Edits irrep section

main.tex

@@ -311,7 +311,7 @@ The allowed combinations can be obtained by a careful symmetry analysis.
 %         $E_1$  & $[\hat{\mathbf{d}}_x, \hat{\mathbf{d}}_y]$ \\ \hline
 %     \end{tabular}
 %     \caption{
-%         \captiontitle{Two spin basis functions of the irreducible 
+%         \captiontitle{two-spin basis functions of the irreducible 
 %         representations of \texorpdfstring{$C_{6v}$}{C6v}.} The basis 
 %         functions are naturally expressed in terms of the $d$-vector terminology 
 %         developed for to describe superconducting pairing. 
@@ -332,51 +332,57 @@ The allowed combinations can be obtained by a careful symmetry analysis.
 %     e^{-i \bvec a_2 \cdot \bvec k})
 % \end{equation}
 
-\todo{This needs more work}
-
 As mentioned above, when analysing the results of the FRG flow, we classify the 
-superconducting instability that diverges in 
-our flow by its symmetry behaviour\,\cite{sigrist1987}. The divergent eigenstate 
-will belong to one of the irreducible representations, or irreps, of the point
-symmetry group of the lattice. This is easiest done in systems where the $SU(2)$
-symmetry of the spins remains preserved, as we may then ignore the symmetric spin 
-degree of freedom of our eigenstate, and the irrep of the entire state can be fully 
-understood by symmetry of the momentum space pairing. For example, the triangular 
-lattice has a point symmetry group $C_{6v}$, and the irrep breakdown of superconducting
-states can be classified by matching the momentum space pairing with one of the 
-spatial irrep. A breakdown of these momentum space irreps shown in Table 
-\ref{tab:mom-irreps} .
-
-When the $SU(2)$ symmetry of the spins is broken by spin-oribtal coupling, 
-the representation of 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 both the momentum and the two spins of our 
-superconducting states transform under the group symmetry actions, both shown in
-Tables \ref{tab:mom-irreps} \& \ref{tab:spin-irreps}. The details on how the two
-spin irrep basis functions were calculated can be found in Appendix\,\ref{app:spin-irreps}.
-
-The overall basis function irreps can be constructed from the tensor product of
-the two spin and momentum irreps. We only take those products which give us physical
+divergent superconducting instability by its symmetry behaviour\,\cite{sigrist1987}. 
+\todo{better citation for symmetry analysis in FRG?}
+The divergent eigenstate will belong to one of the irreducible representations, 
+or irreps, of the point symmetry group of the lattice. 
+This is easiest done in systems where the $SU(2)$ symmetry of the spins remains preserved, 
+as we may then ignore the symmetric spin degree of freedom of our eigenstate, 
+and the irrep of the entire state can be fully understood by symmetry of the 
+spatial pairing, \textit{i.e.} by analysing the symmetry of the real space bond 
+order or the momentum space pairing.
+For example, the triangular lattice has a point symmetry group $C_{6v}$, and the 
+irrep breakdown 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 
+first coloumn of Table \ref{tab:mom-irreps} .
+
+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 
+second column of Table \ref{tab:mom-irreps} .
+
+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 even-momentum spin-triplet states, which 
-are not physical. Out of all but one of the possible physical products, at 
-least one of the two irreps going into the tensor product is one-dimensional, 
+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 of $E_1 \otimes E_1$ requires a further decomposition.
+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 
 theoretic calculation exploiting the orthogonality of the group characters \todo{source?}.
 The results is the following decomponsition:
 \begin{equation}
-    E_1 \otimes E_1 = A_1 \oplus B_2 \oplus E_2 \, .
+    \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 Table 
 \ref{tab:total-irreps}.
 
-
-
 \begin{table}
     \centering
 \begin{tabular}{cccc} \toprule
@@ -403,6 +409,7 @@ All of the basis functions of the combined spin-momentum system are given in Tab
     \label{tab:total-irreps}
 \end{table}
 
+
 \section{Results for the nearest-neighbor model}
 \label{sec:nn_results}