text:
\section{Introduction}
\label{Intro}
Electrons $5$\textit{f} are source of intriguing physics due to their dual localized-delocalized character \cite{moore}. This can have profound implications for their collective behavior.
Indeed, the uranium $5$\textit{f}-electrons systems shows a variety of ground states which includes
localized and itinerant magnetism \cite{Santini,Miyake},
unconventional superconductivity \cite{Pfleiderer,Aoki}
and the ignematic exotic Hidden Order in  URu$_2$Si$_2$
\cite{Mydosh2011,Oppeneer2014,Wolowiec,Calegari2017}.
Moreover, it is well known that these states
can be tuned by pressure (hydrostatic or chemical) and magnetic field \cite{Magalhaes2020,Faundez2021}.
This imposes a requirement on any comprehensive microscopic model for $5$\textit{f}-electrons that it should describe the appearance and evolution of quantum conventional, unconventional or even exotic collective states \cite{Valiska2018, Correa2012, Niklowitz210}.

The Underscreened Anderson Lattice Model (UALM) has been introduced as a generalization of the Underscreened Kondo Lattice
which successfully described certain aspects of the physics of uranium compounds \cite{Riseborough2012, Perkins, Thomas}. In particular,
the interplay of the Kondo effect and ferromagnetism \cite{Bernard}. The UALM includes the direct hopping between distinct orbitals ($\chi=\alpha $ and $\beta$) which gives rise to two quite narrow $f$-bands. These $f$-bands in turn, are hybridized with a wide $d$-band.  Lastly, there are $f$-electron intra- and inter-orbitals interactions. Remarkably,  this model can host itinerant spins orderings in which time reversal symmetry may or may not be broken \cite{Shah2000,Yuan1}.

In the present work, we study how conventional Spin Density Waves (SDWs) develop in the UALM under simultaneous application of pressure and magnetic fields for both cubic and tetragonal lattices. The applied pressure is mimicked by an increase in the bandwidths.  We also assume an Ising-like anisotropy for the magnetic order parameters (OPs) \cite{Schoenes1981,Havela1992,Maskova2019}. The magnetic field is applied along the axis of anisotropy. Even for the conventional spin ordering,
 the two-orbital nature of model has an unusual role. In particular, the SDW  exist in two distinct phase (with same nesting vectors) AF$_1$ and AF$_2$ that are characterized by the relative magnitudes of the orbital's staggered magnetizations \cite{Magalhaes2020,Faundez2021}. For low pressure, there is the onset of AF$_1$ at lower temperature. When the pressure is increased,  AF$_2$ starts to compete by the stability with AF$_1$.
Eventually, the initial AF$_1$ is abruptly replaced by AF$_2$.

The SDWs also present noteworthy features concerning their metallic characters.
For instance, it is possible that the Fermi energy, can be located inside the two spin gaps or even inside one spin gap but not inside the other.
However, with the magnetic field aligned with the Ising direction, the Zeeman splitting between the spin-up and spin-down sub-bands is strongly affected as the field changes.
As consequence, the level of metallicity can be also affected.
Therefore, an evolution of the Fermi Surface (FS) can be anticipated with some type of reconstruction as the combined application of pressure and magnetic field changes.

We also highlight the possibility of metamagnetic transitions \cite{Miyake,Ken, Solhanek, Mushnikov}. Since, the two SDWs compete for stability when the pressure is increased, magnetic field modifications of the band structure may produce transitions \cite{Bruck1994,Pospisil2018}.
In such cases, one may expect that the induced transitions will be highly sensitive to the lattice structure, such as cubic or tetragonal.

This paper is organized as follow: the model is presented in the section \ref{model_1}. Then, we derive the Green's functions, gap equations and free energy which are presented in section \ref{FGreen}. In section \ref{Results}, we present the phase diagrams for pressure and magnetic fields for cubic and tetragonal lattices.
The conclusions and other remarks are found in section \ref{Conclusion}.

\section{Model}
\label{model_1}

In this work we investigate the effects of an applied magnetic field and pressure in an UALM Hamiltonian for cubic and tetragonal lattices. The Hamiltonian is written as follow
\begin{equation}
   \hat{H} = \hat{H}_f + \hat{H}_d + \hat{H}_{fd}\, .
  \label{hamilt}
\end{equation}

The first term $\hat{H}_f$
represents the $5$\textit{f}-electron and is composed of two parts:
\begin{equation}
\hat{H}_f=\hat{H}_{f,0}+ \hat{H}_{f,int}\,,
\label{int}
\end{equation}
where the noninteracting part $H_ {0, f}$ describes two degenerate narrow $5$\textit{f} bands given as below
\begin{equation}
\hat{H}_{f,0} = \sum_{\vec{k},\sigma}\sum_{\chi} \ E_{f}^{\chi}(\vec{k}) \
f^{\dag \chi}_{\vec{k},\sigma}f^{\chi}_{\vec{k},\sigma}\,.
\label{eq0}
\end{equation}
The $f^{\dag \chi}_{\vec{k},\sigma}$ are the creation operators for electrons with
spin $\sigma(=\uparrow,\downarrow )$ at site $i$. The $\chi$-bands ($\chi=\alpha$ and $\beta$) in Eq. (\ref{eq0}) follow the intraband and interband nesting property $E^{\chi}_{f}(\vec{k}+\vec{Q})=-E^{\chi'}_{f}(\vec{k})$
where $\chi=\chi'$  or $\chi\neq\chi'$. The electron band dispersion relations are described by
$E_{f}^{\chi}(\vec{k})$ and the vector $\vec{Q}$ is a commensurate momentum transfer in the first Brillouin zone.

The second term in Eq. (\ref{int}) represents the local Coulomb and Hund's rule exchange interactions, which are given by
\begin{align
\hat{H}_{f, int} \ = \ \bigg(\frac{U - J}{2 N}\bigg) \
\sum_{\vec{k},\vec{k}',\vec{q},\sigma}\sum_{\chi\ne\chi'}
\ f^{\dag,\chi}_{\vec{k}+\vec{q},\sigma} \ f^{\chi}_{\vec{k},\sigma} \
f^{\dag,\chi'}_{\vec{k}'- \vec{q},\sigma} f^{\chi'}_{\vec{k}',\sigma}
\nonumber \\
+ \ \bigg({ U \over 2 N}\bigg)
\sum_{\vec{k},\vec{k}',\vec{q},\sigma}\sum_{\chi,\chi'}
\ f^{\dag,\chi}_{\vec{k}+\vec{q},\sigma} \ f^{\chi}_{\vec{k},\sigma} \
f^{\dag,\chi'}_{\vec{k}'-\vec{q},-\sigma} f^{\chi'}_{\vec{k}',-\sigma}\nonumber
 \\ +  \bigg({J \over 2 N}\bigg)
\sum_{\vec{k},\vec{k}',\vec{q},\sigma}\sum_{\chi\ne\chi'} \
f^{\dag,\chi}_{\vec{k}+\vec{q},\sigma} \  f^{\chi'}_{\vec{k},\sigma} \
f^{\dag,\chi'}_{\vec{k}'-\vec{q},-\sigma} \ f^{\chi}_{\vec{k}',-\sigma}\,.
\label{Hfintreal}
\end{align}

The second term  in Eq. (\ref{hamilt})  represents the conduction electron term $\hat{H}_d$
\begin{equation}
    \hat{H}_d \ = \ \sum_{\vec{k},\sigma} \ \epsilon_d(\vec{k}) \ d^{\dag}_{\vec{k},\sigma} \ d_{\vec{k},\sigma},
\label{Hd}
\end{equation}
where $\epsilon(\vec{k})$ describes the dispersion relation of
conduction electrons labeled by the Bloch wave vector $\vec{k}$.
The term $\hat{H}_{fd}$
describes the on-site hybridization process in the considered model by
{\small
\begin{equation}
\hat{H}_{fd} =  \sum_{\vec{k},\sigma}\sum_{\chi=\alpha\beta} \ \bigg(V_{\chi}(\vec{k})
\ f^{\dag \chi}_{\vec{k},\sigma}  d_{\vec{k},\sigma}  +
V^*_{\chi}(\vec{k}) \ d^{\dag}_{k,\sigma}  f^{\chi}_{\vec{k},\sigma} \ \bigg)\,.
\label{Hhyb}
\end{equation}}
The effects of an applied magnetic field are taken into account by inclusion of an additional term in Eq. (\ref{hamilt}) given by
\begin{equation}
\hat{H}_{ext}=-  \sum_{\vec{k}} \sum_{\sigma=\pm}  \sigma  [ H_{z} ^{f}f^{\dag}_{ \vec{k},\sigma}  f_{\vec{k},\sigma}+ H_{z}^{d}  d^{\dag}_{\vec{k},\sigma}  d_{\vec{k},\sigma}]
\label{campo}
\end{equation}
with
\begin{equation}
H_z^{f(d)}= g_{f(d)}\mu_B h_z.
\label{eqHz}
\end{equation}
The value $\sigma=1$ and $-1$ correspond to the up and down spin projections, respectively.

The dispersion relations for the $5f$ band, $E_f^{\chi}(\vec{k})=\epsilon_{f}+\epsilon_f(\vec{k})$, and the conduction band, $\epsilon_d(\vec{k})$,  refer to a simple tetragonal
lattice. Thus
\begin{equation}
\epsilon_{A}(\vec{k})=-2t_{A,a}[\cos(k_xa)+\cos(k_ya)]-2t_{A,c}\cos(k_zc)
\label{Ek}
\end{equation}
in which $A=f$ or $d$, and $a$ and $c$ are the lattice parameters. If $a=c$, we have a cubic lattice.

