states with opposite chiralities are
induced in different gaps ($[W_{0},W_{\pi}]=[\pm 1,\mp 1]$) supported by the
same two bands with finite Chern number, which cannot be obtained in
equilibrium. We investigate the wave packet dynamics in various topological,
and in particular anomalous, phases. We study populating edge modes at
different quasienergies and with different winding numbers by applying kicks,
controlling the shape of the wave packet, and examine their robustness and
efficiency of preparation. Addressing chiral edge dynamics in different phases
where only a single gap or both gaps harbour edge modes, we show that an
anomalous Floquet topological phase can give rise to much more robust edge
transport than equilibrium Chern insulating phases. We further analyze the
effect of the Floquet gauge and the sublattice character of the edge states.

##  II The Model

We consider a honeycomb lattice in two dimensions with a Hamiltonian given by

$\hat{H}=-\sum_{\langle
i,j\rangle}J\hat{a}^{\dagger}_{i}\hat{a}^{\phantom{{\dagger}}}_{j}+\dfrac{\Delta}{2}\sum_{i\in
A}\hat{a}^{\dagger}_{i}\hat{a}^{\phantom{{\dagger}}}_{i}-\frac{\Delta}{2}\sum_{i\in
B}\hat{a}^{\dagger}_{i}\hat{a}^{\phantom{{\dagger}}}_{i},$ (1)

where $\hat{a}^{\dagger}_{i}(\hat{a}^{\phantom{{\dagger}}}_{i})$ creates
(annihilates) a particle on lattice site $i$, with a nearest-neighbor
tunnelling strength $J$ and an energy offset $\Delta$ between the two
sublattices $A$ and $B$. We will introduce the periodic driving via the
modulation of the hopping amplitudes $J_{m}$ for $m=1,2,3$ along the three
nearest-neighbor vectors $\bm{d}_{1}=(0,-1)a,\,\bm{d}_{2}=(-1/2,\sqrt{3}/2)a$
and $\bm{d}_{3}=(1/2,\sqrt{3}/2)a$, where $a$ is the nearest-neighbor
distance. The Hamiltonian at a given time instance is diagonal with respect to
quasimomentum k, and hence can be written as

$\hat{H}(\textbf{{k}},t)=-\sum_{m=1}^{3}J_{m}(t)\big{(}\cos(d_{m}\textbf{{k}})\sigma_{x}+\sin(d_{m}\textbf{{k}})\sigma_{y}\big{)}+\frac{\Delta}{2}\sigma_{z},$
(2)

where $\boldsymbol{\sigma}$ are the Pauli matrices.

The driving protocols that we implement are of step-wise nature Kitagawa  _et
al._ (2010); Rudner  _et al._ (2013), which not only offers conceptual
simplicity for our theoretical characterisation, but are also experimentally
relevant as they have been recently implemented in optical lattices
Wintersperger  _et al._ (2020) with a smoothed modulation Quelle  _et al._
(2017). Namely, one period of the drive is divided in three even steps of
length $T/3$. For the first driving scheme, the tunnelling is allowed
cyclically only along one of the three directions during each stage with
amplitude $J$ Rudner  _et al._ (2013); Ünal  _et al._ (2019a). Secondly, we
employ another protocol, where during the $m^{th}$-step, the tunneling
$J_{m}=\lambda J$ is enhanced by a factor of $\lambda$, while the tunneling
along the other two directions are kept fixed at $J$ Kitagawa  _et al._
(2010). The first drive can be seen as a limiting case of the second one for
$\lambda\rightarrow\infty,J\rightarrow 0$, while keeping $\lambda J$ fixed. We
will illustrate in detail the difference between the two schemes in the
following.

Since during each step of the driving cycle, the Hamiltonian (2) becomes time-
independent, $\hat{H}_{m}(\textbf{{k}})$, for both driving protocols, the
time-evolution operator at the end of one period can be written as

$\hat{\mathcal{U}}(\textbf{{k}},T)=e^{-i\hat{H}_{3}(\textbf{{k}})\frac{T}{3}}e^{-i\hat{H}_{2}(\textbf{{k}})\frac{T}{3}}e^{-i\hat{H}_{1}(\textbf{{k}})\frac{T}{3}}=e^{-i\hat{\mathcal{H}}_{F}(\textbf{{k}})T},$
(3)

where we set $\hbar=1$ throughout this paper. This stroboscopic evolution is
captured by the Floquet Hamiltonian $\hat{\mathcal{H}}_{F}(\textbf{{k}})$,
defining the quasienergy spectrum through
$\hat{\mathcal{H}}_{F}(\textbf{{k}})|\phi_{n}(\textbf{{k}})\rangle=\varepsilon_{n}|\phi_{n}(\textbf{{k}})\rangle$.
The Berry curvature and the Chern numbers of these quasienergy bands are
calculated using the Floquet eigenstates $\phi_{n}(\textbf{{k}})$. These
topological invariants in the 2D momentum-space BZ have, however, been shown
to be insufficient to capture the Floquet topology Rudner  _et al._ (2013).
One instead needs to consider also the time evolution within the period,
$\hat{\mathcal{U}}(\textbf{{k}},t)$.

Figure 1:  Phase diagrams of the step-wise driven honeycomb lattice with
corresponding winding numbers $[W_{0},W_{\pi}]$. (a) For the first driving
protocol with switching tunnelling amplitudes along the three directions
on/off completely, for modulation frequency $\omega$ and sublattice offset
$\Delta$. (b) The second protocol where tunnelling amplitudes are cyclically
enhanced by a factor of $\lambda$, here $\Delta=2J$.

In Floquet settings, the two bands can close and re-open in two distinct ways;
in the quasienergy gaps at zero but also at $\pi$, corresponding to a change
in the branch cut for defining $\hat{\mathcal{H}}_{F}(\textbf{{k}})$, as
opposed to one possibility in a static system. This quasienergy gap labelling
originating from the time-periodicity requires a topological characterisation
that employs winding numbers defined in the $(k_{x},k_{y},t)$-space. Each gap
closing induces a transfer of Berry curvature between the bands, leaving
chiral edge states behind in their respective gaps characterized by finite
winding numbers. When the transitions in zero and $\pi$ gaps trivialise each
other, we arrive at an anomalous Floquet topological phase with a vanishing
Chern number Rudner  _et al._ (2013). The latter can in general be expressed
as the difference of the winding numbers (net number of edge states in a gap
factoring in their chiralities) above and below a band,
$C_{n}=W_{n,above}-W_{n,below}$. In the Floquet case, the extra $\pi$-gap
renders the spectrum unbounded and, hence, offers more interesting
possibilities.

