# Floquet Nonadiabatic Mixed Quantum-Classical Dynamics in Laser-Dressed Solid
Systems

Jingqi Chen  Fudan University, 220 Handan Road, Shanghai 200433, China
Department of Chemistry, School of Science, Westlake University, Hangzhou
310024 Zhejiang, China  Institute of Natural Sciences, Westlake Institute for
Advanced Study, Hangzhou 310024 Zhejiang, China  Yu Wang
<EMAIL_ADDRESS>Department of Chemistry, School of Science, Westlake
University, Hangzhou 310024 Zhejiang, China  Institute of Natural Sciences,
Westlake Institute for Advanced Study, Hangzhou 310024 Zhejiang, China  Wenjie
Dou<EMAIL_ADDRESS>Department of Chemistry, School of Science,
Westlake University, Hangzhou 310024 Zhejiang, China  Institute of Natural
Sciences, Westlake Institute for Advanced Study, Hangzhou 310024 Zhejiang,
China  Department of Physics, School of Science, Westlake University, Hangzhou
310024 Zhejiang, China

###### Abstract

In this paper, we introduce the Floquet Ehrenfest and Floquet surface hopping
approaches to study the nonadiabatic dynamics in the laser-dressed solid
systems. We demonstrate that these two approaches can be formulated in both
real and reciprocal spaces. Using these approaches, we are able to simulate
the interaction between electronic carriers and phonons under periodic
drivings, such as strong light-matter interactions. Employing the Holstein and
Peierls models, we show that the strong light-matter interactions can
effectively modulate the dynamics of electronic population and mobility.
Notably, our study demonstrates the feasibility and effectiveness of modeling
low-momentum carriers’ interactions with phonons using a truncated reciprocal-
space basis, an approach impractical in real-space frameworks. Moreover, we
reveal that even with significant truncation, carrier populations derived from
surface hopping maintain greater accuracy compared to those obtained via
meanfield dynamics. These results underscore the potential of our proposed
methods in advancing the understanding of carrier-phonon interactions in
various laser-dressed materials.

††preprint: AIP/123-QED

##  I Introduction

In solid-state materials, various interactions such as electron-electron,
electron-phonon, electron-hole, electron-impurity, and electron-photon
interactions play pivotal roles in determining material properties and
designing functional materials for diverse applications. Among these
interactions, electron-phonon and electron-photon interactions are
particularly significant for understanding carrier dynamics in extended
systems.

Electron-phonon interactions significantly influence the electronic and
thermal characteristics of solid-state materials. For example, in hybrid metal
halide perovskites, interactions between charge carriers and optical phonons
dictate the broadening of emission lines at room temperatureWright  _et al._
(2016). Intense electron-phonon interactions in blue-emitting perovskites
result in rapid non-radiative decay, thereby reducing the photoluminescence
quantum yieldGong  _et al._ (2018). Furthermore, electron-phonon coupling is
also a key mechanism for conventional superconductivity, where Cooper pairs of
electrons are formed due to attractive interactions mediated by lattice
vibrations (phonons)Kulić (2000); Shen  _et al._ (2002). Experimental evidence
suggests that electron-phonon coupling strongly influences the electron
dynamics in high-temperature superconductorsLanzara  _et al._ (2001); Reznik
_et al._ (2006).

Nowadays, strong light-matter interaction have become prominent, leading to
significant and observable effects. With the development of modern techniques,
properties of solids have been modified by light-matter strong coupling or
laser fields Garcia-Vidal, Ciuti, and Ebbesen (2021). For example, in the
field of quantum computation, researchers have proposed a scheme to accelerate
the nontrivial two-qubit phase gate by ultrastrongly coupling superconducting
flux qubits to a circuit quantum electrodynamics systemWang  _et al._ (2017).
Experiments have demonstrated that materials subjected to intense terahertz
pulses exhibit transient superconducting properties at significantly higher
temperatures than at equilibrium Mankowsky  _et al._ (2014); Yang  _et al._
(2019). Applying electromagnetic radiation to a topologically trivial
insulator, such as a non-inverted HgTe/CdTe quantum well with no edge state in
the static limit, can induce novel topological edge states Lindner, Refael,
and Galitski (2011). Suitably chosen irradiation parameters can open gaps at
the Dirac point of graphene and even drive graphene into the topological
Haldane phase Kitagawa  _et al._ (2010, 2011).

From theoretical perspective, the analysis of periodically driven systems
often relies on Floquet theorem Bukov, D’Alessio, and Polkovnikov (2015).
Floquet formalism allows for the reduction of the periodic or quasi-periodical
time-dependent Hamiltonian into time-independent Floquet matrix Chu and Telnov
(2004). Several theoretical methods have been developed under the Floquet
representation to solve problems related to periodically driven solid states.
For example, Kitagawa $et$ $al$. Kitagawa  _et al._ (2011) demonstrated that
quantum transport properties can be controlled in materials such as graphene
and topological insulators by applying light using Floquet Green’s function.
Tsuji $et$ $al$. Tsuji, Oka, and Aoki (2008) studied correlated electron
systems that are periodically driven by external fields via Floquet dynamical
mean-field theory and showed that photoinduced midgap states emerge from
strong ac fields, triggering an insulator-metal transition.

In this study, we develop two trajectory-based nonadiabatic mix quantum-
classical dynamics methods, namely the Floquet meanfield dynamics method (FMD)
and the Floquet surface hopping (FSH) algorithm, for solid states under
external periodic drivings. We consider interactions between electronic
carriers and phonons both in real and reciprocal spaces. Real space
simulations are generally more suitable for disordered solids, while
reciprocal space simulations are more suitable for ordered solids. Previously,
several theoretical methods have been proposed to describe lattice-based
electron-phonon couplings beyond perturbative regimes, such as mixed quantum-
classical dynamicsKrotz, Provazza, and Tempelaar (2021), and surface
hoppingKrotz and Tempelaar (2022); Xie  _et al._ (2022). Using a tight-binding
model involving Holstein-type Holstein (1959); Freericks, Jarrell, and
Scalapino (1993); Zhang, Jeckelmann, and White (1999) and Peierls-type Su,
Schrieffer, and Heeger (1979) electron-phonon couplings, we calculate
electronic dynamics and charge mobility under different drivings. We expect
that the FMD and FSH algorithms, both in real- and reciprocal-space, can
provide theoretical predictions on solid state property modified by external
periodic drivings.

The structure of this paper is organized as follows: In Sec. II, we present
the theory of general FMD and FSH in real- and reciprocal-space. In Sec. III,
we employ Holstein and Peierls models to simulate electronic dynamics and
charge mobility under different periodic drivings using FMD and FSH. We
consider the truncation in the Brillouin zone as well. Finally, we conclude in
Sec. IV.

##  II Theory

###  II.1 Model Hamiltonian

Without loss of generality, we focus on the one-dimensional lattice models
that feature a solitary quantum state at each site within a unit cell. The
total Hamiltonian for the coupled electron-phonon system gives as:

$\hat{H}_{\text{tot}}(t)=\hat{H}_{\text{ph}}(\mathbf{R},\mathbf{P})+\hat{H}_{\text{el}}(t)+\hat{H}_{\text{el-
ph}}(\mathbf{R}).$ (1)

In this formulation, the term $\hat{H}_{\text{ph}}(\mathbf{R},\mathbf{P})$
denotes the classical Hamiltonian for the phonon energy, with
$\mathbf{R}=(r_{1},r_{2},...,r_{N})$ ($N$ represents the total number of
sites) and $\mathbf{P}=(p_{1},p_{2},...,p_{N})$ being the classical position
and momentum vectors, respectively. The operator
$\hat{H}_{\text{el}}(t)=\hat{H}_{\text{el}}(t+T)$ is the electronic
Hamiltonian subjected to periodic drivings with a time period $T$. In
addition, $\hat{H}_{\text{el-ph}}(\mathbf{R})$ represents the electron-phonon
coupling.

For a one-dimensional lattice of harmonic, noninteracting, and dispersionless
phonons, the real space Hamiltonian is given by:

$H_{\text{ph}}=\sum_{n=1}^{N}\left(\frac{1}{2}p_{n}^{2}+\frac{1}{2}\omega^{2}r_{n}^{2}\right),$
(2)

where $\omega$ indicates the phonon energy. The transformation from real to
reciprocal space gives as:

$\tilde{f}_{k}=\frac{1}{\sqrt{N}}\sum_{n}e^{ikn}f_{n},$ (3)

Here, we set the lattice constant to be unit, and $k$ is the wavevector
ranging from $-\pi$ to $\pi$. Henceforth, a tilde will be used to denote
reciprocal-space coordinates. After introducing classical equivalent of the
ladder operator:

$\tilde{a}_{k}=\sqrt{\frac{\omega}{2}}(\tilde{r}_{k}+i\frac{\tilde{p}_{k}}{\omega}),$
(4)

we can get the position $\tilde{r}_{k}$ and momentum $\tilde{p}_{k}$ in the
reciprocal space as:

$\begin{split}\tilde{r}_{k}&=\frac{1}{\sqrt{2\omega}}(\tilde{a}_{k}+\tilde{a}_{k}^{*})\\\
&=\frac{1}{\sqrt{N}}\sum_{n=1}^{N}\left(r_{n}\cos(kn)-\frac{p_{n}}{\omega}\sin(kn)\right),\end{split}$
(5)
$\begin{split}\tilde{p}_{k}&=-i\sqrt{\frac{\omega}{2}}(\tilde{a}_{k}-\tilde{a}_{k}^{*})\\\
&=\frac{\omega}{\sqrt{N}}\sum_{n=1}^{N}\left(\frac{p_{n}}{\omega}\cos(kn)+r_{n}\sin(kn)\right).\end{split}$
(6)

Note that under the transformation from real to reciprocal space, classical
position and momentum coordinates become scrambled. Then, we obtain the phonon
Hamiltonian in the reciprocal space:

$H_{\text{ph}}=\sum_{k=-\pi}^{\pi}\left(\frac{1}{2}\tilde{p}_{k}^{2}+\frac{1}{2}\omega^{2}\tilde{r}_{k}^{2}\right).$
(7)

For the electronic part, we consider a simple tight-binding model with
periodic driving in real and reciprocal space:

$\begin{split}\hat{H}_{\text{el}}(t)=&-(J_{0}+J_{1}\cos(\Omega
t))\sum_{n}\left(\hat{c}_{n+1}^{\dagger}\hat{c}_{n}+\mathrm{h.c.}\right)\\\
=&-2(J_{0}+J_{1}\cos(\Omega
t))\sum_{k}\hat{c}_{k}^{\dagger}\hat{c}_{k}\cos(k),\end{split}$ (8)

where $\hat{c}_{n}^{(\dagger)}$ and $\hat{c}_{k}^{(\dagger)}$ are real- and
reciprocal-space electronic annihilation (creation) operators, respectively.
$(J_{0}+J_{1}\cos(\Omega t))$ is the nearest-interaction term. Here, we assume
that lasers can affect the hopping probability between two sides. $J_{0}$
signifies the coupling strength inherent to the electronic states, while
$J_{1}\cos(\Omega t)$ provides a semi-classical portrayal of light-matter
interactions: $J_{1}$ is the coupling strength, and $\Omega$ is the driving
frequency. By placing this term in off-diagonal terms, we highlight how light
influences the couplings among different electron states.