\section{Green's functions, Gaps and Free energy}
\label{FGreen}

The temporal and spatial Fourier transform of one-particle $f$-electron Green's function
satisfy the equations of motion
given by:
\begin{align}
[ \ \omega \ - \ \tilde{E}_{f,\sigma}^{\alpha}(\vec{k}) \ ] \ \langle\langle f^{\alpha}_{\vec{k},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega} =\delta^{\alpha,\chi'}\ \delta_{\vec{k},\vec{k}'}+ V_{\alpha}(\vec{k})\times  \nonumber\\
    \langle\langle d_{\vec{k},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}
+\bigg({{U-J} \over 2}\bigg)
[\langle\langle\hat{\rho}^{\beta}_{-\vec{Q},\sigma} f^{\alpha}_{\vec{k}-\vec{Q},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}   \nonumber\\+\langle\langle\hat{\rho}^{\beta}_{\vec{Q},\sigma} f^{\alpha}_{\vec{k}+\vec{Q},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}]\nonumber \\ +\bigg({U \over 2 }\bigg)
[\sum_{\chi''}\langle\langle\hat{\rho}^{\chi''}_{-\vec{Q},-\sigma} f^{\alpha}_{\vec{k}-\vec{Q},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}   \nonumber\\+\sum_{\chi}\langle\langle\hat{\rho}^{\chi}_{\vec{Q},-\sigma} f^{\alpha}_{\vec{k}+\vec{Q},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}]&\
\label{Galpha}
\end{align}
and

\begin{align}
[ \ \omega \ - \ \tilde{E}_{f,\sigma}^{\beta}(\vec{k}) \ ] \ \langle\langle f^{\beta}_{\vec{k},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega} = \delta^{\beta,\chi'}  \ \delta_{\vec{k},\vec{k}'}+ V_{\beta}(\vec{k}) \times \nonumber\\
 \  \langle\langle d_{\vec{k},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}\
+\bigg(\frac{U-J}{2}\bigg)
[\langle\langle\hat{\rho}^{\alpha}_{-\vec{Q},\sigma} f^{\beta}_{\vec{k}-\vec{Q},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}   \nonumber\\+\langle\langle\hat{\rho}^{\alpha}_{\vec{Q},\sigma} f^{\beta}_{\vec{k}+\vec{Q},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}]\nonumber \nonumber\\  +\bigg(\frac{U}{2}\bigg)
[\sum_{\chi''}\langle\langle\hat{\rho}^{\chi''}_{-\vec{Q},-\sigma} f^{\beta}_{\vec{k}-\vec{Q},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}  \nonumber\\+\sum_{\chi}\langle\langle\hat{\rho}^{\chi}_{\vec{Q},-\sigma} f^{\beta}_{\vec{k}+\vec{Q},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}]&\
\label{Gbeta}
\end{align}
where $\tilde{E}_{f,\sigma}^{\chi}(\vec{k})=E_f^{\chi}(\vec{k}) -\sigma H_{z}^{f}$ and where
\begin{equation}
  \hat{\rho}^{\chi}_{\vec{Q},\sigma} = \bigg(\frac{1}{ N}\bigg) \sum_{\vec{k}'} f^{\dag,\chi}_{\vec{k}'+\vec{Q},\sigma}  f^{\chi}_{\vec{k}',\sigma}\,.
\end{equation}

We use the Hartree-Fock approximation to decouple
the two-particle Green's functions terms in Eqs. (\ref{Galpha})-(\ref{Gbeta}). Thus
\begin{equation}
\langle\langle \hat{\rho}^{\chi}_{\vec{Q},\sigma} f^{\chi'}_{\vec{k}-\vec{Q},\sigma};f^{\dag\chi''}_{ \vec{k'},\sigma} \rangle\rangle_{\omega} \simeq n^{\chi}_{\vec{Q},\sigma}\langle\langle  f^{\chi'}_{\vec{k}-\vec{Q},\sigma};f^{\dag\chi''}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}\,,
\label{aprox1}
\end{equation}
where $\vec{Q}$ is the nesting vector.
Therefore, the Green's function equations of motion become:
\begin{align}
&&[ \ \omega \ - \ E_{f,\sigma}^{\alpha}(\vec{k}) \ ] \ \langle\langle f^{\alpha}_{\vec{k},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega} =\delta^{\alpha,\chi'}  \ \delta_{\vec{k},\vec{k}'}
 \nonumber\\&& +V_{\alpha}(\vec{k})   \langle\langle d_{\vec{k},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}\
-\phi^{\alpha}_{\sigma}  \langle\langle f_{\vec{k}+\vec{Q},\sigma}^{\alpha};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}\
\label{Galpha_apHF}
\end{align}
and
\begin{align}
&&[ \ \omega \ - \ E_{f,\sigma}^{\beta}(\vec{k}) \ ] \ \langle\langle f^{\beta}_{\vec{k},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega} =\delta^{\beta,\chi'}  \ \delta_{\vec{k},\vec{k}'}
 \nonumber\\&& +V_{\beta}(\vec{k})   \langle\langle d_{\vec{k},\sigma};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}\
-\phi^{\beta}_{\sigma}  \langle\langle f_{\vec{k}+\vec{Q},\sigma}^{\beta};f^{\dag\chi'}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}\,,
\label{Gbeta_apHF}
\end{align}
where the
spin-dependent Hartree-Fock dispersion relation in Eqs. (\ref{Galpha_apHF})-(\ref{Gbeta_apHF})
is given by
\begin{equation}
    E_{f, \sigma}^{\chi}(\vec{k})=
\sum_{\chi'} \bigg( U  n^{\chi'}_{-\sigma}+(U - J)  n^{\chi'}_{\sigma}
     (1-\delta^{\chi,\chi'}) \bigg)
     +\tilde{E}_{f,\sigma}^{\chi}(\vec{k})
\label{Erenorm}
\end{equation}
and the spin gap for the $\chi$-orbital is
\begin{equation}
\phi^{\chi}_{\sigma}
= \sum_{\chi'} (U \ n^{\chi'}_{\vec{Q},-\sigma} +
 (U - J) \ n^{\chi'}_{\vec{Q},\sigma} (1-\delta^{\chi,\chi'}))\,.
\label{GapNeel}
\end{equation}

The mixed $f-d$ Green's function satisfies the following equation
\begin{align}
&&     [ \ \omega \ - \ \epsilon(\vec{k}) \ ] \ \langle\langle d_{\vec{k},\sigma};f^{\dag\chi'}_{\vec{k'},\sigma}\rangle\rangle_{\omega} \ =
 \ V_{\alpha}(\vec{k})^* \
\langle\langle f^{\alpha}_{\vec{k},\sigma};f^{\dag\chi'}_{\vec{k'},\sigma}\rangle\rangle_{\omega}+ \nonumber\\
&&\ V_{\beta}(\vec{k})^* \
\langle\langle f^{\beta}_{\vec{k},\sigma};f^{\dag\chi'}_{\vec{k'},\sigma}\rangle\rangle_{\omega}\,.
\label{Gmix}
\end{align}
We will choose a basis set for the $f$ orbitals, such that $V_{\beta}({\vec{k}})=0$ and $V_{\alpha}({\vec{k}})=V_{\alpha}$
simply to avoid the transformation to a new basis set. The choice of basis states should not change the main physical results, as discussed in ref. \cite{Riseborough2012}. Thus, we obtain the Green's functions that can be recast in the
form
\begin{equation}
\langle\langle f_{\vec{k}+\vec{Q},\sigma}^{\beta};f^{\dag\beta}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}=\sum_{\pm}
\frac{Z^{\beta}_{\pm,\vec{k}}}{\omega-\omega^{\beta}_{\pm,\vec{k}}}
\label{Gb}
\end{equation}
and
\begin{equation}
\langle\langle f_{\vec{k}+\vec{Q},\sigma}^{\alpha};f^{\dag\alpha}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}=\sum_{i=1}^{4}
\frac{Z^{\alpha}_{i,\vec{k}}}{\omega-\omega^{\alpha}_{i,\vec{k}}}\,,
\label{Ga}
\end{equation}
where the weights for the $\beta$ character are given by
\begin{equation}
Z^{\beta}_{\pm,\vec{k}}=\frac{\pm \phi^{\beta}_{\sigma} }{\omega^{\beta}_{+,\vec{k}}-\omega^{\beta}_{-,\vec{k}}}
\end{equation}
and $\omega^{\beta}_{\pm,\vec{k}}$ are the dispersion relations of the bands given by the zeros of the denominator
$\varphi_{\vec{Q}}^{\beta}(\omega)$ of the Green's function \newline$\langle\langle f_{\vec{k}+\vec{Q},\sigma}^{\beta};f^{\dag\beta}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}$ :
\begin{equation}
\varphi_{\vec{Q}}^{\beta}(\omega)=(\omega-E_{f,\sigma}^{\alpha}(\vec{k}))
(\omega-E_{f,\sigma}^{\alpha}(\vec{k}+\vec{Q}))-(\phi^{\beta}_{\sigma})^{2}\,.
\label{Dbeta}
\end{equation}
Also, the weights for the $\alpha$ character are expressed by
\begin{equation}
Z^{\alpha}_{1,3,\vec{k}}=\pm\frac{[(\omega^{\alpha}_{1,\vec{k}})^{2}-(\epsilon(\vec{k}))^{2}]\phi^{\alpha}_{\sigma}}{2\omega^{\alpha}_{1,\vec{k}}[(\omega^{\alpha}_{1,\vec{k}})^{2}-(\omega^{\alpha}_{2,\vec{k}})^{2}]}\
\end{equation}
and
\begin{equation}
Z^{\alpha}_{2,4,\vec{k}}=\mp\frac{[(\omega^{\alpha}_{2,\vec{k}})^{2}-(\epsilon(\vec{k}))^{2}]\phi^{\alpha}_{\sigma}}{2\omega^{\alpha}_{2,\vec{k}}[(\omega^{\alpha}_{1,\vec{k}})^{2}-(\omega^{\alpha}_{2,\vec{k}})^{2}]}.
\end{equation}
Similarly to the $\beta$ character case, the  dispersion  relations $\omega^{\alpha}_{i,\vec{k}}$ of  the  $\alpha$ character bands are  given  by the  zeros  of  the  denominator  $\varphi_{\vec{Q}}^{\alpha}(\omega)$ of the Green's function $\langle\langle f_{\vec{k}+\vec{Q},\sigma}^{\alpha};f^{\dag\alpha}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}$  :
\begin{align}
\label{Dalpha}
\varphi_{\vec{Q}}^{\alpha}(\omega)&=&-(\phi^{\alpha}_{\sigma})^{2}(\omega-\epsilon(\vec{k}))(\omega-\epsilon(\vec{k}+\vec{Q}))\nonumber\\&&+[(\omega-E_{f,\sigma}^{\alpha}(\vec{k}))(\omega-\epsilon(\vec{k}))-|V_{\alpha}(\vec{k})|^{2}]\times \nonumber\\&&
[(\omega-E_{f,\sigma}^{\alpha}(\vec{k}+\vec{Q}))(\omega-\epsilon(\vec{k}+\vec{Q}))-|V_{\alpha}(\vec{k})|^{2}]\,.
\end{align}

The Green's function given in Eqs. (\ref{Gb})-(\ref{Ga}) form a closed set of equations, which can be solved exactly.
Thus, the spin gap in Eq. (\ref{GapNeel}) can be calculated directly from
\begin{equation}
n^{\chi}_{\vec{Q},\sigma}=
\frac{1}{N} \sum_{\vec{k}}
 \oint \frac{d\omega}{2\pi i} f(\omega) \
 \langle\langle f_{\vec{k}+\vec{Q},\sigma}^{\chi};f^{\dag\chi}_{ \vec{k'},\sigma} \rangle\rangle_{\omega}\,.
\label{GF}
\end{equation}

In Eq. (\ref{GF}), the contour of the path integral encircles the real axis without enclosing any poles  of the Fermi-Dirac distribution $f(\omega)$.
Therefore, we can explore
a scenario where
the instability of the paramagnetic phase towards to two distinct
phases
SDWs occurs at the same nesting vector $\vec{Q}$ given by the spin gaps in distinct orbitals.
However, it is important to be noted that  the spin gaps $\phi^{\alpha}_{\sigma}$ and $\phi^{\beta}_{\sigma}$ are, indeed, coupled.
Moreover, the spin gaps are proportional to the magnetic  order  parameters and present exactly same behavior.

