\section{Machine learning exchange interactions}
\label{sec:ml-exc-inter}

% Slide PhD project flowchart % Section phd-project
\begin{frame}
  \frametitle{Vision: Electronic structure learning}
  % frametitle notes: PhD project flowchart
  \framesubtitle{as integrated, high-level multiscale workflows}
  \vspace*{0em}

  \includegraphics[width=1.0\textwidth]{../resources/fig/presentation-2023-02/atomistic-ml/classification-of-atomistic-ml_presentation-2023-02_02-emph-both_ktikz.pdf}

  \vspace*{2em}
  \begin{columns}[t]
    \hspace{1em}
    \begin{column}{0.6\linewidth}
      \begin{center}
        Better \enquote{initial guess}

        for fast SCF convergence
      \end{center}
    \end{column}
    \vrule{}
    \hspace{1em}
    \begin{column}{0.39\linewidth}
      \begin{center}
        Magnetic property prediction

        (ML-Exc) for spin dynamics
      \end{center}
    \end{column}
  \end{columns}
\end{frame}


% Slide MTIs (magnetic topological insulators) % Section co-doping
{
  \setbeamercolor{background canvas}{bg=}
  \includepdf[pages=1]{../resources/fig/presentation-2023-03/ruess/ruess-TIs.pdf}
}

% Slide study design % Section co-doping
\begin{frame}[plain,c]
  \frametitle{Project \enquote{ML-Exc}}
  \framesubtitle{Magnetic co-doping of topological insulators}

  \hspace*{1em} \ce{Bi2Te3} \hspace*{3em} Dimer clusters of \(3d\), \(4d\) transition
  metal defects\vspace*{-1em}

  \begin{columns}[c]
    \hspace*{1em}
    \begin{column}{0.6\textwidth}
      \begin{center}
        \includegraphics[width=0.35\textheight]{../resources/fig/external/papers/mozumderHighthroughputMagneticCodoping2024/processed/Fig1-a.pdf}%
      \hspace*{-0.5em}
        \includegraphics[width=0.80\textwidth]{../resources/fig/external/papers/mozumderHighthroughputMagneticCodoping2024/processed/FigA3-extract-6.pdf}%
      \end{center}
    \end{column}
    \hspace*{3em}
    \begin{column}{0.4\textwidth}
      {\small
        Single-impurity dabase, N=2'000.
        \href{https://go.fzj.de/judit}{go.fzj.de/judit}\vspace*{1em}

        Dimer database, N=2'000\footcite{mozumderHighthroughputMagneticCodoping2024}.\vspace*{3em}

        Co-doping can help to control

        \begin{itemize}
        \item critical \(T_c\) of QAHE
        \item exchange splitting \(\Delta_{xc}\)
        \item long-range magnetic ordering
        \end{itemize}\vspace*{1em}

        for applications in spintronics and

        fault-tolerant quantum computing.
      }
    \end{column}
  \end{columns}
\end{frame}

% Slide AiiDA-KKR workflows % Section co-doping
\begin{frame}[plain,c]
  \frametitle{Project \enquote{ML-Exc}}
  \framesubtitle{\logoAiida{}-KKR workflows\footcite{russmannAiiDAKKRPluginIts2021}}
  \vspace*{-1em}
  \begin{columns}[t]
    \hspace*{-2em}
    \begin{column}{0.3\linewidth}
      % {\footnotesize Single impurity}
      \begin{center}
        \includegraphics[width=0.8\linewidth]{../resources/fig/presentation-2023-03/ruess/ruess-aiida-kkr-paper-workflow-c.pdf}
      \end{center}
    \end{column}
    \begin{column}{0.5\linewidth}
    \hspace*{-3em}
      \begin{center}
        % {\footnotesize \(N+1\) impurities}
        \includegraphics[width=0.8\linewidth]{../resources/fig/external/papers/mozumderHighthroughputMagneticCodoping2024/processed/FigA2-no-subfigure-labels.pdf}%
      \end{center}
    \end{column}
  \end{columns}
  \vspace*{-0em}
  \begin{flushright}
    Extended Heisenberg Hamiltonian.
    \(H = -\frac{1}{2}\sum_{i,j}J_{ij} \, \vec{S}_i \cdot \vec{S}_j
    -\frac{1}{2}\sum_{i,j}\vec{D}_{ij} \cdot \left( \vec{S}_i \times \vec{S}_j \right)
    \)

    Exchange constants from method of infinitesimal rotations\footcite{liechtensteinLocalSpinDensity1987}.
    \( % Liechtenstein infinitesimal roation Jij from KKR-GF
    \mathcal{J}_{ij} = -\frac{1}{\pi} \operatorname{Im} \int_{-\infty}^{E_F}
    \mathrm{d} E \operatorname{Tr}[\delta t_i G_{ij} \delta t_j G_{ji}]
    \)
  \end{flushright}
\end{frame}

% Slide results Jijs 1 % Section co-doping
{
  \setbeamercolor{background canvas}{bg=}
  \includepdf[pages=1]{../resources/fig/external/papers/mozumderHighthroughputMagneticCodoping2024/originals/Fig4.pdf}
}