##  III Phase diagrams

Fig. 1 demonstrates the phase diagrams that we numerically calculate in our
driven honeycomb models for a representative parameter range. The winding
numbers can in general be computed using the time evolution operator at every
point in the $(2+1)$D parameter space Rudner  _et al._ (2013), although this
may prove computationally and experimentally demanding in most cases. Instead,
we here employ an approach based on tracking the change of the winding numbers
in each gap as introduced in Ref. Ünal  _et al._ (2019a) and successfully
implemented in Munich Wintersperger  _et al._ (2020) to measure anomalous
winding numbers. In particular, in the high-frequency regime we utilise the
equilibrium topological classification based on Chern numbers with a trivial
winding number in the $\pi$-gap Ünal  _et al._ (2019a); Nathan and Rudner
(2015). In the case of the second driving protocol, $\lambda=1$ automatically
satisfies the static definition. We depart from the topological invariants
that we calculate for these initial parameters at high frequencies for both
driving protocols. As model parameters are being tuned, we compute the winding
numbers, which change via band touching points in each gap, by evaluating the
charge of these topological singularities in a gap-specific way (see
Supplementary Material for details See ). The band singularities (hence, edge
modes) at the FBZ edge involve a change in the branch cut by $\pi$. We further
confirm these winding numbers by computing the Hopf invariant at
representative points in a given phase, which has been shown to equal the
winding numbers Ünal  _et al._ (2019b).

Figure 2: Quasienergy spectra for a ribbon with armchair (upper panel) and
zigzag (lower panel) terminations, with crystal momentum $k_{\parallel}$ along
the periodic direction. In the $[1,1]$ phase, here given for the first driving
protocol introduced in the main text, zero- and $\pi$-gap edge states occur
well-separate in momentum at the fine-tuned point $\omega=4J/3,\Delta=0$ (a,d)
and $\omega=1.5J,\Delta=0.5J$ (b,e). (c,f) In the $[-1,1]$ phase with the
second driving scheme, the two edge states appear closer in momentum, for
$\omega=4.5J$, $\Delta=2J$, $\lambda=3$.

For the first driving protocol where the tunneling amplitudes are cyclically
turned on and off completely, the relevant tuning parameters are the
sublattice offset and frequency. Indeed, this simple model illustrates a rich
phase diagram including the previously predicted and observed anomalous
Floquet topological phase $([1,1])$ (see Fig. 1a), which can be understood by
considering the limit of $\Delta=0$ and $\omega=4J/3$. For particles starting
from one of the sublattices, this fine-tuned point corresponds to a complete
population transfer to the other sublattice at the end of each step with
tunnelling allowed for a time of $JT/3=\pi/2$. As we follow the driving cycle,
it can be easily seen that particles in the bulk remain localized and only
circularly move around each hexagon, ending up in alternating sublattice
flavor at the end of each period and, hence, mixing the pseudospin character.
However, in a finite system, particles move along the edge in a direction set
by the chirality of the drive, corresponding to unit winding numbers of same
sign in both gaps despite the trivial invariant of the bulk bands.

The second driving scheme on the other hand provides one more knob to tune,
namely the driving amplitude $\lambda$. This allows for reaching more exotic
phases as we present in Fig. 1b as a function of $\lambda$ and $\omega$ for a
fixed sublattice offset $\Delta=2J$. We identify a phase with winding numbers
$[W_{0},W_{\pi}]=[-1,1]$, which we numerically verify to be inaccessible using
the first driving protocol owing to the less number of tuning parameters in
that case. Distinct from the previously introduced anomalous phase, this is a
hitherto-unexplored anomalous Floquet topological phase that harbours edge
states in both zero and the anomalous–$\pi$ gaps with opposite chiralities,
supported by the same two bands. Hence, the lower (upper) band carries a Chern
number $C_{1}=-2\;(C_{2}=+2)$. This anomalous phase is unique to the driven
system, since in equilibrium there is only one gap where the topological
transition can occur between the two bands (i.e. the zero-gap). This gap could
in principle host two edge modes of opposite chiralities also in the static
case, provided that they occur at two different quasimomenta. This would
however correspond to vanishing Chern numbers of the two bands, making the
anomalous $[-1,1]$ phase exclusively emerging in the Floquet setting.
Interestingly, these subtle differences also reflect on the edge transport and
wave-packet dynamics as will be illustrated subsequently.

##  IV Anomalous edge states

In order to investigate transport properties and chiral edge dynamics in
different Floquet (anomalous) topological phases, we consider a ribbon
geometry extended along the $x$-direction, with $N_{y}$ layers along the
finite $y$-direction. We present the edge spectra in Fig. 2 for both armchair
and zigzag terminations. At the fine-tuned point within the $[1,1]$ phase of
the first driving protocol, the Floquet spectrum features completely flat
bands corresponding to the localised bulk motion with extended edge modes
crossing the entire FBZ, see Fig. 2(a,d). Although the armchair termination
folds these two edge states to the same point, the zigzag spectrum reveals
that the edge modes are well separated in momentum: While the zero-gap states
form at the $K$-point with finite momentum $\pi/\sqrt{3}$ in units of the
lattice constant, $\pi$-gap states appear at the $\Gamma$-point. We find that
this is true in general in the $[1,1]$ phase owing to the nature of band
inversions required in this phase, also away from the fine-tuned case as
illustrated in Fig. 2(b,e) with dispersive bands, as well as for the second
driving protocol. Since armchair and zigzag terminations correspond to
projecting along perpendicular directions in the momentum space,
$k\rightarrow-k$ symmetry is naturally broken in the presence of a finite
sublattice offset for the latter (see Fig. 2(b,e)). The edge modes nonetheless
still carry a large momentum difference. On the contrary, in the $[-1,1]$
phase in Fig. 2(c,f), the two edge modes with opposite chiralities appear
closer in momentum, which will bring about an important distinction for the
dynamics in the two anomalous phases.

Figure 3: Distribution of edge states at quasienergy zero (left panel) and
$\pi$ (right panel), marked by dots in the insets on their corresponding
quasienergy spectra, on a cylinder periodic along the $x$-direction. (a,b) In
the $[1,1]$ phase with the same parameters as Fig. 2b, both edge states are
counterclockwise and localised at the same sublattices. (c,d) In the $[-1,1]$
phase for the parameters given in Fig. 2c, while the $\pi$-gap state localises
on $A$ at the top edge, the zero energy state with the opposite chirality
localises on the $B$ sublattice.