Then we give the Hamiltonian of electron-phonon coupling $\hat{H}_{\text{el-
ph}}$ for two models. Firstly, as for the Holstein model:

$\displaystyle\begin{aligned} \hat{H}_{\text{el-
ph}}=&g\sqrt{2\omega^{3}}\sum_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n}r_{n}\\\
=&g\sqrt{\frac{\omega}{2N}}\sum_{k,\kappa}\hat{c}_{k+\kappa}^{\dagger}\hat{c}_{k}(\omega(\tilde{r}_{-\kappa}+\tilde{r}_{\kappa})-i(\tilde{p}_{-\kappa}-\tilde{p}_{\kappa})),\end{aligned}$
(9)

while for the Peierls model:

$\displaystyle\begin{aligned} \hat{H}_{\text{el-
ph}}=&g\sqrt{2\omega^{3}}\sum_{n}(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+\hat{c}_{n+1}^{\dagger}\hat{c}_{n})(r_{n}-r_{n+1})\\\
=&g\sqrt{\frac{2\omega}{N}}\sum_{k,\kappa}\hat{c}_{k+\kappa}^{\dagger}\hat{c}_{k}(i\omega(\tilde{r}_{-\kappa}+\tilde{r}_{\kappa})+(\tilde{p}_{-\kappa}-\tilde{p}_{\kappa}))\\\
&\times(\sin(k+\mathbf{\kappa})-\sin(k)).\end{aligned}$ (10)

Here, $g$ is the dimensionless coupling parameter, which is related to the
vibrational reorganization energy as $g^{2}\omega$.

###  II.2 Floquet Theory

Floquet theory is a useful tool to address the complexities arising from the
time-periodic Hamiltonian $\hat{H}(\mathbf{R},t)=\hat{H}(\mathbf{R},t+T)$.
Consider a general periodic Hamiltonian and the time-dependent Schrödinger
equation:

$\displaystyle\begin{aligned}
i\hbar\ket{\dot{\Psi}(\mathbf{R},t)}=\hat{H}(\mathbf{R},t)\ket{\Psi(\mathbf{R},t)}.\end{aligned}$
(11)

Then we can expand our wavefunction in the following form, which is commonly
used to describe energy eigenstates and their evolution:

$\Psi(\mathbf{R},t)=u(\mathbf{R},t)e^{-i\epsilon t/\hbar}.$ (12)

Here, $u(\mathbf{R},t)$ is a complex function that represents the time-varying
amplitude of the wave function. $e^{-i\epsilon t/\hbar}$ is a complex
exponential function that represents the time-varying phase of the wave
function, $\epsilon$ is a constant representing an energy eigenvalue. If we
take the expanded form of this wave function into Eq. (11), we can get:

$H^{\text{F}}u(\mathbf{R},t)=\epsilon u(\mathbf{R},t),$ (13)

where
$H^{\text{F}}(\mathbf{R},t)=\hat{H}(\mathbf{R},t)-i\hbar\frac{\partial}{\partial
t}$ is called Floquet Hamiltonian. We aim to make Hamiltonian time-
independent; the first step is to use the periodicity:
$u(\mathbf{R},t)=u(\mathbf{R},t+T)$, so it can be Fourier transformed as:

$u(\mathbf{R},t)=\sum_{\alpha=-n_{\text{F}}}^{n_{\text{F}}}c_{\alpha}(\mathbf{R})e^{i\alpha\Omega
t},$ (14)

where the frequency is $\Omega=\frac{2\pi}{T}$. In the next step, we do the
unitary transform on the time-dependent Hamiltonian, and we have:

$\displaystyle\begin{aligned}
&u^{\dagger}(\mathbf{R},t)H^{\text{F}}(\mathbf{R},t)u(\mathbf{R},t)=\sum_{\beta=-m_{\text{F}}}^{m_{\text{F}}}\sum_{\alpha=-n_{\text{F}}}^{n_{\text{F}}}\\\
&(c^{\dagger}_{\beta}(r)e^{-i\beta\Omega
t}\hat{H}(\mathbf{R},t)e^{i\alpha\Omega
t}c_{\alpha}(r)-c^{\dagger}_{\beta}(r)c_{\alpha}(r)\alpha\hbar\Omega
e^{i(\alpha-\beta)\Omega t})\end{aligned}$ (15)

We need to do the integration in one time period to get the results:

$\begin{split}[(\sum_{\beta=-m_{\text{F}}}^{m_{\text{F}}}\sum_{\alpha=-n_{\text{F}}}^{n_{\text{F}}}&\frac{1}{T}\int_{0}^{T}dte^{-i\beta\Omega
t}\hat{H}(\mathbf{R},t)e^{i\alpha\Omega t})\\\
&-\alpha\hbar\Omega\delta_{\alpha\beta}]c_{\alpha}(r)=\epsilon
c_{\beta}(r).\end{split}$ (16)

Finally, we can get the time-independent Floquet Hamiltonian:

$\hat{H}^{\text{F}}(\mathbf{R})=\sum_{\alpha}H_{\beta\alpha}(\mathbf{R})-\alpha\hbar\Omega\delta_{\alpha\beta},$
(17)

Now, we can transform the Hamiltonian into the Floquet representation. As for
the Holstein model, we can construct each single block $H_{\beta\alpha}$ as:

$\begin{split}H_{\beta\alpha}=&[-J_{0}\sum_{n}\left(\hat{c}_{n+1}^{\dagger}\hat{c}_{n}+\text{h.c.}\right)+g\sqrt{2\omega^{3}}\sum_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n}r_{n}\delta_{\alpha\beta}\\\
&-\frac{1}{2}J_{1}\sum_{n}\left(\hat{c}_{n+1}^{\dagger}\hat{c}_{n}+\text{h.c.}\right)(\delta_{\alpha,\beta+1}+\delta_{\alpha,\beta-1})\\\
=&[-2J_{0}\sum_{k}\hat{c}_{k}^{\dagger}\hat{c}_{k}\cos(k)+g\sqrt{\frac{\omega}{2N}}\sum_{k,\kappa}\hat{c}_{k+\kappa}^{\dagger}\hat{c}_{k}(\omega(\tilde{r}_{-\kappa}+\tilde{r}_{\kappa})\\\
&-i(\tilde{p}_{-\kappa}-\tilde{p}_{\kappa}))]\delta_{\alpha\beta}-J_{1}\sum_{k}\hat{c}_{k}^{\dagger}\hat{c}_{k}\cos(k)(\delta_{\alpha,\beta+1}+\delta_{\alpha,\beta-1}),\end{split}$

while, as for the Peierls model:

$\displaystyle\begin{aligned}
H_{\beta\alpha}=&[-J_{0}\sum_{n}\left(\hat{c}_{n+1}^{\dagger}\hat{c}_{n}+\mathrm{h.c.}\right)\\\
&+g\sqrt{2\omega^{3}}\sum_{n}(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+\hat{c}_{n+1}^{\dagger}\hat{c}_{n})(r_{n}-r_{n+1})]\delta_{\alpha\beta}\\\
&-\frac{1}{2}J_{1}\sum_{n}\left(\hat{c}_{n+1}^{\dagger}\hat{c}_{n}+\text{h.c.}\right)(\delta_{\alpha,\beta+1}+\delta_{\alpha,\beta-1})\\\
=&[-2J_{0}\sum_{k}\hat{c}_{k}^{\dagger}\hat{c}_{k}\cos(k)+g\sqrt{\frac{2\omega}{N}}\sum_{k,\kappa}\hat{c}_{k+\kappa}^{\dagger}\hat{c}_{k}(i\omega(\tilde{r}_{-\kappa}+\tilde{r}_{\kappa})\\\
&+(\tilde{p}_{-\kappa}-\tilde{p}_{\kappa}))(\sin(k+\mathbf{\kappa})-\sin(k))]\delta_{\alpha\beta}\\\
&-J_{1}\sum_{k}\hat{c}_{k}^{\dagger}\hat{c}_{k}\cos(k)(\delta_{\alpha,\beta+1}+\delta_{\alpha,\beta-1}).\end{aligned}$

It is easier to understand the band structure in the reciprocal space
according to its mathematical form. Therefore, we plot the multiple Floquet
quasi-surfaces with different driving conditions in the first Brillouin zone,
which are shown in Fig.1.

Figure 1: Floquet replicas without electron-phonon coupling in reciprocal
space. Here, we set the Floquet levels $n_{\text{F}}=2$, so there are $5$
quasi-surfaces in total. Other parameters: $J=0.3$, $\hbar\omega=0.3$,
$g=0.5$, and $kT=0.5$.

The original band has been extended into multiple Floquet bands due to the
Floquet drivings. The energy gap between the nearest two Floquet bands is
proportional to the driving frequency.

###  II.3 Floquet Meanfield Dynamics (FMD)

In this subsection, we will introduce the Floquet meanfield approach in real
and reciprocal space. Firstly, according to Schrödinger’s equation, we can
propagate the wavefunction as:

$\frac{\partial}{\partial
t}\ket{\Psi^{\text{F}}(t)}=-i\hbar\hat{H}^{\text{F}}(\mathbf{R})\ket{\Psi^{\text{F}}(t)}.$
(18)

Here, $\hat{H}^{\text{F}}(\mathbf{R})$ including pure electron Hamiltonian and
electron-phonon coupling Hamiltonian. We can express the Ehrenfest dynamics in
real space as follows:

$\displaystyle\begin{aligned} &\dot{r}_{n}=p_{n},\\\
&\dot{p}_{n}=-\omega^{2}r_{n}-\langle\Psi^{\text{F}}\mid\nabla_{r_{n}}\hat{H}^{\text{F}}\mid\Psi^{\text{F}}\rangle.\end{aligned}$
(19)

Next, we will extend this approach to the dynamical evolution in reciprocal
space as well. The evolution of wavefunctions and nuclear coordinates in
reciprocal space follows a similar approach as in real space. However, the
classical position coordinates in real space depend both on position and
momentum coordinates in reciprocal space as shown in Eqs. 5 and 6. Therefore,
the equations of motion involve both position and momentum-derivative terms in
reciprocal space are given by:

$\displaystyle\begin{aligned}
&\dot{r}_{k}=p_{k}+\langle\Psi^{\text{F}}\mid\nabla_{p_{k}}\hat{H}^{\text{F}}\mid\Psi^{\text{F}}\rangle,\\\
&\dot{p}_{k}=-\omega^{2}r_{k}-\langle\Psi^{\text{F}}\mid\nabla_{r_{k}}\hat{H}^{\text{F}}\mid\Psi^{\text{F}}\rangle.\end{aligned}$
(20)

###  II.4 Floquet Surface Hopping(FSH)

Within the surface hopping algorithm, one propagates a trajectory along the
adiabatic surface, and integrates the electronic density matrix. Here, we
propagate a trajectory along the Floquet adiabatic surface. The hopping rate
from surface $J$ to surface $I$ can be determined bySubotnik  _et al._ (2016);
Wang, Akimov, and Prezhdo (2016)

$k_{J\rightarrow I}=\Theta(\Gamma_{J\rightarrow I}),$ (21)

where $\Theta$ is the Heaviside function

$\Theta(x)=\begin{cases}x&\text{x>0}\\\ 0&\text{x<0}\end{cases}$ (22)

and

$\Gamma_{J\rightarrow
I}=-2\Re\left(\mathbf{P}\cdot\bm{d}_{IJ}\frac{\hat{\rho}^{\text{F(ad)}}_{JI}}{\hat{\rho}^{\text{F(ad)}}_{JJ}}\right).$
(23)

Here, we set nuclear mass and $\hbar$ equal to $1$.
$\hat{\rho}^{\text{F(ad)}}=\ket{\Psi^{\text{F(ad)}}}\bra{\Psi^{\text{F(ad)}}}$
is the density matrix in adiabatic representation. $\bm{d}_{IJ}$ is the
derivative coupling

$\bm{d}_{IJ}=\frac{-\bra{\Psi_{I}^{\text{F(ad)}}}\frac{\partial\hat{H}^{\text{F}}}{\partial
r_{n}}\ket{\Psi_{J}^{\text{F(ad)}}}}{E_{I}-E_{J}},$ (24)

where $\ket{\Psi^{\text{ad}}}$ and $E$ are the eigenvector and eigenvalue of
$\hat{H}^{\text{F}}$, respectively.

