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Commit ea1bfd27 authored by Jacob Beyer's avatar Jacob Beyer
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Merge branch 'master' of rwth_git:jacob.beyer/triangular_rashba

parents f574b67a 8f76e473
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...@@ -79,13 +79,13 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2} ...@@ -79,13 +79,13 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2}
(A) at (0,0) {$\phantom{\Delta_{x^2-y^2}}$}; (A) at (0,0) {$\phantom{\Delta_{x^2-y^2}}$};
\node[regular polygon, inner sep = -2pt, regular polygon sides=6] \node[regular polygon, inner sep = -2pt, regular polygon sides=6]
(B) at (A) {$\phantom{\Delta_{x^2-y^2}}$}; (B) at (A) {$\phantom{\Delta_{x^2-y^2}}$};
\node[draw=none, fill=white, opacity=0.6, text opacity=1, inner sep=0] at (A) {#1}; \node[draw=none, fill=white, opacity=1, text opacity=1, inner sep=0] at (A) {#1};
\node[] at (B.corner 1) {#2}; \node[] at (B.corner 1) {$#2$};
\node[] at (B.corner 2) {#3}; \node[] at (B.corner 2) {$#3$};
\node[] at (B.corner 3) {#4}; \node[] at (B.corner 3) {$#4$};
\node[] at (B.corner 4) {#5}; \node[] at (B.corner 4) {$#5$};
\node[] at (B.corner 5) {#6}; \node[] at (B.corner 5) {$#6$};
\node[] at (B.corner 6) {#7}; \node[] at (B.corner 6) {$#7$};
\end{tikzpicture} \end{tikzpicture}
} }
} }
...@@ -298,20 +298,49 @@ The allowed combinations can be obtained by a careful symmetry analysis. ...@@ -298,20 +298,49 @@ The allowed combinations can be obtained by a careful symmetry analysis.
\begin{table} \begin{table}
\centering \centering
\begin{tabular}{ccc} \toprule \begin{tabular}{cccc} \toprule
{$\mu$} & Spatial & Spin \\ \hline {$\mu$} & Spatial & Spin & Total \\ \hline
\sym A 1 & \numberedHexagon{$\Delta_s$}{1}{1}{1}{1}{1}{1} & $\hat{\mathbf{d}}_0$ \\ \sym A 1 & \numberedHexagon{$\Delta_s$}{1}{1}{1}{1}{1}{1}
\sym A 2 & - & $\hat{\mathbf{d}}_z$ \\ & $\hat{\mathbf{d}}_0$
\sym B 1 & \numberedHexagon{$\Delta_f$}{-1}{1}{-1}{1}{-1}{1} & \\ \hline \\ & $\begin{gathered}
\Delta_{s} \hat{\mathbf{d}}_0 \\
\Delta_{p_x} \hat{\mathbf{d}}_x - \Delta_{p_y} \hat{\mathbf{d}}_y
\end{gathered}$ \\
\sym A 2 & - & $\hat{\mathbf{d}}_z$
& $\Delta_{p_y} \hat{\mathbf{d}}_x + \Delta_{p_x} \hat{\mathbf{d}}_y$\\
\sym B 1 & \numberedHexagon{$\Delta_f$}{-1}{1}{-1}{1}{-1}{1} & - & - \\
\sym B 2 & - & - & $\Delta_{f} \hat{\mathbf{d}}_z$ \\ \hline \\
\sym E 1 & \sym E 1 &
$\left[\numberedHexagon{$\Delta_{p_x}$}{1}{-1}{-2}{-1}{1}{2} $\begin{bmatrix}
\numberedHexagon{$\Delta_{p_y}$}{1}{1}{0}{-1}{-1}{0} \right] $ \numberedHexagon{$\Delta_{p_x}$}{1}{-1}{-2}{-1}{1}{2} \\
& $[\hat{\mathbf{d}}_x, \hat{\mathbf{d}}_y]$ \\[3em] \numberedHexagon{$\Delta_{p_y}$}{1}{1}{0}{-1}{-1}{0}
\end{bmatrix} $
& $\begin{bmatrix}
\hat{\mathbf{d}}_x \\
\hat{\mathbf{d}}_y
\end{bmatrix}$
& $\begin{bmatrix}
\Delta_{p_x} \\
\Delta_{p_y}
\end{bmatrix}$ \\[3em]
\sym E 2 & \sym E 2 &
$\left[\numberedHexagon{$\Delta_{d_{x^2-y^2}}$}{-1}{-1}{2}{-1}{-1}{2} $\left[
\begin{aligned}
\numberedHexagon{$\Delta_{d_{x^2-y^2}}$}{-1}{-1}{2}{-1}{-1}{2} \\
\numberedHexagon{$\Delta_{d_{xy}}$}{1}{-1}{0}{1}{-1}{0} \numberedHexagon{$\Delta_{d_{xy}}$}{1}{-1}{0}{1}{-1}{0}
\end{aligned}
\right] $ \right] $
& \\[3em] \hline & -
& $\begin{gathered}
[\Delta_{d_{xy}},\Delta_{d_{x^-y^2}}] \hat{\mathbf{d}}_0 \\
\Delta_f [\hat{\mathbf{d}}_x, \hat{\mathbf{d}}_y] \\
\begin{aligned}
[\Delta_{p_y} \hat{\mathbf{d}}_x
- \Delta_{p_x} \hat{\mathbf{d}}_y, \\
\Delta_{p_x} \hat{\mathbf{d}}_x
+ \Delta_{p_y} \hat{\mathbf{d}}_y] \end{aligned}
\end{gathered}$
\\[3em] \hline
\end{tabular} \end{tabular}
\caption{ \caption{
\captiontitle{Basis functions of irreducible representations} \captiontitle{Basis functions of irreducible representations}
...@@ -321,6 +350,33 @@ The allowed combinations can be obtained by a careful symmetry analysis. ...@@ -321,6 +350,33 @@ The allowed combinations can be obtained by a careful symmetry analysis.