In the Hartree-Fock approximation, since $\phi^{\alpha(\beta)}_{\uparrow\downarrow}=\mp\phi^{\alpha(\beta)}$, the free energy can be expressed in terms of the gaps by:
\begin{eqnarray}
f_{HF}&=&\Omega(T,\mu)+ \mu N_{tot} +\frac{N}{U^2-J^2} \  [U\left((\phi^{\alpha})^2 + (\phi^{\beta})^2\right)\nonumber\\&& -2J\phi^{\alpha}\phi^{\beta}]\,,
    \label{freeenergy}
\end{eqnarray}
in which $\mu$ is the chemical potential,  $N_{tot}=n_f^{\alpha}+n_f^{\beta}+n_d$ ($n_d$ is the average occupation of the conduction electrons) and
\begin{equation}
 \Omega(T,\mu)= -k_{B} T \frac{1}{N}\sum_{\vec{k}}\sum_{\gamma} \ln[1+ e^{-\frac{(E^{\gamma}-\mu)}{k_{B} T}}],
     \label{freeenergy0}
\end{equation}
where $N$ is the number of sites in the lattice. The quantity $E^{\gamma}$ represents the dispersion relation of the $\beta$ and $\alpha$ bands that are defined by the conditions $\varphi_{\vec{Q}}^{\beta}(\omega)=0$ and $\varphi_{\vec{Q}}^{\alpha}(\omega)=0$, in Eqs. (\ref{Dbeta}) and (\ref{Dalpha}), respectively.

\section{Numerical results}\label{Results}

The numerical results presented in this section were calculated
for $J=5U$ with $U=0.165$ eV  and the total occupancy $N_{tot} = 1.609$. This choice for $N_{tot}$ places the $5f$ bands close to half-filling, which favors the instability of the paramagnetic state with respect to Néel antiferromagnetism. It is important to make clear that the above parameters were chosen in order to emphasize the existence of the competing AF phases. We did not intend to describe any particular compound.

\begin{figure}[!htb]
    \centering
    \includegraphics[scale=1.,angle=0]{GAPSABRC110821.pdf}
    \caption{The Neel gaps $\phi^{\alpha}$ and $\phi^{\beta}$

    for the simple cubic lattice as a function of

    $H_z$ for zero temperature and different values for the band width $W$. The dotted lines denote the AF$_{1}$-AF$_{2}$ and AF$_{2}$-PM first-order transitions while the dashed lines are associated with metamagnetic-like transitions.  }
    \label{POHZC}
\end{figure}
\begin{figure}[!bh]
    \centering
    \includegraphics[scale=01.0,angle=0]{GAPsABRCW1190921.pdf}
    \caption{The Neel gaps $\phi^{\alpha}$ and $\phi^{\beta}$
    for the simple cubic lattice as a function of the magnetic field $h_z$ for $W=1.00$ eV and different values of temperature. The dotted and the dashed lines play the same role as in Fig. \ref{POHZC}.}
    \label{POHZCWfixo}
\end{figure}

\begin{figure}[!th]
    \centering
    \includegraphics[scale=0.7,angle=0]{ENcubica.pdf}
    \caption{The energy per unit cell at zero temperature. The
projection of the ground state energy on the plan of the gaps show dark regions, in which the self-consistent solutions for the gaps
$\phi^{\alpha}$ and $\phi^{\beta}$ are to be found. The diagonal dotted line is defined by the condition $\phi^{\alpha}=\phi^{\beta}$.}
    \label{FNRC}
\end{figure}

The value of the effective $5f$ level energy was chosen to be $\epsilon_{f} = 0.3$ eV. The width of the conduction ($5f$) band is defined as $2W_{d(f)}$ with $W_{f}/W_{d}=0.3$. It has been assumed that the bandwidths $W_{d(f)}$ are sensitive to external pressure.
We also considered a $\vec{k}$-independent hybridization $V_{\alpha}(\vec{k})=V_{\alpha}= W_{d}/10$. With the purpose to simplify the notation, from now on, we use $W_{d} = W$.

\subsection{Simple Cubic Lattice}
Firstly, we present results for a simple cubic lattice, for which $a=c$ and $t_{A,c}=t_{A,a}$, in the dispersion relation given in Eq. \ref{Ek}.

\begin{figure*}[!t]
    \centering
    \includegraphics[scale=1.0,angle=-90]{diagramasVkv121021.pdf}
    \caption{The  phase diagram  for a simple cubic lattice with the band width $W$ versus $H_z$ for several temperatures. The solid and the dotted lines denote second-order and first-order
transition, respectively. The dashed lines mark metamagnetic-like transitions.}
    \label{WHZC}
\end{figure*}

\begin{figure}[!hb]
    \centering
    \includegraphics[scale=0.85,angle=-90]{DOSAB030821.pdf}
    \caption
    The $\alpha$ and $\beta$ partial densities of states for
$W=1.20$ eV, $T=0$, and different
values of $H_z$. The values of $H_z$ have been chosen in order to show the densities of states behavior inside each phase of the diagram presented in the Fig. \ref{WHZC}.}
    \label{DOS_RCWfixo}
\end{figure}

The Neel gaps $\phi^{\alpha}$ and $\phi^{\beta}$
at $T=0$, are shown in Fig. \ref{POHZC} as a function of $H_z$ which is directly proportional to the magnetic field $h_z$ (see Eq. (\ref{eqHz})). Both gaps exhibit discontinuities which indicate the occurrence of first-order phase transitions. The discontinuities denoted by the dotted lines mark first order transitions between two competing antiferromagnetic phases, AF$_{1}$ and AF$_{2}$, or, at higher magnetic fields, between a antiferromagnetic and a paramagnetic phase (PM). The phase AF$_{1}$ is characterized by   $\phi^{\beta}>\phi^{\alpha}>0$, while AF$_{2}$ denotes the phase where $\phi^{\alpha}>\phi^{\beta}>0$. When the system evolves to the PM phase we have $\phi^{\alpha}$= $\phi^{\beta}$= 0. The discontinuities marked by the dashed lines, at lower magnetic field, suggests metamagnetic-like transitions which resemble transitions reported in antiferromagnetic systems \cite{Ken}.

Fig. \ref{POHZCWfixo} displays the Neel gaps
as a function of
$H_z$, for $W=1.00$ eV and for different temperatures. These results show that the effect of increasing of the temperature is to suppress the discontinues found at low magnetic fields. For $k_BT=0.004$ eV, the transition between the phases AF$_2$ and PM change its nature from first to second order.
On the other hand, the nature of the transition AF$_{1}$-AF$_2$, is unaffected.
Nevertheless, at even higher temperature, for $k_{B}T = 0.008$ eV, the AF$_{1(2)}\rightarrow$ PM phase transition becomes a first-order transition again.
This behavior suggest the existence of tricritical points.

In order to attain better understanding on the discontinuities seen in the Neel gaps shown in Fig. \ref{POHZC} and \ref{POHZCWfixo}, we have analyzed the ground state energy per unit cell E$_{N}$, close to the discontinuities.
The results for $T=0$ and $W=1.00$ eV are shown in Fig. \ref{FNRC} as a function of $\phi^{\alpha}$ and $\phi^{\beta}$. In the figure, we have also included a contour plot of the energy, E$_{N}$. The dark regions are associated to the minima of the ground state energy E$_{N}$, and indicate the places where the self-consistent solutions for the gaps
$\phi^{\beta}$ and $\phi^{\alpha}$, are to be found. A discontinuous transition occurs when the local and the global minima exchange positions.
The dotted line on the contour plot represents the diagonal $\phi^{\alpha}=\phi^{\beta}$. We can distinguish between AF$_1$ - AF$_2$ and the metamagnetic-like transitions based on the position of both local and global minimum relative to the diagonal line.
If both local and global minimum are in the same side of the dotted line when they switch positions, then we have a metamagnetic-like transition. This is the case shown in Figs. \ref{FNRC}(a) and \ref{FNRC}(b). Nevertheless, if the local and the global minimum are on opposite sides of the dotted line, when they switch positions, then there is a first order transition between the AF$_{1}$ and AF$_{2}$ phases, as shown in Figs. \ref{FNRC}(c) and \ref{FNRC}(d).
The first order transition between the phases AF$_2$ and PM is illustrated in Figs. \ref{FNRC}(e) and \ref{FNRC}(f).  In Fig. \ref{FNRC}(e) the global minimum is located in the region where  $\phi^{\alpha}<\phi^{\beta}$ (phase AF$_2$) while in \ref{FNRC}(d) the global minimum is shifted to $\phi^{\alpha}=\phi^{\beta}=0$, characterizing a first order transition to the PM phase. The behavior of $E_{N}$ shown in Fig. \ref{FNRC} is in agreement with the results presented in Fig. \ref{POHZC}.

The effect of the temperature on the boundary of the phases AF$_1$, AF$_2$, and PM, is summarized in the phase diagrams
shown in Fig. \ref{WHZC}.
The dotted lines indicate first order transitions while the solid lines represent second order transitions. The Helmholtz free energy has been used to find the correct positions of the first order transitions on the phase diagrams. In the panel \ref{WHZC}(a), it can be seen that the phase AF$_1$
occurs mainly for low values of $W$ while the AF$_2$ phase is predominant found at higher values of
$H_z$. However, the combination of high values of $W$ and $H_z$, favors the AF$_2$ phase.
For $T=0$, we observe two lines representing transitions between the phases AF$_{1}$ and AF$_{2}$
which end at two critical points localized at the region of $W\approx 1.6$ eV, between $H_z=0.03$ eV and $H_z=0.04$ eV. The critical points are denoted by black solid circles.
For finite temperatures, the AF$_2$ phase is restricted to a small portion of the phase diagram at high magnetic fields and low $W$, as shown in Figs. \ref{WHZC}(b) and \ref{WHZC}(c). On the other hand, the AF$_1$ phase is much more robust to the effect of temperature. With increasing temperature,
there are regions where the first-order AF$_{1(2)}\rightarrow$ PM phase transitions are replaced by second-order phase transitions, in agreement with the results presented in Fig. \ref{POHZCWfixo}. The red solid circles indicate the positions of the tricritical points.
In addition, the dashed lines denote metamagnetic-like transitions. Such transitions can be observed in both AF phases, however for $k_{B}T=0.004$ eV, the transitions occur only for low values of $W$. The inset in Fig. \ref{WHZC}(b) highlights the region where the metamagnetic-like transitions occurs. For $k_{B}T=0.008$ eV the metamagnetic-like transitions no longer appear in the range of parameters considered.
\begin{figure}[!b]
    \centering
    \includegraphics[scale=1.0,angle=0]{GAPsABRTT0.pdf}
    \caption{The Neel gaps $\phi^{\alpha}$ and $\phi^{\beta}$ for the tetragonal lattice as a function of $H_z$ for $T=0$ and different values for the band width $W$. The regions with dashed lines denotes the AF$_{1}\rightarrow$ AF$_{2}$ and AF$_{2}\rightarrow$ PM first-order transitions.}
    \label{POHZT}
\end{figure}

\begin{figure*}[!ht]
    \centering
    \includegraphics[scale=1.0,angle=-90]{diagrama2DRTMultiPlotHorizv130921.pdf}
    \caption{The  phase diagram  for a tetragonal lattice with the bandwidth $W$ versus $H_z$, for different temperatures. The solid and the dashed lines denote second-order and first-order
transition, respectively. The parameters of the dispersion relation are  $c/a=1.10$ and  $r=0.90$.}
    \label{WHZRT}
\end{figure*}

\begin{figure}[!hb]
    \centering
    \includegraphics[scale=0.85,angle=-90]{DOSABRT090821.pdf}
    \caption{The $\alpha$ and $\beta$ partial densities of states for
W$=1.0$ eV, $T=0$, and different
values of $H_z$. The values of $H_z$ have been chosen in order to show the densities of states behavior inside each phase  of the diagram presented in the Fig. \ref{WHZRT}.    }
    \label{DOS_RTWfixo}
\end{figure}