After a hop from Floquet state $J$ to Floquet state $I$, one should rescale
the nuclear momentum in the direction of derivative coupling
$\bm{d}_{JI}$Subotnik, Ouyang, and Landry (2013); Dou and Subotnik (2017)

$p^{\text{new}}=p-\xi\bm{d}_{JI}/|\bm{d}_{JI}|,$ (25)

where the $\xi$ can be solved by energy conservation

$\frac{(p_{k}^{\text{new}})^{2}}{2}+E_{I}=\frac{(p_{k})^{2}}{2}+E_{J}.$ (26)

Among the two roots satisfying Eq. 26, we choose the one with a smaller
$|\kappa|$. We now proceed to FSH method in reciprocal space, and we know that
after this canonical transformation from real to reciprocal space in Eqs. 5
and 6, the classical position and momentum become scrambled. Therefore, we can
deduce that:

$\begin{split}\bra{\Psi_{I}^{\text{F(ad)}}}\frac{\partial}{\partial
t}\ket{\Psi_{J}^{\text{F(ad)}}}=&\dot{r}_{k}\bra{\Psi_{I}^{\text{F(ad)}}}\frac{\partial}{\partial
r_{k}}\ket{\Psi_{J}^{\text{F(ad)}}}+\\\
&\dot{p}_{k}\bra{\Psi_{I}^{\text{F(ad)}}}\frac{\partial}{\partial
p_{k}}\ket{\Psi_{J}^{\text{F(ad)}}}.\end{split}$ (27)

Here, we can define:

$d_{IJ}^{r_{k}}=\bra{\Psi_{I}^{\text{F(ad)}}}\frac{\partial}{\partial
r_{k}}\ket{\Psi_{J}^{\text{F(ad)}}},d_{IJ}^{p_{k}}=\bra{\Psi_{I}^{\text{F(ad)}}}\frac{\partial}{\partial
p_{k}}\ket{\Psi_{J}^{\text{F(ad)}}}.$ (28)

In this way, $\mathbf{P}\cdot\bm{d}_{IJ}$ in real-space hopping rate (see Eq.
23) becomes $(p_{k}\cdot d_{IJ}^{r_{k}}-\omega^{2}r_{k}\cdot d_{IJ}^{p_{k}})$.
After a hop from Floquet state $J$ to Floquet state $I$ in reciprocal space,
one should rescale not only the nuclear momentum but also the position:

$\begin{split}&p^{\text{new}}_{k}=p_{k}-\xi\bm{d}_{JI}^{r_{k}}/|\bm{d}_{JI}^{r_{k}}|,\\\
&r^{\text{new}}_{k}=r_{k}+\xi\bm{d}_{JI}^{p_{k}}/|\bm{d}_{JI}^{p_{k}}|,\end{split}$
(29)

where $\xi$ can be solved by energy conservation,

$\frac{(p^{\text{new}}_{k})^{2}}{2}+\frac{\omega^{2}(r^{\text{new}}_{k})^{2}}{2}+E_{I}=\frac{(p_{k})^{2}}{2}+\frac{\omega^{2}(r_{k})^{2}}{2}+E_{J}.$
(30)

##  III Results and discussions

###  III.1 Electronic populations

In this analysis, we consider a lattice composed of $N=10$ sites while
adhering to the parameters $J_{0}=0.3$, $\hbar\omega=0.3$, $g=0.5$, and
$T=0.5$ in the context of the Holstein model and the Peierls model. To obtain
these numerical results, we employ a 4th-order Runge-Kutta algorithm to
propagate both classical and quantum coordinates. The chosen time step for the
algorithm is $dt=0.01$. For the sake of statistical accuracy, the outcomes
have been averaged across $10,000$ initial thermal conditions that conform to
the Boltzmann distribution for the classical coordinates. Besides, we set the
initial probability that the electron appears in each electronic state to be
exactly equal in real space; thus, the probability is completely gathered in
the $k=0$ state in reciprocal space. Here we graph
$P_{0}(t)\equiv\left|\langle 0|\Psi(t)\rangle\right|^{2}$ as a function of
time.

Figure 2: Electronic population dynamics at $k=0$ under varying periodic
drivings for the Holstein model, employing four methods: FMD and FSH in both
real- and reciprocal-space. In scenario (a), we fix the driving frequency
$\Omega=0.5$ and vary the driving amplitude $J_{1}$ from $0$ to $1$. In
scenario (b), with a constant driving amplitude $J_{1}=0.4$, the driving
frequency $\Omega$ is altered from $0.2$ to $2$. The parameters are set as
$J_{0}=0.3$, $\omega=0.3$, $g=0.5$, $T=0.5$, and $N=10$.

In Fig. 2 (a), we fix the driving frequency $\Omega=0.5$. We observe that with
an increase in the driving amplitude $A$, the steady-states of the electronic
population at momentum zero initially decrease, followed by a subsequent
increase. This observation suggests that periodic drivings have the capacity
to transfer energy to electrons within solid-state materials, thereby
rendering the electrons more dynamically active. In Fig. 2 (b), with a fixed
driving amplitude $A=0.4$, we observe an intriguing pattern as the driving
frequency ($\Omega$) is varied. Initially, the steady-states of $\rho_{0}$
decrease, followed by a subsequent increase until reaching a point where they
converge to the same outcome as in the absence of drivings ($J_{1}=0$). Under
fast drivings, where $\Omega$ is sufficiently large, the system may fail to
respond to such rapid drivings. Consequently, the properties of solids revert
to their original conditions under fast drivings as in the absence of periodic
drivings. Next, we turn to the Peierls model to simulate electronic population
dynamics under various periodic drivings, as shown in Fig. 3. Remarkably, we
observe a consistent trend reminiscent of the findings in the Holstein model.

Figure 3: Electronic population dynamics at $k=0$ under varying periodic
drivings for the Peierls model, employing four methods: FMD and FSH in both
real- and reciprocal-space. In scenario (a), we fix the driving frequency
$\Omega=0.5$ and vary the driving amplitude $J_{1}$ from $0$ to $1$. In
scenario (b), with a constant driving amplitude $J_{1}=0.4$, the driving
frequency $\Omega$ is altered from $0.2$ to $2$. The parameters are set as
$J_{0}=0.3$, $\omega=0.3$, $g=0.5$, and $T=0.5$, and $N=10$.

###  III.2 Charge mobility simulations

By employing our FMD and FSH methods, we also simulate charge mobility in a
short time under different periodic drivings in real space, as shown in Figs.
4 and 5. Short-time charge mobility is essential for semiconductor devices or
high-frequency electronics. For example, gallium nitride (GaN) high electron-
mobility transistors have emerged as excellent power devicesMeneghesso  _et
al._ (2008); Meneghini  _et al._ (2021). Here, we simulate the dynamics of the
mean square displacement $\langle r^{2}\rangle$ of the charge in real space,
which is proportional to the charge mobilityWang  _et al._ (2010). Different
from last subsection, here we initialize the electronic population at the
middle site in real space and take an average of $10,000$ trajectories for
each dynamic result.

For the Holstein model, we find that by adding Floquet drivings, short-time
charge mobility can be enhanced with increasing the strength of the driving
(see Fig. 4 (a)). When the driving frequency becomes sufficiently high, the
system may fail to respond effectively to the rapid driving, resulting in
charge mobility comparable to the condition without periodic drivings. In the
Peierls model, we observe a similar trend to that in the Holstein model, as
shown in Fig. 5.

Figure 4: Mean square displacement in real space ($\langle r^{2}\rangle$) as a
function of time under different periodic drivings for Holstein model. a) We
fix the driving frequency $\Omega=0.5$, and change the driving amplitude
$J_{1}$ from $0$ to $0.8$. We see $\langle r^{2}\rangle$ increases with
enhancing the driving amplitude $A$. b) We fix the driving amplitude
$J_{1}=0.4$, and change driving frequencies $\Omega$ from $0.2$ to $2$. We see
$\langle r^{2}\rangle$ increases first and then decreases to a point where
there is no driving. Here, $J_{0}=0.3$, $\omega=0.3$, $g^{2}=0.25$, $T=0.5$,
and $N=10$. Figure 5: Mean square displacement in real space ($\langle
r^{2}\rangle$) as a function of time under different periodic drivings for
Holstein model. a) We fix the driving frequency $\Omega=0.5$, and change the
driving amplitude $J_{1}$ from $0$ to $0.8$. We see $\langle r^{2}\rangle$
increases with enhancing the driving amplitude $A$. b) We fix the driving
amplitude $J_{1}=0.4$, and change the driving frequencies $\Omega$ from $0.2$
to $2$. We see $\langle r^{2}\rangle$ increases first and then decreases to a
point where there is no driving. Here, $J_{0}=0.3$, $\omega=0.3$,
$g^{2}=0.25$, $T=0.5$, and $N=10$.

###  III.3 Truncation in Brillouin zone

The advantage of using a real-space approach lies in its ability to precisely
capture local interactions within a specific spatial region while keeping
computational demands in check by limiting the scope of local interactions
considered. Similarly, for phenomena restricted to a certain range within the
Brillouin zone, a comparable strategy can be employed in reciprocal space. In
scenarios where electronic carriers are localized due to either the short time
scales not permitting full Brillouin zone traversal or their stabilization at
band minima, such confinement can be effectively modeled. This concept has
been effectively employed in the theoretical analysis of exciton and trion
states in monolayer transition-metal dichalcogenides. In these studies, the
analysis was focused around the K points in the Brillouin zone by implementing
a specific truncation radius, thereby streamlining the computations.

In FMD and FSH dynamics, a reciprocal space formulation allows for effective
truncation when the initial population is concentrated at or near $k=0$. Here
we set $N=20$, $J_{0}=0.6$, $\hbar\omega=0.2$, $J_{1}=0.3$, $\Omega=0.5$,
$g^{2}=5$, and $T=1$. We symmetrically truncate the corner of the Brillouin
zone and keep $4/5$, $3/5$, and $2/5$ sizes of the original one, and the
corresponding size numbers $N$ are $16$, $12$, and $8$, respectively. Figs. 6
and 7 illustrate the temporal evolution of the whole population (Figs. 6 (a)
and 7 (a)) and $P_{0}$ (Figs. 6 (b) and 7 (b)) for the Holstein model under
different truncation scales, as calculated by FMD and FSH respectively.
Remarkably, these methods yield accurate electronic population profiles even
with significant truncation, such as at $k_{0}=\frac{2}{5}\pi$, closely
matching the untruncated results, as can be seen in the Figs. 6 (c) and 7 (c).