The different chiralities of the edge states in the anomalous Floquet
topological phases also affect their sublattice character as shown in Fig. 3.
We here consider a cylinder geometry with periodic boundary conditions
connecting $N_{x}$ layers along the $x$-direction. While in the $[1,1]$ phase,
the counter-clockwise edge states in the zero-gap are localised on the $A(B)$
sublattice on the upper (lower) end of the cylinder, the $\pi$-gap states
support the same chiral motion localised on the same sublattice flavors. In
the case of $[W_{0},W_{\pi}]=[-1,1]$, however, the system harbours both
clockwise and counter-clockwise edge modes. Fig. 3(c,d) demonstrates that this
is facilitated by swapping of the sublattice character of the zero-gap states
along with their chirality. Hence, on the upper/lower end of the cylinder, the
two edge modes support opposite currents in different sublattices.
Interestingly, the layers where zero and $\pi$-gap currents have maximum
density, depicted by the size of the circles, are also different. We emphasize
that these two chiral modes do not hybridize despite being on the same edge as
they are well separated in quasienergy. We now analyse how the interplay
between the edge modes located at different momentum, quasienergy, sublattices
and with different chiralities affect their transport properties, especially
with respect to each other and in different Floquet (anomalous) topological
phases.

Figure 4: Overlap of a wave packet with the Floquet eigenstates (bottom panel)
and its evolution (upper panel) at the edge sites (shaded two layers in the
inset) on a cylinder periodic along $x$ with $N_{x}=104$, $N_{y}=41$ layers.
The initial wave packets have $\sigma_{x}=1$, $\sigma_{y}=0.5$ as depicted in
the inset where the radius of the circles is proportional to the initial
probability at each site. (a, e) The $[1,1]$ phase at the fine-tuned point.
The wave packet initialised without a kick follows a clear chiral motion. For
the $[1,1]$ phase for the same parameters as in Fig. 2b, the wave packet is
initially given (b,f) a small kick $\textbf{{q}}=(-0.17,0)$ and (c, g) a
larger kick $\textbf{{q}}=(1.56,0)$, to target the Dirac cones at the $\pi$\-
and zero-gap respectively. As revealed by the overlaps, applying appropriate
kicks allows us to mainly populate a given gap, where the dynamics evolve
similarly. (d, h) The $[-1,1]$ phase for the same parameters as Fig. 2c. We
now can largely populate both gaps simultaneously without a kick since the
opposite chirality edge modes appear closer in momentum. The wave packet
separates into two, giving rise to topologically protected currents travelling
in both directions at the edge, with the Chern number ($|C|=2$) corresponding
to the difference of them.

##  V Wave packet dynamics in anomalous topological phases

Ultracold atomic systems have recently enjoyed rapid advances in single-site
accessibility and local control with techniques like quantum gas microscopes
and optical tweezers Gross and Bakr (2021); Kaufman and Ni (2021); Braun  _et
al._ (2023); Nixon  _et al._ (2023). Giving access to the creation of sharp
edges, these pose a timely question and offer new opportunities for
investigating wave packet dynamics localised at the edges of a topological
system as a powerful tool in optical lattices. The models that we implement
allow us to retrieve a rich phase diagram within a simple geometry, where we
compare edge transport in the conventional Haldane ($[1,0]$) Haldane (1988)
and the Haldane-like phases ($[0,1]$), which are gauge equivalent, with the
anomalous Floquet topological phase ($[1,1]$). Indeed, we show that edge state
population in a given gap can be mostly controlled by employing appropriate
kicks. Most importantly, we find that the $[1,1]$ phase allows for a more
robust chiral edge motion than the Haldane phases. We analyse the effects of
opposite chiralities with the opportunity provided by the second anomalous
phase ($[-1,1]$) and the simultaneous activation of both edge modes giving
rise to interesting chiral edge dynamics.

We consider a cylindrical geometry of $N_{x}$ by $N_{y}$ layers and a Gaussian
wave packet,
$\Psi(x,y)=\exp{\left\\{-(x-x_{0})^{2}/4\sigma^{2}_{x}-(y-y_{0})^{2}/4\sigma^{2}_{y}+iq_{x}x+iq_{y}y\right\\}}/\mathcal{N}$,
initially localized at the upper edge (see Fig. 4e inset), at $(x_{0},y_{0})$
with a spread given by $(\sigma_{x},\sigma_{y})$ and normalization
$\mathcal{N}$. We allow for an initial kick with momentum
$\textbf{{q}}=(q_{x},q_{y})$ which can be applied to control the overlap of
the wave packet with edge states that are projected from the specified momenta
onto the edge. We numerically calculate the evolution of the wave packet and
present the probability at the edge sites in the upper two layers. In the
$[1,1]$ phase, a wave packet without any kick ($\textbf{{q}}=0$) predominantly
overlaps with the $\pi$-gap states in Fig. 4(e) and (f), at and away from the
fine-tuned point, since these edge states form at $\Gamma$ point (c.f. Fig.
2). The corresponding wave packet dynamics display a clear chiral motion for
long times around the edge of the cylinder which is periodic along the
$x$-direction. While some of the probability naturally disperses into the bulk
for the dispersive $[1,1]$ phase (see Fig. 4b), at the fine-tuned point the
edge states are exclusively localised at $A$ sublattices. The probability at
$B$ sites, hence, follows a chiral motion around each hexagon with completely
flat bulk bands, giving rise to some probability dwelling around the initial
position at all times in Fig. 4a.

Due to the large separation of the zero and $\pi$-gap states in the $[1,1]$
phase, we can populate the edge modes at the zero-gap by applying an initial
kick of amount $|\textbf{{q}}|=|K|$ to the wave packet. Fig. 4(c,g) displays a
similar chiral motion with the not-kicked wave packet that is visibly
indistinguishable, which is now mainly supported by the zero-gap states. We
note that the kick direction (along the chiral edge mode or opposite to it) in
fact does not matter as it only gives an overall phase, where the overlaps
with eigenstates do not change and we observe the same dynamics. Moreover,
since these wave packets are well localised in position space, we still obtain
some spurious probability at the other gap, with or without a kick. This
originates from the extensive overlap of the edge modes from the two gaps in
position space (see Fig. 3(a,b)), despite the fact that they are well
separated in momentum. Nevertheless, in both cases, we can control the chiral
motion to be carried predominantly by the target quasienergy-gap modes. This
shows that although these edge modes form at different gaps and despite the
difference in the branch cuts by $\pi$ while defining them, the wave packets
can be controlled to populate mainly a given edge mode by applying appropriate
kicks depending on their localisation in quasimomentum rather than
quasienergy. We present wider wave packets in the Supplementary Material where
the population of a given gap can be further increased, which could be
experimentally realised for example by scanning a wider region with the laser
initialising the wave packets.

