# Hybrid-order topological odd-parity superconductors via Floquet engineering

Hong Wu* — Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou
University, Lanzhou 730000, China  Lanzhou Center for Theoretical Physics, Key
Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou 730000, China  Jun-Hong An* —<EMAIL_ADDRESS>Key Laboratory of
Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou 730000,
China  Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China

###### Abstract

Having the potential for performing quantum computation, topological
superconductors have been generalized to the second-order case. The
hybridization of different orders of topological superconductors is attractive
because it facilitates the simultaneous utilization of their respective
advantages. However, previous studies found that they cannot coexist in one
system due to the constraint of symmetry. We propose a Floquet engineering
scheme to generate two-dimensional (2D) hybrid-order topological
superconductors in an odd-parity superconductor system. Exotic hybrid-order
phases exhibiting the coexisting gapless chiral edge states and gapped
Majorana corner states not only in two different quasienergy gaps but also in
one single quasienergy gap are created by periodic driving. The generalization
of this scheme to the 3D system allows us to discover a second-order Dirac
superconductor featuring coexisting surface and hinge Majorana Fermi arcs. Our
results break a fundamental limitation on topological phases and open a
feasible avenue to realize topological superconductor phases without static
analogs.

##  I Introduction

As one of the central fields in modern physics, topological phases of matter
not only enrich the paradigm of condensed matter physics and stimulate the
discovery of many novel quantum materials, but also generate various
fascinating applications in quantum technologies [1, 2, 3, 4, 5]. In the
family of topological phases, the topological superconductor has attracted
wide attention. It successfully simulates the mysterious Majorana fermion in a
condensed matter system [6]. It also may be a promising application in
realizing quantum computation due to its non-Abelian statistics [7, 8, 9].
Recently, there is an intense interest in extending the traditional
topological phases to higher-order ones [10, 11, 12, 13]. Being parallel to
the theoretical proposals [14, 15, 16, 17, 18, 19] and experimental
observations [20, 21, 22, 23, 24, 25, 26, 27, 28, 29] on second-order
topological insulators, second-order topological superconductors (SOTSCs) have
been proposed [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. This opens
another avenue toward topological quantum computation [43, 44, 45, 46].

Explorations of physical systems supporting exotic topological features and of
efficient ways to control these features are not only in the mainstream of
condensed matter physics, but are also a demand of quantum technologies.
Symmetry plays an important role in realizing SOTSCs [30, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42]. It is conventionally believed that SOTSCs are
achieved by applying an even-parity term in an odd-parity superconductor to
break the symmetry [47, 43, 48, 49]. Recently, a general approach to realize
SOTSCs in odd-parity superconductors was proposed [50]. An interesting
question is whether the first-order and SOTSCs can coexist in one system.
First, this kind of hybrid-order topological superconductor (HOTSC)
facilitates the simultaneous utilization of the advantages of both corner
states and gapless chiral edge states in designing bifunctional devices [27].
Second, it enriches the family of topological phases and leads to a different
band theory. Nevertheless, according to the general view, first- and second-
order topologies exist in different systems [51, 52, 48] due to their
substantially different features in the energy spectrum at the boundaries [51,
52, 53, 48, 54]. Therefore, it seems that a HOTSC is impossible to realize in
a system under the framework of traditional topological energy-band theory. On
the other hand, coherent control via periodic driving, referred to as Floquet
engineering, has become a versatile tool in artificially creating novel
topological phases in systems of ultracold atoms [55, 56], photonics [57, 58],
superconductor qubits [59], and graphene [60]. Many exotic phases absent in
static systems have been controllably generated by periodic driving [61, 62,
63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74]. One of the interesting
findings is that periodic driving can cause different high-symmetry points to
have different topological charges [75], which lays the foundation to realize
different types of topological phases [76]. Is it possible to realize HOTSCs
by periodic driving?

Addressing these questions, we propose a general scheme to create a two-
dimensional (2D) HOTSC in a two-band odd-parity superconductor system by
Floquet engineering. A complete topological characterization to such a Floquet
quasienergy band structure is established. We discover two kinds of HOTSCs,
where in one of them the coexisting gapless chiral edge states and the gapped
Majorana corner states reside in two different quasienergy gaps, and in the
other they reside in one common gap. The generalization to the three-
dimensional (3D) odd-parity superconductor system reveals another exotic
phase, i.e., a second-order Dirac superconductor, which features coexisting
surface and hinge Majorana Fermi arcs. Breaking the conventional constraint on
realizing hybrid-order topology, our result reveals Floquet engineering as a
useful way to explore different topological phases.

##  II Static system

Conventional studies suggested that superconductors with appropriate mixed-
parity pairings are candidates for SOTSCs [47, 48]. Reference [50] presented a
scheme to realize SOTSCs in purely odd-parity pairing superconductors. We use
this static system as the starting point of our Floquet engineering. We
consider a $p$-wave superconductor whose Hamiltonian reads
$\hat{H}_{1}=\frac{1}{2}\sum_{\mathbf{k}}\hat{\Psi}^{{\dagger}}_{\mathbf{k}}\mathcal{H}_{1}(\mathbf{k})\hat{\Psi}_{\mathbf{k}}$,
with
$\hat{\Psi}_{\mathbf{k}}^{{\dagger}}=(\hat{c}_{\mathbf{k}}^{{\dagger}},\hat{c}_{-\mathbf{k}})$
and

$\mathcal{H}_{1}(\mathbf{k})=\left(\begin{array}[]{ccc}\epsilon(\mathbf{k})&\Delta(\mathbf{k})\\\
\Delta^{*}(\mathbf{k})&-\epsilon(\mathbf{k})\\\ \end{array}\right),$ (1)

where $\epsilon(\mathbf{k})$ is the kinetic energy of the normal state and
$\Delta({\bf k})$ is the pairing potential of the superconductor. Equation (1)
has spatial inversion symmetry $\mathcal{P}=\sigma_{z}$. We focus on the
inversion symmetric normal state $\epsilon(\mathbf{k})$ and the odd-parity
pairings satisfying $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$. Via a Hopf map
[77], a SOTSC model which is trivial in the first-order topology in this odd-
parity superconductor is constructed as $\Delta({\bf k})=2(\cos
k_{x}+\lambda_{x}+i\cos k_{y}+i\lambda_{y})(\sin k_{x}-i\sin k_{y})$ and
$\epsilon({\bf k})=\sum_{j=x,y}[(\cos k_{j}+\lambda_{j})^{2}-\sin^{2}k_{j}]$.
The SOTSC is formed when $|\lambda_{x,y}|<1$, where four gapped zero-mode
Majorana corner states are formed [see Figs. 1(a) and 1(b)]. Different from
conventional topological phases, this SOTSC is not caused by the closing and
reopening of the bulk-band gap. The energy spectra under the $x$-direction
open boundary condition in Figs. 1(b)-1(d) indicate that, when the parameter
runs from the SOTSC to the topologically trivial regimes, the bulk-band gap is
persistently open, however, the boundary-state gap occurs a closing and
reopening. Therefore, we cannot find a topological invariant of the bulk bands
to characterize the SOTSC. This type of SOTSC is referred to as the boundary
obstructed topological phase [54]. A counterexample arises when
$\lambda_{x}=\lambda_{y}$, where the system has a mirror-rotation symmetry
$\sigma_{y}\mathcal{H}_{1}(k_{x},k_{y})\sigma_{y}=-\mathcal{H}_{1}(k_{y},k_{x})$.
Its topology can be described by the mirror-graded winding number defined in
the bulk bands as $\mathcal{W}=\frac{i}{2\pi}\int_{0}^{2\pi}{\langle
u(k)\lvert\partial_{k}\lvert u(k)\rangle}dk$, where $\lvert u(k)\rangle$ is
the eigenstate of $\mathcal{H}_{1}({k})$, along the high-symmetry line
$k_{x}=k_{y}\equiv k$ [see Fig. 1(a)].

Figure 1: (a) Phase diagram in the $\lambda_{x}$-$\lambda_{y}$ plane. The red
and white lines mark the regimes with $\mathcal{W}=1$ and $0$, respectively.
Energy spectra under the $x$-direction open boundary condition when (b)
$\lambda_{y}=0.8$, (c) 1.0, and (d) 1.2. The inset in (a)-(c) shows the energy
spectrum under the $x$\- and $y$-direction open boundary condition. We use
$\lambda_{x}=0.3$ and lattice sizes $L_{x}=L_{y}=50$.

##  III Floquet engineering

To generate HOTSC, we apply a periodic driving

$\mathcal{H}({\bf k},t)=\mathcal{H}_{1}({\bf k})+\mathcal{H}_{2}({\bf
k})\delta(t/T-n),$ (2)

where $\mathcal{H}_{2}=m\sigma_{z}$, $T$ is the driving period, and $n$ is an
integer. The periodic system $\mathcal{H}({\bf k},t)=\mathcal{H}({\bf k},t+T)$
does not have an energy spectrum because its energy is not conserved.
According to the Floquet theorem, the one-period evolution operator
$U(T)=\mathbb{T}\exp[-i\int_{0}^{T}\mathcal{H}({\bf k},\tau)d\tau]$ defines
$\mathcal{H}_{\text{eff}}({\bf k})\equiv iT^{-1}\ln U(T)$ whose eigenvalues
are called quasienergies [75]. The topological phases of our system are
defined in such a quasienergy spectrum. Applying the Floquet theorem to Eq.
(2), we have $\mathcal{H}_{\text{eff}}({\bf
k})=iT^{-1}\ln[e^{-i\mathcal{H}_{2}({\bf k})T}e^{-i\mathcal{H}_{1}({\bf
k})T}]$ [78]. Different from the static case, our topological phase transition
occurs not only at zero quasienergy but also at $\pi/T$ due to the periodicity
of the quasienergies, which may cause the Floquet anomaly in pure-order
topological phases [79].

We first focus on Floquet engineering when
$\lambda_{x}=\lambda_{y}\equiv\lambda$, where the phases are characterized by
the bulk bands. First, we can derive from $\mathcal{H}_{\text{eff}}({\bf k})$
that the phase transition occurs when the parameters satisfy either

$\displaystyle 2(1-\lambda^{2})-m=n\pi/T$ (3)
$\displaystyle\text{or}~{}~{}2[1+\lambda^{2}+\lambda(e^{i\alpha_{x}}+e^{i\alpha_{y}})]+m=n_{\alpha_{x},\alpha_{y}}\pi/T$
(4)

at the quasienergy zero ($\pi/T$) for even (odd) $n$ and
$n_{\alpha_{x},\alpha_{y}}$, with $\alpha_{x,y}=0$ or $\pi$, (see Appendix A).
Giving the phase boundaries of our system, Eqs. (3) and (4) offer us
sufficient space to artificially synthesize exotic topological phases absent
in the static system (1) by periodic driving. Second, we can establish a
complete topological characterization on $\mathcal{H}_{\text{eff}}({\bf k})$
to the rich emergent topological phases in our periodic system. Besides the
intrinsic SOTSC in the static case, our periodic driving also induces a first-
order Chern superconductor phase. We have to find appropriate topological
invariants to characterize the first- and second-order superconductor phases
occurring at both zero and $\pi/T$ quasienergies. The overall topology of the
first- and second-order phases at the $\alpha/T$ quasienergy, with $\alpha=0$
or $\pi$, is described by
$\mathcal{W}_{\alpha/T}=(\mathcal{W}_{1}+e^{i\alpha}\mathcal{W}_{2})/2$, where
$\mathcal{W}_{j}$ are the mirror-graded winding numbers defined in
$\tilde{\mathcal{H}}_{j}({\bf k})=iT^{-1}\ln[G_{j}U(T)G_{j}^{\dagger}]$, with
$G_{j}=e^{i(-1)^{j}\mathcal{H}_{j}({\bf k})T/2}$, along the high-symmetry line
$k_{x}=k_{y}=k$ [80]. The first-order topology at the $\alpha/T$ quasienergy
is described the dynamical winding number [81, 82]

$\displaystyle\mathcal{V}_{\alpha/T}=\int_{0}^{T}dt\frac{d^{2}{\bf
k}}{24\pi^{2}}\epsilon_{ijk}{\rm
Tr}(\mathcal{U}^{{\dagger}}_{\alpha}\partial_{i}\mathcal{U}_{\alpha}\mathcal{U}^{{\dagger}}_{\alpha}\partial_{j}\mathcal{U}_{\alpha}\mathcal{U}^{{\dagger}}_{\alpha}\partial_{k}\mathcal{U}_{\alpha}).~{}~{}$
(5)

Here $\epsilon_{ijk}$, with $\\{i,j,k\\}\in\\{t,k_{x},k_{y}\\}$, is the
completely antisymmetric tensor and a sum over repeated indices has been used,
$\mathcal{U}_{\alpha}=U(t)\sum_{l=1}^{2}e^{i\mathcal{E}^{(\alpha)}_{l,{\bf
k}}t}|u_{l}({\bf k},T)\rangle\langle u_{l}({\bf k},T)|$, where
$\mathcal{E}^{(\alpha)}_{l,\mathbf{k}}$ is the quasienergy in the $l$th band
chosen in the regime $[\alpha/T,(\alpha+2\pi)/T)$ and $|u_{l}({\bf
k},T)\rangle$ is the corresponding Floquet eigenstate. The number of the
first-order gapless edge states and the second-order gapped Majorana corner
states at the $\alpha/T$ quasienergy is equal to $|\mathcal{V}_{\alpha/T}|$
pairs and $2(|\mathcal{W}_{\alpha/T}|-|\mathcal{V}_{\alpha/T}|)$,
respectively.

Figure 2: Phase diagram in the $\lambda$-$m$ plane characterized by
$\mathcal{W}_{\alpha/T}$ and $\mathcal{V}_{\alpha/T}$ ($\alpha=0,\pi$). The
black dotted-dashed line is from Eq. (3) with $n=0$. The red dashed, blue
dotted, and white solid lines are from Eq. (4) with $n_{0,0}=1$, $2$, and
$n_{\pi,\pi}=1$, respectively. NSCα/T denotes the topologically trivial normal
superconductor with $\mathcal{W}_{\alpha/T}=\mathcal{V}_{\alpha/T}=0$, CSCα/T
denotes the first-order Chern superconductors with
$|\mathcal{W}_{\alpha/T}|=1$ and $|\mathcal{V}_{\alpha/T}|=1$, and SOTSCα/T is
the SOTSC with $|\mathcal{W}_{\alpha/T}|=2$ and $|\mathcal{V}_{\alpha/T}|=0$.
We use $T=1$.