Figure 6: Truncated Electronic Population Dynamics $P_{k}(t)$ for the Holstein
Model Using the FMD Method. a) Comparison of the truncated overall population
dynamics over time against the untruncated results. b) Time evolution of the
truncated $P_{0}(t)$ at the center of the Brillouin zone, juxtaposed with the
untruncated counterpart. c) Analysis of the error $\Delta P_{0}(t)$,
quantifying the deviation from the full Brillouin zone results, across various
truncation radii $k_{0}$. Parameters: $J=0.6$, $\omega=0.2$, $J_{1}=0.3$,
$\Omega=0.5$, $g^{2}=5$, $T=1$, and $N=20$. Figure 7: Truncated Electronic
Population Dynamics $P_{k}(t)$ for the Holstein Model Using the FSH Method. a)
Comparison of the truncated overall population dynamics over time against the
untruncated results. b) Time evolution of the truncated $P_{0}(t)$ at the
center of the Brillouin zone, juxtaposed with the untruncated counterpart. c)
Analysis of the error $\Delta P_{0}(t)$, quantifying the deviation from the
full Brillouin zone results, across various truncation radii $k_{0}$.
Parameters: $J_{0}=0.6$, $\omega=0.2$, $J_{1}=0.3$, $\Omega=0.5$, $g^{2}=5$,
$T=1$, and $N=20$.

However, as observed in Fig. 8, the accuracy of large truncations
($k_{0}=\frac{2}{5}\pi$) in the FMD method for the Peierls model is less
satisfactory. In contrast, as shown in Fig. 9, the FSH method demonstrates
superior performance under similar truncation conditions, highlighting its
advantages.

Importantly, truncation calculations in reciprocal space offer significant
computational benefits. By reducing the size of the computational domain, they
enable more efficient use of computational resources while still maintaining a
high degree of accuracy. This approach is particularly beneficial when dealing
with large systems, especially for the Floquet Hamiltonian, making it an
attractive strategy for efficiently exploring electronic dynamics in complex
models.

Figure 8: Truncated Electronic Population Dynamics $P_{k}(t)$ for the Peierls
Model Using the FMD Method. a) Comparison of the truncated overall population
dynamics over time against the untruncated results. b) Time evolution of the
truncated $P_{0}(t)$ at the center of the Brillouin zone, juxtaposed with the
untruncated counterpart. c) Analysis of the error $\Delta P_{0}(t)$,
quantifying the deviation from the full Brillouin zone results, across various
truncation radii $k_{0}$. Parameters: $J=0.6$, $\omega=0.2$, $J_{1}=0.3$,
$\Omega=0.5$, $g^{2}=5$, $T=1$, and $N=20$. Figure 9: Truncated Electronic
Population Dynamics $P_{k}(t)$ for the Peierls Model Using the FSH Method. a)
Comparison of the truncated overall population dynamics over time against the
untruncated results. b) Time evolution of the truncated $P_{0}(t)$ at the
center of the Brillouin zone, juxtaposed with the untruncated counterpart. c)
Analysis of the error $\Delta P_{0}(t)$, quantifying the deviation from the
full Brillouin zone results, across various truncation radii $k_{0}$.
Parameters: $J=0.6$, $\omega=0.2$, $J_{1}=0.3$, $\Omega=0.5$, $g^{2}=5$,
$T=1$, and $N=20$.

##  IV Conclusions

Our study successfully constructs Floquet Hamiltonian for the Holstein and
Peierls models, showing the huge impact of Floquet driving on Floquet
replicas. The derived FMD and FSH, explored in both real and reciprocal space,
yield consistent results through numerical simulations. This agreement
validates the reliability of our theoretical framework, emphasizing the
equivalence of real and reciprocal space outcomes.

Significantly, our results showcase that light-matter interactions,
facilitated by Floquet driving, can markedly enhance electronic mobility. This
finding points towards practical applications, suggesting the possibility of
engineering materials with improved conductivity. Moreover, our study
demonstrates the potential for computational efficiency through truncation
techniques, which are particularly effective when applied to well-chosen
initial conditions. This insight opens avenues for reducing computational
costs in simulating Floquet-driven systems.

In conclusion, our research contributes valuable insights into the Floquet-
driven dynamics of fundamental models, bridging theoretical understanding with
practical applications. These findings hold promise for the future design and
optimization of materials with tailored electronic properties.

###### Acknowledgements.

This material is based upon work supported by National Natural Science
Foundation of China (NSFC No. 22273075). W.D. acknowledges start-up funding
from Westlake University.

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# Contextual Client Selection
for Federated Learning in Vehicular Networks

Rui Song1,2, Lingjuan Lyu3, Wei Jiang4, Andreas Festag1,5 and Alois Knoll2
This work was supported by the German Federal Ministry for Digital and
Transport (BMVI) in the projects “KIVI – KI im Verkehr Ingolstadt” and ”5GoIng
– 5G Innovation Concept Ingolstadt”.1Rui Song and Andreas Festag are with
Fraunhofer Institute for Transportation and Infrastructure Systems IVI,
Ingolstadt, Germany, e-mail: {rui.song<EMAIL_ADDRESS>Song and Alois Knoll are with Technical University of Munich, Robotics,
Artificial Intelligence and Real-Time Systems, Garching, Germany, e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Lyu is with Sony AI, Tokyo, Japan,
e-mail<EMAIL_ADDRESS>Jiang is with German Research Center for
Artificial Intelligence (DFKI), Kaiserslautern, Germany, e-mail:
<EMAIL_ADDRESS>Festag is with Technische Hochschule Ingolstadt,
CARISSMA Institute for Electric, COnnected, and Secure Mobility (C-ECOS),
Ingolstadt, Germany, e-mail<EMAIL_ADDRESS>
# V2X-Boosted Federated Learning for Cooperative Intelligent Transportation
Systems with Contextual Client Selection

Rui Song1,2, Lingjuan Lyu3, Wei Jiang4, Andreas Festag1,5 and Alois Knoll2
This work was supported by the German Federal Ministry for Digital and
Transport (BMVI) in the projects “KIVI – KI im Verkehr Ingolstadt” and ”5GoIng
– 5G Innovation Concept Ingolstadt”.1Rui Song and Andreas Festag are with
Fraunhofer Institute for Transportation and Infrastructure Systems IVI,
Ingolstadt, Germany, e-mail: {rui.song<EMAIL_ADDRESS>Song and Alois Knoll are with Technical University of Munich, Robotics,
Artificial Intelligence and Real-Time Systems, Garching, Germany, e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Lyu is with Sony AI, Tokyo, Japan,
e-mail<EMAIL_ADDRESS>Jiang is with German Research Center for
Artificial Intelligence (DFKI), Kaiserslautern, Germany, e-mail:
<EMAIL_ADDRESS>Festag is with Technische Hochschule Ingolstadt,
CARISSMA Institute for Electric, COnnected, and Secure Mobility (C-ECOS),
Ingolstadt, Germany, e-mail<EMAIL_ADDRESS>
###### Abstract

Machine learning (ML) has revolutionized transportation systems, enabling
autonomous driving and smart traffic services. Federated learning (FL)
overcomes privacy constraints by training ML models in distributed systems,
exchanging model parameters instead of raw data. However, the dynamic states
of connected vehicles affect the network connection quality and influence the
FL performance. To tackle this challenge, we propose a contextual client
selection pipeline that uses Vehicle-to-Everything (V2X) messages to select
clients based on the predicted communication latency. The pipeline includes:
(i) fusing V2X messages, (ii) predicting future traffic topology, (iii) pre-
clustering clients based on local data distribution similarity, and (iv)
selecting clients with minimal latency for future model aggregation.
Experiments show that our pipeline outperforms baselines on various datasets,
particularly in non-iid settings.

##  I Introduction

Machine learning (ML), a subfield of artificial intelligence, focuses on
developing learning algorithms and inference models that enable digital
systems to make decisions and predictions in terms of the knowledge learned
from data. Over the past years, ML-based approaches exhibited great potential
to revolutionize various scientific, engineering, economic, and cultural
fields with outstanding technological advancements such as Google AlphaGo and
Open AI’s ChatGPT. In the filed of road transportation, ML is possible to
empower numerous new applications for realizing Intelligent Transportation
System (ITS), e.g., environmental perception, road traffic flow optimization,
and trajectory planning, which can significantly enhance the safety and
efficiency of transportation systems [1, 2, 3, 4, 5].

Recently, a new ITS concept referred to as Cooperative Intelligent
Transportation System (C-ITS) attacked a lot of interests from both academia
and industry [6]. In C-ITS, the cooperation between two or more ITS sub-
systems (personal, vehicle, roadside and central) offers better quality and an
enhanced service level, compared to that of the conventional ITS. As
illustrated in Fig. 1, road participants –- specifically, connected automated
vehicles (CAVs) –- can share information with one another through vehicle-to-
everything (V2X) networks, which encompass vehicle-to-vehicle (V2V), vehicle-
to-infrastructure (V2I), vehicle-to-network(V2N), and infrastructure-to-
network (I2N).

The European standards for C-ITS define several types of V2X messages to
facilitate decentralized information sharing. Specifically, for cooperative
awareness and perception, dedicated message types – the Cooperative Awareness
Message (CAM) and the Collective Perception Message (CPM)111European
Telecommunications Standards Institute (ETSI) http://etsi.org/standards,
specifically EN 302 637-2 for the cooperative awareness service and TS 103 324
for the collective perception service. – are periodically exchanged among CAVs
and with roadside infrastructure[7]. By sending and receiving V2X messages,
enriched and improved environmental data of road traffic can be made available
within vehicular networks.

Figure 1: An overview of vehicular networks in C-ITS, including vehicle-to-
vehicle (V2V), vehicle-to-infrastructure (V2I), vehicle-to-network (V2N), and
infrastructure-to-network (I2N) communication.

In centralized model training using ML, CAV clients transmit data to a
centralized system through vehicle-to-network communications. This process can
generate an enormous volume of data, potentially exceeding the network’s
capacity. Moreover, data collected from CAV clients for ML model training
cannot be directly shared due to privacy concerns. Differing from conventional
ML, federated learning (FL) trains ML models using data from distributed
systems, such as devices or clients, without centralizing the data [8]. In FL,
connected clients share a model trained on their local data with a server,
which aggregates the local models and updates the global model. The updated
global model is then shared back with the clients. This process is repeated
for a sufficient number of communication rounds until FL converges.

The deployment of 5G-V2X vehicular networks has further facilitated the use of
FL in C-ITS by providing higher data rates and greater reliability for data
exchange. This allows for the training of larger ML models for C-ITS
applications and services, such as [9, 10, 11]. Although FL has great
potential to preserve privacy and utilize a broader range of data resources
[12], the employment of FL in C-ITS has to address major challenges due to
heterogeneity in data and networks, which can not only limit the performance
but also lead to FL failures.

