diff --git a/main.tex b/main.tex
index 39a08e85bee76130ef2e71151ba4f4fef540805e..e73065d2e8a7ef4efc795d922abee8a002f4b04d 100644
--- a/main.tex
+++ b/main.tex
@@ -138,7 +138,7 @@ Rashba Hubbard model}
Triplet superconductors have recently risen to prominence as possible building
blocks of topological superconductors.
-The addition of Rashba SOC, with the mixing of spin subspaces promises
+The addition of Rashba SOC, with the mixing of spin subspaces promises
\cite{hauck2023}
\todo{TODO}
@@ -304,28 +304,28 @@ presented in the following.
\begin{tabular}{cccc} \toprule
{$\gamma$} & Spatial & Spin & Total \\ \hline
\sym A 1 & \numberedHexagon{$\Delta_s$}{1}{1}{1}{1}{1}{1}
- & $\hat{\mathbf{d}}_0$
+ & $d_0$
& $\begin{gathered}
- \Delta_{s} \hat{\mathbf{d}}_0 \\
- \Delta_{p_x} \hat{\mathbf{d}}_x - \Delta_{p_y} \hat{\mathbf{d}}_y
+ \Delta_{s} d_0 \\
+ \Delta_{p_x} d_x - \Delta_{p_y} d_y
\end{gathered}$ \\
- \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 A 2 & --- & $d_z$
+ & $\Delta_{p_y} d_x + \Delta_{p_x} d_y$\\
+ \sym B 1 & \numberedHexagon{$\Delta_f$}{-1}{1}{-1}{1}{-1}{1} & --- & --- \\
+ \sym B 2 & --- & --- & $\Delta_{f} d_z$ \\ \hline
\sym E 1 &
$\begin{bmatrix}
\numberedHexagon{$\Delta_{p_x}$}{1}{-1}{-2}{-1}{1}{2} \\
\numberedHexagon{$\Delta_{p_y}$}{1}{1}{0}{-1}{-1}{0}
\end{bmatrix} $
& $\begin{bmatrix}
- \hat{\mathbf{d}}_x \\
- \hat{\mathbf{d}}_y
+ d_x \\
+ d_y
\end{bmatrix}$
& $\begin{bmatrix}
\Delta_{p_x} \\
\Delta_{p_y}
- \end{bmatrix} \hat{\mathbf{d}}_z $ \\[3em]
+ \end{bmatrix} d_z $ \\[3em]
\sym E 2 &
$\left[
\begin{aligned}
@@ -339,16 +339,16 @@ presented in the following.
\begin{bmatrix}
\Delta_{d_{xy}} \\
\Delta_{d_{x^-y^2}}
- \end{bmatrix} \hat{\mathbf{d}}_0 \\
+ \end{bmatrix} d_0 \\
\Delta_f \begin{bmatrix}
- \hat{\mathbf{d}}_x \\
- \hat{\mathbf{d}}_y
+ d_x \\
+ d_y
\end{bmatrix} \\
\begin{bmatrix}
- \Delta_{p_y} \hat{\mathbf{d}}_x
- - \Delta_{p_x} \hat{\mathbf{d}}_y \\
- \Delta_{p_x} \hat{\mathbf{d}}_x
- + \Delta_{p_y} \hat{\mathbf{d}}_y
+ \Delta_{p_y} d_x
+ - \Delta_{p_x} d_y \\
+ \Delta_{p_x} d_x
+ + \Delta_{p_y} d_y
\end{bmatrix}
\end{gathered}$
\hspace{0.5cm}
@@ -360,7 +360,7 @@ presented in the following.
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 $\hat{\mathbf{d}}$. The fourth column is the total
+ vector formulation $d$. The fourth column is the total
spatial and two-spin basis function, calculated as decribed in the text.}
\label{tab:irrep-basis-fns}
\end{table}
@@ -380,39 +380,39 @@ of superconducting states can be classified by matching the spatial pairing
with one of the spatial irrep basis functions. A breakdown of these functions
shown in the second coloumn of Table \ref{tab:irrep-basis-fns}.
-When the $SU(2)$ symmetry of the spins is broken by the introduction of
-spin-oribtal coupling, the transformation behaviour of
-the spins under symmetry group actions can no longer be ignored\,\cite{kaba2019}.
-Instead, we aim to build an irrep basis by understanding the symmetry of the
-spatial component of our state in combination with the two spins of our
-superconducting states, \textit{i.e.} how the whole state transforms under
-group symmetry actions.