To reveal the topological characters of our periodic system, we plot in Fig. 2
the phase diagram characterized by $\mathcal{W}_{\alpha/T}$ and
$\mathcal{V}_{\alpha/T}$ ($\alpha=0,\pi$). We find that the system exhibits
more colorful topological phases than the static one in Fig. 1(a). Its
topological features are not only carried by the quasienergy gaps at both zero
and $\pi/T$, but also become diverse for a given gap. We discover that rich
phases, which may be any possible pair combination of the topologically
trivial normal superconductor, first-order Chern topological superconductor,
and SOTSC coexisting in the two quasienergy gaps, are created by periodic
driving. Each phase boundary perfectly obeys either Eq. (3) or (4). The result
indicates that, permitting us to realize diverse exotic HOTSCs absent in its
static counterpart, periodic driving supplies us with another dimension to
change the topology orders and realize different topological phases at will.

Figure 3: (a) Quasienergy spectrum, (b) $\mathcal{W}_{\alpha/T}$, (c)
$\mathcal{V}_{\alpha/T}$, and probability distributions of the (d) zero- and
$\pi/T$-mode states in different $m$. We use $T=1$ and $\lambda=0.3$.

Figure 3(a) shows the quasienergy spectrum in different $m$ for a given
$\lambda$. We observe three typical regimes from the zero quasienergy, i.e.,
the regimes with gapped zero-mode states when $m<1.82$, with an open zero-mode
gap when $1.82<m<2.9$, and with a closed gap when $m\geq 2.9$. The topological
invariants $\mathcal{W}_{0}$ and $\mathcal{V}_{0}$ in Figs. 3(b) and 3(c)
reveal that these regimes correspond exactly to the SOTSC with four Majorana
corner states, the normal superconductor, and the first-order Chern
superconductor with one pair of gapless edge states, respectively. At the
$\pi/T$ quasienergy, the gap is closed when $m\leq 2.16$ and the gapped
$\pi/T$-mode states are formed when $m>2.16$. The two regimes, according to
$\mathcal{W}_{\pi/T}$ and $\mathcal{V}_{\pi/T}$ in Figs. 3(b) and 3(c),
correspond to the first-order Chern superconductor and SOTSC, respectively.
The features in this quasienergy spectrum match well with the phase diagram in
Fig. 2. We show the probability distributions of the zero- and $\pi/T$-mode
states in Figs. 3(d) and 3(e). A clear coexistence of the first-order chiral
edge and the second-order Majorana corner states at the zero and $\pi/T$
quasienergies consistent with Fig. 3(a) is found. It confirms the hybrid-order
nature of the topological odd-parity superconductor phases generated by our
periodic driving protocol.

A generalization of our result to the case of $\lambda_{x}\neq\lambda_{y}$ can
be made. An exotic HOTSC in one single quasienergy gap is realizable. Applying
periodic driving on the critical phase in Fig. 1(c), we find in Fig. 4(a) that
two extra gapless chiral edge states are present near $k_{y}=0$ in the
quasienergy gap of zero. By decreasing the system parameter $\lambda_{y}$, the
boundary-state gap at $k_{y}=\pi$ is opened and four gapped boundary states
are formed [see Fig. 4(b)]. The probability distributions of the gapless edge
state at $k_{y}=0$ in Fig. 4(c) and the gapped boundary state at $k_{y}=\pi$
in Fig. 4(d) confirm that they are first-order edge and second-order Majorana
corner states, respectively. The evidence proves that it is a HOTSC with
coexisting SOTSC and Chern superconductors in the same quasienergy gap.
Although it is not protected by bulk topology, it also exhibits a certain
robustness to the static and temporal disorders due to the inversion symmetry
(see Appendix B).

Figure 4: Quasienergy spectra under the $x$-direction open boundary condition
when (a) $\lambda_{y}=1$ and (b) $0.95$ (b). The inset of (b) shows the
quasienergies under the $x$\- and $y$-direction open boundary condition. (c)
Probability distributions of the gapless chiral zero-mode edge state and (d)
the gapped Majorana corner state near the zero quasienergy when
$\lambda_{y}=0.95$. We use $\lambda_{x}=0.3$, $m=0.6$, and $T=1.1$.

##  IV 3D second-order Dirac superconductor

We generalize our result to the 3D odd-parity superconductor system with a
spin degree of freedom. We consider the same driving protocol as Eq. (2) but
with $\mathcal{H}_{1}({\bf
k})=d_{x}s_{z}\sigma_{x}+d_{y}\sigma_{y}+d_{z}\sigma_{z}$ and
$\mathcal{H}_{2}({\bf k})=(t_{z}\sin k_{z}+m)\sigma_{z}$, where $s_{i}$ and
$\sigma_{i}$ are the Pauli matrices, $d_{x}+id_{y}=\Delta({\bf k})$, and
$d_{z}=\epsilon({\bf k})$. It is found that $\mathcal{H}_{1}({\bf k})$ and
$\mathcal{H}_{2}({\bf k})$ are $\mathcal{PT}$ invariant, with
$\mathcal{T}=is_{2}\sigma_{0}\mathcal{K}$ and $\mathcal{P}=s_{0}\sigma_{z}$
being the time-reversal and spatial-inversion operations, which are inherited
by $\mathcal{H}_{\text{eff}}({\bf k})$. The 3D phase is sliced into a family
of 2D $k_{z}$-parametrized phases. If they are a SOTSC or a normal
superconductor for all $k_{z}$, then the 3D phase is SOTSC or normal
superconductor. If a 2D phase transition occurs at certain $k_{z}$, then the
3D phase is a Dirac superconductor [83, 84]. The Dirac superconductor is
conventionally classified into first and second order [see Figs. 5(a) and
5(b)]. Here, we can realize a distinct second-order Dirac superconductor.
Being independent of $k_{z}$, $\mathcal{H}_{1}({\bf k})$ possesses the same
energy spectrum as the 2D case in Eq. (1). Thus, the static phase of
$\mathcal{H}_{1}({\bf k})$ is a 3D SOTSC or normal superconductor depending on
the parameters in Fig. 1(a). Using the parameter of $\mathcal{H}_{1}({\bf k})$
in the SOTSC regime and switching on the periodic driving, it is remarkable to
find that a 3D second-order Dirac superconductor is realized [see Fig. 5(c)].
Such a unique topological phase is featured with coexisting surface and hinge
Majorana Fermi arcs [see Fig. 5(d)]. Although a similar coexistence has been
found in Weyl semimetals [85, 86], it has yet to be found in Dirac semimetals
and superconductors. It reveals that Floquet engineering offers us a useful
tool to manipulate the types of Majorana Fermi arcs in
$\mathcal{PT}$-symmetric systems.

Figure 5:  Schematic illustration of conventional (a) first- and (b) second-
order and (c) our second-order Dirac superconductors. The phase transitions
between the 2D sliced $Z_{2}$ topological superconductor ($Z_{2}$ TSC) and the
SOTSC occur at certain $k_{z}$. (d) Numerical result of the coexisting surface
and hinge Majorana Fermi arcs contributed by the $\pi/T$-mode first-order
gapless chiral edge and second-order gapped Majorana corner states. We use
$\lambda_{x}=\lambda_{y}=0.3$, $m=1.87$, $t_{z}=0.45$, and $T=1$.

##  V Discussion and conclusion

It is noted that the delta-function driving protocol is considered just for
the convenience of analytical calculation. Our scheme is generalizable to
other driving forms. Our HOTSC phases are generalizable to other static
systems constructed by the Hopf map [50], which reveals the universality of
our result. Boundary obstructed topological high-$T_{c}$ superconductivity is
predicted in Ca1-xLaxFeAs2 [40]. Floquet engineering has been used to design
novel topological phases in several platforms [55, 56, 57, 58, 59, 60]. The
higher-order topological phases have been simulated in Josephson junction
platforms [87, 88] and circuit QED systems [89], where in the latter our
static model has been realized. Based on these developments, we believe that
our proposal can be realized in state-of-the-art quantum simulation platforms.

In summary, we have investigated the periodic-driving-induced topological
phase transition in a two-band odd-parity superconductor system. A general
Floquet engineering scheme to create exotic HOTSCs exhibiting coexisting
first-order chiral edge and second-order Majorana corner states not only in
two different quasienergy gaps but also in one single quasienergy gap has been
established. The generalization of this scheme to a 3D system in the presence
of $\mathcal{PT}$ symmetries permits us to realize a second-order Dirac
superconductor. This phase features coexisting surface and hinge Majorana
Fermi arcs. Our result indicates that periodic driving offers us a feasible
way to explore exotic topological phases by adding time periodicity as an
extra control dimension in the system. This significantly expands the scope of
topological materials and enriches their controllability.

## Acknowledgment

The work is supported by the National Natural Science Foundation (Grants No.
12275109, No. 12247101, and No. 11834005) and the Supercomputing Center of
Lanzhou University.

##  Appendix A Boundaries of topological phase transition

Applying the Floquet theorem to our periodically driven system
$\mathcal{H}({\bf k},t)=\mathcal{H}_{1}({\bf k})+\mathcal{H}_{2}({\bf
k})\delta(t/T-N)$, we obtain

$\mathcal{H}_{\text{eff}}({\bf k})=iT^{-1}\ln[e^{-i\mathcal{H}_{2}({\bf
k})T}e^{-i\mathcal{H}_{1}({\bf k})T}].$ (6)

Using $\mathcal{H}_{j}({\bf k})=\mathbf{h}_{j}\cdot{\boldsymbol{\sigma}}$ and
the Euler’s formula of the Pauli matrices
$e^{-i\mathbf{h}_{j}\cdot\boldsymbol{\sigma}T}=\cos(h_{j}T)-i\sin(h_{j}T)\underline{\mathbf{h}}_{j}\cdot\boldsymbol{\sigma}$,
with $h_{j}=|\mathbf{h}_{j}|$ and
$\underline{\mathbf{h}}_{j}=\mathbf{h}_{j}/h_{j}$, we have

$\displaystyle{U}_{T}$ $\displaystyle=$ $\displaystyle
e^{-i\mathbf{h}_{2}\cdot\boldsymbol{\sigma}T}e^{-i\mathbf{h}_{1}\cdot\boldsymbol{\sigma}T}=\epsilon
I_{2\times 2}+i\mathbf{r}\cdot\boldsymbol{\sigma}\equiv
e^{-i\mathcal{H}_{\text{eff}}T},$ $\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\cos(h_{1}T)\cos(h_{2}T)-\underline{\mathbf{h}}_{1}\cdot\underline{\mathbf{h}}_{2}\sin(h_{1}T)\sin(h_{2}T),~{}~{}~{}~{}$
(7) $\displaystyle\mathbf{r}$ $\displaystyle=$
$\displaystyle\underline{\mathbf{h}}_{1}\times\underline{\mathbf{h}}_{2}\sin(h_{1}T)\sin(h_{2}T)-\underline{\mathbf{h}}_{2}\cos(h_{1}T)$
(8)
$\displaystyle\times\sin(h_{2}T)-\underline{\mathbf{h}}_{1}\sin(h_{1}T)\cos(h_{2}T).$

Again using the Euler’s formula, we can infer $\mathcal{H}_{\text{eff}}$ from
Eq. (A) as

$\displaystyle\mathcal{H}_{\text{eff}}({\bf k})$ $\displaystyle=$
$\displaystyle-\arccos(\epsilon)\mathbf{r}\cdot\boldsymbol{\sigma}/[T\sin(\arccos\epsilon)]$
(9) $\displaystyle=$
$\displaystyle-\arccos(\epsilon)\mathbf{r}\cdot\boldsymbol{\sigma}/[T\sqrt{1-\epsilon^{2}}]$
$\displaystyle=$
$\displaystyle-\arccos(\epsilon)\underline{\mathbf{r}}\cdot\boldsymbol{\sigma}/T,$

where $\epsilon^{2}+|\mathbf{r}|^{2}=1$ has been used. Thus we obtain
$\mathbf{h}_{\text{eff}}=-{\arccos\epsilon\over T}\underline{\mathbf{r}}$. The
quasienergy bands touch at $0$ and $\pi/T$, which occurs when $\epsilon=+1$
and $-1$, respectively. Therefore, the phase transition is associated with the
closing of the quasienergy bands, which occurs for ${k}$ and driving
parameters satisfying any one of the following conditions.

  1. 1.

$\sin(h_{1}T)\sin(h_{2}T)=0$: In this case,
$\epsilon=\cos(h_{1}T)\cos(h_{2}T)$. Then the bands of
$\mathcal{H}_{\text{eff}}({\bf k})$ close when

$h_{j}T=n_{j}\pi,~{}n_{j}\in\mathbb{Z}.$ (10)

  2. 2.

$\underline{\mathbf{h}}_{1}\cdot\underline{\mathbf{h}}_{2}=\pm 1$: In this
case, $\epsilon=\cos(h_{1}T\pm h_{2}T)$. Then the bands of
$\mathcal{H}^{\alpha}_{\text{eff}}({\bf k})$ close when

$h_{1}T\pm h_{2}T=n\pi,~{}n\in\mathbb{Z},$ (11)

at the zero quasienergy (or $\pi/T$) if $n$ is even (or odd).

As a sufficient condition for the topological phase transition, Eqs. (10) and
(11) supply a guideline to manipulate the driving parameters for Floquet
engineering to various topological phases at will.

Figure 6: (a) The quasienergy spectrum as a function of the disorder strength
$d$ when a static disorder is considered. Probability distributions of the
gapless zero-mode state when (b) $d=1$, (c) $2$, and (d) $3$. We use
$\lambda_{x}=0.3$, $\lambda_{y}=0.4$, $m=0.38$, and $T=1$.