Data heterogeneity. Data across clients is non-iid (non-identically
independently distributed), resulting from various sensor types, combinations,
poses, road scenes, traffic scenarios, climate and weather conditions, and
more.

Network heterogeneity. The diverse connection qualities of clients can slow
down model sharing and cause communication delays for global model
aggregation, which impedes the FL process.

To address these challenges and enhance the application of FL in C-ITS, we
propose a novel FL framework. The main idea is to select clients for upcoming
communication rounds based on (_i_) the prediction of connection qualities in
the context of road traffic status and (_ii_) the similarity of local data
distribution in clients.

##  II Background and Related Work

Figure 2: Contextual client selection pipeline: (1) V2X message fusion; (2)
Road traffic topology graph (RTTG) prediction; (3) Data-level client grouping;
(4) Network-level client selection.

We discuss greedy and gossip client selection in FL, data- and network-based
strategies, and FL in vehicular networks considering road traffic features.

Greedy and gossip client selection. FL, initially proposed by McMahan et al.
[13], suffers from the straggler effect due to varying connection qualities
[14]. Greedy client selection includes all clients in each communication
round, while gossip (stochastic greedy) selection randomly selects connected
clients. Both strategies struggle to avoid the straggler effect.

Network-based client selection. Strategies focusing on network quality [15,
16, 17, 18] reduce the straggler effect but aren’t specifically designed for
vehicular networks with dynamic connection qualities and high-priority traffic
services. Inspired by [19], we optimize client selection by predicting
communication latency in vehicular networks.

Data-based client selection. Client selection based on data distribution
tackles heterogeneity. The approaches in [20, 21, 22, 23, 24, 25] consider
data heterogeneity but overlook network parameters. Our work addresses both
data distribution and network quality in vehicular networks. A comparison of
the client selection paradigms is shown in Tab. II.

##  III Framework

Our _contextual client selection_ framework is illustrated in Fig. 2,
comprising V2X information sharing, traffic topology prediction, data-level
client clustering, and network-level client clustering.

V2X message fusion. We first fuse V2X messages. Continuously receiving CAM and
CPM enables dynamic road maps with traffic object states. Road-side
infrastructure collects and forwards V2X messages to a server via V2I and I2N
networks. The server filters and fuses messages, obtaining traffic object
states, such as position, speed, and acceleration. Fused results form an road
traffic topology graph (RTTG), with each CAV characterized by a node with
attributes. The RTTG digitizes C-ITS and recreates vehicular networks
virtually.

RTTG prediction. We predict future RTTGs. After V2X message fusion, we
initialize a prediction instance for each CAV to estimate its trajectory.
Predicted trajectories build future RTTGs. Predicted RTTGs integrate with
digital C-ITS, providing possible connection quality for each CAV. We simulate
networks in digital twin and calculate FL communication latency based on
predictive transport scenarios.

Data-level client grouping. We cluster clients into groups considering data
heterogeneity. Our goal is to group clients with similar data distribution,
ensuring each subset represents the whole group’s data features. We observe
model updates, considering gradient similarity as a data similarity criterion
[20]. We group clients based on model parameter similarity. Clients must
report gradient updates before a deadline for inclusion in data-level client
grouping. After grouping, each subset represents its cluster. Selecting at
least one client per cluster ensures satisfactory training performance.

Network-level client election. We elect clients in each group based on
contextual communication latency. Using predicted RTTG latency, we determine
efficient client contributions for upcoming communication rounds. We employ
the _Fast- $\gamma$_ rule, selecting the $\gamma$ clients with the lowest
communication delay ($0<\gamma<1$) per cluster.

Through these stages, representative clients with minimal contextual
communication latency are chosen for model aggregation. This process increases
FL communication efficiency by optimizing communication rounds and round
duration. De-selected clients save computational resources by not training
models locally.

##  IV Performance evaluation

Figure 3: FL training performance (testing accuracy changes over time in seconds) using various client selection strategies on non-iid MNIST (left), CIFAR-10 (middle) and SVHN (right) data distributed in 100 clients. TABLE I: Required time to reach 0.5 of test accuracy using various paradigms of client selection strategies with decreasing connection rate (CR). CR |  Strategy |  Time (s) |  Reduction rate
---|---|---|---
- |  Gossip |  3 891.14 |  1$\times$
1.0 |  Data-based |  213.50 |  18.23$\times$
|  Network-based |  620.47 |  6.27$\times$
|  Contextual |  l93.77 |  20.08$\times$
0.5 |  Data-based |  2 446.75 |  1.59$\times$
|  Network-based |  690.29 |  5.64$\times$
|  Contextual |  l79.54 |  21.67$\times$
0.2 |  Data-based |  2 563.20 |  1.52$\times$
|  Network-based |  634.12 |  6.14$\times$
|  Contextual |  l86.47 |  20.87$\times$

We implement and demonstrate FL with our pipeline as well as other four
baselines, i.e. greedy, gossip, data-based and network-based client selection
strategy, on a computer cluster with 4$\times$ NVIDIA-A100-PCIE-40GB GPUs and
4$\times$ 32-Core-AMD-EPYC-7513 CPUs. The environment is a Linux system with
Pytorch 1.8.1 and Cuda 11.1.

###  IV-A Experiment setup

We conduct the experimental evaluation by training models on three widely used
open datasets MNIST [26], CIFAR-10 [27] and SVHN [28] distributed into 100 CAV
clients in non-iid setting.222In our default non-iid setting, each client owns
only 2 out of 10 classes.

We compare our pipeline with four other client selection strategies as
baselines, i.e., greedy, gossip, data-based and network-based, as described in
Sec. II. The learning rate is 0.001 and the batch size is 64. The number of
the local epochs is set as 3 for training on MNIST, and 1 for training on
CIFAR-10 and SVHN, respectively. Except the greedy strategy (all clients are
selected in each communication round), the general selection rate for FL
clients is defined as 10%, i.e. around 10 clients are selected in each
communication round.

###  IV-B Performance results

We show the general performance of FL with contextual client selection for
training models on three datasets distributed in 100 vehicle clients with
respect to default non-iid setting. We train deep learning models with
different sizes on MNIST, CIFAR-10 and SVHN as FL tasks. As the experimental
results in Fig. 3 show for all three tasks, FL with our contextual client
selection can outperform the other four baselines. Generally, the FL with
contextual client selection can achieve remarkable higher test accuracy than
the other four strategies. Even though the network-based strategy allows the
ML-model to be trained to a comparable test accuracy on SVHN, the contextual
client selection results showcases much more stable convergence, as the data
heterogeneity across CAVs are taken into account.

We conduct the experiments with various connection rates and evaluate the
performance of FL. We take the required time to reach 0.5 of test accuracy for
FL with gossip client selection as a baseline, and evaluate the time reduction
rate of FL with other strategies. As the comparison results show in Tab. I, FL
with contextual client selection always needs less time than other two
strategies at each connection rate. The time reduction rates are robustly over
20$\times$ even when only 20% of clients are connected in networks.

##  V Conclusion

In this work, we reviewed the existing client selection strategies for FL and
introduced a novel four-stage V2X-Boosted FL pipeline for C-ITS. The approach
tackles both data and network heterogeneity in vehicular networks, boosting
communication efficiency by reducing the number of communication rounds and
shortening the time required for each round. Compared to other strategies, FL
with contextual client selection achieves higher accuracy and more stable
convergence performance by leveraging V2X messages disseminated in vehicular
networks. Future work will further consider the analytical model of
communication networks and conduct more validation in traffic scenario data,
such as [29, 30, 31].

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##  VI Appendix

TABLE II: Comparison of various paradigms of client selection strategies. Greedy |  Gossip |  Data-based |  Network-based |  Contextual (ours)
---|---|---|---|---
All clients should be selected in each FL communication round. |  A random subset of all connected clients is selected in each FL communication round. |  Clients are selected according to the similarity of local data distribution. |  Clients are selected according to the connection quality and availability in networks. |  Clients are selected in consideration of both data and network heterogeneity.
Figure 4: Test accuracy of FL with baselines and contextual client selection
on CIFAR-10 data distributed in different class ratio in each client.

We also demonstrate various client selection strategies under different class
ratio in each client to evaluate the performance in non-iid settings. We
consider a scenario for training a ML-model with 100 CAVs for three minutes.
Note that the class ratio 100% indicates iid setting. As Fig. 4 shows, the
pure network-based strategy can make the test accuracy of FL higher than
others, because the data heterogeneity is not needed to be considered under
ideal iid settings. However, in non-iid settings, the contextual client
selection enhances FL and leads to a better test accuracy than other
strategies. For instances, when the class ratio is 20%, it achieves
2.85$\times$ test accuracy compared to a pure data-based and 1.67$\times$ to a
pure network-based client selection strategy. Even in extremely non-iid
setting with only 1 class in each client, the contextual client selection can
reach 38% test accuracy within three minutes, while the FL with network- or
data-based strategies cannot converge.


# A metric deformation on Fiber bundles and applications

Leonardo F. Cavenaghi† Instituto de Matemática, Estatística e Computação
Cinetífica – Unicamp, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas,
SP, Brazil  and  Llohann D. Sperança∗ Instituto de Ciência e Tecnologia –
Unifesp, Avenida Cesare Mansueto Giulio Lattes, 1201, 12247-014, São José dos
Campos, SP, Brazil

###### Abstract.

We develop the concept of a Cheeger deformation on the context of fiber
bundles with compact structure group. This procedure is used to provide
metrics with positive Ricci curvature on the total space of such bundles,
generalizing the results in Nash [10] and Poor [11].

###### Key words and phrases:

Cheeger deformantions, Fiber Bundles, Positive Ricci curvature, Compact
structure group.

##  1\. Introduction

The metric deformation known as Cheeger deformation was firstly introduced on
[3]. Its main goal was to produce metrics with non-negative sectional
curvature on manifolds with symmetries. Since then, Cheeger deformations were
used in [6, 5] to produce new examples of manifolds with non-negative and
positive sectional curvature; in [12, 13, 7] to study curvature properties on
homogeneous spaces and biquotients and on [14] to lift positive Ricci
curvature from a metric quotient $M/G$ to $M$.

On the other hand Nash [10] and Poor [11] proved that the total space of some
classes of fiber bundles admit metrics of positive Ricci curvature: for
instance, linear sphere bundles, vector bundles and principal bundles over
manifolds with positive Ricci curvature. In particular, their results provide
metrics of positive Ricci curvature on exotic spheres. The main idea of the
present paper is to establish an analogous to the Cheeger deformation on a
specific class of metrics on fiber bundles. We use such deformation to unify
and extend the works of Nash [10] and Poor [11], providing positive Ricci
curvature on a broader class of fiber bundles:

######  Theorem 1.1.

Let $F\hookrightarrow M\stackrel{{\scriptstyle\pi}}{{\to}}B$ be a fiber bundle
from a compact manifold $M$ with fiber $F$, compact structure group $G$ and
base $B$. Suppose that

  1. (1)

$B$ has a metric of positive Ricci curvature,

  2. (2)

The principal orbit of the $G$-action on $F$ has finite fundamental group,

  3. (3)

$F$ has a $G$-invariant metric such that $F/G$, with the orbit distance
metric, has $\operatorname{Ric}_{F/G}\geq 1$.

Then $M$ carries a metric of positive Ricci curvature.