\label{tab:mom-irreps} \label{tab:mom-irreps}
\end{table} \end{table}
\begin{table}
\centering
\begin{tabular}{cccc} \toprule
{$\mu$} & \multicolumn{3}{c}{Basis functions} \\ \hline
\sym A 1 & $\Delta_{s} \hat{\mathbf{d}}_0$
& $\Delta_{p_x} \hat{\mathbf{d}}_x - \Delta_{p_y} \hat{\mathbf{d}}_y$ & \\
\sym A 2 & & $\Delta_{p_y} \hat{\mathbf{d}}_x + \Delta_{p_x} \hat{\mathbf{d}}_y$ & \\
\sym B 1 & & & \\
\sym B 2 & $\Delta_{f} \hat{\mathbf{d}}_z$ & & \\ \hline
\sym E 1 & $[\Delta_{p_x},\Delta_{p_y}] \hat{\mathbf{d}}_z$ & & \\
\sym E 2 & $[\Delta_{d_{xy}},\Delta_{d_{x^-y^2}}] \hat{\mathbf{d}}_0$
& $\Delta_f [\hat{\mathbf{d}}_x, \hat{\mathbf{d}}_y]$
& $\begin{aligned}
[\Delta_{p_y} \hat{\mathbf{d}}_x
- \Delta_{p_x} \hat{\mathbf{d}}_y, \\
\Delta_{p_x} \hat{\mathbf{d}}_x
+ \Delta_{p_y} \hat{\mathbf{d}}_y] \end{aligned}$
\\ \hline
\end{tabular}
\caption{
\captiontitle{Basis functions of irreducible representation}
We show the possible basis functions for two spins that are
antisymmetric under exchange of quantum indices.}
\label{tab:total-irreps}
\end{table}
% \begin{table} % \begin{table}
% \centering % \centering
% \begin{tabular}{ccc} \toprule % \begin{tabular}{ccc} \toprule
...@@ -402,31 +458,6 @@ The results is the following decomponsition: ...@@ -402,31 +458,6 @@ The results is the following decomponsition:
All of the basis functions of the combined spin-momentum system are given in Table All of the basis functions of the combined spin-momentum system are given in Table
\ref{tab:total-irreps}. \ref{tab:total-irreps}.
\begin{table}
\centering
\begin{tabular}{cccc} \toprule
{$\mu$} & \multicolumn{3}{c}{Basis functions} \\ \hline
\sym A 1 & $\Delta_{s} \hat{\mathbf{d}}_0$
& $\Delta_{p_x} \hat{\mathbf{d}}_x - \Delta_{p_y} \hat{\mathbf{d}}_y$ & \\
\sym A 2 & & $\Delta_{p_y} \hat{\mathbf{d}}_x + \Delta_{p_x} \hat{\mathbf{d}}_y$ & \\
\sym B 1 & & & \\
\sym B 2 & $\Delta_{f} \hat{\mathbf{d}}_z$ & & \\ \hline
\sym E 1 & $[\Delta_{p_x},\Delta_{p_y}] \hat{\mathbf{d}}_z$ & & \\
\sym E 2 & $[\Delta_{d_{xy}},\Delta_{d_{x^-y^2}}] \hat{\mathbf{d}}_0$
& $\Delta_f [\hat{\mathbf{d}}_x, \hat{\mathbf{d}}_y]$
& $\begin{aligned}
[\Delta_{p_y} \hat{\mathbf{d}}_x
- \Delta_{p_x} \hat{\mathbf{d}}_y, \\
\Delta_{p_x} \hat{\mathbf{d}}_x
+ \Delta_{p_y} \hat{\mathbf{d}}_y] \end{aligned}$
\\ \hline
\end{tabular}
\caption{
\captiontitle{Basis functions of irreducible representation}
We show the possible basis functions for two spins that are
antisymmetric under exchange of quantum indices.}
\label{tab:total-irreps}
\end{table}
\section{Results for the nearest-neighbor model} \section{Results for the nearest-neighbor model}
......
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