Remembering $\mathcal{H}_{2}({\bf k})=m\sigma_{z}$ and rewriting
$\mathcal{H}_{1}({\bf k})={\bf h}_{1}({\bf k})\cdot\boldsymbol{\sigma}$ with
${\bf h}_{1}({\bf k})\equiv\text{Tr}[\mathcal{H}_{1}({\bf
k})\boldsymbol{\sigma}]$, we have

$\displaystyle h_{1}^{x}({\bf k})$ $\displaystyle=$ $\displaystyle
2\sum_{j=x,y}(\cos k_{j}+\lambda)\sin k_{j},$ (12) $\displaystyle
h_{1}^{y}({\bf k})$ $\displaystyle=$ $\displaystyle 2[(\cos k_{x}+\lambda)\sin
k_{y}-(\cos k_{y}+\lambda)\sin k_{x}],~{}~{}~{}$ (13) $\displaystyle
h_{1}^{z}({\bf k})$ $\displaystyle=$ $\displaystyle\sum_{j=x,y}[(\cos
k_{j}+\lambda)^{2}-\sin^{2}k_{j}],$ (14) $\displaystyle h_{2}^{x}({\bf k})$
$\displaystyle=$ $\displaystyle h_{2}^{y}({\bf k})=0,~{}h_{2}^{z}({\bf k})=m.$
(15)

It can be checked that the band touching points given by Eq. (10) generally do
not cause a topological phase transition in the phase diagram [75]. Then,
according to the band touching condition in Eq. (11), we obtain the boundaries
of topological phase transition as follows.
Case I: $\underline{\mathbf{h}}_{1}\cdot\underline{\mathbf{h}}_{2}=-1$ needs
$\cos k_{x}=\cos k_{y}=-\lambda$. Then Eq. (11) requires

$2(1-\lambda^{2})T-mT=n\pi.$ (16)

Case II: $\underline{\mathbf{h}}_{1}\cdot\underline{\mathbf{h}}_{2}=1$ needs
$k_{x,y}\equiv\alpha_{x,y}=0$ or $\pi$. Then Eq. (11) requires

$2[1+\lambda^{2}+\lambda(e^{i\alpha_{x}}+e^{i\alpha_{y}})]T+mT=n_{\alpha_{x},\alpha_{y}}\pi.$
(17)

Figure 7: When a temporal disorder is considered, the evolution of the
probability distributions of the $\pi/T$-mode corner state at (a) $t=10T$, (b)
$15T$, (c) $20T$, and the gapless zero-mode edge state (d) $t=10T$, (e) $20T$,
(f) $30T$. We use $\lambda_{x}=0.3$, $\lambda_{y}=0.4$, $m=0.38$, and $T=1$.

##  Appendix B Robustness to disorder

We now show the robustness of our hybrid-order topological superconductor to
the disorders. According to Ref. [90], the existence of class A Floquet
topological phases with gapless edge states does not require any symmetry. The
higher-order topological odd-parity superconductors in our Floquet systems are
protected by inversion symmetry. So our hybrid-order topological odd-parity
superconductors are robust against disorders that preserve inversion symmetry.

After adding a disorder $d\xi$, where $\xi$ is a random number in the regime
$[-0.5,0.5]$ and $d$ is the disorder strength, on the on-site potential $m$,
we plot in Fig. 6(a) the quasienergy spectrum as a function of the disorder
strength $d$. In the clean case when $d=0$, the system is a hybrid-order
topological superconductor phase with a first-order gapless edge state in the
zero mode and the second-order corner state in the $\pi/T$ mode. After turning
on the disorder, we see that the $\pi/T$-mode corner state survives even when
$d$ is as large as $4$. The probability distribution of the zero-mode edge
state in Figs. 6(b)-6(d) reveals that the zero-mode gapless edge state is
present even when $d$ is as large as 2. With further increasing $d$, the two
gaps close and the hybrid-order topological phase vanishes.

To reveal the robustness of our hybrid-order phase to the temporal disorder,
we add a disorder $\mathcal{T}\xi$, where $\xi$ again is a random number in
the regime $[-0.5,0.5]$ and $\mathcal{T}$ is the disorder strength, on the
driving period $T$. It is naturally expected that the zero and the
$\pi/T$-mode states would no longer be in the stationary state as long as
$\mathcal{T}\xi$ is added on the driving period $T$. Therefore, they are not
dynamically stable. Figures 7(a)-7(c) show the evolution of the $\pi/T$-mode
corner state when the disorder strength is $\mathcal{T}=0.5$. It is seen that
the corner feature is stably presented even when the evolution time is as
large as $20T$. Similar behavior is observed in the gapless zero-mode edge
state [see Figs. 7(d)-7(f)].

Therefore, our hybrid-order topological phases exhibit robustness to both of
the static and temporal disorders.

## References

  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * Bansil  _et al._ [2016] A. Bansil, H. Lin, and T. Das, Colloquium: Topological band theory, Rev. Mod. Phys. 88, 021004 (2016).
  * Chiu  _et al._ [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  * Sato and Ando [2017] M. Sato and Y. Ando, Topological superconductors: a review, Reports on Progress in Physics 80, 076501 (2017).
  * Armitage  _et al._ [2018] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  * Wilczek [2009] F. Wilczek, Majorana returns, Nature Physics 5, 614 (2009).
  * Ivanov [2001] D. A. Ivanov, Non-Abelian statistics of half-quantum vortices in $\mathit{p}$-wave superconductors, Phys. Rev. Lett. 86, 268 (2001).
  * Kitaev [2003] A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003).
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  * Benalcazar  _et al._ [2017] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357, 61 (2017).
  * Schindler  _et al._ [2018] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Higher-order topological insulators, Science Advances 4, eaat0346 (2018).
  * Song  _et al._ [2017] Z. Song, Z. Fang, and C. Fang, $(d-2)$-dimensional edge states of rotation symmetry protected topological states, Phys. Rev. Lett. 119, 246402 (2017).
  * Liu and Wakabayashi [2017] F. Liu and K. Wakabayashi, Novel topological phase with a zero berry curvature, Phys. Rev. Lett. 118, 076803 (2017).
  * Ren  _et al._ [2020] Y. Ren, Z. Qiao, and Q. Niu, Engineering corner states from two-dimensional topological insulators, Phys. Rev. Lett. 124, 166804 (2020).
  * Ezawa [2018] M. Ezawa, Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices, Phys. Rev. Lett. 120, 026801 (2018).
  * Wang  _et al._ [2021a] J.-H. Wang, Y.-B. Yang, N. Dai, and Y. Xu, Structural-disorder-induced second-order topological insulators in three dimensions, Phys. Rev. Lett. 126, 206404 (2021a).
  * Chen  _et al._ [2020a] C. Chen, Z. Song, J.-Z. Zhao, Z. Chen, Z.-M. Yu, X.-L. Sheng, and S. A. Yang, Universal approach to magnetic second-order topological insulator, Phys. Rev. Lett. 125, 056402 (2020a).
  * Chen  _et al._ [2020b] R. Chen, C.-Z. Chen, J.-H. Gao, B. Zhou, and D.-H. Xu, Higher-order topological insulators in quasicrystals, Phys. Rev. Lett. 124, 036803 (2020b).
  * Benalcazar and Cerjan [2022] W. A. Benalcazar and A. Cerjan, Chiral-symmetric higher-order topological phases of matter, Phys. Rev. Lett. 128, 127601 (2022).
  * Chen  _et al._ [2019] X.-D. Chen, W.-M. Deng, F.-L. Shi, F.-L. Zhao, M. Chen, and J.-W. Dong, Direct observation of corner states in second-order topological photonic crystal slabs, Phys. Rev. Lett. 122, 233902 (2019).
  * Yang  _et al._ [2020] Z.-Z. Yang, X. Li, Y.-Y. Peng, X.-Y. Zou, and J.-C. Cheng, Helical higher-order topological states in an acoustic crystalline insulator, Phys. Rev. Lett. 125, 255502 (2020).
  * Wu  _et al._ [2020a] J. Wu, X. Huang, J. Lu, Y. Wu, W. Deng, F. Li, and Z. Liu, Observation of corner states in second-order topological electric circuits, Phys. Rev. B 102, 104109 (2020a).
  * Cerjan  _et al._ [2020] A. Cerjan, M. Jürgensen, W. A. Benalcazar, S. Mukherjee, and M. C. Rechtsman, Observation of a higher-order topological bound state in the continuum, Phys. Rev. Lett. 125, 213901 (2020).
  * Zhang  _et al._ [2021] W. Zhang, D. Zou, Q. Pei, W. He, J. Bao, H. Sun, and X. Zhang, Experimental observation of higher-order topological anderson insulators, Phys. Rev. Lett. 126, 146802 (2021).
  * Fan  _et al._ [2019] H. Fan, B. Xia, L. Tong, S. Zheng, and D. Yu, Elastic higher-order topological insulator with topologically protected corner states, Phys. Rev. Lett. 122, 204301 (2019).
  * Xie  _et al._ [2019] B.-Y. Xie, G.-X. Su, H.-F. Wang, H. Su, X.-P. Shen, P. Zhan, M.-H. Lu, Z.-L. Wang, and Y.-F. Chen, Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals, Phys. Rev. Lett. 122, 233903 (2019).
  * Yang  _et al._ [2021] Y. Yang, J. Lu, M. Yan, X. Huang, W. Deng, and Z. Liu, Hybrid-order topological insulators in a phononic crystal, Phys. Rev. Lett. 126, 156801 (2021).
  * Chen  _et al._ [2021] Z.-G. Chen, W. Zhu, Y. Tan, L. Wang, and G. Ma, Acoustic realization of a four-dimensional higher-order chern insulator and boundary-modes engineering, Phys. Rev. X 11, 011016 (2021).
  * Wang  _et al._ [2021b] Y. Wang, Y. Ke, Y.-J. Chang, Y.-H. Lu, J. Gao, C. Lee, and X.-M. Jin, Constructing higher-order topological states in higher dimensions, Phys. Rev. B 104, 224303 (2021b).
  * Hsu  _et al._ [2020] Y.-T. Hsu, W. S. Cole, R.-X. Zhang, and J. D. Sau, Inversion-protected higher-order topological superconductivity in monolayer WTe2, Phys. Rev. Lett. 125, 097001 (2020).
  * Yan  _et al._ [2018] Z. Yan, F. Song, and Z. Wang, Majorana corner modes in a high-temperature platform, Phys. Rev. Lett. 121, 096803 (2018).
  * Khalaf [2018] E. Khalaf, Higher-order topological insulators and superconductors protected by inversion symmetry, Phys. Rev. B 97, 205136 (2018).
  * Zhu [2019] X. Zhu, Second-order topological superconductors with mixed pairing, Phys. Rev. Lett. 122, 236401 (2019).
  * Langbehn  _et al._ [2017] J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Reflection-symmetric second-order topological insulators and superconductors, Phys. Rev. Lett. 119, 246401 (2017).
  * Pan  _et al._ [2019] X.-H. Pan, K.-J. Yang, L. Chen, G. Xu, C.-X. Liu, and X. Liu, Lattice-symmetry-assisted second-order topological superconductors and Majorana patterns, Phys. Rev. Lett. 123, 156801 (2019).
  * Geier  _et al._ [2018] M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, Second-order topological insulators and superconductors with an order-two crystalline symmetry, Phys. Rev. B 97, 205135 (2018).
  * Shapourian  _et al._ [2018] H. Shapourian, Y. Wang, and S. Ryu, Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop materials, Phys. Rev. B 97, 094508 (2018).
  * Kheirkhah  _et al._ [2020] M. Kheirkhah, Y. Nagai, C. Chen, and F. Marsiglio, Majorana corner flat bands in two-dimensional second-order topological superconductors, Phys. Rev. B 101, 104502 (2020).
  * Ahn and Yang [2020] J. Ahn and B.-J. Yang, Higher-order topological superconductivity of spin-polarized Fermions, Phys. Rev. Research 2, 012060(R) (2020).
  * Wu  _et al._ [2020b] X. Wu, W. A. Benalcazar, Y. Li, R. Thomale, C.-X. Liu, and J. Hu, Boundary-obstructed topological high-${\mathit{t}}_{c}$ superconductivity in iron pnictides, Phys. Rev. X 10, 041014 (2020b).
  * Plekhanov  _et al._ [2021] K. Plekhanov, N. Müller, Y. Volpez, D. M. Kennes, H. Schoeller, D. Loss, and J. Klinovaja, Quadrupole spin polarization as signature of second-order topological superconductors, Phys. Rev. B 103, L041401 (2021).
  * Li  _et al._ [2022] M.-H. Li, T. Neupert, S. A. Parameswaran, and A. Tiwari, Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry, Phys. Rev. B 106, 125121 (2022).
  * Zhu [2018] X. Zhu, Tunable Majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields, Phys. Rev. B 97, 205134 (2018).
  * Ezawa [2019] M. Ezawa, Braiding of Majorana-like corner states in electric circuits and its non-Hermitian generalization, Phys. Rev. B 100, 045407 (2019).
  * Pahomi  _et al._ [2020] T. E. Pahomi, M. Sigrist, and A. A. Soluyanov, Braiding Majorana corner modes in a second-order topological superconductor, Phys. Rev. Research 2, 032068(R) (2020).
  * Zhang  _et al._ [2020a] S.-B. Zhang, W. B. Rui, A. Calzona, S.-J. Choi, A. P. Schnyder, and B. Trauzettel, Topological and holonomic quantum computation based on second-order topological superconductors, Phys. Rev. Research 2, 043025 (2020a).
  * Wu  _et al._ [2019] Z. Wu, Z. Yan, and W. Huang, Higher-order topological superconductivity: Possible realization in Fermi gases and ${\mathrm{sr}}_{2}{\mathrm{ruo}}_{4}$, Phys. Rev. B 99, 020508(R) (2019).
  * Wang  _et al._ [2018a] Y. Wang, M. Lin, and T. L. Hughes, Weak-pairing higher order topological superconductors, Phys. Rev. B 98, 165144 (2018a).
  * Wang  _et al._ [2018b] Q. Wang, C.-C. Liu, Y.-M. Lu, and F. Zhang, High-temperature Majorana corner states, Phys. Rev. Lett. 121, 186801 (2018b).
  * Yan [2019] Z. Yan, Higher-order topological odd-parity superconductors, Phys. Rev. Lett. 123, 177001 (2019).
  * Qi  _et al._ [2009] X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Time-reversal-invariant topological superconductors and superfluids in two and three dimensions, Phys. Rev. Lett. 102, 187001 (2009).
  * Qi  _et al._ [2010] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Chiral topological superconductor from the quantum Hall state, Phys. Rev. B 82, 184516 (2010).
  * Viyuela  _et al._ [2018] O. Viyuela, L. Fu, and M. A. Martin-Delgado, Chiral topological superconductors enhanced by long-range interactions, Phys. Rev. Lett. 120, 017001 (2018).
  * Khalaf  _et al._ [2021] E. Khalaf, W. A. Benalcazar, T. L. Hughes, and R. Queiroz, Boundary-obstructed topological phases, Phys. Rev. Research 3, 013239 (2021).
  * Eckardt [2017] A. Eckardt, Colloquium: Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys. 89, 011004 (2017).
  * Meinert  _et al._ [2016] F. Meinert, M. J. Mark, K. Lauber, A. J. Daley, and H.-C. Nägerl, Floquet engineering of correlated tunneling in the Bose-Hubbard model with ultracold atoms, Phys. Rev. Lett. 116, 205301 (2016).
  * Rechtsman  _et al._ [2013] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic Floquet topological insulators, Nature 496, 196 (2013).
  * Cheng  _et al._ [2019] Q. Cheng, Y. Pan, H. Wang, C. Zhang, D. Yu, A. Gover, H. Zhang, T. Li, L. Zhou, and S. Zhu, Observation of anomalous $\pi$ modes in photonic Floquet engineering, Phys. Rev. Lett. 122, 173901 (2019).
  * Roushan  _et al._ [2017] P. Roushan, C. Neill, A. Megrant, Y. Chen, R. Babbush, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus, P. J. J. OHuo Heng alley, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, E. Kapit, H. Neven, and J. Martinis, Chiral ground-state currents of interacting photons in a synthetic magnetic field, Nature Physics 13, 146 (2017).
  * McIver  _et al._ [2020] J. W. McIver, B. Schulte, F.-U. Stein, T. Matsuyama, G. Jotzu, G. Meier, and A. Cavalleri, Light-induced anomalous Hall effect in graphene, Nature Physics 16, 38 (2020).
  * Tong  _et al._ [2013] Q.-J. Tong, J.-H. An, J. Gong, H.-G. Luo, and C. H. Oh, Generating many majorana modes via periodic driving: A superconductor model, Phys. Rev. B 87, 201109(R) (2013).
  * Rodriguez-Vega  _et al._ [2019] M. Rodriguez-Vega, A. Kumar, and B. Seradjeh, Higher-order Floquet topological phases with corner and bulk bound states, Phys. Rev. B 100, 085138 (2019).
  * Peng [2020] Y. Peng, Floquet higher-order topological insulators and superconductors with space-time symmetries, Phys. Rev. Research 2, 013124 (2020).
  * Nag  _et al._ [2021] T. Nag, V. Juričić, and B. Roy, Hierarchy of higher-order Floquet topological phases in three dimensions, Phys. Rev. B 103, 115308 (2021).
  * Plekhanov  _et al._ [2019] K. Plekhanov, M. Thakurathi, D. Loss, and J. Klinovaja, Floquet second-order topological superconductor driven via ferromagnetic resonance, Phys. Rev. Research 1, 032013(R) (2019).
  * Huang and Liu [2020] B. Huang and W. V. Liu, Floquet higher-order topological insulators with anomalous dynamical polarization, Phys. Rev. Lett. 124, 216601 (2020).
  * Jangjan and Hosseini [2020] M. Jangjan and M. V. Hosseini, Floquet engineering of topological metal states and hybridization of edge states with bulk states in dimerized two-leg ladders, Scientific Reports 10, 14256 (2020).
  * Nag  _et al._ [2019] T. Nag, V. Juričić, and B. Roy, Out of equilibrium higher-order topological insulator: Floquet engineering and quench dynamics, Phys. Rev. Research 1, 032045(R) (2019).
  * Ghosh  _et al._ [2021a] A. K. Ghosh, T. Nag, and A. Saha, Floquet generation of a second-order topological superconductor, Phys. Rev. B 103, 045424 (2021a).
  * Ghosh  _et al._ [2021b] A. K. Ghosh, T. Nag, and A. Saha, Floquet second order topological superconductor based on unconventional pairing, Phys. Rev. B 103, 085413 (2021b).
  * Wu and An [2022] H. Wu and J.-H. An, Non-Hermitian Weyl semimetal and its Floquet engineering, Phys. Rev. B 105, L121113 (2022).
  * Dehghani  _et al._ [2021] H. Dehghani, M. Hafezi, and P. Ghaemi, Light-induced topological superconductivity via Floquet interaction engineering, Phys. Rev. Research 3, 023039 (2021).
  * Ghosh  _et al._ [2022a] A. K. Ghosh, T. Nag, and A. Saha, Systematic generation of the cascade of anomalous dynamical first- and higher-order modes in Floquet topological insulators, Phys. Rev. B 105, 115418 (2022a).
  * Ghosh  _et al._ [2022b] A. K. Ghosh, T. Nag, and A. Saha, Dynamical construction of quadrupolar and octupolar topological superconductors, Phys. Rev. B 105, 155406 (2022b).
  * Xiong  _et al._ [2016] T.-S. Xiong, J. Gong, and J.-H. An, Towards large-chern-number topological phases by periodic quenching, Phys. Rev. B 93, 184306 (2016).
  * Wang  _et al._ [2021c] B.-Q. Wang, H. Wu, and J.-H. An, Engineering exotic second-order topological semimetals by periodic driving, Phys. Rev. B 104, 205117 (2021c).
  * Yan  _et al._ [2017] Z. Yan, R. Bi, and Z. Wang, Majorana zero modes protected by a Hopf invariant in topologically trivial superconductors, Phys. Rev. Lett. 118, 147003 (2017).
  * Kundu  _et al._ [2014] A. Kundu, H. A. Fertig, and B. Seradjeh, Effective theory of Floquet topological transitions, Phys. Rev. Lett. 113, 236803 (2014).
  * Rudner  _et al._ [2013] M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X 3, 031005 (2013).
  * Asbóth  _et al._ [2014] J. K. Asbóth, B. Tarasinski, and P. Delplace, Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems, Phys. Rev. B 90, 125143 (2014).
  * Yu  _et al._ [2021] J. Yu, Y. Ge, and S. Das Sarma, Dynamical fragile topology in Floquet crystals, Phys. Rev. B 104, L180303 (2021).
  * Roy and Harper [2017] R. Roy and F. Harper, Periodic table for floquet topological insulators, Phys. Rev. B 96, 155118 (2017).
  * Ghorashi  _et al._ [2019] S. A. A. Ghorashi, X. Hu, T. L. Hughes, and E. Rossi, Second-order dirac superconductors and magnetic field induced majorana hinge modes, Phys. Rev. B 100, 020509(R) (2019).
  * Zhang  _et al._ [2020b] R.-X. Zhang, Y.-T. Hsu, and S. Das Sarma, Higher-order topological dirac superconductors, Phys. Rev. B 102, 094503 (2020b).
  * Wang  _et al._ [2020] H.-X. Wang, Z.-K. Lin, B. Jiang, G.-Y. Guo, and J.-H. Jiang, Higher-order Weyl semimetals, Phys. Rev. Lett. 125, 146401 (2020).
  * Ghorashi  _et al._ [2020] S. A. A. Ghorashi, T. Li, and T. L. Hughes, Higher-order Weyl semimetals, Phys. Rev. Lett. 125, 266804 (2020).
  * Yang  _et al._ [2018] Z. Yang, S. Qin, Q. Zhang, C. Fang, and J. Hu, $\pi$/2-josephson junction as a topological superconductor, Phys. Rev. B 98, 104515 (2018).
  * Kalenyuk  _et al._ [2018] A. A. Kalenyuk, A. Pagliero, E. A. Borodianskyi, A. A. Kordyuk, and V. M. Krasnov, Phase-sensitive evidence for the sign-reversal ${s}_{\pm{}}$ symmetry of the order parameter in an iron-pnictide superconductor using $\mathrm{Nb}/{\mathrm{ba}}_{1-x}{\mathrm{na}}_{x}{\mathrm{fe}}_{2}{\mathrm{as}}_{2}$ josephson junctions, Phys. Rev. Lett. 120, 067001 (2018).
  * Niu  _et al._ [2021] J. Niu, T. Yan, Y. Zhou, Z. Tao, X. Li, W. Liu, L. Zhang, H. Jia, S. Liu, Z. Yan, Y. Chen, and D. Yu, Simulation of higher-order topological phases and related topological phase transitions in a superconducting qubit, Science Bulletin 66, 1168 (2021).
  * Yao  _et al._ [2017] S. Yao, Z. Yan, and Z. Wang, Topological invariants of floquet systems: General formulation, special properties, and floquet topological defects, Phys. Rev. B 96, 195303 (2017).