For the exposition, we follow [9, 16], introducing a suitable
reparametrization of $2$-planes and a non-decreasing formula for the deformed
sectional curvature (see (3.1)). The class of admissible metrics for the
present deformation includes any submersion metric whose fibers metrically are
homogeneous spaces with respect to the structure group (which is assumed to be
compact).

On Section 2 we recall the basic concepts and notaion of Cheeger deformations
following [16]. On Section 3 we introduce the new deformation and prove
Theorem 1.1.

##  2\. Cheeger deformations

Here we quickly recall the procedure according to Ziller [17] and M. Müter
[9].

Let $(M,g)$ be a Riemannian manifold with an isometric $G$-action, where $G$
is a compact Lie group and $Q$ is a biinvariant metric on $\mathfrak{g}$, the
Lie algebra of $G$. For each point $p\in M$ the orbit $Gp$ through $p$ is a
submanifold of $M$ satisfying $Gp\cong G\big{/}G_{p}$, where $G_{p}$ is the
isotropy subgroup of the $G$-action at $p$. Let $\mathfrak{m}_{p}$ be the
$Q$-orthogonal complement of $\mathfrak{g}_{p}$, the Lie algebra of $G_{p}$.
Observe that $\mathfrak{m}_{p}$ is isomorphic to the tangent space to the
orbit $Gp$. The isomorphism is induced by computing action fields: given an
element $U\in\mathfrak{g}$, define

$U^{*}_{p}=\frac{d}{dt}\Big{|}_{t=0}e^{tU}p.$

The map $U\mapsto U^{*}_{p}$ is a linear morphism whose kernel is
$\mathfrak{g}_{p}$. We call the tangent space $T_{p}Gp$ as the _vertical
space_ at $p$, $\mathcal{V}_{p}$, and its orthogonal complement,
$\mathcal{H}_{p}$, as the _horizontal space_. A tangent vector
$\overline{X}\in T_{p}M$ can be uniquely decomposed as
$\overline{X}=X+U^{\ast}_{p}$, where $X$ is horizontal and
$U\in\mathfrak{m}_{p}$.

The main idea in the Cheeger deformation is to consider the product manifold
$M\times G$, observing that the action

(1) $r(p,g):=(rp,gr^{-1})$

is free and isometric for the product metric $g\times\frac{1}{t}Q$, $t>0$. Its
quotient space is diffeomorphic to $M$ since the projection

$\displaystyle\pi^{\prime}:M\times G$ $\displaystyle\to M$
$\displaystyle(p,g)$ $\displaystyle\mapsto gp$

is a principal bundle for action (1). It induces a family of metrics $g_{t}$
on $M$.

Next, we define useful tensors:

  1. (1)

The _orbit tensor_ at $p$ is the linear map
$P:\mathfrak{m}_{p}\to\mathfrak{m}_{p}$ defined by

$g(U^{\ast},V^{\ast})=Q(PU,V),\quad\forall
U^{\ast},V^{\ast}\in\mathcal{V}_{p}$

  2. (2)

For each $t>0$ we define $P_{t}:\mathfrak{m}_{p}\to\mathfrak{m}_{p}$ as

$g_{t}(U^{\ast},V^{\ast})=Q(P_{t}U,V),\quad\forall
U^{\ast},V^{\ast}\in\mathcal{V}_{p}$

  3. (3)

The _metric tensor of $g_{t}$_, $C_{t}:T_{p}M\to T_{p}M$ is defined as

$g_{t}(\overline{X},\overline{Y})=g(C_{t}\overline{X},\overline{Y}),\quad\forall\overline{X},\overline{Y}\in
T_{p}M$

All the three tensors above are symmetric and positive definite. The next
proposition shows how they are related:

######  Proposition 2.1 (Proposition 1.1 in [17]).

The tensors above satisfy:

  1. (1)

$P_{t}=(P^{-1}+t1)^{-1}=P(1+tP)^{-1}$,

  2. (2)

If $\overline{X}=X+U^{\ast}$ then
$C_{t}(\overline{X})=X+((1+tP)^{-1}U)^{\ast}$.

The metric tensor $C_{t}^{-1}$ can be used to define an suitable
reparametrization of $2$-planes. Using this reparametrization we can observe
that Cheeger deformations do not create ‘new’ planes with zero sectional
curvature. More generally:

######  Theorem 2.2.

Let $\overline{X}=X+U^{\ast},~{}\overline{Y}=Y+V^{\ast}$ be tangent vectors.
Then
$\kappa_{t}(\overline{X},\overline{Y}):=R_{g_{t}}(C_{t}^{-1}\overline{X},C_{t}^{-1}\overline{Y},C_{t}^{-1}\overline{Y},C_{t}^{-1}\overline{X})$
satisfies

(2)
$\kappa_{t}(\overline{X},\overline{Y})=\kappa_{0}(\overline{X},\overline{Y})+\frac{t^{3}}{4}\|[PU,PV]\|_{Q}^{2}+z_{t}(\overline{X},\overline{Y}),$

where $z_{t}$ is non-negative.

(See [16, Proposition 1.3] for details.)

Next we provide a formula for the Ricci curvature of Cheeger deformed metric
(compare with [15, Equation (57)]).

###  2.1. Ricci curvature

Let $\\{v_{1},\ldots,v_{k}\\}$ be a $Q$-orthonormal basis of eigenvectors of
$P:\mathfrak{m}_{p}\to\mathfrak{m}_{p}$, with eigenvalues
$\lambda_{1}\leq\ldots\leq\lambda_{k}$. Given a $g$-orthonormal basis
$\\{e_{k+1},..,e_{n}\\}$ for $\mathcal{H}_{p}$, we fix the $g$-orthonormal
basis $\\{e_{1},\ldots,e_{k},e_{k+1},\ldots,e_{n}\\}$ for $T_{p}M$, where
$e_{i}=\lambda_{i}^{-1/2}v^{\ast}_{i}$ for $i\leq k$.

It follows that:

######  Claim 1.

The set $\\{C_{t}^{-1/2}e_{i}\\}_{i=1}^{n}$ is a $g_{t}$-orthonormal basis for
$T_{p}M$. Moreover, $C_{t}^{-1/2}e_{i}=(1+t\lambda_{i})^{1/2}e_{i}$ for $i\leq
k$ and $C_{t}^{-1/2}e_{i}=e_{i}$ for $i>k.$

Define the horizontal Ricci curvature as

(3)
$\operatorname{Ric}^{\mathbf{h}}(\overline{X}):=\sum_{i=k+1}^{n}R(\overline{X},e_{i},e_{i},\overline{X}).$

######  Lemma 1.

For $\\{e_{1},...,e_{n}\\}$ as above,

(4)
$\operatorname{Ric}_{g_{t}}(\overline{X})=\operatorname{Ric}_{g}^{\mathbf{h}}(C_{t}\overline{X})+\sum_{i=1}^{n}z_{t}(C_{t}^{1/2}e_{i},C_{t}\overline{X})\\\
+\sum_{i=1}^{k}\frac{1}{1+t\lambda_{i}}\Big{(}\kappa_{0}(\lambda_{i}^{-1/2}v^{\ast}_{i},C_{t}\overline{X})+\frac{\lambda_{i}t}{4}\|[v_{i},tP(1+tP)^{-1}\overline{X}_{\mathfrak{g}}]\|_{Q}^{2}\Big{)}.$

###### Proof.

A straightforward computation following equation (2) gives

$\operatorname{Ric}_{g_{t}}(C_{t}^{-1}\overline{X})=\sum_{i=1}^{n}R_{g_{t}}(C_{t}^{-1/2}e_{i},C_{t}^{-1}\overline{X},C_{t}^{-1}\overline{X},C_{t}^{-1/2}e_{i})=\sum_{i=1}^{n}\kappa_{t}(C_{t}^{1/2}e_{i},\overline{X})\\\
=\sum_{i=1}^{n}\kappa_{0}(C_{t}^{1/2}e_{i},\overline{X})+\sum_{i=1}^{n}z_{t}(C_{t}^{1/2}e_{i},\overline{X})+\frac{t^{3}}{4}\sum_{i=1}^{k}\|[PC_{t}^{1/2}\lambda_{i}^{-1/2}v_{i},P\overline{X}_{\mathfrak{g}}]\|_{Q}^{2}\\\
=\operatorname{Ric}_{g}^{\mathbf{h}}(\overline{X})+\sum_{i=1}^{n}z_{t}(C_{t}^{1/2}e_{i},\overline{X})+\sum_{i=1}^{k}\frac{1}{1+t\lambda_{i}}\Big{(}\kappa_{0}(\lambda_{i}^{-1/2}v^{\ast}_{i},\overline{X})+\frac{\lambda_{i}t^{3}}{4}\|[v_{i},P\overline{X}_{\mathfrak{g}}]\|_{Q}^{2}\Big{)}.$

Equation (4) now follows by replacing $\overline{X}$ by $C_{t}\overline{X}$. ∎

##  3\. Cheeger deformations on fiber bundles

We use Cheeger deformation to produce deformed metrics on fiber bundles with
compact structure group.

Let $F\hookrightarrow M\stackrel{{\scriptstyle\pi}}{{\to}}B$ be a fiber bundle
from a compact manifold $M$, with fiber $F$, compact structure group $G$ and
base $B$. We start by recalling that the structure group of a fiber bundle is
the group where some choice of transition functions on $M$ take values.
Precisely, if $G$ acts effectively on $F$, then $G$ is a structure group for
$\pi$ if there is a choice of local trivializations
$\\{(U_{i},\phi_{i}:\pi^{-1}(U_{i})\to U_{i}\times F)\\}$ such that, for every
$i,j$ with $U_{i}\cap U_{j}\neq\emptyset$, there is a continuous function
$\varphi_{ij}:U_{i}\cap U_{j}\to G$ satisfying

(5) $\phi_{i}\circ\phi_{j}^{-1}(p,f)=(p,\varphi_{ij}(p)f),$

for all $p\in U_{i}\cap U_{j}$.

The existence of $\\{\varphi_{ij}\\}$ allows us to construct a principal
$G$-bundle over $B$ (see [8, Proposition 5.2] for details) that we shall
denote by $\mathcal{P}$. Furthermore, there exists a principal $G$-bundle
$\bar{\pi}:\mathcal{P}\times F\to M$ whose principal action is given by

(6) $r(p,f):=(rp,rf).$

(For details see the construction on the proof of [4, Proposition 2.7.1].)

For each pair $g$ and $g_{F}$ of $G$-invariant metrics on $\mathcal{P}$ and
$F$, respectively, there exists a metric $h$ on $M$ induced by $\bar{\pi}$.
Denote by $\mathcal{M}$ the set of all metrics obtained in this way (for
instance, if $G$ is transitive on $F$ then every metric on $M$ such that the
holonomy acts by isometries on each fiber belongs to $\mathcal{M}$). The set
$\mathcal{M}$ is the set of admissible metrics for our deformation:

######  Definition 1 (The deformation).

Given $h\in\mathcal{M}$, consider $g\times g_{F}$ a product metric on
$\mathcal{P}\times F$ such that $\bar{\pi}:(\mathcal{P}\times F,g\times
g_{F})\to(M,h)$ is a Riemannian submersion. We define $h_{t}$ as the
submersion metric induced by $g_{t}\times g_{F}$, where $g_{t}$ is the time
$t$ Cheeger deformation associated to $g$.