On the other hand, in the [-1,1] phase achievable with the second driving
protocol, both zero and $\pi$ gaps harbour edge modes but their separation in
momentum is less pronounced. Both edge modes can then be extensively populated
with the same wave packet as demonstrated in Fig. 4h without applying any
kick. Distinctively, since the winding numbers have opposite signs, we observe
that the wave packet separates into two (see Fig. 4d), with a topologically
protected current going in both directions at the edge of the cylinder, where
the $\pi$-modes travel slightly faster in accordance with the spectrum (c.f.
Fig. 2(c,f)). We emphasise that in this phase, the quasienery bands carry a
Chern number $|C|=2$, which is visible in the difference of the chiral
currents at the edge of the system, rather than two topologically protected
channels travelling along the same direction that would be expected in a
static setting.

Figure 5: Edge dynamics of a wave packet (a) with an initial kick
$\textbf{{q}}=(\pi/\sqrt{3},0)$ in the $[1,0]$ phase for $\omega=2.2J$,
$\Delta=0$, and (b) without a kick in the $[0,1]$ phase for $\omega=2J$,
$\Delta=1J$. Both dynamics in the Haldane phases show qualitative differences
with Fig. 4a-c, as they are less pronounced. (c) Percentage of the total
probability carried by edge states in each gap, along the $\Delta=0$ cut of
Fig. 1a crossing different phases. The edge state population is much higher in
the $[1,1]$ (red shaded area) than in the $[1,0]$ phase (green shaded area),
for a wave packet with an initial kick as in (a) to target zero-gap states.
(d) Similarly, the total probability per gap along a diagonal cut on the phase
diagram crossing from $[1,1]$ to $[0,1]$, where the wave packet is given a
small initial momentum to target the $\pi$-gap states forming close to
$\Gamma$. A greater probability is supported by the two edge channels in the
$[1,1]$ phase along both cuts (c,d), quantifying that edge transport is
overall more robust in the anomalous phase.

The anomalous Floquet topological phases carry distinct features arising from
their out of equilibrium nature. First of all, both edge modes behave
effectively as one single channel in the $[1,1]$ phase due to their similar
chirality. It is, hence, instructive to explore whether this anomalous Floquet
topological phase (with two edge states of the same chirality at different
quasienergy gaps) gives rise to a different edge transport than the Haldane
phases with a single chiral mode, i.e. to contrast anomalous Floquet
topological phases with the equilibrium Chern insulating phases. We
demonstrate the wave packet dynamics in the $[1,0]$ and $[0,1]$ phases in Fig.
5a and b respectively. While we apply a kick ($\textbf{{q}}=K$) to the wave
packet to target the zero-gap states in $[1,0]$ as they form at the $K$ point,
we generally obtain a chiral transport at the edge as expected in both phases.

Most importantly, upon comparing with the Haldane phases as visible in their
color scales, we observe that the wave packet dynamics in the anomalous
Floquet phase $[1,1]$ is much more robust, with the edge separating more from
the bulk. To quantify this finding, we consider cuts on the phase diagram
(Fig. 1a) across different phases and evaluate the total overlap of a wave
packet with the edge states in Fig. 5(c,d). Targeting the zero-gap states with
a kick, we indeed find that the total percentage of the edge state population
is overall much higher in the $[1,1]$ than in the $[1,0]$ phase, where the
former comprises mostly zero-gap states that are further enhanced by the
$\pi$-gap contribution (see Fig. 5c). Similarly, Fig. 5d shows that the wave
packet, now initialised without a kick, overlaps with the $\pi$-gap states
much more in $[1,1]$ than $[0,1]$, where the zero-gap states further
strengthen edge dynamics in the former. We observe that this behaviour is in
general present across the phase diagram and also for the second driving
protocol.

Remarkably, this demonstrates that a wave packet can be prepared to populate
the edge modes more easily and efficiently in the anomalous phase than in
Haldane phases. The anomalous topological phase then supports a more
pronounced chiral edge motion with less leakage into the bulk. The more robust
edge transport in the anomalous Floquet phase stems from the fact that the
edge of the system accommodate two different channels rather than one,
increasing the relative weight of the edge channels compared to the bulk so
that they better separate spatially. Compared to the topological phases with
equilibrium counterparts, there is one more gap available to harbour edge
states in the anomalous Floquet topological phase, which renders stronger edge
transport possible supported with more edge states in the driven system. This
can also aid novel anomalous Floquet Anderson phases under disorder Titum  _et
al._ (2016). We note that this qualitative analysis naturally depends on
system details such as the bulk gap width and the properties of the phase.
Indeed, we also obtain that the total edge population by a wave packet is
overall larger for the $[-1,1]$ phase than the Haldane-like phase (see
Supplementary Material). Although the trend is less pronounced due to
different chiralities and parameter dependencies, the effect is still clearly
discernible when the anomalous phase harbours two edge channels.

Figure 6: In the $[-1,1]$ phase with the parameters in Fig. 2c, the wave
packet is initialised without a kick on a smaller cylinder with $N_{x}=52$,
$N_{y}=41$. (a) Opposite going currents circle the cylinder and cross without
disruption. (b) Two wave packets initialised at the edge pass through each
other without hybridising as they are well separate in quasienergy.

The anomalous Floquet phase $[-1,1]$ manifests appealing features as well with
the edge transport of opposite chiralities supported by the same bands. Since
these edge modes are located at different quasienergy gaps and have support on
different sublattices, they do not hybridize as demonstrated in Fig. 6a. When
we consider a cylinder with a smaller size, we confirm that the opposite going
currents at the edge circle the cylinder and pass through each other without
disruption for several cycles. Similarly, when we introduce two wave packets,
the crossing of the edge currents is clearly visible in Fig. 6b. We note that
introducing two wave packets is an experimentally promising route for probing
these anomalous topological dynamics with opposite chiralities, where
localised particles can be created around circular-shaped sharp edges punched
in a system with laser potentials.

Although the analysis carried out here is generic for any parameters in a
given phase, the particular details of different edge currents depend on
various aspects. For example, we observe some tails developing and travelling
with different velocities in some dynamics. This generally arises when a lot
of momentum states are stimulated due to the finite size of the wave packet,
where the velocity of a given edge mode might be different at different
quasimomenta (c.f. Fig. 2) as well as different velocities that can be
associated to different edge modes. We analyse some important details that can
give rise to varying edge dynamics in a real system next.

##  VI Floquet gauge and sublattice dependence

For the wave packet dynamics presented in previous sections, we numerically
calculate the time evolution by employing the Floquet Hamiltonian given in
Eq.(3). While such stroboscopic definitions are useful, we verify that these
dynamic features are overall reproduced also by following the exact time
evolution under the two driving protocols, justifying the employment of an
effective description neglecting the micromotion Eckardt (2017). Although we
here set the initial time to zero ($t_{0}=0$) for simplicity, the time
evolution operator and hence the Floquet Hamiltonian in Eq.(3) actually depend
on this so-called Floquet gauge,
$\hat{\mathcal{H}}^{t_{0}}_{F}(\textbf{{k}})$, as well as the Floquet
eigenstates Eckardt (2017); Bukov  _et al._ (2015).