# Imaging Atomic-Level Random Walk of a Point Defect in Graphene

Jani Kotakoski,1,2∗, Clemens Mangler1, Jannik C. Meyer1

1University of Vienna, Faculty of Physics,
Boltzmanngasse 5, A-1090 Vienna, Austria
2University of Helsinki, Department of Physics,
P.O. Box 43, FI-00014 Helsinki, Finland

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
## Abstract

Deviations from the perfect atomic arrangements in crystals play an important
role in affecting their properties. Similarly, diffusion of such deviations is
behind many microstructural changes in solids. However, observation of point
defect diffusion is hindered both by the difficulties related to direct
imaging of non-periodic structures and by the time scales involved in the
diffusion process. Here, instead of imaging thermal diffusion, we stimulate
and follow the migration of a divacancy through graphene lattice using a
scanning transmission electron microscope operated at 60 kV. The beam-
activated process happens on a timescale that allows us to capture a
significant part of the structural transformations and trajectory of the
defect. The low voltage combined with ultra-high vacuum conditions ensure that
the defect remains stable over long image sequences, which allows us for the
first time to directly follow the diffusion of a point defect in a crystalline
material.

## Introduction

High resolution electron microscopy has recently exposed the atomic structure
of two-dimensional materials such as graphene [1, 2], hexagonal boron nitride
[3, 4] and transition metal dichalcogenide [5] monolayers, and a two-
dimensional silica glass structure [6] for direct observation. Modern imaging
techniques are also able to directly discern between different chemical
elements in these structures, even for atoms which are neighbors in the
periodic table [7, 8, 9, 10]. In addition to structural and chemical analysis,
electron microscopy has also led to advances in the understanding of
dynamical, mostly beam-driven, processes in graphene and similar materials.
For example, imaging with 80 keV electrons has been shown to excite structural
changes in pristine graphene [2], and at its point defects [11, 12, 13, 14],
grain boundaries [15] and dislocations [16, 17], whereas 60 keV imaging has
revealed dynamics of a 6-atomic Si cluster [18] in graphene. However, beam-
induced knock-on damage with less-than-ideal vacuum conditions have until the
current state-of-the-art instruments prevented direct observation of point
defect migration in pristine graphene over long image sequences. Here, using
60 keV imaging, we reveal at the atomic resolution a random walk performed by
a defect in an otherwise perfect graphene crystal.

## Results

Image sequences of divacancy migration. Our data consist of two images
sequences of a divacancy defect in monolayer graphene (with 57 and 143 frames,
recorded over about 5 min 32 s and 12 min 59 s, respectively), obtained with a
Nion UltraSTEM 100 [19] electron microscope operated at 60 kV. Due to the low
voltage and the ultra-high vacuum conditions ($1.3\times 10^{-9}$ mbar at the
sample), the divacancy in graphene is extraordinarily stable: it neither
converts into higher-order vacancies nor traps carbon and converts back to a
pristine lattice, for long sequences of images. However, the defect rapidly
moves via beam-driven bond rotations during observation, and constantly
changes shape between four different configurations, which have also been
identified in earlier images [20]. As a result, the defect performs a random
walk through the lattice. A sequence of 10 consecutive frames and a final
frame of one image sequence is presented in Fig. 1a as an example. Fig. 1b
shows the first frame from another image sequence. Fig. 1c is a
“superposition” of all frames highlighting the trace of the defect (by showing
the minimum of intensity from the sequence at every pixel), and Fig. 1d
depicts the defect trajectory from the same sequence, obtained by locating the
center of the divacancy in every frame. Complete sequences are provided as
supplementary movie 1 for the first image sequence and as supplementary movie
2 for the second image sequence. Non-treated versions of all images are
provided as supplementary movie 3 and 4 for the first and the second sequence,
respectively.

Atomic configurations of the divacancy. Fig. 2 a–d shows the defect in three
different configurations: V2(585) in panel a, V2(5555-6-7777) in panels b and
c, and V2(555-777) in panel d. (In this notation, the carbon rings contained
within the defects are listed, when possible, along the longest axis through
the defect.) Due to the symmetry of the lattice, V2(555-777) can appear with
two distinct orientations, whereas the other two have three possible
conformations. Throughout the data, these defects have occurrancies of 50.3%
for V2(585), 14.1% for V2(555-777) and 18.8% for V2(5555-6-7777).
Additionally, in three frames (1.8%) the defect appears in the $2\times(57)$
configuration [21]. The rest (14.7%) of the frames contain either unclear
structures or combinations of two or more of the above-mentioned
configurations. Examples of these are shown in Fig. 2 e–h. In two frames the
defect has completely eluded detection although it is visible both in the
previous and the following frames (an example is presented in Fig. 2i-k).

Defect transformations. As mentioned, less than 15% of the scan images showed
any indications of the structure changing during the scan. This is surprising,
because the consecutive scans nevertheless revealed the defect in other
locations and often in another configuration. It is almost as if the scans
would correspond to photographs taken in a busy but dark room, only
momentarily illuminated by the flash of the camera freezing the moment in
time. However, it is well understood that the bond rotations, which are
responsible for the observed structural changes, are associated with an energy
barrier of 5–10 eV [22] and can thus not be driven by thermal activation in
our room temperature experiments. Indeed, they must be caused by collisions
between individual imaging electrons and individual target nuclei [11].
Therefore, one could expect that the transformations always occur when the
electron probe is placed atop the defect, which would necessarily lead to
detection of the transformation, as in Fig. 2e-h. Perhaps even more mysterious
are the two frames where the defect has completely avoided detection (for an
example, see Fig. 2i-k).

Determination of the probe shape. The answer to the transformation puzzle lies
in the shape of the electron probe, which is pixel by pixel and line by line
scanned over the area within the field of view (in our case $512\times 512$
pixels within $4\times 4$ nm2 for sequence 1 and $5\times 5$ nm2 for sequence
2). While the full width at half maximum (FWHM) of the probe must be in the
order of 1 Å for atomic resolution imaging, the actual shape of the probe, and
especially its tail [7] further away from the point of the maximum intensity
is not exactly known. In order to understand the role of the probe tails in
our observations, we estimated the shape of our probe experimentally based on
an intensity profile recorded over a graphene edge, as illustrated in Fig. 3.
A good match between the recorded profile and a convolution of a step function
with a model probe consisting of three 2D Gaussians (see Fig. 3b) was obtained
for standard deviations of $\sigma_{1}\approx 0.06$ nm, $\sigma_{2}\approx
0.25$ nm and $\sigma_{3}\approx 0.30$ nm. The estimated accuracy of the manual
fit is ca. 10%. A one-dimensional profile of the measured probe is presented
in Fig. 3c. The FWHM for the probe is about 0.14 nm and the vacuum level is
reached at a distance of about 1.5 nm. Less than 21% of the beam intensity is
contained within the FWHM, which shows that a significant dose is deposited
outside the beam maximum position. Based on this, and taking into account that
the probe spends much more time at a distance of $0.07~{}\mathrm{nm}<r\leq
1.5$ nm from the defect than on the defect itself [the middle sized defect
V2(555-777) has an area roughly 5% of that of $\pi(1.5~{}\mathrm{nm})^{2}$],
it becomes clear that the effect of scanning near the defect can alter its
atomic configuration. As noted, in our experiment this effect accounted for up
to $>85$% of all of the transformations. When the transformations happen to
drive the defect towards the already imaged area, this effect leads to its
apparent disappearance for the duration of one or more scans.