In general, the discontinuities in the gaps as a function of
$H_z$ (see Fig. \ref{POHZC}), are related to the position of the Fermi energy $\varepsilon_F$ relative to the gaps in the partial densities of states (DOS). In Fig. \ref{DOS_RCWfixo}, the DOS
associated with the sequence of transitions AF$_1$ $\rightarrow$AF$_2$ $\rightarrow$PM, are shown for $T=0$ and $W=1.20$ eV. The vertical dashed red lines indicate the position of the Fermi energy, for each case. The first and second columns of the panels shown the partial DOS for the AF$_1$ and AF$_2$ phases, respectively.  The third column shown the partial DOS in the PM phase of the system.
When
$H_z$ increases from 0.030 eV to 0.032 eV, the Fermi energy moves out of the gap of the $\beta$-band partial DOS, $\rho^{\beta}_{\sigma}$, resulting in a discontinuity in the gap (see Fig. \ref{POHZC}) what gives rise to the AF$_1$ $\rightarrow$AF$_2$ phase transition. The positions of $\varepsilon_F$ in both cases, are shown in Figs. \ref{DOS_RCWfixo}(g) and \ref{DOS_RCWfixo}(h). In general, every time that $\varepsilon_F$ moves out of a gap in the DOS, due to an increase of either $H_z$ or $W$, the gaps change discontinuously (see Fig. \ref{POHZC}) and are accompanied by a phase transition or a metamagnetic-like transition.
\subsection{Tetragonal Lattice}

In this section we present results for the tetragonal lattice, i.e., $a\neq c$ and $r=t_{A,c}/t_{A,a}$. The crystalline symmetry lifts the degeneracy of the dispersion relations given in Eq. \ref{Ek}. In order to stay relatively close to the cubic lattice case, most of the results presented in this section were obtained using $c/a=1.10$, and $r=0.90$.

In Fig. \ref{POHZT}, it is seen that behavior of the Neel gaps for $W=0.80$ eV and $W=1.00$ eV,
is very similar to the behavior observed for the cubic lattice in Fig. \ref{POHZC}.
However, in the tetragonal case, a higher magnetic field is required to close the Neel gaps. For $W=1.20$ eV,
with an increase of $H_z$, the system leaves the phase AF$_1$ and enters in the PM phase in which the gaps are zero, for small values of $H_z$. If the magnetic field and therefore $H_z$ is further increased, the system reaches the AF$_1$ phase again.
When the magnetic field is increased to higher values, the system undergoes a first-order transition to AF$_{2}$ phase at $H_{z}\approx$ 0.028 eV and another first-order transition is found at $H_{z}\approx$ 0.05 eV where the system enters the PM phase.

The $W$ versus $H_z$ phase diagrams and their evolution with temperature, are shown in Fig. \ref{WHZRT}. For $T=0$, the region where the AF$_1$ phase occurs is similar to that of the cubic lattice. However, the AF$_2$ phase is concentrated in the region of higher magnetic field while in the cubic lattice the AF$_2$ phase also occurs for intermediate values of $H_z$. Indeed, the $\beta$ DOS for the tetragonal lattice is asymmetric relative to $\omega =0$ which results in the phase AF$_2$ being favored. The asymmetry can be seen, for example, in Fig. \ref{DOS_RTWfixo}(l). As in the case of the cubic lattice, two critical points (black solid circles) are present in the $T=0$ phase diagram.  For $k_{B}T=0.004$ eV, we observe the presence of four tricritial points (red solid circles) while in the cubic lattice the four tricritical points first occur at $k_{B}T=0.008$ eV. Furthermore, the critical point observed for $k_{B}T=0.004$ eV is still present for $k_{B}T=0.008$ eV. These facts indicate that the existence of critical and tricritical points is favored in the tetragonal lattice. On the other hand, the metamagnetic-like transitions represented by the dashed lines in Fig. \ref{WHZRT}(a), are less favored than in the cubic lattice.
 %
\begin{figure}[!bth]
    \centering
    \includegraphics[scale=.80,angle=-90]{diagramaT0RTetra121021.pdf}
    \caption{The phase diagram for the tetragonal lattice with $H_z$ and  $W$ versus  $c/a$ in the first column and versus $r$ in the second column.  All the transitions shown in the phase diagrams, have first order nature.}
    \label{HZ_Wvsr}
\end{figure}

The $\alpha$ and $\beta$ partial densities of states, $\rho^\alpha$ and $\rho^\beta$, are shown in Fig. \ref{DOS_RTWfixo} for $T=0$,
$W=1.0$ eV and different values of $H_z$. The values of $H_z$ in the AF$_1$ and AF$_2$ phases, were chosen in order to be close to the AF$_1$ $\rightarrow$ AF$_2$ phase transition.
The behavior of the Neel gaps for this set of parameters has been shown in Fig. \ref{POHZT}. By comparing the results in Fig. \ref{DOS_RTWfixo} with those for the cubic lattice shown in Fig. \ref{POHZC}, it is possible to see that the results are slightly different, mainly for $\rho_{-\sigma}^{\beta}$.
For the phase AF$_1$ with $H_z$=0.033 eV, the position of the  Fermi energy, which is represented by the vertical dashed red line in Fig. \ref{DOS_RTWfixo}, is found inside the gap for both $\rho_{\sigma}^{\beta}$ and $\rho_{-\sigma}^{\beta}$ partial densities, while for the cubic lattice the Fermi energy is found inside the gap only for $\rho_{\sigma}^{\beta}$ (see Figs. \ref{DOS_RCWfixo}(g) and \ref{DOS_RCWfixo}(j)). This feature is related to the asymmetry of the partial DOS for the tetragonal lattice. For instance, in Fig. \ref{DOS_RTWfixo}(j), the area of the partial DOS $\rho_{-\sigma}^{\beta}$ below the Fermi energy (colored in red) is slightly larger than the area above the Fermi energy. In order to keep the total occupation of the bands constant, the Fermi energy has been moved to lower energies, i.e. into the gap of the DOS $\rho_{-\sigma}^{\beta}$, which results in a phase AF$_1$ that is less metallic when compared with the cubic case shown in Fig. \ref{DOS_RCWfixo}, for which the partial DOS $\rho^{\beta}$ is symmetric.

The results presented so far in this section have been obtained considering small deviations from the cubic lattice, for the parameters $c/a$ and $r$. Now, we investigate how the boundaries of the phases AF$_1$, AF$_2$ and PM behave when the parameters $c/a$ and $r$, are changed.
Fig. \ref{HZ_Wvsr}(a) exhibit the phase diagram with $H_z$ versus $c/a$, for fixed $W$ and $r$, while Fig. \ref{HZ_Wvsr}(c) exhibit the phase diagram with $W$ versus $c/a$, for $H_z$=0.0 and $r$ fixed. While the phase AF$_1$ is robust to the effects of $H_z$ and $W$ when $c/a$ is enhanced, the phase AF$_2$ is significantly affected by the increasing of $H_z$ or $W$, in this same situation.
Nevertheless, while the phase AF$_2$ is favored by the increasing of $c/a$ when $H_z$ is enhanced, the same phase is suppressed by the increasing of $c/a$, when $W$ is enhanced.
Such feature is related to the way that $H_z$ and $W$ affect the partial DOS, maily the $\rho_{\pm\sigma}^{\beta}$. While $H_z$ shifts $\rho_{\sigma}^{\beta}$ to lower energies and $\rho_{-\sigma}^{\beta}$ to higher energies, the main effect of $W$ is to increases the wide of the bands. Therefore,  the effects of $H_z$ combined with the asymmetry of the DOS $\rho_{\pm\sigma}^{\beta}$, relative to the gap (see Fig. \ref{DOS_RTWfixo}), are the main reasons for the features
present in the phase diagrams of Figs. \ref{HZ_Wvsr}(a) and
\ref{HZ_Wvsr}(c).
In Figs. \ref{HZ_Wvsr}(b) and
\ref{HZ_Wvsr}(d) it can be noted that the effect of to increasing $r$ keeping $c/a$ fixed, is similar to that one in which $r$ is kept fix while $c/a$ varies. However, the effects of varying $r$ are much less intense. The dashed lines in Figs.\ref{HZ_Wvsr}(b) and
\ref{HZ_Wvsr}(d) indicate the metamagnetic-like transitions.

\section{Conclusions}
\label{Conclusion}
In this work, we have investigated the effects of pressure and magnetic field $h_z$ on two distinct itinerant Neel phases using the underscreened Anderson Lattice Model (UALM).
The version of the UALM that we considered is composed by two narrow $f$-bands (of either $\alpha$ or $\beta$ character)
that hybridize with a single conduction band.
Besides the direct Coulomb interaction
between electrons in the same $5f$ band, we include a Hund's rule
exchange interaction between electrons in the different bands \cite{Niklowitz210}.
We assume that application of pressure produces a variation of the bandwidth.
Moreover, given that the order parameter has an Ising-like
anisotropy, the magnetic field is considered parallel to this
anisotropy direction.
The Hund's rule exchange interaction $J$ couples the gaps $\phi^{\alpha}$ and $\phi^{\beta}$ of the different bands and give rise to two competing antiferromagnetic phases AF$_1$ and AF$_2$. The phase  AF$_1$ is characterized by $\phi^{\beta}>\phi^{\alpha}>0$ while in the phase AF$_2$ we have $\phi^{\alpha}>\phi^{\beta}>0$. The transition between these phases is first order. We analysed the UALM model for a cubic and a tetragonal lattice.