% Slide AiiDA-Spirit workflows % Section jij-prediction
\begin{frame}[plain,c]
  \frametitle{Spin dynamics with \logoAiida{}-\raisebox{-0.4em}{\logoSpiritWithText{}}\footcite{russmannAiiDASpiritPluginAutomated2022}}
  \vspace*{-2em}
  \begin{columns}[t]
    \begin{column}{0.4\linewidth}
      \begin{center}
        \includegraphics[width=1.0\linewidth]{../resources/fig/external/papers/mozumderHighthroughputMagneticCodoping2024/processed/FigA2-no-subfigure-labels.pdf}%
        % \includegraphics[width=1.0\linewidth]{../resources/fig/presentation-2023-03/ruess/ruess-aiida-kkr-paper-workflow-c.pdf}\footcite{russmannAiiDAKKRPluginIts2021}

        \vspace*{1em}
        Liechtenstein formula
        \[
          \mathcal{J}_{ij} = -\frac{1}{\pi} \operatorname{Im} \int_{-\infty}^{E_F}
          \mathrm{d} E \operatorname{Tr}[\delta t_i G_{ij} \delta t_j G_{ji}]
        \]

        ML-IAP approach. \(E_k = \sum_k E_k \longrightarrow J_{ij} =
    \sum_k \left( J_{ij} \right)_k\)


      \end{center}
    \end{column}
    \begin{column}{0.55\linewidth}
      \begin{center}
        \includegraphics[width=1.0\linewidth]{../resources/fig/presentation-2023-03/ruess/ruess-aiida-spirit-paper-workflow.pdf}%

        Landau-Lifshitz-Gilbert equation
        \[
          \frac{\partial \vec{S}_i}{\partial t}=-\gamma^{\prime} \vec{S}_i \times \vec{B}_i^{\text {eff }}-\lambda \vec{S}_i \times\left(\vec{S}_i \times \vec{B}_i^{\text {eff }}\right)
        \]

        % \vspace*{2em}
          \normalsize{\href{https://juspin.de}{juspin.de}}
      \end{center}
    \end{column}
  \end{columns}
\end{frame}






% Slide project design % Section jij-prediction
\begin{frame}[plain,c]
  \frametitle{Project \enquote{ML-Exc}}
  \framesubtitle{Model selection}

  \begin{table}[H]
    \resizebox{\columnwidth}{!}{%
      \centering
      \input{../resources/table/jij-prediction-model-selection}%
      % \caption[short caption]{Long caption.}
      % \label{tab:model-selection}
    }
  \end{table}
\end{frame}


% Slide exc-int tensor prediction % Section jij-prediction
\begin{frame}
  \frametitle{Project \enquote{ML-Exc}}
  \framesubtitle{Tensorial interaction}
  Heisenberg Hamiltonian in tensor form.

  \[\mathcal{H}_H=-\sum_{j>i} \vec{m}_i \cdot \mathcal{J}_{i j} \vec{m}_j\]

  Tensor components: isotropic, anti-symmetric (DMI) and anisotropic or
  traceless symmetric part (neglected so far).

  \[\mathcal{J}_{i j}=J_{i j} \mathbb{1} + \mathcal{J}_{i j}^A+\mathcal{J}_{i
      j}^S\]

  with \(J_{i j}^{x x}=J_{i j}^{y y}=J_{i j}^{z z}=\frac{1}{3} J_{i j}\) and

  \[\mathcal{J}_{i j}^A=\left[\begin{array}{ccc}
    0 & J_{i j}^{x y} & J_{i j}^{x z} \\
    J_{i j}^{y x} & 0 & J_{i j}^{y z} \\
    J_{i j}^{z x} & J_{i j}^{z y} & 0
  \end{array}\right]=\left[\begin{array}{ccc}
    0 & J_{i j}^{x y} & -J_{i j}^{z x} \\
    -J_{i j}^{x y} & 0 & J_{i j}^{y z} \\
    J_{i j}^{z x} & -J_{i j}^{y z} & 0
  \end{array}\right]=\left[\begin{array}{ccc}
    0 & -D_{i j}^z & D_{i j}^y \\
    D_{i j}^z & 0 & -D_{i j}^x \\
    -D_{i j}^y & D_{i j}^x & 0
  \end{array}\right]\]

\end{frame}

% Slide higher-order exchange interactions
{
  \setbeamercolor{background canvas}{bg=}
  \includepdf[pages=1]{../resources/fig/external/presentations/spinqx23/hkatsumoto/spinqx23-hkatsumoto-page24.pdf}
}

% Slide relativistic exchange interactions
{
  \setbeamercolor{background canvas}{bg=}
  \includepdf[pages=1]{../resources/fig/external/presentations/spinqx23/hkatsumoto/spinqx23-hkatsumoto-page25.pdf}
}


% Slide best-of-aml % Section odds-and-ends
\begin{frame}[plain,c]
  \frametitle{Community resources}
  \framesubtitle{Best of atomistic machine learning}
  \vspace*{-4em}
  \begin{columns}[c]
    \begin{column}{0.5\linewidth}
      \vspace*{3em}
      \begin{center}
        % \includegraphics[width=0.7\textwidth]{../resources/fig/github/best-of-aml-thumbnail-2023-09-05.png}%
        \includegraphics[width=0.9\textwidth]{../resources/fig/github/best-of-aml-thumbnail-2024-07-08.png}%

        Largest list of atomistic ML tools on the web (400+), auto-ranked,
        regular updates\footcite{wasmerBestAtomisticMachine2023}

        \vspace*{2em}
          \Large{\href{https://go.fzj.de/best-of-aml}{go.fzj.de/baml}}
      \end{center}

    \end{column}

    \begin{column}{0.4\linewidth}
      \includegraphics[width=0.85\textwidth]{../resources/fig/github/best-of-aml-contents-2024-07-08.png}%
    \end{column}
  \end{columns}
\end{frame}


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