Statistical analysis of the random walks. Histograms of all of the jumps
between recorded frames in both of the image sequences are plotted in Fig. 4a.
Assuming normal distribution, the average jump length is $\delta_{1}\approx
0.23$ nm for sequence 1 and $\delta_{2}\approx 0.26$ nm for sequence 2, which
is close to the hexagon-hexagon distance in the carbon lattice. From the
definition of diffusivity in two dimensions

$D\equiv\frac{\delta^{2}}{4\tau},$ (1)

where $\tau$ is the jump time, we get $D\approx 3.10\times
10^{-3}~{}\mathrm{nm}^{2}\mathrm{s}^{-1}=3.1\times 10^{-21}$
m${}^{2}\mathrm{s}^{-1}$ (using the value for the longer sequence), which is
well in line with typical diffusivity values in solids. However, we stress
that the migration is in the case of our experiment driven by the knock-on
collisions between individual electrons and individual target nuclei, and the
measured diffusivity is thus not directly comparable to values describing
thermally driven point defect diffusion in solids.

Although some of the jumps between scans are considerably longer than others,
the overall total cumulative distance ($d$) traveled by the defect increases
linearly as a function of time (see Fig. 4b), yielding an estimated migration
speed of 3.61 nm min-1 during the 13 min long experiment. For a random walk,
the root-mean-square distance, which is a measure of the average distance of
the walker from the start after $n$ steps, is

$\sqrt{<r(n)^{2}>}\equiv\sqrt{\frac{1}{N}\sum_{k=1}^{N}r_{k}(n)^{2}}=\delta\sqrt{n},$
(2)

where $k$ runs over the $N$ different random walks (in our case the two image
sequences) and $\delta$ is the average jump length. As can be seen in Fig. 4c,
the random walks analyzed here follow this behaviour. A fit to the data
reveals $\delta\approx 0.25$ nm, which is in between the above-estimated
values of $\delta_{1}$ and $\delta_{2}$, as can be expected.

We can take the analogue between thermal and electron-beam-driven diffusion
one step further to compare our experimental observations with surface
diffusion, where

$\Gamma(T)=\nu\exp(-E_{b}/k_{B}T).$ (3)

Here $\Gamma(T)$ is the jump rate at temperature $T$, $\nu$ the attempt
frequency, $E_{b}$ the diffusion energy barrier, and $k_{B}$ the Boltzmann
constant. For the area of the middle-sized divacancy, V2(555-777), and
$E_{b}=5$ eV as the energy barrier associated with a bond rotation [22], we
get an estimate for the conditions which we simulate with the electron beam.
The apparent temperature of the system is $T\approx 3050$ K, with the caveat
that every recorded image is here assumed to represent exactly one migration
step, whereas in reality we know that often several bond rotations have taken
place between two subsequent frames. We stress that this is a virtual
temperature, since the actual heat brought in by the electron beam is quickly
dissipated away, and only modest if any actual heating of the sample is
expected during the experiment [23]. Nevertheless, the observed defect
migration is otherwise indistinquishable from what would be expected to occur
at elevated temperatures: The energy input from the beam helps to overcome an
activation barrier; in a similar manner as heat would
but—importantly—resulting in a much slower process.

## Discussion

As a conclusion, we have demonstrated for the first time that electron
irradiation at 60 kV can be used to stimulate an atomic-scale random walk of a
point defect in an otherwise pristine crystal, and to record it over several
minutes. Despite the atomically small probe size, most of the transformations
occur when the beam is situated away from the actual defect due to irradiation
dose accumulation from the low-intensity tail of the electron beam. In rare
cases, the defect can even completely avoid detection during a scan. Via
analysis of the defect trajectory during the experiment, we estimate that the
beam-stimulated migration of the divacancy corresponds to a virtual
temperature of about 3050 K, establishing atomic resolution transmission
electron microscopy as a method for simultaneous imaging and driving diffusion
of point defects in low-dimensional materials.

## Methods

Scanning transmission electron microscopy. The experiments were carried out
with a Nion UltraSTEM 100 device [19] recently installed at the University of
Vienna. The device is equipped with a cold field emission gun, which was
operated at 60 kV in ultrahigh vacuum. Medium angle annular dark-field
detector was used to record two image sequences of the same defect. The first
sequence contains 57 frames recorded over about 5 min and 32 s with a field of
view of $4\times 4$ nm2. The second sequence contains 143 frames and was
recorded over about 12 min and 59 s. Sample drift is less than the lattice
spacing during the image sequences (the image sequences provided in
supplementary movies 1–4 were not compensated for drift). The time the beam
was held at each pixel was 16 $\mu$s for both sequences and all images contain
$512\times 512$ pixels. A typical camera current for the device is in the
order of $5\times 10^{-11}$ A, from which a dose of circa $8\times
10^{6}e^{-}$Å-2 was estimated per recorded frame for sequence 1 and $5\times
10^{6}e^{-}$Å-2 for sequence 2.

Image processing. The images were processed to reduce noise by applying a
Gaussian filter with a radius of 6 px (for field of view of $4\times 4$ nm2)
and 4 px (for field of view of $5\times 5$ nm2) after which the processed
image was multiplied with the original image. The process was carried out
twice for the second image sequence due to lower signal-to-noise ratio.

Determination of the defect position. To track the migration of the defect, we
first aligned the image sequences, and then marked the middle of the defect
manually to obtain the coordinates for each frame. The accuracy of this
positioning procedure is estimated to be better than the interatomic
separation in the lattice.

Sample preparation. The graphene sample was grown via chemical vapor
deposition and suspended on a TEM grid by a commercial supplier.

## References

  * [1] Hashimoto, A., Suenaga, K., Gloter, A., Urita, K. & Iijima, S.  Direct evidence for atomic defects in graphene layers.  _Nature_ 430, 870–873 (2004).
  * [2] Meyer, J. C. _et al._ Direct imaging of lattice atoms and topological defects in graphene membranes.  _Nano Lett._ 8, 3582–3586 (2008).
  * [3] Jin, C., Lin, F., Suenaga, K. & Iijima, S.  Fabrication of a freestanding boron nitride single layer and its defect assignments.  _Phys. Rev. Lett._ 102, 195505 (2009).
  * [4] Meyer, J. C., Chuvilin, A., Algara-Siller, G., Biskupek, J. & Kaiser, U.  Selective sputtering and atomic resolution imaging of atomically thin boron nitride membranes.  _Nano Lett._ 9, 2683–2689 (2009).
  * [5] Komsa, H.-P. _et al._ Two-dimensional transition metal dichalcogenides under electron irradiation: defect production and doping.  _Phys. Rev. Lett._ 109, 035503 (2012).
  * [6] Huang, P. Y. _et al._ Direct imaging of a two-dimensional silica glass on graphene.  _Nano Lett._ 12, 1081–1086 (2012).
  * [7] Krivanek, O. L. _et al._ Atom-by-atom structural and chemical analysis by annular dark-field electron microscopy.  _Nature_ 464, 571–574 (2010).
  * [8] Meyer, J. C. _et al._ Experimental analysis of charge redistribution due to chemical bonding by high-resolution transmission electron microscopy.  _Nature Mater._ 10, 209–215 (2011).
  * [9] Zhou, W. _et al._ Direct determination of the chemical bonding of individual impurities in graphene.  _Phys. Rev. Lett._ 109, 206803 (2012).
  * [10] Ramasse, Q. M. _et al._ Probing the bonding and electronic structure of single atom dopants in graphene with electron energy loss spectroscopy.  _Nano Letters_ 13, 4989–4995 (2013).
  * [11] Kotakoski, J. _et al._ Stone-wales–type transformations in carbon nanostructures driven by electron irradiation.  _Phys. Rev. B_ 83, 245420 (2011).
  * [12] Robertson, A. W. _et al._ Spatial control of defect creation in graphene at the nanoscale.  _Nature Comm._ 3, 1144 (2012).
  * [13] Robertson, A. W. _et al._ Structural reconstruction of the graphene monovacancy.  _ACS Nano_ 7, 4495–4502 (2013).
  * [14] Wang, W. L. _et al._ Direct observation of a long-lived single-atom catalyst chiseling atomic structures in graphene.  _Nano Letters_ 14, 450–455 (2014).
  * [15] Kurasch, S. _et al._ Atom-by-atom observation of grain boundary migration in graphene.  _Nano Lett._ 12, 3168–3173 (2012).
  * [16] Warner, J. H. _et al._ Dislocation-driven deformations in graphene.  _Science_ 337, 209–212 (2012).
  * [17] Lehtinen, O., Kurasch, S., Krasheninnikov, A. V. & Kaiser, U.  Atomic scale study of the life cycle of a dislocation in graphene from birth to annihilation.  _Nature Comm._ 4, 2098 (2013).
  * [18] Lee, J., Zhou, W., Pennycook, S. J., Idrobo, J.-C. & Pantelides, S. T.  Direct visualization of reversible dynamics in a si6 cluster embedded in a graphene pore.  _Nature Comm._ 4, 1650 (2013).
  * [19] Krivanek, O. L. _et al._ An electron microscope for the aberration-corrected era.  _Ultramicroscopy_ 108, 179–195 (2008).
  * [20] Banhart, F., Kotakoski, J. & Krasheninnikov, A. V.  Structural defects in graphene.  _ACS Nano_ 5, 26–41 (2011).
  * [21] Kotakoski, J., Krasheninnikov, A. V., Kaiser, U. & Meyer, J. C.  From point defects in graphene to two-dimensional amorphous carbon.  _Phys. Rev. Lett._ 106, 105505 (2011).
  * [22] Li, L., Reich, S. & Robertson, J.  Defect energies of graphite: Density-functional calculations.  _Phys. Rev. B_ 72, 184109 (2005).
  * [23] Egerton, R., Li, P. & Malac, M.  Radiation damage in the TEM and SEM.  _Micron_ 35, 399–409 (2004).

## Acknowledgments

We acknowledge funding from the Austrian Science Fund (FWF) through projects M
1481-N20 and P 25721-N20 and from University of Helsinki Funds.

## Author contributions

J.K. initiated the study, carried out the experiment, analyzed data and wrote
the manuscript. C.M. contributed to establishing the experimental setup and
J.C.M. contributed to the analysis. All authors commented on the manuscript.

## Additional information

Supplementary Information: accompanies this paper at
http://www.nature.com/naturecommunications

Competing financial interests: The authors declare no competing financial
interests.

## Figures

Figure 1: Traveling divacancy in graphene. (a) Ten consecutive frames and the
final frame from one image sequence showing the movement of the defect through
the lattice. (b) First frame of another image sequence. (c) ”Superposition” of
all of the frames from the second sequence highlighting the trace of the
defect by showing the minimum intensity from the sequence at every pixel. (d)
Actual trajectory of the defect in the second sequence, determined by locating
the approximate middle point of the defect in every frame. Only those images
where the location of the defect was clearly identifiable have been included.
The start position is marked with a black star and the last location with a
diamond. All scale bars are 1 nm. Figure 2: Example exposures of the defect.
(a-d) Four subsequent frames from the second image sequence. The bonds
associated with the defects are highlighted with an overlay. The structure has
undergone at least four bond rotations between panels a and b and one between
c and d, as marked with the black arrows. (e–h) Examples of image scans where
the structure changed while scanning exactly at the location of the defect.
White and black overlays mark the structure of the defect before and after the
change, respectively. (i–k) Three subsequent frames from the second image
sequence. The defect appears in the V2(585) configuration in panel i, but
disappears for the duration of the next scan resulting in panel j, before
appearing again in panel k, in the V2(5555-6-7777) configuration. The darker
area within panel j presumably corresponds to the area where the defect is
located, although it avoids detection (locations of all atoms belonging to the
pristine lattice can be identified). In panel k, a circle with radius of 1.5
nm is drawn for scale with the experimentally obtained probe shape. In these
panels the complete field of view is shown. The scale bars are 1 nm. Figure 3:
Determination of the probe shape. (a) Unprocessed (but colored) image of a
graphene edge. (b) Line profile obtained from the area shaded in panel a in
the direction of the arrow along with a simulated line profile calculated
assuming that the graphene edge is a step function and convoluting it with a
probe that consists of three 2D Gaussians. The graphene edge position was set
to the zero of $x$-axis. The standard deviations for the Gaussians
($\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$) were obtained via manual
fitting, with an estimated accuracy of ca. 10%. (c) One dimensional profile of
the probe consisting of the three Gaussians. Vacuum intensity was set to zero.
Figure 4: Statistical analysis of a random walk. (a) Histogram of all of the
jump distances by the defect in the two image sequences. The lines show normal
distributions fitted to the data. (b) Corresponding cumulative total distance
traveled by the defect as a function of time. The lines are fits to the data.
(c) Root-mean-square distance of the defect from the starting position as a
function of time. The solid line is a fit to the data.


the complement of which, according to the preceding arguments, is
$O_{\delta}(1)$-sparse. For each $n\in\mathbb{N}$ we write

$A(n):=\prod_{\begin{subarray}{c}p^{k}||n\\\
p\notin\tilde{\mathcal{G}}\end{subarray}}p^{k}\cdot\prod_{\begin{subarray}{c}p^{k}||n\\\
p\in\tilde{\mathcal{G}}\\\ k\geq 2\end{subarray}}p^{k},\quad
s(n):=\prod_{\begin{subarray}{c}p||n\\\
p\in\tilde{\mathcal{G}}\end{subarray}}p.$

Define also the $\pm 1$-valued multiplicative function $h(n)$ on prime powers
by

$h(p^{k})=\begin{cases}f(p)/p^{\alpha_{0}}&\text{ if
}p\in\tilde{\mathcal{G}},\,k=1\\\ 1&\text{ otherwise.}\end{cases}$