In order to investigate the effects of a magnetic field $h_z$ for different band widths $W$, we constructed $W$ versus $H_z$ phase diagrams for various temperatures. The $H_z$ and $h_z$ are related as in Eq. (\ref{eqHz}). The results show rich variety at $T=0$ for both lattices. In a previous work \cite{Magalhaes2020}, we investigated the effects of a
magnetic field oriented transverse to the $z$ axis with the same UALM model.
In that case, for a cubic lattice, the results showed that the increase of the transverse magnetic field suppresses the phase AF$_2$ while the phase AF$_1$ persists even at higher magnetic fields. In the present work, we find the opposite situation, i.e., the AF$_1$ phase is replaced by the AF$_2$ phase at higher magnetic fields
while the phase AF$_1$ occurs for lower values of $W$ and lower and intermediate values of $H_z$.
This dissemblance is related to the fact that the transverse field produces a spin-dependent momentum shift of the quasi-particles bands. On the other hand, the magnetic field $h_z$ splits the bands generating a spin-up and a spin-down sub-band \cite{Calegari2017}. An increase of $H_z$ shifts the spin-up and the spin-down sub-bands in opposite directions. The analysis of the partial density of states $\rho_{\pm\sigma}^{\alpha}$ and $\rho_{\pm\sigma}^{\beta}$ at $T=0$, helps us better understand why the field $h_z$ favors the phase AF$_2$. We demonstrated in Fig. \ref{DOS_RCWfixo} that in the AF$_1$ phase, the Fermi energy is inside the gaps of $\rho_{-\sigma}^{\alpha}$ and  $\rho_{\sigma}^{\beta}$, at least. On the other hand, if the Fermi energy is outside the gaps of $\rho_{\pm\sigma}^{\beta}$, but is still inside the gap of   $\rho_{-\sigma}^{\alpha}$, the system is found in the phase AF$_2$. Considering the fact that $H_z$ shifts the spin-up and the spin-down sub-bands in opposite directions, the configuration in which the Fermi energy is outside the gaps of both $\rho_{-\sigma}^{\beta}$ and $\rho_{\sigma}^{\beta}$, is favored when $H_z$ increases. Moreover, duo to the hybridization gap present in  $\rho_{\pm\sigma}^{\alpha}$, the gap
$\phi^{\alpha}$ is less affected by the magnetic field (see Fig. \ref{POHZC}), allowing the Fermi energy to remain inside the gap of $\rho_{-\sigma}^{\alpha}$, until higher values of $H_z$.
Indeed, these are the main reasons why the AF$_2$ phase is favored by the magnetic field $h_z$.

The phase diagram for the tetragonal lattice at $T=0$, shows that the phase AF$_2$ occurs at higher magnetic fields while in the cubic lattice it is present at moderate magnetic fields. Moreover, the phase AF$_2$ tends to be concentrated at lower values of $W$ and higher values of magnetic fields, in comparison with the cubic lattice case. The main reason for that, is an asymmetry present in the partial density of states of the tetragonal lattice, which is enhanced with the increasing of $c/a$, where $c$ and $a$ are the lattice parameters.
As a result, we observed that for a tetragonal lattice, the AF$_2$ phase is enhanced by the increasing of both $H_z$ and $c/a$. On the other hand, the increasing of $W$ and $c/a$ at the same time, is detrimental for the AF$_2$ phase. Otherwise, the phase AF$_1$ is much less sensitive to the variation of such parameters, including the temperature. In addition, another difference relative to the application of a transverse field \cite{Magalhaes2020}, is the presence of metamagnetic-like transitions which occurs in both AF phases under the application of a magnetic field $h_z$. We highlight that such phenomenology,  the metamagnetic-like transitions inside the antferromagnetic phases have been reported in some antiferromagnetic heavy fermions \cite{Ken} which also presents a competition between two distinct antiferromagnetic phases.

\section*{Acknowledgments}
The present study was partially supported by the brazilian agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and Fundação de Amparo à pesquisa do Estado do RS (FAPERGS).
ACL acknowledges the hospitality of the Temple University, where part of this work was done.

\medskip

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
Deep learning based on convolution neural networks (CNN) is considered as one of the most successful methods in modern machine learning and computer vision \yky{fields}. It \yky{generally} requires a large amount of data samples for real world applications, yet there exist physical limitation \yky{and privacy issues in acquiring more and more data}. \yky{Meanwhile, recent researches suggest that utilizing more data is crucial for better performance}. Some recent researches \citep{kolesnikov2019large, xie2020self} show that just giving an additional big stream of data outperforms a model designed by a machine (AutoML) \citep{zoph2018learning}, or carefully hand-crafted models \citep{tan2019efficientnet}.

Federated learning (FL) \citep{yang2019federated, ludwig2020ibm}, a recently suggested concept, is one of the \ky{many} solutions to the challenge of acquiring \ky{vast amount of} data without invasion of privacy. FL assumes that many distributed \nj{devices exist for their own data collection}. \ky{The main goal is} the parallel or distributed training of those \nj{device-specific} datasets \ky{by} one central model. \ky{Since its proposal with deep neural nets, many works have focused on model averaging of the client models to acquire a centralized \nj{server} model. However, this standardized practice of model averaging necessitates communicating with all or subsets of clients for a single update of the centralized model, which may lead to a bottleneck in case of a network delay. Furthermore, this method of heavy aggregation also entails communication costs that linearly increases with the number of aggregated clients. These two main assumptions of the current FL formulation restrict its application to a real-world environment that fledgling services face. Namely, the environment is characterized by dynamic inflow and outflow of users, limited amount of users, and scarce resources allocated for communication. Thereby, in this paper we propose an amended formulation of FL, called \textit{asynchronous edge learning}, which intends to extract as much information from a single interaction with a user \nj{as possible}. In addition, we propose an effective method that can asynchronously update the centralized model without any bottleneck via knowledge distillation \citep{hinton2015distilling}.}

\ky{In our edge learning, we call the centralized service provider as the "core" and the customer's device as the "edge". Edge learning characteristically assumes that the service provider has at most one round of interaction with a given edge and that only a single edge is used to update the centralized core model to minimize the communication cost. These assumptions resemble the realistic environment with dynamic inflow and outflow of users and minimal resource constraint to communicate with users. Thus, the main goal is to extract as much information from a single interaction with a given edge, while preventing catastrophic forgetting of the core model. To achieve this, we utilize knowledge distillation (KD) as a communication method between the core and edges and devise a concept called \textit{prediction consensus score} to explain when KD is appropriate. Few works \citep{xie2020self} have exploited KD in a similar manner to ours in different fields of study. We call this variant of KD as ``cloned distillation" and explain when this type of distillation is appropriate with the prediction consensus score. Note that although the restrictions of single core-edge interaction followed by edge churning may be too rigorous, it nevertheless sets a lower bound of a real-world environment. Thus, demonstrating feasibility in this environment is meaningful.}

Our contributions are as follows: \\\
(1) We \nj{reformulate} the federated learning problem to \nj{a} more realistic problem \nj{of dynamic user pool change} called asynchronous edge learning. \\
(2) \ky{We define ``cloned distillation" as a variant of KD and explain when and why this is useful for a certain class of problems.} \\
(3) In utilizing cloned knowledge distillation for asynchronous edge learning, we suggest a method that can \ky{effectively learn from the decentralized edge datasets and \nj{alleviate} the \nj{catastrophic} forgetting problem of the core model.}

\begin{figure}[t]
    \centering
        \begin{subfigure}[t]{0.45\textwidth}
            \centering
            \includegraphics[width=0.8\textwidth]{fig_intro_a.pdf}
            \caption{Heavy aggregation of FL}
        \end{subfigure}
            \hfill
        \begin{subfigure}[t]{0.5\textwidth}
            \centering
            \includegraphics[width=\textwidth]{fig_intro_b.pdf}
            \caption{Our lightweight edge learning (EL)}
        \end{subfigure}
    \caption{
    Schematic comparison of heavy aggregation of the conventional FL and our proposed edge learning scenario.
    }
    \label{fig:edgelearning}
\end{figure}

\section{Related Works}
\ky{With the prevalence of mobile devices that can collect rich variety of data, the concept of training a robust model with decentralized data has emerged as an important research area in both industry and academia (\cite{ludwig2020ibm, Wang2020Federated_matavg, chen2019communication}). The first recent proposal (\cite{mcmahan2017communication}) applied to deep networks used model averaging of the client models to acquire a centralized model.
Since then, many variants of model averaging in the aggregation step have been proposed. \cite{li2020federated} \nj{add} a proximal penalty \yky{between the client model and the core model}. \cite{yurochkin2019bayesian} \nj{solve} the permutation invariant property of neural networks that causes a detrimental effect in model averaging by matching neurons of the client models. \cite{Wang2020Federated_matavg} further \nj{extend} this idea to deep neural networks by layer-wise matching and averaging. While all the aforementioned methods improve upon the standardized aggregation method of model averaging, this aggregation practice is only applicable when multiple client models are simultaneously available for communication. Meanwhile, \cite{li2019fedmd} \nj{approach} the task \nj{by} using knowledge distillation between the averaged model and the client model. \cite{he2020group} also approach the problem with knowledge distillation to solve different model size issue between the server and the edge.}

\section{\nj{Proposed Method: Asynchronous Edge Learning}}
\label{sec:methodology}

\ky{In the standard federated learning problem, a single round, which is characterized by a heavy communication with multiple sub-sampled or entire clients, is iterated for a server-specified \nj{number of iterations}. In contrast, rather than a server-specified number of rounds, \nj{our} edge learning assumes that an edge communicates with the core at most once. Single communication between the core and the edge constitutes a round. This difference is depicted in Figure \ref{fig:edgelearning}}. Under this condition, our solution is \ky{to} send a certain deep model as a local learner to each edge device, learn from it, and re-collect it as a teacher to the core model \ky{in parallel}. \sh{The re-collected teacher performs knowledge distillation with a small, pre-prepared core dataset.}
The model sent to each edge device is called as an ``edge model".
From now on, consider the core dataset $C$ whose size is $N_C$, and $K$ edge datasets $\{E_k\}_{k=1}^K$, each with the size of $N_{E_{k}}$. For a data sample $x$, the output \nj{logit vector} of the core model is expressed as $F(x)$ and that of the $k$-th edge model is denoted as $f_k(x)$.

Algorithm~\ref{alg:main} is used to train the core model using the core dataset \ky{without} directly seeing the edge datasets. However, the core model learns from the returned edge models \nj{that had previously been} sent to \ky{the} edge devices to learn from the corresponding edge dataset. The algorithm is composed of 3 phases of which Phase 1 and Phase 2 are iterated. This iteration \ky{constitutes a} round. In detail, in Phase 0 the core model learns from the core dataset, and each round is \ky{executed} for each edge \ky{that participates in the core service}. Note that the rounds could run asynchronously, \ky{not sequentially}. \sh{The effect of lagged response is discussed in \ky{the} experiment section.}

\begin{algorithm}[t]
\SetAlgoLined
\KwResult{Incrementally increasing \yky{core model performance} }
 $K$: the number of currently available edges\;
 $i$: the index of round, initially set to 0\;
 \textsc{Phase 0}: Core initialization -- core is trained with the core dataset $C$ \sh{with $L_{core}$}\;
 \While{edge $E_{i+1}$ exists}{
  $i$++\;
  \nj{\textsc{Downlink}:} Preparation of an edge model -- send the core model to the edge\;
  \While{edge model \nj{has not reached} a learning threshold}{
   \textsc{Phase 1}: Expedition and learning on edge -- edge model is trained with the edge dataset \nj{$E_i$} \sh{with $L_{edge}$}\;
  }
  \nj{\textsc{Uplink}:} Re-collection of the edge model -- send the edge model  to the core\;
  \While{knowledge distillation \nj{has not converged}}{
  \textsc{Phase 2}: Knowledge distillation -- transfer edge's knowledge to the core \sh{with $L_{distill}$}\;
  }
  \If{$i=K$}{wait until another edge is appended\;}
 }
 \caption{\ky{Asynchronous Edge Learning}}
 \label{alg:main}
\end{algorithm}

In Algorithm~\ref{alg:main}, the learning in each phase (Phase 0 through 2) can be expressed \nj{as follows:}
\begin{equation}
\begin{split}
    L_{core} = \sum_{i=1}^{N_C} L(F(x_i), y_i)&, \quad L_{edge} = \sum_{k=1}^{K}\sum_{j=1}^{N_{E_k}} L(f_k(x_j), y_j), \\
    L_{distill} = L_{core} &+ t^2 \sum_{i=1}^{N_C} \mathrm{KL}(F(\ky{x_i}), f_k(\ky{x_i})/t). \label{eq:distill}
    \end{split}
\end{equation}
\nj{Here, $L(\cdot, \cdot)$ and $KL(\cdot, \cdot)$ are the cross entropy loss and KL divergence, respectively. }
Note that $C$ and $E_k$ \nj{are} fully disjoint sets, and each of those functions must be performed only in the specified phase, i.e, in Phase 0, $L_{core}$ is used, while $L_{edge}$ and $L_{distill}$ are used in Phase 1 and Phase 2 \ky{respectively}.

