diff --git a/main.tex b/main.tex index 3ac0d41e8fe33d4c5892ee52cba3b72a72bab0e7..7509ef0e90ff4c96346af00cb62b8077c5090ec7 100644 --- a/main.tex +++ b/main.tex @@ -1,5 +1,5 @@ \documentclass[aps, prb, twocolumn, showpacs, amsmath, amssymb, amscd, -superscriptaddress, floatfix, longbibliography]{revtex4-2} +superscriptaddress, floatfix, longbibliography, booktabs]{revtex4-2} \usepackage{braket} @@ -15,6 +15,9 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2} \usepackage[capitalize]{cleveref} \usepackage{csquotes} +\usepackage{diagbox} +\usepackage{colortbl} + \usepackage{bbold} \usepackage{pifont} @@ -44,7 +47,7 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2} \node[shape=circle,draw,inner sep=1pt] (char) {#1};}} \MakeRobustCommand\circled -\newcommand{\su}{SU(2)~} +\newcommand{\su}[1]{SU(2)~} \newcommand{\bvec}[1]{\boldsymbol #1} \newcommand{\Ucrit}{U_\mathrm{c}} @@ -310,10 +313,12 @@ presented in the following. \Delta_{s} \Psi \\ \Delta_{p_x} d_x - \Delta_{p_y} d_y \end{gathered}$ \\ - \sym A 2 & --- & $d_z$ + \sym A 2 & \textbf{---} & $d_z$ & $\Delta_{p_y} d_x + \Delta_{p_x} d_y$\\ - \sym B 1 & \numberedHexagon{$\Delta_f$}{-1}{1}{-1}{1}{-1}{1} & --- & --- \\ - \sym B 2 & --- & --- & $\Delta_{f} d_z$ \\[0.5em] \hline \\ + \sym B 1 & \numberedHexagon{$\Delta_f$}{-1}{1}{-1}{1}{-1}{1} + & \textbf{---} & \textbf{---} \\ + \sym B 2 & \textbf{---} & \textbf{---} + & $\Delta_{f} d_z$ \\[0.5em] \hline \\ \sym E 1 & \hspace{0.15cm} % Fix spacing in the table $\begin{bmatrix} @@ -333,16 +338,16 @@ presented in the following. \sym E 2 & $\left[ \begin{aligned} - \numberedHexagon{$\Delta_{d_{x^2-y^2}}$}{-1}{-1}{2}{-1}{-1}{2} \\ + \numberedHexagon{$\Delta_{d_{x^2-y^2}}$}{-1}{-1}{2\phantom{-}}{-1}{-1}{2} \\ \numberedHexagon{$\Delta_{d_{xy}}$}{1}{-1}{0}{1}{-1}{0} \end{aligned} \right] $ - & --- + & \textbf{---} & \hspace{0.5cm} $\begin{gathered} \begin{bmatrix} \Delta_{d_{xy}} \\ - \Delta_{d_{x^-y^2}} + \Delta_{d_{x^2-y^2}} \end{bmatrix} \Psi \\ \Delta_f \begin{bmatrix} d_x \\ @@ -369,6 +374,51 @@ presented in the following. \label{tab:irrep_basis_fns} \end{table} +\begin{table} + \definecolor{cell}{gray}{0.8} +\centering +\begin{tabular}{cccc} \toprule + \diagbox{$\Delta$}{$\sigma$} + & \sym A 1 & \sym A 2 & \sym E 1 \\ \hline + \sym A 1 + & \sym A 1 : $\Delta_s \Psi$ + & \cellcolor{cell} + & \cellcolor{cell} \\ + \sym E 2 + & \sym E 2 : $\begin{bmatrix} \Delta_{d_{xy}} \\ + \Delta_{d_{x^2-y^2}}\end{bmatrix} \Psi$ + & \cellcolor{cell} + & \cellcolor{cell} \\ \hline + \sym B 1 + & \cellcolor{cell} + & \sym B 2 : $\Delta_f d_z $ + & \sym E 2 : $\Delta_f \begin{bmatrix} d_x \\ d_y \end{bmatrix}$ \\ + \sym E 1 + & \cellcolor{cell} + & \sym E 1 : $\begin{bmatrix} \Delta_{p_x} \\ \Delta_{p_y} \end{bmatrix}d_z$ + & $\begin{gathered} \sym A 1 \oplus \sym B 2 \oplus \sym E 2 : \\ + \Delta_{p_x}d_x - \Delta_{p_y}d_y \\ + \oplus \\ + \Delta_{p_y}d_x + \Delta_{p_x}d_y \\ + \oplus \\ + \begin{bmatrix} + \Delta_{p_y}d_x - \Delta_{p_x}d_y \\ + \Delta_{p_x}d_x + \Delta_{p_y}d_y \end{bmatrix} + \end{gathered}$ + \\ \hline + +\end{tabular} + \caption{ + \captiontitle{Basis functions of irreducible representations} + The first column lists the irreducible representation (irrep) $\gamma$. + The second column is the basis function in real space, for example, as + the bond pairing on nearest neighbour bonds. The third column is the + two-spin basis function, in terms of the typical superconducting psuedo- + vector formulation $(\Psi, \bvec d)$. The fourth column is the total + spatial and two-spin basis function, calculated as decribed in the text.} + \label{tab:irrep_combinations} +\end{table} + The divergent eigenstate will belong to one of the irreps of the point symmetry group of the lattice (\textit{i.e.} \sym C {6v}). When the Hamiltonian has an \su 2 spin symmetry, the point symmetry irreps