As it can be seen, the deformation itself is well-defined for a broader class
of metrics (for instance, $\mathcal{P}\times F$ could have any metric such
that each slice $\\{p\\}\times F$ has a $G$-invariant metric), however, if the
metric is not a product metric, the deformed curvature is harder to control
and Theorem 3.1 below does not hold.

Fix $(p,f)\in\mathcal{P}\times F$. Any $\overline{X}\in
T_{(p,f)}(\mathcal{P}\times F)$ can be written as
$\overline{X}=(X+V^{\vee},X_{F}+W^{*})$, where $X$ is orthogonal to the
$G$-orbit on $\mathcal{P}$, $X_{F}$ is orthogonal to the $G$-orbit on $F$ and,
for $V,W\in\mathfrak{g}$, $V^{\vee}$ and $W^{\ast}$ are the action vectors
relative to the $G$-actions on $\mathcal{P}$ and $F$ respectively. Let
$P,~{}P_{F}$ and $P_{t}$ be the orbit tensors associated to $g,~{}g_{F}$ and
$g_{t}$, respectively.

We claim that $\overline{X}$ is $g_{t}{\times}g_{F}$-orthogonal to the
$G$-orbit of (6) if and only if

(7) $\overline{X}=(X-(P^{-1}_{t}P_{F}W)^{\vee},X_{F}+W^{\ast}).$

for some $W\in\mathfrak{m}_{f}$. A vector $(X+{V^{\vee}},X_{F}+W^{\ast})$ is
horizontal if and only if, for every $U\in\mathfrak{g}$:

$0=g_{t}\times
g_{F}((X+V^{\vee},X_{F}+W^{\ast}),(U^{\vee},U^{\ast}))=g(V^{\vee},U^{\vee})+g_{F}(W^{\ast},U^{\ast})\\\
=Q(P_{t}V+P_{F}W,U).$

Since $U$ is arbitrary, we conclude that $V=-P^{-1}_{t}P_{F}W$.

Keeping in mind that the point $(p,f)$ is fixed, we abuse notation and denote

(8) $d\bar{\pi}_{(p,f)}(X,X_{F}+U^{*}):=X+X_{F}+U^{*}.$

Define the tensors
$\tilde{P}_{t},\tilde{C}_{t}:\mathfrak{m}_{f}\to\mathfrak{m}_{f}$,

(9) $\displaystyle\tilde{P}_{t}$
$\displaystyle:=P_{F}(1+P_{t}^{-1}P_{F})^{-1}=(P_{F}^{-1}+P_{t}^{-1})^{-1},$
(10) $\displaystyle\tilde{C}_{t}$
$\displaystyle:=-C_{t}P_{t}^{-1}\tilde{P}_{t}=-P^{-1}\tilde{P}_{t}.$

######  Claim 2.

Let $\mathcal{L}_{\bar{\pi}}:T_{\overline{\pi}(p,f)}M\to
T_{(p,f)}(\mathcal{P}\times F)$ be the horizontal lift associated to
$\bar{\pi}$. Then,

(11)
$\mathcal{L}_{\bar{\pi}}(X+X_{F}+U^{*})=(X-(P_{t}^{-1}\tilde{P}_{t}U)^{\vee},X_{F}+(P_{F}^{-1}\tilde{P}_{t}U)^{*}).$

###### Proof.

First observe that the right-hand-side of (11) is of the form (7): take
$W=P_{F}^{-1}\tilde{P}_{t}U$, so $P_{t}^{-1}\tilde{P}_{t}U=P^{-1}_{t}P_{F}W$.
Therefore, it is sufficient to verify that

$d\bar{\pi}(X-(P_{t}^{-1}\tilde{P}_{t}U)^{\vee},X_{F}+(P_{F}^{-1}\tilde{P}_{t}U)^{*})=X+X_{f}+U^{*}.$

Since $\ker d\bar{\pi}=\\{(U^{\vee},U^{*})~{}|~{}U\in\mathfrak{g}\\}$,
convention (8) gives $d\bar{\pi}(U^{\vee},0)=-U^{*}$, thus

$d\bar{\pi}(X-(P_{t}^{-1}\tilde{P}_{t}U)^{\vee},X_{F}+(P_{F}^{-1}\tilde{P}_{t}U)^{*})=X+X_{F}+((P_{t}^{-1}+P_{F}^{-1})\tilde{P}_{t}U)^{*}=X+X_{f}+U^{*}$

since $\tilde{P}_{t}=(P_{t}^{-1}+P_{F}^{-1})^{-1}$. ∎

Next, we prove that the unreduced sectional curvature is non-decreasing for
reparametrized planes (Theorem 3.1). For the reparametrization, extend
$\tilde{C}_{t}$ to $T_{\bar{\pi}(p,f)}M$ via

(12) $\tilde{C}_{t}(X+X_{F}+U^{\ast}):=X+X_{F}+(\tilde{C}_{t}U)^{\ast}.$

Denote
$\tilde{\kappa}_{t}(\tilde{X},\tilde{Y})=R_{h_{t}}(\tilde{C}_{t}^{-1}X,\tilde{C}_{t}^{-1}Y,\tilde{C}_{t}^{-1}Y,\tilde{C}_{t}^{-1}X)$.

######  Theorem 3.1 (Sectional curvature).

Let $h\in\mathcal{M}$ and $g_{t}\times g_{F}$ be as in Definition 1. Then, for
every pair $\tilde{X}=X+X_{F}+U^{*}$, $\tilde{Y}=Y+Y_{F}+V^{*}$ ,

$\tilde{\kappa}_{t}(\tilde{X},\tilde{Y})=\kappa_{t}(X{+}U^{\vee},Y{+}V^{\vee}){+}K_{g_{F}}(X_{F}-(P_{F}^{-1}PU)^{*},Y_{F}-(P_{F}^{-1}PV)^{*})+\tilde{z}_{t}(\tilde{X},\tilde{Y}),$

where $\kappa_{t}$ is as in Theorem 2.2 and $\tilde{z}_{t}$ is non-negative.

###### Proof.

The proof follows from a direct use of Gray–O’Neill curvature formula and
Claim 2. Observe that

$\mathcal{L}_{\bar{\pi}}(\tilde{C}_{t}^{-1}\tilde{X})=(C_{t}^{-1}(X+U^{\vee}),X_{F}-(P_{F}^{-1}PU)^{\ast}).$

Let $\tilde{z}_{t}$ be three times the norm squared of the integrability
tensor of $\bar{\pi}$ applied to
$\mathcal{L}_{\bar{\pi}}\tilde{C}_{t}^{-1}\tilde{X},\mathcal{L}_{\bar{\pi}}\tilde{C}_{t}^{-1}\tilde{Y}$
(see [4] for details). Then,

$R_{h_{t}}(\tilde{C}_{t}^{-1}X,\tilde{C}_{t}^{-1}Y,\tilde{C}_{t}^{-1}Y,\tilde{C}_{t}^{-1}X)=K_{g_{t}}(C_{t}^{-1}(X+U^{\vee}),C_{t}^{-1}(Y+V^{\vee}))\\\
+K_{g_{F}}(X_{F}-(P_{F}^{-1}PU)^{\ast},Y_{F}-(P_{F}^{-1}PV)^{\ast})+\tilde{z}_{t}(\tilde{X},\tilde{Y})\\\
=\kappa_{t}(X{+}U^{\vee},Y{+}V^{\vee}){+}K_{g_{F}}(X_{F}-(P_{F}^{-1}PU)^{*},Y_{F}-(P_{F}^{-1}PV)^{*})+\tilde{z}_{t}(\tilde{X},\tilde{Y}).$

∎

Although, the term $\tilde{z}_{t}$ plays no role on the applications in this
paper, we claim that it is, actually, non-decreasing with respect to $t$. This
is a crucial observation since $\tilde{z}_{0}$ is an essential part of the
initial curvature: since $\bar{\pi}:(\mathcal{P}\times F,g\times
g_{F})\to(M,h)$ is chosen to be a Riemannian submersion, $\tilde{z}_{0}$ is
the $A$-tensor term on the submersion formula. Or, equivalently, taking $t=0$
in Theorem 3.1,

$K_{h}(\tilde{X},\tilde{Y})=\tilde{\kappa}_{0}(\tilde{X},\tilde{Y})\\\
=\kappa_{0}(X{+}U^{\vee},Y{+}V^{\vee}){+}K_{g_{F}}(X_{F}-(P_{F}^{-1}PU)^{*},Y_{F}-(P_{F}^{-1}PV)^{*})+\tilde{z}_{0}(\tilde{X},\tilde{Y})\\\
=K_{g}(X{+}U^{\vee},Y{+}V^{\vee}){+}K_{g_{F}}(X_{F}-(P_{F}^{-1}PU)^{*},Y_{F}-(P_{F}^{-1}PV)^{*})+\tilde{z}_{0}(\tilde{X},\tilde{Y}).$

###  3.1. Applications to Ricci curvature

We use of the theory developed so far to prove Theorem 1.1.

We begin by constructing an appropriate basis for the horizontal space of
$\bar{\pi}$ with respect to $g_{t}\times g_{F}.$ Consider a $Q$-orthonormal
basis $\\{v_{k}(0)\\}$ of $\mathfrak{m}_{f}$ and define

$v_{k}(t)=\tilde{P}_{t}^{-1/2}v_{k}(0).$

######  Lemma 2.

The set

(13)
$\\{(-{P_{t}^{-1}\tilde{P}_{t}v_{k}(t)}^{\vee},{P_{F}^{-1}\tilde{P}_{t}v_{k}(t)}^{*})\\}=\\{(-P_{t}^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)^{\vee},P_{F}^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)^{*})\\}$

is $g_{t}\times g_{F}$-orthonormal and $g_{t}\times g_{F}$ orthogonal to
$(U^{\vee},U^{*})$, for every $U\in\mathfrak{g}$.

###### Proof.

Note that the elements in (13) are of the form (11). Thus, it is sufficient to
show that the set (13) is orthonormal. A straightforward computation gives:

$g_{t}\times
g_{F}((-{P_{t}^{-1}\tilde{P}_{t}v_{i}(t)}^{\vee},{P_{F}^{-1}\tilde{P}_{t}v_{i}(t)}^{*}),(-{P_{t}^{-1}\tilde{P}_{t}v_{j}(t)}^{\vee},{P_{F}^{-1}\tilde{P}_{t}v_{j}(t)}^{*}))\\\
=Q(\tilde{P}_{t}v_{i}(t),P_{t}^{-1}\tilde{P}_{t}v_{j}(t))+Q(\tilde{P}_{t}v_{i}(t),P_{F}^{-1}\tilde{P}_{t}v_{j}(t))\\\
=Q(\tilde{P}_{t}v_{i}(t),(P_{t}^{-1}+P_{F}^{-1})\tilde{P}_{t}v_{j}(t))\\\
=Q(\tilde{P}_{t}v_{i}(t),v_{j}(t))=Q(\tilde{P}_{t}^{1/2}v_{i}(0),\tilde{P}_{t}^{-1/2}v_{j}(0))=\delta_{ij},$

where we have used that $(P_{t}^{-1}+P_{F}^{-1})=\tilde{P}_{t}^{-1}$ and that
$\tilde{P}_{t}$ is symmetric. ∎

Let $\\{e_{i}^{B}\\}$ and $\\{e_{j}^{F}\\}$ be orthonormal bases for the
spaces normal to the orbits on $\mathcal{P}$ and on $F$, respectively. We
complete the set on Lemma 2 to a $g_{t}\times g_{F}$-othornormal basis for the
$\bar{\pi}$-horizontal space:

(14)
$\mathcal{B}_{t}:=\left\\{(e_{i}^{B},0),(-{P_{t}^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)}^{\vee},{P_{F}^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)}^{*}),(0,e_{j}^{F})\right\\}.$

Denote by $e_{1},...,e_{n}$ the elements in $\mathcal{B}_{t}$. Theorem 1.1
follows from the next two lemmas.