\section{\nj{Detailed Methodology}}
\label{sec:method}

\begin{figure}[tb]
    \centering
        \vspace{-2mm}
        \includegraphics[width=\textwidth]{fig_method.pdf}
    \caption{
    \nj{We incorporate model ensemble at each edge node and temporal memory at the core to store maximum $m$ proceeding edge models. In Phase 1, to obtain an ensemble of edge models from one identical copy of the core model, a different input batch sampling is applied for each edge model (M1 to M3). In the KD phase (Phase 2), the returned ensemble of edge models and the edge models from the previous (M-1) rounds stored in the memory are used as teachers.}

    }
    \label{fig:method}
\end{figure}

\nj{To boost the performance of the proposed asynchronous edge learning, as shown in Figure \ref{fig:method}, we incorporate model ensemble and temporal memory in the edge model training (Phase 1) and knowledge distillation (Phase 2), respectively. The details are described in the following subsections. Before moving on, we introduce the unique property of cloned knowledge distillation that constitutes the gist of our method, which is different from the conventional knowledge distillation.}

 \subsection{Property of cloned Knowledge Distillation}
 \ky{In the previous section, we mentioned that the core model is dispatched to the edge and trained using the edge dataset. Once the training is done, the edge model becomes the teacher model \nj{for the training of} the core model. Using the cloned core model as the initial point \nj{of edge learning} rather than \nj{using an independent random initial point} highlights a fundamental difference between cloned KD and vanilla KD (independent KD hereafter).}

In our method, a specific form of cloned KD is applied. In detail, \ky{a cloned core model that has been trained on the edge dataset teaches the core model} \nj{with the initial high} \ky{learning} rate of the original learning schedule.
\ky{In essence}, cloned KD performs knowledge distillation \ky{without} deviating significantly from the solution that \nj{was obtained by training with the first learning rate schedule}. \ky{In contrast}, the independent knowledge distillation involves teacher and student models \ky{trained} with two different \ky{initial} seeds. \ky{Accordingly, no restrictions exist in the proximity of the two models in the parameter space, which is depicted in Figure \ref{fig:stars}.}
\ky{The motivation of maintaining a desirable initial point as in }cloned distillation is not \ky{a} newly suggested concept in the \ky{fields of }deep learning. \ky{\cite{xie2020self} utilized a similar idea for self-training by using an identical teacher model trained with perturbation. Lottery Ticket Hypothesis \citep{frankle2018lottery} is also an effort to take advantage of a good initial point referred to as the lottery ticket, although KD is not used.}
Additionally, many federated learning \ky{methods operate in a} similar way (even if they do not use knowledge distillation), because there is no need to train another core model or edge model from scratch.

\ky{Then why is cloned KD useful in our scenario?}
In KD, the \ky{student} model receives the learned knowledge by \ky{imitating} the teacher model. But at the same time, \yky{the student attempts to retain its own knowledge exclusive to the teacher.}
\ky{Thus, a trade-off exists between two traits of the student: to attend to the teacher or to retain its exclusive knowledge.}
\nj{General research} on KD will be interested \ky{in both traits rather than one particular trait}, because it needs to \ky{boost} accuracy.
However, we need a way to focus more \nj{on} the \ky{attending trait} for \nj{our} scenario, \ky{because only through the dispatched teacher model can the student model in the core learn about the edge dataset.}
\ky{Cloned distillation does this by preparing a teacher \nj{that shares the same knowledge} with the student, \nj{which is done by initializing the teacher with the student's weight, then training it with its exclusive edge dataset.} \yky{This} allows the student to focus more on the exclusive side of the teacher's knowledge learned on the edge dataset.}
\nj{Fig. \ref{fig:stars} compares the difference between cloned KD and independent KD. In the parameter space, consider the two initial settings of the teacher model $W_0$ and $W_S$ for independent and cloned teacher respectively. After learning the edge data, the final teacher model is denoted by the pentagons ($W_I$ and $W_C$) in the figure. In the distillation stage at the core, the distance between the student ($W_S$) and the teacher for independent KD ($W_I$) is typically so large that the final student's weight after KD will be located somewhere in the middle between $W_S$ and $W_I$. On the other hand, in cloned KD, the teacher-student distance ($||W_C-W_S||$) is small and the resultant student's weight after KD will be close to the original student's weight ($W_S$). This means that the student did not forget about the original knowledge, while also absorbing teacher's new knowledge on the edge dataset. This explains the larger intersection area in Fig.~\ref{fig:venn_edge0}(b) compared to that of Fig.~\ref{fig:venn_edge0}(a).  }

\begin{figure}[t]
    \centering
    \includegraphics[width=\textwidth]{fig_stars.pdf}
    \caption{Intuitive interpretation of the difference between cloned KD and independent KD on an \yky{imaginary 2-dimensional parameter space. The properties of the two types of distillation are written in text.} }
    \label{fig:stars}
\end{figure}

\begin{figure}[t]
        \vspace{-5mm}
        \centering
        \begin{subfigure}[t]{0.48\textwidth}
            \centering
            \includegraphics[width=0.6\textwidth]{fig_venn_ind_edge0.pdf}
            \caption{\nj{Independent distillation}}
        \end{subfigure}
        \vspace{3mm}
        \begin{subfigure}[t]{0.48\textwidth}
            \centering
            \includegraphics[width=0.6\textwidth]{fig_venn_clone_edge0.pdf}
            \caption{Cloned distillation}
        \end{subfigure}
        \vspace{-5mm}
    \caption{\nj{Venn diagrams of correctly classified samples after KD with a fixed teacher. In both cases (a) and (b), the teacher model was dispatched to and re-collected from the edge. \yky{Results on CIFAR-100 with ResNet-32.}}}
    \label{fig:venn_edge0}
\end{figure}

\ky{To quantify this underlying effect,} we propose a measurement that we name ``prediction intersection", which can be \ky{seen in} the Venn diagram of Figure \ref{fig:venn_edge0}.
This is represented by the intersection ($\cap$) \nj{in} the Venn diagram \nj{of} the \ky{correct} prediction result \nj{for} an untrained dataset such as a test set.
It also \ky{yields} a score called prediction consensus, which is defined as \nj{the intersection over union}.
The teacher-student intersection area and the student's exclusive area \ky{are the regions of interest} referred to as \ky{mutual} areas and exclusive areas, respectively.
As long as the prediction consensus is less then 1, knowledge distillation will \ky{have an effect on the student}, and both the \ky{mutual} area and the exclusive area \yky{may change.}
That is, if ($\Delta$mutual  + $\Delta$exclusive) is positive after distillation, there is a performance improvement.
\ky{For} the case of \nj{cloned} distillation, the exclusive area is almost unchanged, while the \ky{mutual} area is \ky{enlarged}.
With \ky{this interpretation}, we can get an intuition as to why there is an improvement in accuracy after KD, even when an equivalent-sized or smaller model is used as a teacher.

 \subsection{cloned perturbation ensemble}
As mentioned earlier, \nj{we} need a way to maximize efficiency with only one round \ky{of interaction with an edge}.
So to boost the performance at the edge, \nj{we} propose an ensemble methodology.
\ky{In order to obtain} a \ky{meaningful} ensemble of models, \ky{individual models have to provide exclusive domains of knowledge not learned by the other models.}
\ky{In this paper, we achieve this by enforcing a} small perturbation to \ky{induce the models to yield different solutions.}
By simply performing different batch sampling, the cloned models are trained differently and their ensemble becomes meaningful.
These ensembled \ky{edge models} \nj{(M1, M2, M3 in Fig.~\ref{fig:method})} are re-collected and passed to Phase 2.

 \subsection{Temporal memory}
As mentioned above, in our scenario each edge is used once and discarded.
\ky{If catastrophic forgetting happens, this may be a serious problem as no edge is left.}
\ky{To prevent this, }we propose a method of storing up to $M$ number of edge models that are nearest to the current round and use them together. \ky{We call this temporal memory.}
At this time, $L_{distill}$ in Eq.~(\ref{eq:distill}) for Phase 2 is converted to:
\begin{equation}
\begin{split}
    L_{distill} = L_{core} &+ t^2 \sum_{m=1}^{M} \sum_{i=1}^{N_C} \mathrm{KL}(F(x_i), f_m(x_i)/t).
    \end{split}
\end{equation}
We will show in the experiment that using the re-collected edge models from the past \ky{benefits the training of the core model.}

\section{Experiments}
\sh{In all our experiments, we used the CIFAR-100 dataset \ky{with the ResNet-32 \citep{he2016deep} architecture} and assumed 9 edges (clients) and one core. \ky{The} 50K training samples were evenly split into ten subsets, each of which consisting of 50$\pm$6.9 samples per class. \nj{All the hyper-parameters such as the training epochs of the edge and core models} were not fine-tuned in our experiment. Instead, we maintained the original ResNet schedule for both edge and core models.}

\begin{figure}[tb]
    \begin{minipage}[b]{0.48\linewidth}
    \begin{adjustbox}{max width=1.1\textwidth}
        \centering
    \begin{tabular}[b]{c | c c c | c} \hline
          \scriptsize{Experiment}     & \scriptsize{\makecell{Edge Model\\Preparation}}  & \scriptsize{Phase1}    &  \scriptsize{Phase2}   & \scriptsize{Final Acc.}  \\ \hline
             \multicolumn{5}{c}{Independent KD (dashed line)} \\ \hline
          IsPP       & scratch      & -            & -             & 39.99       \\
          IPP        & indep.        & -            & -             & 44.97       \\
          IEP        & indep.        & ensemble     & -             & 47.04       \\
          IPM        & indep.        & -            & memory        & 49.05       \\ \hline
          \multicolumn{5}{c}{Cloned KD (solid line)} \\ \hline
          CPP        & clone       & -             & -             & 53.37       \\
          CEP        & clone       & ensemble      & -             & 54.26       \\
          CPM        & clone       & -             & memory        & 55.11       \\

          CEM (Ours)        & clone       & ensemble      & memory        & \textbf{56.67} \\ \hline
    \end{tabular}
    \end{adjustbox}
    \vspace{0.2mm}
    \end{minipage}
    \qquad
    \quad
  \begin{minipage}[b]{0.44\linewidth}
     \includegraphics[width=\linewidth]{fig_9node_plot.pdf}
  \end{minipage}
\caption{\ky{The left table shows four variants of independently trained edge model (I--) and variants of cloned edge model (C--). The next two letters indicate whether proposed methods are applied or not with 'P' (plain) indicating no method is applied. The final accuracy of the table and the accuracy curve all indicate that cloned distillation (solid line) is crucial. Refer to Section 5.1 for more details. \yky{Results on CIFAR-100 with ResNet-32.}}
}
\label{fig:9node_plot}
\end{figure}

\subsection{\ky{Cloned Distillation for Asynchronous Edge Learning}}
\ky{The main purpose of this experiment is to demonstrate the promising effect of cloned distillation in incrementally enhancing the accuracy of the core model as opposed to independent distillation. }
In Figure \ref{fig:9node_plot}, we compare several variants of independent KD (Baselines) and cloned KD (Our method) \ky{by applying cloned perturbation ensemble in Phase 1 and temporal memory in Phase 2.}
\ky{For instance,} the first letter I in the table indicates that the edge model was initialized independently and trained with the edge dataset. In other words, it is trained by scratch \ky{(Is-)} at the edge or trained at the core \ky{(I-)} using different seeds with the core model, and sent to the edge for \ky{local training}. On the other hand, cloned distillation (C-) is our method, which \ky{sends the cloned core model to the edge.}

\ky{We first compare plain variants indicated by the first and the second lines of Independent KD (IsPP, IPP) and the first line of cloned KD (CPP).}
\ky{The final accuracies of models trained independently all fall short of CPP, even though the same amount of datasets and \yky{training resources} was used for IPP.}
\ky{This result indicates that cloned distillation is crucial for edge learning.}
Also, from Figure \ref{fig:scores_cons}, we can see the benefit of cloned KD compared to the independent KD.