Figure 7: (a) Illustration of the relation between the upper/lower edge
dynamics and the sublattice character by considering the fine-tuned point of
the first driving protocol. Starting from $t_{0}=0$ and following the chiral
motion with $J_{1}\rightarrow J_{2}\rightarrow J_{3}$ turned on cyclically, a
particle localised on $A$ sublattice at the upper edge moves along the edge
leftwards (red arrow). Starting from a different initial time of $t_{0}=T/3$
induces a bulk motion around the hexagon (purple). At the lower edge, the
sublattice characters giving rise to the edge and bulk motions are swapped,
where starting from $B$ at $t_{0}=0$ generates edge motion (blue). (b,c)
Dynamics on a cylinder in the $[-1,1]$ phase for the same parameters as in
Fig. 2c, where we localise the wave packet without a kick at the same $x$
position on the upper and lower edges respectively. We average over $20$
equally spaced initial times $t_{0}$ within a period, which produce similar
chiral motion.

The Floquet gauge can influence the wave packet dynamics in experiments
especially at lower frequencies where the details within a period become more
crucial. We elucidate this by considering the extreme limits of a wave packet
completely localised on an $A$ site at the upper edge in Fig. 7a and the
fined-tuned point in the first driving protocol, where we obtain complete
population transfer between sublattices at the end of each step of the drive.
When we start from $t_{0}=0$ with the tunneling amplitudes are turned on/off
cyclically as $J_{1}\rightarrow J_{2}\rightarrow J_{3}$, particles move along
the edge leftwards. However, if we consider another initial time e.g.
$t_{0}=T/3$ within the same sequence (i.e. $J_{2}\rightarrow J_{3}\rightarrow
J_{1}$), particles would follow a chiral bulk motion around the hexagon. Away
from the fine-tuned point, eigenstates mix sublattice flavors at varying
degrees, but the Floquet gauge dependence remains. While this can give rise to
some modifications in exact dynamics observed, it does not alter our general
conclusions. Furthermore, this is naturally less pronounced for more balanced
or wider wave packets.

The upper and lower edges of the system may give rise to different dynamics as
well, since the pseudospin character reflects onto these two physical edges
oppositely. Namely, the chiral edge motion observed at the upper edge when we
start from the sublattice $A$ as depicted in Fig. 7a can be obtained at the
bottom edge starting from the $B$ sublattice (and similarly for the bulk
localised states). When $\Delta=0$, the Hamiltonian can be cast into a
diagonal form in the pseudospin. The up spin in the upper edge correspond to
the down spin at the lower edge, which hence can bring about different
dynamics depending on their population by the wave packet. While in general
for $\Delta\neq 0$, the spin characters are mixed, they are mixed in the same
but opposite way at the upper and lower edges, i.e. some “up-down” mixture on
one end corresponds to the opposite “down-up” mixture on the other end.
Changing the sign of the offset $\Delta\rightarrow-\Delta$ then inverts the
upper/lower edge character, which we confirm in our numerics.

Overall, when we average out over various dynamics, the upper and lower ends
of the system naturally give rise to the same behaviour as it samples through
different initial times and sublattice characters. We consider a wave packet
localised at the same way at the upper and lower edges of a cylinder, at the
same $x$ positions in the geometry depicted in Fig. 7a. We numerically
calculate the chiral edge currents starting from different $t_{0}$ initial
times (for 20 data points equally spaced over one period) which correspond to
different eigenstates. We demonstrate the average wave packet dynamics at the
respective edges in Fig. 7(b,c). While the upper/lower edge dynamics might be
very different at a given $t_{0}$, we observe that the $\pi$-gap states
generate the similar leftward/rightward currents at the top/bottom edges on
average as expected, as well as the zero-gap modes supporting opposite
chirality. We note that the negligible difference between the upper/lower
dynamics in the figure stems from the coarse sampling for averaging, where
more data points wash out these differences further.

##  VII Conclusion

In this work, we focus on the topological edge transport and chiral motion at
the edge of a periodically driven system, particularly in anomalous Floquet
topological phases unique to these out of equilibrium settings Kitagawa  _et
al._ (2010); Rudner  _et al._ (2013). In light of recent developments in
single-site accessibility in optical lattices, which offers the possibility to
probe the dynamics locally at sharp edges, we investigate wave packet dynamics
as a versatile tool in exploring topologically protected Floquet edge modes.
We consider a honeycomb lattice under experimentally relevant and conceptually
insightful step-wise modulated driving protocols Ünal  _et al._ (2019a);
Wintersperger  _et al._ (2020), which allow us to retrieve a rich phase
diagram, involving the conventional Haldane (-like) phases that can be
realised in equilibrium as well as two different anomalous Floquet topological
phases with no static counterparts that harbour edge states of different
chiralities in both quasienergy gaps.

We show that, in the $[1,1]$ phase, the edge modes living in different
quasienergies behave predominantly like a single channel with the same
chirality and sublattice character, where wave packets can be controlled to
mostly populate one of the gaps by applying appropriate kicks as they are well
separate in momentum. While this behaviour is similar to the Haldane phases
with a topologically protected edge mode present only in a single gap Haldane
(1988), we find that the anomalous $[1,1]$ phase can generally support a more
robust edge transport with edge states separating more from the bulk. We
explain this with having two different channels coming from both gaps
accommodated at the edge of the system, which enhances the population of the
edge modes by the wave packets, providing an overall advantage over Haldane
phases with equilibrium counterparts.

We further show a hitherto-unexplored anomalous phase with opposite winding
numbers ($[-1,1]$) that can be achieved using the second driving protocol in a
honeycomb lattice, where the system features both clockwise and anti-clockwise
currents now carried mainly by different sublattices at a given edge. Since
the edge modes of opposite chiralities form much closer in momentum, we
observe that wave packets localised at the edge largely populate both
channels, which generates two independent currents going in opposite ways that
can transverse through one another without hybridising. We also analyse the
dependence of the dynamics on the details of the drive and the exact position
where the wave packets are localised. We discuss that these effects can be
controlled by changing the shape of the wave packet or averaging over
different initial times. Our results demonstrate that investigating Floquet
topological edge modes by using wave packets in optical lattices can reveal
unique out-of-equilibrium features. These insights can be also employed in
phases involving more number of edge states and with different chirality
combinations Zhang  _et al._ (2022), which might reveal more interesting edge
dynamics and more robust edge transport engaging multiple edge channels.
Photonic lattices employing quantum walks offer another promising route to
study various anomalous Floquet phases with different winding number
combinations Adiyatullin  _et al._ (2023).