Since every integer can be written as

$n=A(n)s(n)$

with $(A(n),s(n))=1$, we always have the identity

$f(n)=f(A(n))f(s(n))=f(A(n))h(n)s(n)^{\alpha_{0}}=\frac{f(A(n))}{A(n)^{\alpha_{0}}}h(n)n^{\alpha_{0}}=:G(n)n^{\alpha_{0}}.$

Thus, for each $n\in\mathcal{N}_{f,a,b}$,

(57) $(n+a)^{\alpha_{0}}G(n+a)=n^{\alpha_{0}}G(n)+b.$

Now, applying Lemma 6.10 with $\varepsilon=\delta^{2}$ and the
$O_{\delta}(1)$-sparse set

$S=\\{p^{\nu}:p\notin\tilde{\mathcal{G}},\,\nu\geq
1\\}\cup\\{p^{\nu}:p\in\tilde{\mathcal{G}},\,\nu\geq 2\\},$

there is a choice $N_{0}=O_{\delta}(1)$ such that the set

$\mathcal{N}_{f,a,b}^{\prime}:=\\{n\in\mathcal{N}_{f,a,b}:\,A(n),A(n+a)\leq
N_{0}\\}$

satisfies $\overline{\delta}(\mathcal{N}_{f,a,b}^{\prime})\geq\delta/2$, when
$\delta$ is small enough. Since also $h(n),h(n+a)\in\\{-1,+1\\}$, by the
pigeonhole principle, there are integers $1\leq A,B\leq N_{0}$ and elements
$\eta_{1},\eta_{2}\in\\{-1,+1\\}$ for which

$\mathcal{N}_{f,a,b}^{\prime\prime}:=\\{n\in\mathcal{N}_{f,a,b}^{\prime}:\,A(n+a)=A,\,A(n)=B,\,(h(n),h(n+a))=(\eta_{1},\eta_{2})\\}$

has $\overline{\delta}(\mathcal{N}_{f,a,b}^{\prime\prime})\gg_{\delta}1$. In
particular, for each $n\in\mathcal{N}_{f,a,b}^{\prime\prime}$,

$G(n+a)=\eta_{2}f(A)/A^{\alpha_{0}},\quad G(n)=\eta_{1}f(B)/B^{\alpha_{0}}.$

Writing

$\tilde{A}:=\eta_{2}f(A)B^{\alpha_{0}},\quad\tilde{B}:=\eta_{1}f(B)A^{\alpha_{0}},\quad\tilde{C}:=b(AB)^{\alpha_{0}}$

and rearranging (57), we obtain

$\tilde{A}(n+a)^{\alpha_{0}}=\tilde{B}n^{\alpha_{0}}+\tilde{C}\text{ for all
}n\in\mathcal{N}_{f,a,b}^{\prime\prime}.$

Since $\overline{\delta}(\mathcal{N}_{f,a,b}^{\prime\prime})>0$, dividing both
sides by $n^{\alpha_{0}}$ and taking $n\rightarrow\infty$ along an infinite
increasing subsequence of $\mathcal{N}_{f,a,b}^{\prime\prime}$ yields
$\tilde{A}=\tilde{B}$, from which we obtain

$\tilde{A}((n+a)^{\alpha_{0}}-n^{\alpha_{0}})=\tilde{C}.$

Note that $\tilde{C}\neq 0$ by definition, so $\tilde{A}\neq 0$ as well. Since
$a\geq 1$ we have
$(n+a)^{\alpha_{0}}-n^{\alpha_{0}}\geq\alpha_{0}an^{\alpha_{0}-1}$. As this
holds for infinitely many $n$ we conclude that $\alpha_{0}=1$, as required.

Step (c): When $f$ is completely multiplicative we can make the following
reduction, with the effect that every element of $\mathcal{N}_{f,a,b}$ may be
assumed to be coprime to $a$.
For each $d|a$ let us define

$\mathcal{N}_{f,a,b}(d):=\\{n\in\mathcal{N}_{f,a,b}:(n,a)=d\\}=\\{n\in\mathbb{N}:f(n+a)=f(n)+b,\,(n,a)=d\\}.$

We thus obtain the partition

$\mathcal{N}_{f,a,b}=\bigcup_{d|a}\mathcal{N}_{f,a,b}(d),$

and by the pigeonhole principle, there is a divisor $d_{0}|a$ for which
$\overline{\delta}(\mathcal{N}_{f,a,b}(d_{0}))>\delta/d(a)$, where $d(m)$
denotes the divisor function.
Now, for each $n\in\mathcal{N}_{f,a,b}(d_{0})$, writing $n=md_{0}$ and
$a^{\prime}=a/d_{0}$ we get by complete multiplicativity that

$f(d_{0})f(m+a^{\prime})=f(n+a)=f(md_{0})+b=f(d_{0})f(m)+b.$

We thus deduce that $f(d_{0})|b$. We now set $b^{\prime}:=b/f(d_{0})$,
$\eta:=\delta/(d_{0}d(a))>0$ and

$\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}}:=\\{m\in\mathbb{N}:md_{0}\in\mathcal{N}_{f,a,b}(d_{0})\\}=\mathcal{N}_{f,a^{\prime},b^{\prime}}(1).$

Then $\overline{\delta}(\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}})>\eta$,
and $(n,a^{\prime})=1$ for each
$n\in\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}}$.
By analogy with the definitions in Step (b), we define

$\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}}^{\prime}:=\\{n\in\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}}:A(n),A(n+a^{\prime})\leq
N_{0}\\}.$

Choosing $N_{0}$ larger in terms of $\delta$ if needed, we may assume that
$\overline{\delta}(\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}}^{\prime})\geq\eta/2$.
Rearranging (57) (with $a^{\prime},b^{\prime}$ in place of $a,b$) we get

$n\left(G(n+a^{\prime})-G(n)\right)=b^{\prime}-a^{\prime}G(n),\quad
n\in\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}}.$

Note that the bracketed expression, if non-zero, is of size $\asymp_{\delta}1$
for all $n\in\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}}^{\prime}$, since
each of $G(n)$ and $G(n+a^{\prime})$ belongs to a set bounded at most in terms
of $\delta$. Thus, if
$n\in\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}}^{\prime}$ is sufficiently
large then

$h(n+a^{\prime})\frac{f(A(n+a^{\prime}))}{A(n+a^{\prime})}=h(n)\frac{f(A(n))}{A(n)}=\frac{b^{\prime}}{a^{\prime}}.$

Clearing denominators, we see from here that $a^{\prime}|A(n)b^{\prime}$. But
since $A(n)|n$ and $(n,a^{\prime})=1$ for all
$n\in\tilde{\mathcal{N}}_{f,a^{\prime},b^{\prime}}$, we must conclude that
$a^{\prime}|b^{\prime}$, as claimed. ∎

##  7\. Proof of Theorem 1.6

We now complete the proof of Theorem 1.6. Thus, let
$f:\mathbb{N}\rightarrow\mathbb{Z}$ be a completely multiplicative function
for which

$\sum_{p:|f(p)|\neq 1}\frac{1}{p}=\infty,$

and suppose that the set

$\mathcal{N}_{f,a,b}=\\{n\in\mathbb{N}:f(n+a)=f(n)+b\\}$

satisfies $\delta(\mathcal{N}_{f,a,b})\neq 0$ for some non-zero
$a,b\in\mathbb{Z}$. Thus, there is a $\delta\in(0,1)$ such that
$\overline{\delta}(\mathcal{N}_{f,a,b})>\delta$. Assume also that the set

$S_{f}:=\\{\ell\text{ prime}:\,\ell|f(\ell)\\}$

has positive Dirichlet density,

$\lim_{X\rightarrow\infty}\frac{1}{\log\log X}\sum_{\begin{subarray}{c}p\leq
X\\\ p\in S_{f}\end{subarray}}\frac{1}{p}>0.$

We begin with some reductions.
First of all, suppose $a<0$ and let $n\in\mathcal{N}_{f,a,b}$ with $n>|a|$.
Making the change of variables $m=n+a$ we see that

$\text{if }n\in\mathcal{N}_{f,a,b},\,n>|a|\text{ then
}m\in\mathcal{N}_{f,|a|,-b}.$

Thus, as $a$ is fixed, upon replacing $\mathcal{N}_{f,a,b}$ by
$\mathcal{N}_{f,|a|,-b}$ if necessary we may assume in the sequel that $a\geq
1$.
Furthermore, by Remark 4.2,

$\text{if }\sum_{p:f(p)=0}\frac{1}{p}=\infty\text{ then
}\delta(\mathcal{N}_{f,a,b})=0,$

contradicting our hypotheses. Thus, in the sequel we may assume that $f$
satisfies _both_ conditions

$\sum_{p:|f(p)|\neq
1}\frac{1}{p}=\infty,\quad\sum_{p:f(p)=0}\frac{1}{p}<\infty.$

Applying Proposition 4.1, there exists $0<r\ll_{\delta}1$ such that the set of
primes

$\mathcal{S}_{\delta}:=\\{p\in\mathcal{P}:|f(p)|/p^{r}\notin(1/2,2)\\}$

is $O_{\delta}(1)$-sparse, i.e.,

$\sum_{\begin{subarray}{c}p\leq X\\\
p\in\mathcal{S}_{\delta}\end{subarray}}\frac{1}{p}\ll_{\delta}1,\quad
X\rightarrow\infty.$

By Proposition 6.3,we see that

$\\{\ell\in\mathcal{P}:\ell^{\prime}\nmid f(\ell)\text{ for all
}\ell^{\prime}\neq\ell\\}$

is a sparse set. Consequently, Proposition 6.4 allows us to conclude that
$|f(\ell)|=\ell$ except on an $O_{\delta}(1)$-sparse set of primes $\ell$, and
that when $f$ is completely multiplicative there is a divisor $d|a$ such that
$f(d)|b$, and $a/d$ divides $b/f(d)$, as claimed.

######  Remark 7.1.

It is worthwhile speculating whether a condition on the size of $S_{f}$ is
also necessary in Conjecture 1.5. Certainly, our method requires that we at
least assume that

(58) $\sum_{p\in S_{f}}\frac{1}{p}=\infty,$

since we needed to discard sparse subsets of $S_{f}$ at various points in the
argument.
Conversely, one can show that if $f$ satisfies
$\delta(\mathcal{N}_{f,a,b})\neq 0$ then one can produce many infinite sets
$S$ of primes that satisfy $\sum_{p\in S}1/p<\infty$, and associated
multiplicative functions $g=g_{S}$ with $S_{g}\supseteq\\{p:f(p)\neq
g(p)\\}=S$ such that $\delta(\mathcal{N}_{g,a,b})\neq 0$. For example, if
$\overline{\delta}(\mathcal{N}_{f,a,b})>\delta$ and $S$ is a set of primes
with151515In fact, one can argue more carefully and obtain such $S$ with
$\sum_{p\in S}p^{-1}\in[C,C+1)$ for any fixed $C$, by employing Lemma 5.1 to
better sieve out those $n\in\mathcal{N}_{f,a,b}$ with $d|n(n+a)$ and $d$
composed of primes $p\in S$; we leave this argument to the interested reader.
However, our corresponding lower bound for $\mathcal{N}_{g,a,b}$ also worsens
in the process, at least heuristically, i.e.
$\overline{\delta}(\\{n\in\mathcal{N}_{f,a,b}:(n(n+a),S)=1\\})\approx\overline{\delta}(\mathcal{N}_{f,a,b})\prod_{p\in
S}(1-2/p)\gg e^{-2C}\delta.$ $\sum_{p\in S}p^{-1}<\delta/3$, then we may
define $g$ to be the multiplicative function defined at prime powers by

$g(p^{k})=\begin{cases}f(p^{k})&\text{ if }p\notin S,\,k\geq 1,\\\
p^{k}&\text{ if }p\in S,\,k\geq 1.\end{cases}$

Clearly, $g(n+a)=g(n)+b$ whenever $n\in\mathcal{N}_{f,a,b}$ with
$p|n(n+a)\Rightarrow p\notin S$. Moreover, along a suitable subsequence we
find

$\displaystyle\frac{1}{\log X}\sum_{\begin{subarray}{c}n\leq X\\\
n\in\mathcal{N}_{f,a,b}\\\ p|n(n+a)\Rightarrow p\notin
S\end{subarray}}\frac{1}{n}\geq\frac{1}{\log X}\sum_{\begin{subarray}{c}n\leq
X\\\ n\in\mathcal{N}_{f,a,b}\end{subarray}}\frac{1}{n}-\sum_{p\in
S}\frac{1}{\log X}\sum_{\begin{subarray}{c}n\leq X\\\
p|n(n+a)\end{subarray}}\frac{1}{n}$ $\displaystyle\geq\delta-2\sum_{p\in
S}\frac{1}{p}-o(1)$ $\displaystyle>\delta/3-o(1).$

Therefore, if $f$ does not satisfy (58) then there is not much more that
$S_{f}$ can tell us about $f$.

##  8\. Gaps and the proof of Corollary 1.2

Conjecture 1.5, Theorem 1.6 and the main theorem of [19] imply the following
result from which Corollary 1.2 follows.

######  Corollary 8.1.

Suppose $f:\mathbb{N}\rightarrow\mathbb{Z}$ is a multiplicative function for
which

(59) $\sum_{p:|f(p)|\neq p}\frac{1}{p}=\infty,\quad\sum_{p:|f(p)|\neq
1}\frac{1}{p}=\infty\text{ and }\sum_{p:f(p)=0}\frac{1}{p}<\infty.$

Assume one of the following:

  1. (a)

Conjecture 1.5 holds,

  2. (b)

ERH holds and $\sum_{p|f(p)}p^{-1}=\infty$, or

  3. (c)

$d_{\text{Dir}}(\\{p\in\mathcal{P}:p|f(p)\\})>0$.