\ky{Next, note the variants that applied \textit{cloned perturbation ensemble} mentioned in Section \ref{sec:method} are indicated as IEP and CEP. \lsh{Note that the individual network's accuracy is about 40\%s, but their ensemble shows 43.8\% performance.} The ensembles were created by perturbing the batch sampling and sent to the core for Phase 2.}
These experiments \ky{demonstrate the effect of the improved edge models on the core model}.
Interestingly, this \yky{method} contributes performance improvement for Cloned KD, whereas for Independent KD, the ensemble only improved the performance at the first round, but did not show sustained performance improvement.
In other words, without cloned KD, even if the edge model \ky{is improved by the ensemble, the core model at Phase 2 cannot attend enough to the teacher to benefit from it.}

\ky{We also compare} IPM and CPM, \ky{which are the variants with} \textit{temporal memory}.
\ky{The results demonstrate that Cloned KD benefits from temporal memory of past models.}
We actually observed that the prediction consensus between the core model and the first edge model (edge-\yky{1}) during the ninth round was higher when \ky{temporal memory was used.}
Note that \textit{cloned perturbation ensemble} method and \textit{temporal memory} method show higher synergy when combined (CEM).

\begin{figure}[tb]
        \begin{subfigure}[t]{0.3\textwidth}
            \centering
            \includegraphics[width=\textwidth]{fig_cons_int.pdf}
            \caption{Prediction intersection between edge model and core model ($T \cap S$)}
        \end{subfigure}
            \hfill
                \begin{subfigure}[t]{0.3\textwidth}
            \centering
            \includegraphics[width=\textwidth]{fig_cons_int_uni.pdf}
            \caption{Prediction consensus between edge model and core model\\ ($T \cap S)/(T \cup S$)}
        \end{subfigure}
            \hfill
                \begin{subfigure}[t]{0.3\textwidth}
            \centering
            \includegraphics[width=\textwidth]{fig_cons_edge0.pdf}
            \caption{Prediction intersection between edge-1 model and core model ($T \cap S$)}
        \end{subfigure}
            \hfill
    \caption{Scores that show cloned distillation is more easy-to-follow than independent knowledge distillation. \yky{Results on CIFAR-100.}}
    \label{fig:scores_cons}
    \vspace{-2mm}
\end{figure}

\subsection{lagged edge model}
\ky{In this experiment, we show that our proposed method is robust to lagged response of the edges.
\sh{Some delayed responses are still inevitable in a realistic federated learning scenario.} For instance, some users may have a slower training process on the edge device or not be able to communicate with the server even after training has completed. Such delays may bottleneck the learning process if the federated learning scheme is seriously hampered by omitting the delayed responses. Suppose we wish to aggregate all the feedbacks from the edges at time $t$, but a feedback (edge model) from a certain edge is delayed and hence not available. After aggregation has already been completed, simply rejecting the delayed response would be wasteful, but using it may found to be detrimental to the core model.}

\ky{We claim that since our method use knowledge distillation as the aggregation method, it is robust to such delayed responses as knowledge distillation has both the attending trait and the retaining trait. In Figure \ref{table:lagged}, we show the results of the core model for three scenarios : 1. Omitting the delayed response, 2. Normal learning without any delays, 3. Using the delayed response once it has reached the core. In Case 3, an edge model model was dispatched to Edge 2 at time $t$, but was unable to return at the specified time. Hence, at time $t+1$ an edge model was dispatched to another edge, Edge 1 and returned to the core model to distill knowledge. Then, the delayed response of Edge 2 returned and proceeded to the knowledge distillation process. Rather than having a detrimental effect, using the delayed response enhanced the core model's accuracy than omitting it (63.37 vs. 62.65), although it slightly fell short of the normal learning process (64.25).}

\begin{figure}[t]
\includegraphics[width=\textwidth]{fig_lag.pdf}
\caption{
\sh{\ky{Diagram and results of} using one core and two edges with lagged response \ky{on CIFAR-100}.}
\ky{The results indicate that simply using the lagged response (63.37\%) is beneficial than aborting it as in Case \#1 (62.65\%).}
}
 \label{table:lagged}
 \vspace{-2mm}
\end{figure}

\subsection{\yky{Noisy Edge Problem}}
\ky{
Anticipating malignant users that produce corrupted data is valid considering how much damage it can provoke to the federated learning process.
Thereby in this experiment, we shuffle the input-label pairs of an entire edge batch with probability $p$. The edge model using such corrupted data and the subsequent core model, to which knowledge is distilled, will be harmed by this. The results shown in Figure \ref{fig:wronglabel} have noisy edge models on the \yky{4th} and \yky{7}th edge. As expected, we can observe from Fig. \ref{fig:wronglabel} (a,b) that the performance drastically drops for both the core model and the edge model after training from the corrupted labels. However, once a regular edge model joins the learning process to teach the core model, the core model of \sh{ours} recovers its performance and nearly returns to the regular learning trajectory.}

\sh{What is particularly interesting is the tendency of the \textit{training accuracy} of the core sharply jumping when corrupted labels are used.
These phenomena are typical cases of over-fitting to the corrupted labels.
This implies that distilling false information induces the core model to memorize the inputs rather than learn semantic patterns via training, which can be supported by other findings (\citet{zhang2016understanding, zhang2017interpretable}).
This observation hints at a possible scheme unique to our method to prevent malignant data from disturbing the federated learning process.
 Applying certain threshold to detect abnormally high training accuracy may be a possible key to detecting malignant dataset.
}

\begin{figure}[t]
        \begin{subfigure}[t]{0.3\textwidth}
            \centering
            \includegraphics[width=\textwidth]{fig_noise_core.pdf}
            \caption{Test accuracy of core model after \# round}
        \end{subfigure}
            \hfill
                \begin{subfigure}[t]{0.3\textwidth}
            \centering
            \includegraphics[width=\textwidth]{fig_noise_edge.pdf}
            \caption{Test accuracy of edge model after \# round}
        \end{subfigure}
            \hfill
                \begin{subfigure}[t]{0.3\textwidth}
            \centering
            \includegraphics[width=\textwidth]{fig_noise_train.pdf}
            \caption{Train accuracy of core model after \# round}
        \end{subfigure}
            \hfill
    \caption{\yky{Robustness of the method} when noisy edge model is included on CIFAR-100.}
    \vspace{-2mm}
    \label{fig:wronglabel}
\end{figure}

\section{Conclusion}
In this \nj{paper}, we formulate a modified federated learning \ky{scenario} \nj{and propose} the \ky{asynchronous} edge learning \nj{which is} more suitable for situations \ky{characterized by dynamic inflow and outflow of edges.}
\ky{As a solution}, we \ky{define a variant of} KD that we call cloned KD. We explain how it works and where \ky{it is appropriate}, and show that this is suitable for edge learning.
We proposed an improved edge learning method through a method of ensemble and a method of using knowledge of past edge models together.

\newpage


text:
\section{Introduction}
Supercooled liquids undergo an exponential (Arrhenius) or even larger increase of their viscosity when the temperature decreases.
This large modification of the transport properties appear while the structure changes only slightly with temperature.
While several theories\cite{gt2,gt3} have been proposed to solve that long standing glass-transition problem, it is still open\cite{anderson,gt1,gt2,gt3,gt4}.
Interestingly while the reasons for the strange behavior of supercooled liquids are still not understood, molecular dynamics simulations\cite{md1} reproduce the unexplained phenomena\cite{gt1,gt2}.
Consequently molecular dynamics simulation is an invaluable tool\cite{md1,visco,prl,w1,w2,w3} to study the glass-transition problem, and more generally\cite{md1,md3,md4,md5,md6,cage,md7} for the study of condensed matter physics.
Due to the universality of the glass-transition  phenomenology\cite{gt1,gt2,gt3,gt4} one is tempted to chose the simplest existing potential function in order to simplify as much as possible the complexity of the problem.
Unfortunately when the potential is too simple, the liquid crystallizes rapidly.
Thus one is conducted to search for the simplest potential that prevents crystallization for supercooled liquids simulations while displaying as strongly as possible the dynamical behavior of supercooled liquids. Various potentials\cite{potG,potDz1,potDz2,potDz3,potKL1,potKL2,potKA1,potKA2} have been proposed in that purpose. However some eventually undergo partial crystallization for long runs, while others are not so simple.
The most popular potential to date that hinders the crystallisation is the Kob-Andersen potential\cite{potKA1,potKA2}. The corresponding liquid is a mixture of two different Lennard-Jones atoms (A and B)  with a proportion of $20\%$ of atoms A and $80\%$ of atoms B.
If that potential is one of the simplest it also creates  an unnecessary increase of the complexity of the problem due to the mixture of atoms.
However liquids constituted with only unmixed atoms A or B do crystallize very fast.
A simple idea to overcome that problem is to bound the two atoms A and B creating a diatomic molecule which liquid may be expected not to crystallize. Unfortunately after long runs that liquid also crystallizes partially.

In this work we study a simple and relatively versatile potential function, based nonetheless on that idea of two bounded Lennard-Jones atoms which parameters are chosen to prevent crystallization. Due to its simple Lennard-Jones structure we expect that potential to be a good candidate to model the universal physics of molecular liquids.
We show that the liquid constituted by these molecules doesn't crystallize at low temperatures and that it follows the typical behavior of non-Arrhenius supercooled liquids.