######  Lemma 3.

For any $(p,f)\in\mathcal{P}\times F$ and $X+X_{F}+U^{*}\in
T_{\bar{\pi}(p,f)}M$,

(15)
$\lim_{t\to\infty}\operatorname{Ric}_{h_{t}}(X+X_{F}+U^{*})\geq\operatorname{Ric}_{g}^{\mathbf{h}}(X)+\operatorname{Ric}_{g_{F}}^{\mathbf{h}}(X_{F})+\sum_{k}\frac{1}{4}\|[v_{k}(0),U]\|_{Q}^{2}.$

###### Proof.

Using the basis $\mathcal{B}_{t}$, from (14), and Theorem 3.1, we have:

$\operatorname{Ric}_{h_{t}}(\tilde{X})=\sum_{i=1}^{n}\kappa_{t}(\tilde{C}_{t}\tilde{X},\tilde{C}_{t}e_{i})\geq\\\
\sum_{i}\kappa_{t}(X-(C_{t}P_{t}^{-1}\tilde{P}_{t}U)^{\vee},e_{i}^{B})+\sum_{k}\kappa_{t}(C_{t}X-(C_{t}P_{t}^{-1}\tilde{P}_{t}U)^{\vee},-C_{t}P_{t}^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)^{\vee})\\\
+\sum_{j}K_{g_{F}}(X_{F}+(P_{F}^{-1}\tilde{P}_{t}U)^{*},e^{F}_{j})+\sum_{k}K_{g_{F}}(X_{F}+(P_{F}^{-1}\tilde{P}_{t}U)^{*},P_{F}^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)^{*}).$

On the other hand, the $\tilde{P}_{t}$ satisfy:

(16) $\displaystyle\lim_{t\to\infty}t\tilde{P}_{t}$ $\displaystyle=1,$ (17)
$\displaystyle\underset{t\to\infty}{\lim}P_{t}^{-1}\tilde{P}_{t}$
$\displaystyle=1.$

In particular, $\tilde{P}_{t}\to 0$ as $t\to\infty$. Both (16) follows since
$t\tilde{P}_{t}=P_{F}(P_{t}+P_{F})^{-1}tP_{t}$, $P_{t}\to 0$ and $tP_{t}\to
1$. Equation (17) follows since

$\lim_{t\to\infty}P_{t}^{-1}\tilde{P}_{t}=\lim_{t\to\infty}(tP_{t})^{-1}\lim_{t\to\infty}t\tilde{P}_{t}=1.$

Using (16), we observe that

$\displaystyle\lim_{t\to\infty}\left\\{\sum_{j}K_{g_{F}}(X_{F}+(P_{F}^{-1}\tilde{P}_{t}U)^{*},e^{F}_{j})+\sum_{k}K_{g_{F}}(X_{F}+(P_{F}^{-1}\tilde{P}_{t}U)^{*},P_{F}^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)^{*})\right\\}$
(18) $\displaystyle=\operatorname{Ric}_{g_{F}}^{\mathbf{h}}(X_{F}).$

Moreover, using equation (2) and that $C_{t}P_{t}^{-1}=P^{-1}$,

(19)
$\lim_{t\to\infty}\sum_{i}\kappa_{t}(X-(C_{t}P_{t}^{-1}\tilde{P}_{t}U)^{\vee},e_{i})\geq\lim_{t\to\infty}\operatorname{Ric}_{g}^{\mathbf{h}}(X-(P^{-1}\tilde{P}_{t}U)^{\vee})=\operatorname{Ric}_{g}^{\mathbf{h}}(X).$

For the remaining term:

$\kappa_{t}(C_{t}X-(C_{t}P_{t}^{-1}\tilde{P}_{t}U)^{\vee},-C_{t}P_{t}^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)^{\vee})\\\
\geq
K_{g}(X-(P^{-1}\tilde{P}_{t}U)^{\vee},-P^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)^{\vee})+\frac{t^{3}}{4}\|[\tilde{P}_{t}U,\tilde{P}_{t}^{1/2}v_{k}(0)]\|^{2}_{Q}$

Using (16) and (17), we obtain

(20)
$\lim_{t\to\infty}\kappa_{t}(C_{t}X-(C_{t}P_{t}^{-1}\tilde{P}_{t}U)^{\vee},-C_{t}P_{t}^{-1}\tilde{P}_{t}^{1/2}v_{k}(0)^{\vee})\geq\frac{1}{4}\|[v_{k}(0),U]\|_{Q}^{2}.$

Lemma 3 follows by putting together (18), (19) and (20). ∎

######  Lemma 4.

Let $F$ be a $G$-manifold. Assume that the orbits of the $G$-action on $F$
have finite fundamental group and that the orbital distance metric on
$F\big{/}G$ has Ricci curvature $\operatorname{Ric}_{F/G}\geq 1$. Then there
is a $G$-invariant $g_{F}$ on $F$ with
$\operatorname{Ric}_{g_{F}}^{\mathbf{h}}>0$.

###### Proof.

Let $g_{F}$ be a $G$-invariant metric on $F$ where the orbits of $G$ have
finite fundamental group. Then, $g_{F}$ has positive horizontal Ricci
curvature if and only if the Ricci curvature of any arbitrarily large Cheeger
deformation of $g_{F}$ is positive. This is decoded in the following way:

(21)
$\lim_{t\to\infty}\operatorname{Ric}_{(g_{F})_{t}}(X)=\operatorname{Ric}^{\mathbf{h}}(X),$

where $(g_{F})_{t}$ is the Cheeger defomormation of $g_{F}$.

Note that the hypothesis on Lemma 4 are the same as the ones in [14, Theorem
A]. Under such hypothesis, the authors constructed a $G$-invariant metric
$g_{F}$ whose Ricci curvature was positive after arbitrarily large Cheeger
deformation (see[14, Theorems 6.1, 6.2]), proving Lemma 4. ∎

###### Proof of Theorem 1.1.

Let $g_{F}$ be as in Lemma 4. Assume by contradiction that, for each $t>0$,
there is a point $x_{t}\in M$ and a $h$-unitary vector
$X_{t}+(X_{F})_{t}+U_{t}^{*}$ such that
$\operatorname{Ric}_{h_{t}}(X_{t}+(X_{F})_{t}+U_{t}^{*})\leq 0$. Then,
considering $t=n\in\mathbb{N}$, we get a sequence
$X_{n}+(X_{F})_{n}+U^{*}_{n}$ such that

$\lim_{n\to\infty}\operatorname{Ric}_{h_{n}}(X_{n}+(X_{F})_{n}+U^{*}_{n})\leq
0.$

Since $\mathcal{P}$ and $F$ are compact and the horizontal lift is an
isometry, there is a limit point $x$ and a non-zero vector $X+X_{F}+U^{*}$
such that

(22)
$\operatorname{Ric}_{g}^{\mathbf{h}}(X)+\frac{1}{4}\sum_{j}\|[v_{j}(0),U]\|^{2}_{Q}+\operatorname{Ric}_{g_{F}}^{\mathbf{h}}(X_{F})\leq
0.$

Recall that the sum $\sum_{k}\frac{1}{4}\|[v_{k}(0),U]\|_{Q}^{2}$ is the Ricci
curvature of the $Q$-normal homogeneous space $G/G_{f}$ in the direction $U$.
We claim that $G/G_{f}$ has finite fundamental group, whenever $f$ is on the
principal part or not.

Let $H<G$ be a principal isotropy group for the $G$-action on $F$. By
hypothesis, $\pi_{1}(G/H)$ is finite. Following Bredon [2, section IV.3], we
can assume (up to conjugation) that $H<G_{f}$. The long homotopy sequence of
$G_{f}/H\hookrightarrow G/H\to G/G_{f}$ gives:

$\cdots\rightarrow\pi_{1}(G/H)\rightarrow\pi_{1}(G/G_{f})\to\pi_{0}(G_{f}/H)\to\cdots$

Therefore, $\pi_{1}(G/G_{f})$ is finite if and only if $G_{f}/H$ has a finite
number of connected components. However, $G_{f}$ is a compact manifold, since
it is a closed subgroup of the compact Lie group $G$. Therefore,
$\pi_{1}(G/G_{f})$ is finite.

We conclude that there is a $C>0$ such that
$\frac{1}{4}\sum_{j}\|[v_{j}(0),U]\|^{2}_{Q}\geq C\|U\|_{Q}^{2}$ (see e.g.
Proposition 3.4 in Nash [10], Proposition 6.6 in [14] or Theorem 1 in [1]).
Therefore,

$\operatorname{Ric}_{g}^{\mathbf{h}}(X)+\operatorname{Ric}_{g_{F}}^{\mathbf{h}}(X_{F})+C\|U\|_{Q}^{2}\leq
0,$

which is a contradiction. ∎

Next, we list the results from Nash [10] and Poor [11] which are Corollary to
Theorem 1.1.

######  Corollary 3.2 ([10] Theorem 3.5.).

Let $(E,M,G/H,G,\pi)$ be a fibre bundle such that $M$ is compact and admits a
metric $g$ with $\operatorname{Ric}_{g}>0$. If $G$ is compact and $G\big{/}H$
admits a metric with $\operatorname{Ric}\geq 1$ (equivalently, if
$\pi_{1}(G/H)$ is finite), then $E$ admits a metric of positive Ricci
curvature.

######  Corollary 3.3 ([10] Corollary 3.6.).

Let $(E,M,S^{n},O(n+1),\hat{\pi})$, $n\geq 2,$ be a sphere bundle whose basis
is compact and possesses a metric of positive Ricci curvature. Then $E$ admits
a metric of positive Ricci curvature.

######  Corollary 3.4 ([10] Theorem 3.8.).

Let $(E,M,F,G,\hat{\pi})$ be a bundle with $M$ compact admitting a metric $g$
with $\operatorname{Ric}_{g}>0$. If $G$ is compact, possesses finite
fundamental group and acts freely on $F$, which is compact such that $F/G$
admits a metric with $\operatorname{Ric}\geq 1$, then $E$ admits a metric of
positive Ricci curvature.

######  Corollary 3.5 ([11] Main Theorem).

Let $M$ and $F$ be compact Riemannian manifolds such that the Ricci curvature
of $M$ and the sectional curvature of $F$ are positive. Suppose $\pi:FM\to M$
is a bundle with fiber $F$ for which the structural group $G$ acts on F by
isometries. Then $FM$ admits a metric of positive Ricci curvature in which the
fibers of $\pi$ are totally geodesic and mutually isometric in $FM$.

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