Note added–We have became aware of a relevant experimental work during the
completion of this manuscript, which has been announced following our work in
Ref. Braun  _et al._ (2023), studying and probing wave packet dynamics in an
anomalous Floquet topological phase.

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

###### Acknowledgements.

We thank Nigel Cooper and Robert-Jan Slager, for insightful discussions and
comments on the manuscript, as well as the experimental team Monika
Aidelsburger, Raphaël Saint-Jalm, Christoph Braun, Alexander Hesse and
Johannes Arceri. F.N.Ü. acknowledges funding from the Royal Society under a
Newton International Fellowship, the Marie Skłodowska-Curie programme of the
European Commission Grant No 893915, and Trinity College Cambridge. This work
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation program (Grant Agreement No.
101001902).

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# A problem for Hankel measures on Hardy space

Guanlong Bao and Fangqin Ye  Guanlong Bao
Department of Mathematics
Shantou University
Shantou, Guangdong 515063, China<EMAIL_ADDRESS>Fangqin Ye
Business School
Shantou University
Shantou, Guangdong 515063, China<EMAIL_ADDRESS>
###### Abstract.

In this note, we investigate a condition related to the characterization of
Hankel measures on Hardy space. We address a problem mentioned by J. Xiao in
2000.

###### Key words and phrases:

Hankel measure; Hardy space; Hankel matrix.

###### 2010 Mathematics Subject Classification:

30H10, 47B35

The work was supported by NNSF of China (No. 11720101003 and No. 11571217),
Department of Education of Guangdong Province (No. 2017KQNCX078) and STU
Scientific Research Foundation for Talents (No. NTF17020 and No. STF17005).

##  1\. Introduction

Let $\mathbb{D}$ be the open unit disk in the complex plan $\mathbb{C}$ and
let $H(\mathbb{D})$ be the space of analytic functions in $\mathbb{D}$. Recall
that for $0<p<\infty$ the Hardy space $H^{p}$ consists of functions $f\in
H(\mathbb{D})$ such that

$\|f\|_{H^{p}}=\sup_{0<r<1}\left(\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}d\theta\right)^{1/p}<\infty.$

It is well known that every function $f\in H^{p}$ has non-tangential limits
$f(\zeta)$ for almost every $\zeta$ on the unit circle $\mathbb{T}$. See [8]
for the theory of Hardy spaces.

Let $BMOA$ be the space of analytic functions of bounded mean oscillation. It
is well known (cf. [4, 9]) that the space $BMOA$ can be defined as the set of
functions $f\in H(\mathbb{D})$ satisfying

$\|f\|_{BMOA}^{2}=\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}|f^{\prime}(z)|^{2}(1-|\sigma_{a}(z)|^{2})dA(z)<\infty,$

where

$dA(z)=\frac{1}{\pi}dxdy=\frac{r}{\pi}drd\theta,\ \ z=x+iy=re^{i\theta},$

and

$\sigma_{a}(z)=\frac{a-z}{1-\overline{a}z}$

is the Möbius transformation of $\mathbb{D}$ interchanging $a$ and $0$. The
Fefferman-Stein duality theorem tells us that the dual space of $H^{1}$ is
$BMOA$. Also, $BMOA$ is a proper subset of the Bloch space $\mathcal{B}$
consisting of functions $f\in H(\mathbb{D})$ for which

$\|f\|_{\mathcal{B}}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|f^{\prime}(z)|<\infty.$

Given an arc $I$ of $\mathbb{T}$ with arclength $|I|$, the Carleson box $S(I)$
is

$S(I)=\left\\{r\zeta\in\mathbb{D}:1-\frac{|I|}{2\pi}<r<1,\ \zeta\in
I\right\\}.$

A complex measure $\mu$ on $\mathbb{D}$ is called a Carleson measure if

$\sup_{I\subseteq\mathbb{T}}\frac{|\mu|(S(I))}{|I|}<\infty.$

It is well known (cf. [8] ) that $\mu$ is a Carleson measure if and only if
there exists a positive constant $C$ such that

$\int_{\mathbb{D}}|f(z)|^{2}d|\mu|(z)\leq C\|f\|^{2}_{H^{2}}$

for all $f\in H^{2}$.

Following J. Xiao [15], a complex measure $\mu$ on $\mathbb{D}$ is said to be
a Hankel measure  if there exists a positive constant $C$ such that

(1.1) $\left|\int_{\mathbb{D}}f^{2}(z)d\mu(z)\right|\leq C\|f\|^{2}_{H^{2}}$

for all $f$ in $H^{2}$. It is clear that if $\mu$ is a Hankel measure, then
$|\mu(\mathbb{D})|<\infty$. The name of Hankel measure has its root in the
study of Hankel matrices (see also [11, 14]). Clearly, a Carleson measure must
be a Hankel measure. Answering a problem posed by J. Peetre, J. Xiao [15] gave
a series of complete characterizations of Hankel measures involving Carleson
measures, the duality between $H^{1}$ and $BMOA$, and Hankel operators.

We refer to [1, p. 11] and [16] for the study of complex measure inequalities
similar to (1.1) in the setting of the classical Dirichlet space and weighted
Bergman spaces, respectively.

From [15], a necessary condition for a complex measure $\mu$ on $\mathbb{D}$
to be a Hankel measure is

(1.2)
$\sup_{w\in\mathbb{D}}\left|\int_{\mathbb{D}}\frac{1-|w|^{2}}{(1-\overline{w}z)^{2}}d\mu(z)\right|<\infty.$

By this observation, J. Xiao [15, p. 139] mentioned the following problem: Is
condition (1.2) sufficient for $\mu$ to be a Hankel measure too? In this note,
we show that in general the answer to this problem is negative. We give a
complex measure $\mu$ on $\mathbb{D}$ satisfying condition (1.2) but $\mu$ is
not a Hankel measure. A positive Borel measure $\mu$ on [0, 1) can be seen as
a Borel measure on $\mathbb{D}$ by identifying it with the measure
$\tilde{\mu}$ defined by

$\tilde{\mu}(E)=\mu(E\cap[0,1)),$

for any Borel subset $E$ of $\mathbb{D}$. If $\mu$ is a positive Borel measure
on [0, 1), we obtain that condition (1.2) holds if and only if $\mu$ is a
Hankel measure, which also implies some new information of a Hankel matrix
acting on Hardy spaces and Dirichlet type spaces.