-
-To construct the irrep basis functions for the overall states, we first consider
-how just the two spins transform under the point symmetry group $C_{6v}$. The details on how
-the two-spin irrep basis functions were calculated can be found in
-Appendix\,\ref{app:spin-irreps}, and the functions themselves are listed in the
+When the $SU(2)$ symmetry of the spins is broken by the introduction of
+spin-oribtal coupling, the transformation behaviour of
+the spins under symmetry group actions can no longer be ignored\,\cite{kaba2019}.
+Instead, we aim to build an irrep basis by understanding the symmetry of the
+spatial component of our state in combination with the two spins of our
+superconducting states, \textit{i.e.} how the whole state transforms under
+group symmetry actions.
+
+To construct the irrep basis functions for the overall states, we first consider
+how just the two spins transform under the point symmetry group $C_{6v}$. The details on how
+the two-spin irrep basis functions were calculated can be found in
+Appendix\,\ref{app:spin-irreps}, and the functions themselves are listed in the
third column of Table \ref{tab:irrep-basis-fns} .
The overall basis function irreps can then be constructed from the tensor product of
the two-spin and spatial irreps. We only take those products which give us physical
states, \textit{i.e.} those which are antisymmetric under exchange of all quantum indices.
-For example, $E_2 \otimes E_1$ would describe spatially even ($d$-wave)
-spin-triplet states, which are not physical.
-For all but one of the possible physical products, at least one of the two
-irreps going into the tensor product is one-dimensional,
-meaning we can simply multiply the group characters of the two intial irreps, and
-the resulting set of characters will tell us what the resulting irrep is.
+For example, $E_2 \otimes E_1$ would describe spatially even ($d$-wave)
+spin-triplet states, which are not physical.
+For all but one of the possible physical products, at least one of the two
+irreps going into the tensor product is one-dimensional,
+meaning we can simply multiply the group characters of the two intial irreps, and
+the resulting set of characters will tell us what the resulting irrep is.
The one special case, $\sym E 1 \otimes \sym E 1$, requires a further decomposition.
-This can be done by either forming irrep projection operators for the total
-representation, as described in Appendix \ref{app:spin-irreps}, or by a group
+This can be done by either forming irrep projection operators for the total
+representation, as described in Appendix \ref{app:spin-irreps}, or by a group
theoretic calculation exploiting the orthogonality of the group characters \todo{source?}.
The results is the following decomponsition:
\begin{equation}
\sym E 1 \otimes \sym E 1 = \sym A 1 \oplus \sym B 2 \oplus \sym E 2 \, .
\end{equation}
-All of the basis functions of the combined spin-momentum system are given in
+All of the basis functions of the combined spin-momentum system are given in
fourth column of Table \ref{tab:irrep-basis-fns}.
@@ -639,18 +639,18 @@ which we will take to be the mirror w.r.t. the $y$-axis:
\, .
\end{equation}
-Given the symmetry group representation acting on a single spin $S(C_{6v})$, the
-representation for two spins is simply the tensor product $S(C_{6v}) \otimes S(C_{6v})$.
-The irreps are then constructed from the group symmetry matrices by calculating
-the projection operators\,\cite{kaba2019}.
+Given the symmetry group representation acting on a single spin $S(C_{6v})$, the
+representation for two spins is simply the tensor product $S(C_{6v}) \otimes S(C_{6v})$.
+The irreps are then constructed from the group symmetry matrices by calculating
+the projection operators\,\cite{kaba2019}.
For an irrep $\mu$ of a general discrete group $G$, this is given by formula
\begin{equation}
P^{(\mu)} = \sum_g \frac{d_\mu}{|G|} \chi^* (g) S(g)
\end{equation}
-where $\chi(g)$ is the character of group element $g$, $|G|$ is the order of the
+where $\chi(g)$ is the character of group element $g$, $|G|$ is the order of the
group (which is 12 for $C_{6v}$), and $d_\mu$ is the dimension of the irrep $\mu$.
-The basis functions of the irrep are then the non-zero eigenvectors of the projection
-operators. For the two spins, the calculated basis functions are
+The basis functions of the irrep are then the non-zero eigenvectors of the projection
+operators. For the two spins, the calculated basis functions are
listed in Table\,\ref{tab:spin-irreps}.
\bibliographystyle{apsrev4-2}