\section{Calculation}

We model the molecules as constituted of two atoms ($i=1, 2$) that do interact with the following Lennard-Jones potentials:
$V_{ij}=4\epsilon_{ij}((\sigma_{ij}/r)^{12} -(\sigma_{ij}/r)^{6})$ with the parameters: $\epsilon_{11}= \epsilon_{12}=0.5 KJ/mol$, $\epsilon_{22}= 0.4 KJ/mol$, $\sigma_{11}= \sigma_{12}=3.45$\AA\ and $\sigma_{22}=3.28$\AA$ $.
Note that as in the Kob-Andersen model\cite{potKA1,potKA2} we do not use the usual additive mixing rules\cite{md1} for Lennard-Jones potentials.
We make that choice of non-additive mixing rules as it has the property to hinder the crystallization and the formation of plastic crystal phases\cite{pc1,pc2,pc3,M1,M2,M3,M4}.
We use the mass of Argon for each atom of the linear host molecule that we rigidly bonded fixing the interatomic distance to $d=1.73 $\AA$ $.
The reduced shape\cite{M1,M2} of our dumbbell molecule is $L^{*}=d/\sigma=0.5$ a value somehow larger than the limit $L^{*}=0.4$ below which plastic phases are usually created\cite{M1,M2}.
With these parameters the liquid does not crystallize even during long run simulations.
The simulations are first equilibrated during $20$ to $100 ns$ depending on the temperature, then we perform the production run.
 However a few long runs  of $400 ns$ each, have also been realized at low temperature ($T=40 K$) with a smaller simulation box containing $500$ molecules only and we didn't find any sign of crystallization.
We use the Gear algorithm with the quaternion method\cite{md1} to solve the equations of motions with a time step $\Delta t=10^{-15} s$. The temperature is controlled using a Berendsen thermostat\cite{berendsen}. The density is set constant at $\rho=2.24 g/cm^{3}$.
We use in our calculations a cubic simulation box that contains $N=2688$ molecules and has a length $L= 53 $\AA.
The model has the interesting property to be versatile allowing modifications without  crystallizing.
For example one can easily change the mass of the atoms leading to different densities or change the interatomic distance $d$.

\section{Results and discussion}

To verify that there is no crystallization in our liquid, we plot in Figure 1 the radial distribution function (RDF) for various temperatures ranging from above the melting temperature to a deep supercooled liquid.

\includegraphics[scale=0.33]{Fig1.jpg}

{\em \footnotesize  FIG.1. (color online) Radial distribution function $g(r)$ between the molecules center of masses for various temperatures. The structure doesn't change much even at low temperature. We do not see any peak signalling a partial crystallization at low temperatures. Below 150K the liquid is supercooled.\\}

 The radial distribution function $g(r)$ represents the distribution probability to find a molecule a distance $r$ apart from another molecule.
We see on the Figure that the RDF doesn't change much when the temperature decreases from above the melting temperature down to the lowest accessible temperature with our simulations ($T=38 K$). As the temperature decreases, the peaks increase in size, but we do not see any modification of the maxima and minima locations that would have been the signature of a modification of the structure. We do not see ever any sharp peak signalling a partial crystallization of the liquid. Moreover the structure appears to be very simple, quite like the structure of a simple monatomic liquid. The main difference is a small shoulder that we observe in the first peak of the RDF. This shoulder appears due to the difference between the Lennard-Jones potentials of the two atoms constituting the molecule.

\includegraphics[scale=0.33]{Fig2.jpg}

{\em \footnotesize   Fig.2. (color online) Diffusion coefficient (black squares) and inverse of the $\alpha$ relaxation time (red circles) evolution with temperature. The Figure shows an evolution faster than a pure exponential. That evolution called super-Arrhenius is typical of molecular supercooled liquids.\\}

We will now study the dynamic properties of the liquid.
The diffusion coefficient as well as the $\alpha$ relaxation time in Figure 2 display a super-Arrhenius evolution with temperature.
If the diffusive motions are thermally activated processes we may write: $D=D_{0}.e^{-E_{a}/k_{B}T}$, where $E_{a}$ is the activation energy to overpass for the molecule to diffuse.
A super-Arrhenius evolution implies that the activation energy $E_{a}$ is increasing when the temperature decreases and for that reason that behavior has been associated in the past with the emergence of cooperativity.
This super-Arrhenius behavior also show that our liquid is a "fragile" liquid in Angell's classification\cite{angell1,angell2}.
This comportment is typical of molecular liquids, suggesting that our model is well suited for molecular liquids.
We also observe that the $\alpha$ relaxation time evolves more rapidly than the diffusion coefficient at low temperatures.
This result suggests a breakdown of the Stokes-Einstein relation\cite{stokes-einstein}.
Note that the diffusion coefficient $D$ decreases continuously with the temperature showing no sign of a sharp drop of $D$ induced by a crystallization process.

\includegraphics[scale=0.33]{Fig3.jpg}

{\em \footnotesize  FIG.3. (color online) Mean square displacement of the molecules center of masses plotted for various temperatures. The curves display a plateau typical of the supercooled state and that disappears above the melting temperature $T_{m}\approx150 K$.\\}

Figure 3 shows the mean square displacement of the molecules for various temperatures.
We observe the three different time regimes characteristic of supercooled liquids.
For short time scales ($t<10^{-3} ns$) the molecules are in the ballistic regime, between $10^{-3} ns$ and $1 ns$ for the smaller temperature investigated (black curve, T=38 K) we are in the plateau regime and for time scales larger than $1 ns$ we observe the diffusive regime.
The plateau regime appears for temperatures below $T\approx150K$.
The appearance of a plateau is characteristic of supercooled liquids, and we deduce from that emergence that the melting temperature $T_{m}\approx150 K$ in our model liquid. The plateau in the mean square displacement is a signature of the cage effect\cite{cageM} and corresponds to the mean time lapse during which the molecule is trapped inside the cage constituted by its neighbors.
The width of the plateau increases when the temperature drops leading to the increase of the relaxation times.
The diffusion coefficient $D$ plotted in Figure 2 is calculated from these curves using the relation:

$\displaystyle \lim_{t \to \infty} <r^{2}(t)>=6 D t$

\includegraphics[scale=0.33]{Fig4.jpg}

{\em \footnotesize  FIG.4. (color online) Incoherent intermediate scattering function $F_{S}(Q,t)$ for the molecules center of masses at various temperatures. $F_{S}(Q,t)$ is calculated for a wave vector $Q=1.9$\AA$^{-1}$ that corresponds to the location of the first peak of the structure factor $S(Q)$. The functions are normalized so that  $F_{S}(Q,t=0)=1.$The rapid decrease of the very beginning of the curves displaying the ballistic motion of molecules inside their cages is not shown.\\}

Another important correlation function is the incoherent intermediate scattering function, defined as:  $F_{S}^{com}({\bf Q},t)=\displaystyle{ <{1\over N} \sum_{i=1}^{N}{e^{i{\bf Q}.({\bf r}_{i}(t+t_{0})-{\bf r}_{i}(t)) } >}}$.
For an amorphous medium, any direction is equivalent and we may replace the wave vector $\bf Q$ by its modulus.
The incoherent intermediate scattering functions $F_{S}(Q,t)$ are displayed in Figure 4 for various temperatures. As for the mean square displacements in the previous Figure, we observe  the appearance below $T_{m}$ of a plateau that increases when the temperature drops.
The function $F_{S}(Q,t)$ displays the typical evolution of the incoherent intermediate scattering function for supercooled liquids.
We also observe the stretching of the functions for the $\alpha$ relaxation (i.e. the functions do not decrease with a simple Debye relaxation law).
The stretching of the incoherent scattering function has been associated in the past to the appearance of cooperative motions (dynamical heterogeneities) in supercooled liquids, as  that stretching can easily be interpreted as arising from the superposition of different sort of relaxations with different $\alpha$ relaxation times.
The $\alpha$ relaxation time ${{\tau}_{\alpha}}$ plotted in Figure 2 is calculated from these curves using the relation: $F_{S}(Q,{{\tau}_{\alpha}})=e^{-1}$.

\includegraphics[scale=0.3]{Fig5.jpg}

{\em \footnotesize  FIG.5. (color online) Non-Gaussian parameter $\alpha_{2}(t)$ versus time for different temperatures. The maximum value of the non-Gaussian parameter as well as its characteristic time $t^{*}$ increase when the temperature drops. This behavior is induced in supercooled liquids by the appearance of spontaneous cooperative motions called 'dynamic heterogeneity'. We define $t^{*}$ from the relation: $\alpha_{2}(t^{*})=\alpha_{2}^{max}$.\\}

Figure 5 shows the evolution of the non-Gaussian  parameter $\alpha_{2}(t)$ (NGP) when the temperature drops.
The non-Gaussian parameter defined as: $\alpha_{2}(t)=\displaystyle{{3\over 5}{<r^{4}(t)>\over<r^{2}(t)>^{2}} -1}$, measures the deviation of the van Hove correlation function from the Gaussian shape required by Brownian motions.
The main reason for this deviation at low temperature is that the Van Hove develops a tail corresponding to molecules moving cooperatively in motions larger than the average.
As a result $\alpha_{2}(t)$ is, in most cases, a measure of cooperativity in supercooled liquids.
As the temperature drops the maximum value of the NGP increases as well as its characteristic time $t^{*}$ defined from the relation: $\alpha_{2}(t^{*})=\alpha_{2}^{max}$.
This result, typical of supercooled liquids, suggests the presence of cooperative motions in the liquid, that do increase when the temperature drops.

\includegraphics[scale=0.33]{Fig6.jpg}

{\em \footnotesize  FIG.6. (color online) Breakdown of the Stokes-Einstein relation at low temperatures. We see on the Figure that $D.\tau_{\alpha}$ is constant at high enough temperatures, showing that the Stokes-Einstein relation holds for that range of temperatures.  At lower temperatures however we observe a breakdown of the relation as  $D.\tau_{\alpha}$ increases rapidly. This behavior is usually associated with the appearance of cooperativity.\\}

Another test of the appearance of  cooperative motions in supercooled liquids is the breakdown of the Stokes-Einstein relation.
The relation stands that the diffusion coefficient $D$ is inversely proportional to the viscosity $\eta$ of the medium.
 More precisely $D=k_{b}T/( 6\pi  \eta a)$ where $a$ is the characteristic size of the particles (here the molecules) that diffuse.
 As \cite{stokes-einstein} $\tau_{\alpha} \approx \eta/T$ the Stokes-Einstein relation leads to $D. \tau_{\alpha} = constant$.
However for supercooled liquids there is a breakdown of the relation due to the appearance of cooperative motions in the liquid.
This is the behavior that we do observe in Figure 6.
$D. \tau_{\alpha}$ stays approximately constant for $1000/T<20$ $ K^{-1}$ i.e. for $T>50$ $ K$, then it increases, showing that the diffusion coefficient $D$ is decreasing more slowly than  the $\alpha$ relaxation time $\tau_{\alpha}$ (i.e. the viscosity)  increases.

\section{Conclusion}

In summary we have implemented a very simple potential model that doesn't crystallize at low temperatures.
We have shown that this potential reproduces the structural and dynamical behaviors of typical supercooled glass-formers and in particular the appearance of cooperative motions that increase when the temperature drops.
While  these behaviors are universal, some glass-formers do have more pronounced cooperative motions than others making them better candidates as a model, and we have shown that the simple intermolecular potential considered in our study is in that viewpoint a good candidate to model supercooled molecular liquids.