##  2\. Condition (1.2) and Hankel measures

The section is devoted to consider whether condition (1.2) is an equivalent
description of Hankel measure. For a complex measure $\mu$ on $\mathbb{D}$,
the function $P_{\mu}$ is given by

$P_{\mu}(w)=\int_{\mathbb{D}}\frac{1}{1-w\overline{z}}d\mu(z),\ \
w\in\mathbb{D}.$

We will show that the problem considered in this note is equivalent to
investigate those functions $P_{\overline{\mu}}\in\mathcal{B}$ must actually
satisfying $P_{\overline{\mu}}\in BMOA$.

There exist several complete descriptions of Hankel measures in [15]. Now we
cite only some of them as follows.

######  Theorem A.

Let $\mu$ be a complex measure on $\mathbb{D}$. Then the following conditions
are equivalent.

  1. (1)

$\mu$ is a Hankel measure.

  2. (2)

There exists a positive constant $C$ such that

$\left|\int_{\mathbb{D}}f(z)d\mu(z)\right|\leq C\|f\|_{H^{1}},$

for all $f\in H^{1}$.

  3. (3)

$P_{\overline{\mu}}$ is in $BMOA$.

  4. (4)

$\sup_{I\subseteq\mathbb{T}}\frac{1}{|I|}\int_{S(I)}\left|\int_{\mathbb{D}}\frac{\overline{z}d\overline{\mu}(z)}{(1-w\overline{z})^{2}}\right|^{2}(1-|w|^{2})dA(w)<\infty.$

The following theorem characterizes lacunary series in the Bloch space
$\mathcal{B}$ and $BMOA$ (see [9] and [12]).

######  Theorem B.

Let $f\in H(\mathbb{D})$ with the power series expansion
$f(z)=\sum_{k=1}^{\infty}a_{k}z^{n_{k}}$ and suppose there exists $\lambda>1$
such that $n_{k+1}\geq\lambda n_{k}$ for all $k$. Then the following
assertions hold.

  1. (1)

$f\in\mathcal{B}$ if and only if the sequence $\\{a_{k}\\}$ is bounded.

  2. (2)

$f\in BMOA$ if and only if

$\sum_{k=1}^{\infty}|a_{k}|^{2}<\infty.$

In 1995, R. Aulaskari, J. Xiao and R. Zhao [3] introduced $\mathcal{Q}_{p}$
spaces which attracted a lot of attention in recent years. For $0<p<\infty$,
the space $\mathcal{Q}_{p}$ consists of functions $f\in H(\mathbb{D})$ with

$\|f\|_{\mathcal{Q}_{p}}^{2}=\sup_{a\in\mathbb{D}}\,\int_{\mathbb{D}}|f^{\prime}(z)|^{2}\left(1-|\sigma_{a}(z)|^{2}\right)^{p}dA(z)<\infty.$

Clearly, $\mathcal{Q}_{1}=BMOA$. By [2], we know that for $1<p<\infty$, all
$\mathcal{Q}_{p}$ spaces are the same and equal to the Bloch space
$\mathcal{B}$. See J. Xiao’s monographs [17, 18] for more results of
$\mathcal{Q}_{p}$ spaces.

The following result can be founded in [17, p. 29].

######  Theorem C.

Suppose $0<p<\infty$. Let $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ with
$\\{a_{n}\\}$ being a decreasing sequence of nonnegative numbers. Then
$f\in\mathcal{Q}_{p}$ if and only if $a_{n}=O(\frac{1}{n})$.

We first show that the problem considered in this note is associated with
certain self-improving property of functions $P_{\overline{\mu}}$.

######  Theorem 2.1.

Let $\mu$ be a complex measure on $\mathbb{D}$. Then the following two
statements are equivalent.

  1. (1)

If $\mu$ satisfies condition (1.2), then $\mu$ is a Hankel measure.

  2. (2)

If $P_{\overline{\mu}}\in\mathcal{B}$, then $P_{\overline{\mu}}\in BMOA$.

###### Proof.

For $w\in\mathbb{D}$, one gets

$\displaystyle wP_{\overline{\mu}}(w)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{D}}\frac{w}{1-w\overline{z}}d\overline{\mu}(z)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\int_{\mathbb{D}}\overline{z}^{n}d\overline{\mu}(z)\right)w^{n+1}.$

Then

$\displaystyle\sup_{w\in\mathbb{D}}(1-|w|^{2})|\left(wP_{\overline{\mu}}(w)\right)^{\prime}|$
$\displaystyle=$
$\displaystyle\sup_{w\in\mathbb{D}}(1-|w|^{2})\left|\sum_{n=0}^{\infty}(n+1)\left(\int_{\mathbb{D}}\overline{z}^{n}d\overline{\mu}(z)\right)w^{n}\right|$
$\displaystyle=$
$\displaystyle\sup_{w\in\mathbb{D}}\left|\int_{\mathbb{D}}\frac{1-|w|^{2}}{(1-w\overline{z})^{2}}d\overline{\mu}(z)\right|$
$\displaystyle=$
$\displaystyle\sup_{w\in\mathbb{D}}\left|\int_{\mathbb{D}}\frac{1-|w|^{2}}{(1-\overline{w}z)^{2}}d\mu(z)\right|.$

Consequently, the function $w\rightarrow wP_{\overline{\mu}}(w)$ belongs to
$\mathcal{B}$ if and only if condition (1.2) holds for $\mu$.

By the growth estimates of functions in $\mathcal{B}$ (see [17, p. 6] for
example), one gets that there exists a positive $C$ such that

$|f(z)-f(0)|\leq C\|f\|_{\mathcal{B}}\log\frac{1}{1-|z|}$

for all $f\in\mathcal{B}$. Bearing in mind this estimate, we see that
$P_{\overline{\mu}}\in\mathcal{B}$ if and only if the function $w\rightarrow
wP_{\overline{\mu}}(w)$ belongs to $\mathcal{B}$.

Thus we proved that $P_{\overline{\mu}}\in\mathcal{B}$ if and only if $\mu$
satisfies condition (1.2). By Theorem A, $\mu$ is a Hankel measure if and only
if $P_{\overline{\mu}}\in BMOA$. These imply the desired result. ∎

The following result shows that in general the answer to J. Xiao’s problem is
negative.

######  Proposition 2.2.

There exists a complex measure $\mu$ on $\mathbb{D}$ such that condition (1.2)
holds but $\mu$ is not a Hankel measure.

