diff --git a/main.tex b/main.tex
index 68f655b50fd548fb52c73a20779adc3864343f71..73cf045b17d1deb680cf7ac93fba9639c1c24d87 100644
--- a/main.tex
+++ b/main.tex
@@ -79,13 +79,13 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2}
(A) at (0,0) {$\phantom{\Delta_{x^2-y^2}}$};
\node[regular polygon, inner sep = -2pt, regular polygon sides=6]
(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[] at (B.corner 1) {#2};
- \node[] at (B.corner 2) {#3};
- \node[] at (B.corner 3) {#4};
- \node[] at (B.corner 4) {#5};
- \node[] at (B.corner 5) {#6};
- \node[] at (B.corner 6) {#7};
+ \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 2) {$#3$};
+ \node[] at (B.corner 3) {$#4$};
+ \node[] at (B.corner 4) {$#5$};
+ \node[] at (B.corner 5) {$#6$};
+ \node[] at (B.corner 6) {$#7$};
\end{tikzpicture}
}
}
@@ -298,20 +298,49 @@ The allowed combinations can be obtained by a careful symmetry analysis.
\begin{table}
\centering
-\begin{tabular}{ccc} \toprule
- {$\mu$} & Spatial & Spin \\ \hline
- \sym A 1 & \numberedHexagon{$\Delta_s$}{1}{1}{1}{1}{1}{1} & $\hat{\mathbf{d}}_0$ \\
- \sym A 2 & - & $\hat{\mathbf{d}}_z$ \\
- \sym B 1 & \numberedHexagon{$\Delta_f$}{-1}{1}{-1}{1}{-1}{1} & \\ \hline \\
+\begin{tabular}{cccc} \toprule
+ {$\mu$} & Spatial & Spin & Total \\ \hline
+ \sym A 1 & \numberedHexagon{$\Delta_s$}{1}{1}{1}{1}{1}{1}
+ & $\hat{\mathbf{d}}_0$
+ & $\begin{gathered}
+ \Delta_{s} \hat{\mathbf{d}}_0 \\
+ \Delta_{p_x} \hat{\mathbf{d}}_x - \Delta_{p_y} \hat{\mathbf{d}}_y
+ \end{gathered}$ \\
+ \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 &
- $\left[\numberedHexagon{$\Delta_{p_x}$}{1}{-1}{-2}{-1}{1}{2}
- \numberedHexagon{$\Delta_{p_y}$}{1}{1}{0}{-1}{-1}{0} \right] $
- & $[\hat{\mathbf{d}}_x, \hat{\mathbf{d}}_y]$ \\[3em]
+ $\begin{bmatrix}
+ \numberedHexagon{$\Delta_{p_x}$}{1}{-1}{-2}{-1}{1}{2} \\
+ \numberedHexagon{$\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 &
- $\left[\numberedHexagon{$\Delta_{d_{x^2-y^2}}$}{-1}{-1}{2}{-1}{-1}{2}
- \numberedHexagon{$\Delta_{d_{xy}}$}{1}{-1}{0}{1}{-1}{0}
- \right] $
- & \\[3em] \hline
+ $\left[
+ \begin{aligned}
+ \numberedHexagon{$\Delta_{d_{x^2-y^2}}$}{-1}{-1}{2}{-1}{-1}{2} \\
+ \numberedHexagon{$\Delta_{d_{xy}}$}{1}{-1}{0}{1}{-1}{0}
+ \end{aligned}
+ \right] $
+ & -
+ & $\begin{gathered}
+ [\Delta_{d_{xy}},\Delta_{d_{x^-y^2}}] \hat{\mathbf{d}}_0 \\
+ \Delta_f [\hat{\mathbf{d}}_x, \hat{\mathbf{d}}_y] \\
+ \begin{aligned}
+ [\Delta_{p_y} \hat{\mathbf{d}}_x
+ - \Delta_{p_x} \hat{\mathbf{d}}_y, \\
+ \Delta_{p_x} \hat{\mathbf{d}}_x
+ + \Delta_{p_y} \hat{\mathbf{d}}_y] \end{aligned}
+ \end{gathered}$
+ \\[3em] \hline
\end{tabular}
\caption{
\captiontitle{Basis functions of irreducible representations}
@@ -321,6 +350,33 @@ The allowed combinations can be obtained by a careful symmetry analysis.
\label{tab:mom-irreps}
\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}
% \centering
% \begin{tabular}{ccc} \toprule
@@ -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
\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}