diff --git a/main.tex b/main.tex
index a19f9a3b2a341461a2883521cb91be757776d377..74471c734539d9ed6e8ee33fac53a534e31185ca 100644
--- a/main.tex
+++ b/main.tex
@@ -327,7 +327,7 @@ 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.
@@ -348,51 +348,57 @@ 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
@@ -419,6 +425,7 @@ 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}