Then for all $C>0$ we have

$\delta(\\{n\in\mathbb{N}:f(n)\neq 0,\,|f(n+1)-f(n)|\leq C\\})=0.$

###### Proof.

Assume for the sake of contradiction that

$\limsup_{X\rightarrow\infty}\frac{1}{\log X}\sum_{\begin{subarray}{c}n\leq
X\\\ f(n)\neq 0\\\ |f(n+1)-f(n)|\leq C\end{subarray}}\frac{1}{n}=\delta>0.$

By the pigeonhole principle, there is an integer $b\in[-C,C]$ such that

$\limsup_{X\rightarrow\infty}\frac{1}{\log X}\sum_{\begin{subarray}{c}n\leq
X\\\ f(n)\neq 0\\\
f(n+1)-f(n)=b\end{subarray}}\frac{1}{n}\geq\frac{\delta}{2\lfloor
C\rfloor+1},$

and therefore

(60) $\delta(\\{n\in\mathbb{N}:f(n)\neq 0,\,f(n+1)=f(n)+b\\})\neq 0.$

Suppose first that $b\neq 0$. Assuming either (a),(b) or (c) in conjunction
with Theorem 1.6, we find that

$\sum_{p:|f(p)|\neq p}\frac{1}{p}<\infty,$

contradicting the first of the conditions in (59).
On the other hand, if $b=0$ then by [19, Thm. 1.1] there are no multiplicative
functions satisfying (59) for which (60) holds (here the condition $f(n)\neq
0$ is needed). Thus, in either case we obtain a contradiction, and the claim
follows. ∎

Next, we record how the assumption $f(n)\rightarrow\infty$ is related to the
frequency of prime power values $f(p^{\nu})=1$.

######  Lemma 8.2.

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be a multiplicative function. Then

(61) $\limsup_{C\rightarrow\infty}\,\overline{d}\\{n\in\mathbb{N}:1\leq
f(n)\leq C\\}>0$

if and only if

(62) $\sum_{p^{\nu}:f(p^{\nu})\neq 1}\frac{1}{p^{\nu}}<\infty.$

###### Proof.

If (62) fails then Lemma 4.3 (more precisely Remark 4.4) implies that

$d\\{n\in\mathbb{N}:1\leq f(n)\leq C\\}=0$

for all $C>0$, as desired.
Conversely, suppose that (62) holds. For each $y\geq 1$ and $C\geq 1$ we of
course have

$\frac{1}{y}|\\{n\leq y:1\leq f(n)\leq C\\}|\geq\frac{1}{y}|\\{n\leq
y:f(n)=1\\}|\geq\frac{1}{y}|\\{n\leq y:\mu^{2}(n)=1,\,p||n\Rightarrow
f(p)=1\\}|.$

By a zero-dimensional sieve, it is easily verified that as
$y\rightarrow\infty$ this last expression is

$\frac{6}{\pi^{2}}\prod_{\begin{subarray}{c}p\in\mathcal{P}\\\ f(p)\neq
1\end{subarray}}\left(1-\frac{1}{p}\right)+o(1)\geq\frac{6}{\pi^{2}}\exp\left(-\sum_{\begin{subarray}{c}p\in\mathcal{P}\\\
f(p)\neq 1\end{subarray}}\frac{1}{p}\right)+o(1)>0.$

Hence, (61) holds as required. ∎

###### Proof of Corollary 1.2.

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ is multiplicative, such that

$\sum_{\begin{subarray}{c}p\in\mathcal{P}\\\ f(p)\neq
p\end{subarray}}\frac{1}{p}=\infty.$

Further, suppose that for each $C>0$ we have

$d\\{n\in\mathbb{N}:f(n)\leq C\\}=0.$

By Lemma 8.2, it follows that

$\sum_{p:f(p)\neq 1}\frac{1}{p}=\infty.$

Since $f(p)\neq 0$ for all $p$, Corollary 8.1 implies that

$\delta\\{n\in\mathbb{N}:|f(n+1)-f(n)|\leq C\\}=0\text{ for all }C>0$

under the assumption of ERH, assuming furthermore that
$\sum_{p|f(p)}1/p=\infty$, as claimed. ∎

##  9\. Proof of Proposition 1.8

To prove Proposition 1.8 we will construct the prime sets $\mathcal{T}$ via
the probabilistic method.
Let us introduce the following notation. Given a set of primes $\mathcal{T}$
we define the Liouville-type function $\lambda_{\mathcal{T}}$ to be the
completely multiplicative function such that for each prime $p$,

$\lambda_{\mathcal{T}}(p)=\begin{cases}-1\text{ if }p\in\mathcal{T}\\\
+1\text{ if }p\notin\mathcal{T}.\end{cases}$

Fix the sparse set $\mathcal{S}$. Let $(X_{p})_{p\notin\mathcal{S}}$ be a
sequence of i.i.d. random variables on an auxiliary probability space
$(\Omega,\mathcal{B},\mathbb{P})$ with

$\mathbb{P}(X_{p}=0)=\mathbb{P}(X_{p}=1)=1/2.$

We now define a random set of primes

(63) $\mathbf{T}:=\\{p\notin\mathcal{S}:X_{p}=1\\}.$

In the next several lemmas we collect some facts about $\mathbf{T}$.
(In the remainder of this section we follow the convention that the trivial
character is primitive.)

######  Lemma 9.1.

For $k\geq 1$ define the sequence $x_{k}:=e^{e^{k^{3}}}$. Then with the above
scheme, the following statements hold almost surely (i.e. on a set
$A\subseteq\Omega$ with $\mathbb{P}(A)=1$):

  1. (a)

for each $k\geq k_{0}(\omega)$,

$\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{1}{p}\geq\frac{1}{4}\log\log x_{k},$

  2. (b)

for every real primitive Dirichlet character $\chi\pmod{q}$ with $1<q\leq k$,

$\left|\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{\chi(p)}{p}\right|\leq\frac{1}{10}\log\log
x_{k},\quad k\geq k_{0}(\omega).$

###### Proof.

By linearity of expectation we have

$\mathbb{E}\left(\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{1}{p}\right)=\mathbb{E}\left(\sum_{\begin{subarray}{c}p\leq
x_{k}\\\ p\notin\mathcal{S}\end{subarray}}\frac{X_{p}}{p}\right)=\sum_{p\leq
x_{k}}\frac{\mathbb{E}(X_{p})}{p}+O\left(\sum_{\begin{subarray}{c}p\leq
x_{k}\\\ p\in\mathcal{S}\end{subarray}}\frac{1}{p}\right)=\frac{1}{2}\log\log
x_{k}+O(1).$

Therefore, by independence the variance satisfies

$\displaystyle\mathbb{E}\left(\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{1}{p}-\frac{1}{2}\log\log x_{k}\right)^{2}$
$\displaystyle=\sum_{\begin{subarray}{c}p,q\leq x_{k}\\\
p,q\notin\mathcal{S}\end{subarray}}\frac{\mathbb{E}(X_{p}X_{q})}{pq}-\frac{1}{4}(\log\log
x_{k})^{2}+O(\log\log x_{k})$
$\displaystyle=\left(\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\notin\mathcal{S}\end{subarray}}\frac{\mathbb{E}(X_{p})}{p}\right)^{2}+\frac{1}{2}\sum_{p\leq
x_{k}}\frac{1}{p^{2}}-\frac{1}{4}\log\log x_{k}^{2}+O(\log\log x_{k})$
$\displaystyle=O(\log\log x_{k}).$

By Chebyshev’s inequality, we deduce that

$\mathbb{P}\left(\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{1}{p}\leq\frac{1}{4}\log\log
x_{k}\right)\leq 16\frac{\mathbb{E}\left(\sum_{\begin{subarray}{c}p\leq
x_{k}\\\ p\in\mathbf{T}\end{subarray}}\frac{1}{p}-\frac{1}{2}\log\log
x_{k}\right)^{2}}{(\log\log x_{k})^{2}}\ll\frac{1}{\log\log
x_{k}}\ll\frac{1}{k^{3}}.$

Next, suppose $1<q\leq k\leq\log\log x_{k}$ and let $\chi$ be a real primitive
character modulo $q$. Then

$\displaystyle\mathbb{E}\left(\left|\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{\chi(p)}{p}\right|^{2}\right)$
$\displaystyle=\frac{1}{4}\sum_{\begin{subarray}{c}p,q\leq x_{k}\\\
p,q\notin\mathcal{S}\end{subarray}}\frac{\chi(pq)}{pq}+O\left(\sum_{p\leq
x_{k}}\frac{1}{p^{2}}\right)=\left(\frac{1}{2}\sum_{p\leq
x_{k}}\frac{\chi(p)}{p}+O(1)\right)^{2}+O(1).$

By the Siegel-Walfisz theorem and partial summation (see e.g. [8, Cor. 5.29]),

(64) $\sum_{p\leq x_{k}}\frac{\chi(p)}{p}\ll\sum_{p\leq
e^{k}}\frac{1}{p}+\int_{e^{k}}^{x_{k}}\left|\sum_{p\leq
v}\chi(p)\right|\frac{dv}{v^{2}}\ll\log k+k^{1/2}k^{-10}\ll\log\log\log
x_{k}.$

Combining these two estimates and using Chebyshev’s inequality, we find

$\mathbb{P}\left(\left|\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{\chi(p)}{p}\right|\geq\frac{1}{10}\log\log
x_{k}\right)\ll\frac{(\log\log\log x_{k})^{2}}{(\log\log
x_{k})^{2}}\ll\frac{(\log k)^{2}}{k^{4}}\ll\frac{1}{k^{3}}.$

Since there are $\ll k$ real primitive characters with conductor $\leq k$, by
the union bound we deduce that the probability that either (a) fails or that
(b) fails for some $\chi\pmod{q}$, $1<q\leq k$ is

$\ll\frac{1}{k^{3}}+k\cdot\frac{1}{k^{3}}\ll\frac{1}{k^{2}}.$

By the Borel-Cantelli lemma (see [4, VIII.3]), we deduce that, on a set of
$\omega\in\Omega$ with probability $1$, both

$\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}(\omega)\end{subarray}}\frac{1}{p}\geq\frac{1}{4}\log\log
x_{k}+O(1)$

and also

$\max_{1<q\leq k}\left|\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}(\omega)\end{subarray}}\frac{\chi(p)}{p}\right|\leq\frac{1}{10}\log\log
x_{k}$

hold for all $k\geq k_{0}(\omega)$, as claimed. ∎

######  Lemma 9.2.

For every real primitive character $\chi$,

$\lim_{x\rightarrow\infty}\mathbb{D}(\lambda_{\mathbf{T}},\chi;x)=\infty$

almost surely.

###### Proof.

Let $\chi$ be a real primitive character with modulus $q\geq 1$. Given $k\geq
q$ we have

(65)
$\mathbb{D}(\lambda_{\mathbf{T}},\chi;x_{k})^{2}=\sum_{\begin{subarray}{c}p\leq
x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{1+\chi(p)}{p}+\sum_{\begin{subarray}{c}p\leq
x_{k}\\\ p\notin\mathbf{T}\end{subarray}}\frac{1-\chi(p)}{p}=\sum_{p\leq
x_{k}}\frac{1-\chi(p)}{p}+2\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{\chi(p)}{p}.$

If $q=1$ then by Lemma 9.1 the right-hand side of (65) is simply

$2\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{1}{p}\geq\frac{1}{2}\log\log x_{k}$

for all $k\geq k_{0}(\omega)$ almost surely.
If $1<q\leq k$ then combining Lemma 9.1 with the Siegel-Walfisz theorem (as in
(64)), the right-hand side of (65) is bounded from below as

$\displaystyle\geq\log\log x_{k}-\left|\sum_{p\leq
x_{k}}\frac{\chi(p)}{p}\right|-2\left|\sum_{\begin{subarray}{c}p\leq x_{k}\\\
p\in\mathbf{T}\end{subarray}}\frac{\chi(p)}{p}\right|+O(1)$
$\displaystyle\geq\log\log x_{k}+O(\log\log\log x_{k})-\frac{1}{5}\log\log
x_{k}$ $\displaystyle\geq\frac{1}{2}\log\log x_{k}$

for $k\geq k_{0}(\omega)$ almost surely. Thus, we obtain

$\inf_{\begin{subarray}{c}\chi\text{ real primitive}\\\ q\leq
k\end{subarray}}\mathbb{D}(\lambda_{\mathbf{T}},\chi;x_{k})^{2}\geq\frac{1}{2}\log\log
x_{k},\quad k\geq k_{0}(\omega)$

almost surely.
Therefore, for any real primitive character $\chi$ modulo $q_{0}$, any
$k\geq\max\\{k_{0}(\omega),q_{0}\\}$ and any $x\geq x_{k}$,

$\mathbb{D}(\lambda_{\mathbf{T}},\chi;x)^{2}\geq\inf_{q\leq
k}\mathbb{D}(\lambda_{\mathbf{T}},\chi_{q};x_{k})^{2}\geq\frac{1}{2}\log\log
x_{k}+O(1)$

almost surely. Taking $k\rightarrow\infty$ forces $x\rightarrow\infty$, and
the claim follows. ∎

######  Lemma 9.3.

If $\mathcal{C}$ is any countable collection of subsets of primes,

$\mathbb{P}(\mathbf{T}\in\mathcal{C})=0.$

In particular, for any set $A$ with $\mathbb{P}(A)>0$, the collection
$\\{\mathbf{T}(\omega):\omega\in A\\}$ is uncountable.

