Editing irrep section

main.tex

@@ -44,7 +44,7 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2}
     \node[shape=circle,draw,inner sep=1pt] (char) {#1};}}
 \MakeRobustCommand\circled
 
-\newcommand{\su}{SU}
+\newcommand{\su}[1]{SU(#1)}
 
 \newcommand{\bvec}[1]{\boldsymbol #1}
 \newcommand{\Ucrit}{U_\mathrm{c}}
@@ -301,9 +301,9 @@ 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}
-                & $d_0$
+                & $\Psi$
                 &  $\begin{gathered}
-                    \Delta_{s} d_0 \\
+                    \Delta_{s} \Psi \\
                     \Delta_{p_x} d_x  - \Delta_{p_y} d_y
                    \end{gathered}$ \\
     \sym A 2    & ---  & $d_z$
@@ -339,7 +339,7 @@ presented in the following.
             \begin{bmatrix}
                 \Delta_{d_{xy}} \\
                 \Delta_{d_{x^-y^2}}
-            \end{bmatrix} d_0 \\
+            \end{bmatrix} \Psi \\
             \Delta_f \begin{bmatrix}
                 d_x \\
                 d_y
@@ -360,43 +360,45 @@ 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 $d$. The fourth column is the total
+        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_basis_fns}
 \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 state is a spin singlet or \su, this can be identified by matching the
-spacial symmetry behavior with one of the spacial irrep basis functions shown
+When the Hamiltonian has an \su 2 spin symmetry, the point symmetry irreps
+can be identified by matching the symmetry behavior of the spatial pairing with
+one of the spatial irrep basis functions shown
 in \cref{tab:irrep_basis_fns}, second column.
 
-If the \su symmetry is broken by the introduction of SOC, the transformation
+If the \su 2 symmetry is broken by the introduction of SOC, the transformation
 behaviour of the spins under symmetry group operations must be
 considered\,\cite{kaba2019}.
-Here, the total irrep can be understood only as a combination of
-the spatial symmetry and the symmetry behavior of the two spins, \textit{i.e.}
-the total symmetry transformation behavior.
-To construct appropriate basis functions, we first consider the transformation
+Here, the irrep basis functions of the total superconducting state, hereafter
+referred to as the \textit{total irrep}, can be understood only as a combination of
+the spatial pairing symmetry and the symmetry behavior of the two spins,
+\textit{i.e.} the total symmetry transformation behavior.
+To construct these basis functions, we first consider the transformation
 behavior of a pair of spins under \sym C {6v}. The details of this calculation
 can be found in \cref{app:spin_irreps}, the resulting functions are given in
 \cref{tab:irrep_basis_fns}, third column.
 
-The total irreps is constructed as the tensor product of the spin-pair and
-spatial irreps.
+The total irreps basis function are constructed as the tensor product of the
+spin-pair and spatial pairing irreps.
 We retain only the products that yield physical states, \textit{i.e.} those
-which are antisymmetric under exchange of all quantum indices ($E_2 \otimes
-E_1$ would yield spacially even spin-triplet pairing and is discarded).
-Except for the case $\sym E 1 \otimes \sym E 1$, \todo{what about $E2
-\otimes E2$?} either the spacial or the spin-pair space is one-dimensional.
+which are antisymmetric under exchange of all quantum indices (\sym E 2 $\otimes$
+\sym E 1 would yield spatially even spin-triplet pairing and is discarded).
+Except for the case $\sym E 1 \otimes \sym E 1$, either the spacial or the
+spin-pair space is one-dimensional.
 In those cases, multiplication of the irreps' group characters yields the
 total state's characters, and thus the total irrep.
 
 The $\sym E 1 \otimes \sym E 1$ case requires a further decomposition.
 This can be done by either forming irrep projection operators for the total
-representation, as described in \cref{app:spin_irreps} \todo{Is this actually
-in here?}, or by a group theory calculation exploiting the orthogonality of
-the group characters \todo{source?}.
+representation, in the same way as described for the spin-pair representation
+described in \cref{app:spin_irreps}, or by a group theory calculation exploiting
+the orthogonality of the group characters \todo{source?}.
 Either results in:
 \begin{equation}
     \sym E 1 \otimes \sym E 1 = \sym A 1 \oplus \sym B 2 \oplus \sym E 2 \, .