###### Proof.

Set $d\overline{\mu}(z)=f(z)dA(z)$ for $z\in\mathbb{D}$, where
$f(z)=1+\sum_{k=1}^{\infty}(1+2^{k})z^{2^{k}}$. Then
$\overline{\mu}(\mathbb{D})=1$. By the Parseval formula, for any positive
integer $n$,

$\displaystyle\int_{\mathbb{D}}\overline{z}^{n}d\overline{\mu}(z)$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{1}r^{n+1}dr\int_{0}^{2\pi}e^{-in\theta}[1+\sum_{k=1}^{\infty}(1+2^{k})r^{2^{k}}e^{i2^{k}\theta}]d\theta$
$\displaystyle=$ $\displaystyle\begin{cases}&\enspace 0,\ \ \text{if}\ \
n\not\in\\{2^{k}:k=1,2,3,\cdots\\},\\\ &\enspace 1,\ \ \text{if}\ \
n\in\\{2^{k}:k=1,2,3,\cdots\\}.\end{cases}$

Hence, for any $w\in\mathbb{D}$,

$\displaystyle P_{\overline{\mu}}(w)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\int_{\mathbb{D}}\overline{z}^{n}d\overline{\mu}(z)\right)w^{n}$
$\displaystyle=$ $\displaystyle 1+\sum_{k=1}^{\infty}w^{2^{k}}.$

By Theorem B, $P_{\overline{\mu}}\in\mathcal{B}$ but
$P_{\overline{\mu}}\not\in BMOA$. Via Theorem A and the proof of Theorem 2.1,
we see that the measure $\mu$ satisfies condition (1.2), but $\mu$ is not a
Hankel measure. ∎

For a positive Borel measure $\mu$ on [0, 1) and a nonnegative integer $n$,
denote by $\mu[n]$ the moment of order $n$ of $\mu$, that is,

$\mu[n]=\int_{0}^{1}t^{n}d\mu(t).$

The following result shows that there exist measures $\mu$ such that if
$P_{\overline{\mu}}\in\mathcal{B}$, then $P_{\overline{\mu}}\in BMOA$. It
follows from Theorem 2.1 that in this case $\mu$ is a Hankel measure if and
only if $\mu$ satisfies condition (1.2).

######  Theorem 2.3.

Let $\mu$ be a positive Borel measure on [0, 1). Then the following conditions
are equivalent.

  1. (1)

$\mu$ is a Hankel measure.

  2. (2)

$\mu$ satisfies condition (1.2), that is,

$\sup_{w\in\mathbb{D}}\left|\int_{0}^{1}\frac{1-|w|^{2}}{(1-\overline{w}t)^{2}}d\mu(t)\right|<\infty.$

  3. (3)

$P_{\mu}\in\mathcal{Q}_{p}$ for some $0<p<\infty$.

  4. (4)

$P_{\mu}\in\mathcal{Q}_{p}$ for all $0<p<\infty$.

  5. (5)

$\mu[n]=O(\frac{1}{n})$.

###### Proof.

If $\mu$ is a positive Borel measure on [0, 1), then for $w\in\mathbb{D}$,

$P_{\overline{\mu}}(w)=P_{\mu}(w)=\sum_{n=0}^{\infty}\left(\int_{0}^{1}t^{n}d\mu(t)\right)w^{n}=\sum_{n=0}^{\infty}\mu[n]w^{n}.$

Clearly, $\left\\{\mu[n]\right\\}$ is a decreasing sequence with nonnegative
numbers. By Theorem C, we see that (3)$\Leftrightarrow$(4)$\Leftrightarrow$
(5). By the proof of Theorem 2.1, $P_{\overline{\mu}}\in\mathcal{B}$ if and
only if $\mu$ satisfies condition (1.2). From Theorem A,
$P_{\overline{\mu}}\in BMOA$ if and only if $\mu$ is Hankel measure. Bear in
mind that $\mathcal{B}=\mathcal{Q}_{2}$ and $BMOA=\mathcal{Q}_{1}$. Thus both
(1) and (2) are equivalent to (5). The proof is complete. ∎

Let $\mu$ be a positive Borel measure on [0, 1). Denote by $\mathcal{H}_{\mu}$
the Hankel matrix $(\mu[n+k])_{n,k\geq 0}$. This matrix induces formally an
operator, denoted also by $\mathcal{H}_{\mu}$, on $H(\mathbb{D})$ in the
following sense. For $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\in H(\mathbb{D})$, by
multiplication of the matrix with the sequence of Taylor coefficients of the
function $f$,

$\\{a_{n}\\}_{n\geq
0}\mapsto\left\\{\sum_{k=0}^{\infty}\mu[n+k]a_{k}\right\\}_{n\geq 0},$

define

$\mathcal{H}_{\mu}(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\mu[n+k]a_{k}\right)z^{n}.$

If $\mu$ is the Lebesgue measure on [0, 1), the matrix $\mathcal{H}_{\mu}$
reduces to the classical Hilbert matrix $\mathcal{H}=((n+k+1)^{-1})_{n,k\geq
0}$, which induces the classical Hilbert operator $\mathcal{H}$. The Hankel
matrix $\mathcal{H}_{\mu}$ acting on $H^{2}$ was studied in [13, 14]. For all
Hardy spaces $H^{p}$, the operator $\mathcal{H}_{\mu}$ was investigated in [6,
10]. The Dirichlet type space $\mathcal{D}_{\alpha}$, $\alpha\in\mathbb{R}$,
consists of functions $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\in H(\mathbb{D})$
for which

$\|f\|_{\mathcal{D}_{\alpha}}^{2}=\sum_{n=0}^{\infty}(n+1)^{1-\alpha}|a_{n}|^{2}<\infty.$

The Hankel matrix $\mathcal{H}_{\mu}$ acting on $\mathcal{D}_{\alpha}$ spaces
was considered in [5, 7]. It is worth referring to [10] for the recent
development of $\mathcal{H}_{\mu}$ acting on spaces of analytic functions. The
following theorem is from the related literatures mentioned before.

######  Theorem D.

Let $\mu$ be a positive Borel measure on [0, 1). Suppose $1<p<\infty$ and
$0<\alpha<2$. Then the following conditions are equivalent.

  1. (1)

$\mu$ is a Carleson measure.

  2. (2)

$\sup_{w\in\mathbb{D}}\int_{0}^{1}\frac{1-|w|^{2}}{|1-\overline{w}t|^{2}}d\mu(t)<\infty.$

  3. (3)

$\mu[n]=O(\frac{1}{n})$.

  4. (4)

$\mathcal{H}_{\mu}$ is bounded on $H^{p}$.

  5. (5)

$\mathcal{H}_{\mu}$ is bounded on $\mathcal{D}_{\alpha}$.

Combining Theorem 2.3 with Theorem D, we obtain immediately new
characterizations of a Hankel matrix $\mathcal{H}_{\mu}$ acting on Hardy
spaces and Dirichlet type spaces in terms of Hankel measures and
$\mathcal{Q}_{p}$ functions.

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