###### Proof.

For any fixed set of primes $\mathcal{T}$ and any $N\geq 1$ the probability

(66)
$\mathbb{P}(\mathbf{T}=\mathcal{T})\leq\mathbb{P}(\bigcap_{\begin{subarray}{c}p\leq
N\\\ p\notin\mathcal{S}\end{subarray}}\\{X_{p}=1_{p\in T}\\})\leq
2^{-\pi_{\mathcal{S}^{c}}(N)},$

where $\pi_{\mathcal{S}^{c}}(N):=\pi(N)-|\mathcal{S}\cap[2,N]|$. Since
$\mathcal{S}$ is sparse we have
$\pi_{\mathcal{S}^{c}}(N_{k})\geq\tfrac{2}{3}\pi(N_{k})$ on an infinite
increasing sequence $(N_{k})_{k}$. Taking $N\rightarrow\infty$ along
$(N_{k})_{k}$, we deduce that $\mathbb{P}(\mathbf{T}=\mathcal{T})=0$. Thus, by
countable additivity of the measure $\mathbb{P}$ we find

$\mathbb{P}(\mathbf{T}\in\mathcal{C})=\sum_{\mathcal{T}\in\mathcal{C}}\mathbb{P}(\mathbf{T}=\mathcal{T})=0.$

This establishes the first claim.
For the second, observe that if $\\{\mathbf{T}(\omega):\omega\in A\\}$ were
countable then there would exist a collection
$\mathcal{C}=(\mathcal{T}_{j})_{j}$ such that

$\omega\in A\Rightarrow\mathbf{T}(\omega)\in\mathcal{C}.$

By the previous claim,

$0<\mathbb{P}(A)\leq\mathbb{P}(\mathbf{T}\in\mathcal{C})=0,$

which is a contradiction. ∎

###### Proof of Proposition 1.8.

Set $\mathcal{S}_{d}:=\mathcal{S}\backslash\\{p:p|d\\}$. Let $\mathcal{T}$ be
a realisation of the random set $\mathbf{T}$ defined in (63). By Lemma 9.2,
with probability $1$ we have

(67)
$\lim_{X\rightarrow\infty}\mathbb{D}(\lambda_{\mathcal{T}},\chi;X)=\infty$

for each fixed real primitive character $\chi$, and Lemma 9.3 shows that the
collection of $\mathcal{T}$ for which this holds is uncountable. For the
remainder of the proof we fix some realisation $\mathcal{T}$ satisfying (67).
Define now the set

$\mathcal{N}=\mathcal{N}_{\mathcal{T}}:=\\{dpm:\,p\in S_{d},\,p|m\Rightarrow
p\notin\mathcal{S}\text{ and }\lambda_{\mathcal{T}}(m)=+1\\}.$

Let $\tilde{d}=\prod_{\begin{subarray}{c}p||d\end{subarray}}p$. Thus, if $d=1$
or $d$ is squarefull then $\tilde{d}=1$; otherwise, $\tilde{d}>1$. Define also
$k:=db/a\in\mathbb{Z}$, $k\neq 0$. We define the completely multiplicative
function $f=f_{\mathcal{T}}$ at primes via

$f(p)=\begin{cases}\lambda_{\mathcal{T}}(p)p&\text{ if
$p\notin\mathcal{S}$}\\\ P^{+}(|k|)&\text{ if $p=P^{+}(\tilde{d})$}\\\
pk/f(P^{+}(\tilde{d}))&\text{ if $p\in\mathcal{S}_{d}$}\\\ 1&\text{ if $p|d$,
$p\neq P^{+}(\tilde{d})$.}\end{cases}$

Since $k\neq 1$ by assumption, $f$ clearly satisfies items (i) and (ii). Note
furthermore that $f(d)=f(P^{+}(\tilde{d}))$ divides $k$, and is thus a divisor
of $b$ as in item (iii). It thus remains to show that $f$ so-defined satisfies
item (iv).
Consider the set

$\mathcal{N}^{\prime}:=\\{n\in\mathbb{N}:n,n+a\in\mathcal{N}\\}.$

If $n\in\mathcal{N}^{\prime}$ then we can find primes
$p_{1},p_{2}\in\mathcal{S}_{d}$ and $m_{1},m_{2}$ not divisible by any prime
$p\in\mathcal{S}$, such that

$n=m_{1}p_{1}d,\quad n+a=m_{2}p_{2}d=d(m_{1}p_{1}+a/d),$

and $\lambda_{\mathcal{T}}(m_{1})=\lambda_{\mathcal{T}}(m_{2})=+1$. By
construction we have $f(d)=f(P^{+}(\tilde{d}))$. Thus,

$\displaystyle f(n+a)$
$\displaystyle=f(d)f(m_{2})f(p_{2})=f(P^{+}(\tilde{d}))\lambda_{\mathcal{T}}(m_{2})m_{2}(k/f(P^{+}(\tilde{d})))p_{2}=km_{2}p_{2},$
$\displaystyle f(n)+b$
$\displaystyle=f(d)f(m_{1})f(p_{1})+k_{1}k_{2}a/d=k(\lambda_{\mathcal{T}}(m_{1})m_{1}p_{1}+a/d)=k(m_{1}p_{1}+a/d).$

Since $m_{2}p_{2}=m_{1}p_{1}+a/d$, we deduce that $f(n+a)=f(n)+b$, and so

$\mathcal{N}^{\prime}\subseteq\mathcal{N}_{f,a,b}.$

To complete the proof we will show that
$\overline{\delta}(\mathcal{N}^{\prime})>0$.
Since $|\mathcal{S}_{d}|\geq 2$ and the sets

$\mathcal{N}_{q}:=\\{dqm:p|m\Rightarrow p\notin\mathcal{S}\text{ and
}\lambda_{\mathcal{T}}(m)=+1\\},\quad q\in\mathcal{S}_{d}$

are mutually disjoint, it suffices to show that there are
$p_{1},p_{2}\in\mathcal{S}_{d}$ with $p_{1}\neq p_{2}$ and $(p_{1}p_{2},a)=1$
such that

$\\{n\in\mathcal{N}_{p_{1}}:n+a\in\mathcal{N}_{p_{2}}\\}\subseteq\mathcal{N}^{\prime}$

has positive upper logarithmic density. To this end, we will estimate

$\frac{1}{\log x}\sum_{\begin{subarray}{c}n\leq x\\\
n\in\mathcal{N}_{p_{1}},\,n+a\in\mathcal{N}_{p_{2}}\end{subarray}}\frac{1}{n}=\frac{1}{d\log
x}\sum_{\begin{subarray}{c}n^{\prime}\leq x/d\\\
n^{\prime}=m_{1}p_{1},\,n^{\prime}+a/d=m_{2}p_{2}\\\
(m_{1}m_{2},\mathcal{S})=1\\\
\lambda_{\mathcal{T}}(m_{1})=\lambda_{\mathcal{T}}(m_{2})=+1\end{subarray}}\frac{1}{n^{\prime}}.$

By the Chinese remainder theorem, we can find $0\leq u<p_{2}$ so that

$n^{\prime}\equiv 0\pmod{p_{1}},\,n^{\prime}+a/d\equiv 0\pmod{p_{2}}\text{ if
and only if }n^{\prime}\equiv up_{1}\pmod{p_{1}p_{2}}.$

Define the multiplicative function $g(n):=1_{(n,\mathcal{S})=1}$. Using the
changes of variables

$n^{\prime}=mp_{1}p_{2}+up_{1},\quad v:=(up_{1}+a/d)/p_{2}\in\mathbb{Z},$

and the identity
$1_{\lambda_{\mathcal{T}}(m)=+1}=(1+\lambda_{\mathcal{T}}(m))/2$, we can
recast the above expression as

$\displaystyle\frac{1}{4d\log x}\sum_{m\leq
x/dp_{1}p_{2}}\frac{g(mp_{2}+u)g(mp_{1}+v)}{m}(1+\lambda_{\mathcal{T}}(mp_{2}+u))(1+\lambda_{\mathcal{T}}(mp_{1}+v))$
$\displaystyle=\sum_{\eta_{1},\eta_{2}\in\\{0,1\\}}\frac{1}{4d\log
x}\sum_{m\leq
x/dp_{1}p_{2}}\frac{g\lambda_{\mathcal{T}}^{\eta_{1}}(mp_{2}+u)g\lambda_{\mathcal{T}}^{\eta_{2}}(mp_{1}+v)}{m}.$

Since $g(p)=1$ for all $p$ outside the sparse set $\mathcal{S}$, the
pretentious triangle inequality (15) implies that for any fixed, real
primitive character $\chi$,

$\mathbb{D}(g\lambda_{\mathcal{T}},\chi;x)\geq\mathbb{D}(\lambda_{\mathcal{T}},\chi;x)-\mathbb{D}(g,1;x)=\mathbb{D}(\lambda_{\mathcal{T}},\chi;x)-\left(\sum_{\begin{subarray}{c}p\leq
x\\\
p\in\mathcal{S}\end{subarray}}\frac{1}{p}\right)^{1/2}=\mathbb{D}(\lambda_{\mathcal{T}},\chi;x)-O(1).$

Combining (67) with Lemma 3.4, we deduce therefore that

$\lim_{x\rightarrow\infty}\inf_{|t|\leq
x}\mathbb{D}(g\lambda_{\mathcal{T}},\chi;x)=\infty$

for every fixed Dirichlet character $\chi$ (real or complex). As
$d,p_{1},p_{2},u$ and $v$ are all fixed, Theorem 3.5 yields

$\frac{1}{4d\log x}\sum_{m\leq
x/dp_{1}p_{2}}\frac{g\lambda_{\mathcal{T}}^{\eta_{1}}(mp_{2}+u)g\lambda_{\mathcal{T}}^{\eta_{2}}(mp_{1}+v)}{m}=o(1)$

for each $(\eta_{1},\eta_{2})\neq(0,0)$. Thus,

(68) $\frac{1}{\log x}\sum_{\begin{subarray}{c}n\leq x\\\
n\in\mathcal{N}_{p_{1}},\,n+a\in\mathcal{N}_{p_{2}}\end{subarray}}\frac{1}{n}=\frac{1}{4d\log
x}\sum_{\begin{subarray}{c}m\leq x\\\
(mp_{1}+u,\mathcal{S})=(mp_{2}+v,\mathcal{S})=1\end{subarray}}\frac{1}{m}+o(1).$

By partial summation, it now suffices to show that the set

$\\{m\in\mathbb{N}:(mp_{2}+u,\mathcal{S})=(mp_{1}+v,\mathcal{S})=1\\}$

has positive density. By a zero-dimensional sieve this density is

$\frac{1}{p_{1}p_{2}}\left(1-\frac{1}{p_{1}}\right)\left(1-\frac{1}{p_{2}}\right)\prod_{\begin{subarray}{c}p\in\mathcal{S}\\\
p\nmid p_{1}p_{2}\end{subarray}}\left(1-\frac{2}{p}\right),$

which is $>0$ as $2\notin\mathcal{S}$.
When $\mathcal{T}=\emptyset$ the condition $\lambda_{\mathcal{T}}(m)=+1$ in
the definition of $\mathcal{N}$ is trivial, and (68) holds with the prefactor
$1/4$ replaced by $1$. The same conclusion follows. ∎

######  Remark 9.4.

The above proof may easily be modified to allow $2\in\mathcal{S}$, though the
analysis differs according to whether $2$ divides $d$ and/or $a$. We leave
this to the interested reader.

## References

  * [1] P.D.T.A. Elliott.  Probabilistic Number Theory I: Mean value theorems, volume 239 of Grundlehren der mathematischen Wissenschaften.  Springer-Verlag, New York, 1979.
  * [2] P. Erdős.  On the distribution function of additive functions.  Ann. of Math., 47(2):1–20, 1946.
  * [3] J. Fabrykowski and M.V. Subbarao.  On a class of arithmetic functions satisfying a congruence property.  Journal of the Madras University. Section B, Contributions in mathematics, physical and biological sciences, 51(1):48–56, 1988.
  * [4] W. Feller.  An Introduction to Probability Theory and its Applications, Vol. 1, 3rd Ed. Wiley series in probability and mathematical statistics. John Wiley and Sons, 1968.
  * [5] G.A. Freiman.  Groups and the inverse problems of additive number theory.  In Number-theoretic studies in the Markov spectrum and in the structural theory of set addition (Russian), pages 175–183. Kalinin. Gos. Univ., Moscow, 1973.
  * [6] A. Granville, A. Harper, and K. Soundararajan.  A new proof of Halasz’s theorem, and its consequences.  Compos. Math., 155:126–163, 2019.
  * [7] A. Granville and K. Soundararajan.  Large character sums: pretentious characters and the Pólya-Vinogradov theorem.  J. Amer. Math. Soc, 20(2):357–384, 2007.
  * [8] H. Iwaniec and E. Kowalski.  Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications.  American Mathematical Society, Providence, RI, 2004.
  * [9] N. Jones.  A rigidity phenomenon for power maps.  IMRN, 24:7551–7579, 2017.
  * [10] I. Kátai.  Some problems in number theory.  Studia Sci. Math. Hungar., 16(3-4):289–295, 1983.
  * [11] N.M Katz and Z. Zheng.  On the distribution of Gauss sums and Jacobi sums.  In Analytic Number Theory, Vol. 2 (Allerton Park, IL), Progr. Math., vol. 139, pages 537–558, Boston, MA, 1995. Birhäuser Boston.
  * [12] O. Klurman.  Correlations of multiplicative functions and applications.  Compositio Math., 153(8):1622–1657, 2017.
  * [13] O. Klurman and A. P. Mangerel.  Rigidity theorems for multiplicative functions.  Math. Ann., 372(1-2):651–697, 2018.
  * [14] O. Klurman, A. P. Mangerel, and J. Teräväinen.  Multiplicative functions in short arithmetic progressions.  Proc. Lond. Math. Soc., 127(2):366–446, 2023.
  * [15] J.C. Lagarias, H.L. Montgomery, and A.M. Odlyzko.  A bound for the least prime ideal in the Chebotarev density theorem.  Invent. Math., 54(3):271–296, 1979.
  * [16] J.C. Lagarias and A.M. Odlyzko.  Effective versions of the Chebotarev density theorem.  In Algebraic number fields: L-functions and Galois properties, Proc. Sympos. Univ. Durham, pages 409–464, Durham, UK, 1977. London, Academic Press.
  * [17] A. Languasco and A. Zaccagnini.  A note on Mertens’ formula in arithmetic progressions.  J. Number Theory, 127:37–46, 2007.
  * [18] A.P. Mangerel.  Large sums of high order characters.  J. London Math. Soc. (to appear), 2023.  arXiv:2207.14377 [math.NT].
  * [19] A.P. Mangerel.  On equal consecutive values of multiplicative functions.  arXiv:2306.09929 [math.NT], 2023.
  * [20] K. Matomäki, M. Radziwiłł, and T. Tao.  An averaged form of Chowla’s conjecture.  Algebra and Number Theory, 9:2167–2196, 2015.
  * [21] L.B. Pierce, C. Turnage-Butterbaugh, and M.M. Wood.  An effective Chebotarev density theorem for families of number fields, with an application to $\ell$-torsion in class groups.  Invent. Math., 219:707–778, 2020.
  * [22] P. Pollack.  Non-negative multiplicative functions on sifted sets, and the square roots of $-1$ modulo shifted primes.  Glasgow Math. J., 62(1):187–199, 2020.
  * [23] I. Ruzsa.  On the concentration of additive functions.  Acta Math. Acad. Sci. Hung., 36:215–232, 1980.
  * [24] T. Tao.  The logarithmically-averaged Chowla and Elliott conjectures for two-point correlations.  Forum Math. Pi, 4 e8, 2016.
  * [25] G. Tenenbaum.  Introduction to analytic and probabilistic number theory, 3rd Ed. Graduate Studies in Mathematics vol. 163, American Mathematical Society, 2015.
  * [26] J. Thorner and A. Zaman.  A zero density estimate for Dedekind zeta functions.  IMRN, 2023(8):6739–6761, 2023.
  * [27] E. Wirsing, P. Shao, and Y. Tang.  On a conjecture of kátai for additive functions.  J. Number Theory, 56(2):391–395, 1996.