Merge branch 'master' of rwth_git:jacob.beyer/triangular_rashba

main.tex

@@ -387,14 +387,15 @@ presented in the following.
 
 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
-can be identified by matching the symmetry behavior of the spatial pairing with
-one of the spatial irrep basis functions shown
+When the Hamiltonian has an \su spin rotation symmetry, the spin component of
+of the state decouples from the spatial pairing, and the point symmetry irreps
+can be solely 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 2 symmetry is broken by the introduction of SOC, the transformation
-behaviour of the spins under symmetry group operations must be
-considered\,\cite{kaba2019}.
+If the \su symmetry is broken by the introduction of SOC, the spins are `frozen'
+into the lattice, and must transform under the point symmetry group operations
+\,\cite{sigrist1987, kaba2019}.
 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,
@@ -420,8 +421,8 @@ 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 \, .
+\begin{equation} \label{eqn:irrep-breakdown}
+    \sym E 1 \otimes \sym E 1 = \sym A 1 \oplus \sym A 2 \oplus \sym E 2 \, .
 \end{equation}
 We show all possible basis functions for each irrep in
 \cref{tab:irrep_basis_fns}, fourth column.
@@ -434,25 +435,26 @@ We show all possible basis functions for each irrep in
 \label{sec:nn_results}
 
 We analyze the simplest case of the nearest neighbor Rashba Hubbard model on
-the triangular lattice, i.e., $t_1=1, t_{n \neq 1}=0, \alpha_1=\alpha,
-\alpha_{n \neq 1}=0$ where we choose the -- rather strong -- interaction $U_0 =
+the triangular lattice, i.e., $t_1=1, \, t_{n \neq 1}=0, \, \alpha_1=\alpha,
+\, \alpha_{n \neq 1}=0$ where we choose the -- rather strong -- interaction $U_0 =
 8, U_{n \neq 0}=0$.
-By virtue of FRG we are able to resolve both particle-particle and
+By the virtue of FRG we are able to resolve both particle-particle and
 particle-hole phases in an unbiased way.
 We therefore provide the phase diagram in \cref{fig:phases}, showing the
 critical scale as well as phase transition.
 We postpone analysis of the particle-hole instabilities, noting only that the
 dominant instabilities are spin-density waves (SDW).
-As already shown in Ref.~\onlinecite{beyer2022a}, the admixing of charge
+As previously shown in Ref.~\onlinecite{beyer2022a}, the admixing of charge
 density waves into SDWs is prohibited by symmetry in the absence of relative
-momentum dependencies in the eigenstate.
+momentum dependencies in the eigenstate. \todo{I don't understand what this
+sentence is trying to tell me.}
 
 The frayed boundaries between the instabilities are a product of the finite
 resolution of the calculations. As shown on the right side of
 \cref{fig:phases}, the points of seeming inconsistencies are the onsets of new
 pockets.
-The accurate capture of these features of vanishing size is impossible with the
-resolution of calculation.
+These features are vanishing in size, making capturing them impossible
+with the resolution of calculation.
 
 We therefore disregard these points and focus our attention instead on the
 extended superconducting regions around half-filling and around $\nu=0.2,
@@ -493,6 +495,8 @@ the formfactors irreducible representation of the singlet subspace.
 With the singlet corresponding to \sym A 1, the total irreducible
 representation will equal that of the momentum component.
 We thus identify the \sym E 2 irreducible representation.
+\todo{I think it would be cool to talk about the form of the triplet
+component that you see, i.e. which E2 basis function is it?}
 
 Because we would naively expect singlet-triplet mixing under SOC, it is
 surprising that the -- initially triplet -- superconducting
@@ -500,23 +504,22 @@ region at low filling we find negligible singlet-like contributions up to
 $\alpha=0.4$ (where a phase transition to a SDW occurs).
 Determining the irreducible representation is more involved as it necessitates
 a decomposition of the spin-component as presented above.
-For the case of $\alpha=0$, we find \todo{Matt}.
-When we now add SOC, the spin-degeneracy is broken and the system enters the
-$d_x, d_y$ spin subspaces, which follow the $E_1$ irreducible representation.
-
-If we determine the momentum irreducible representation to be $E_1$ as well, we
-are left with a total space of possibilities following
-\begin{equation}
-    \sym A 1 \oplus \sym B 2 \oplus \sym E 2\,.
-\end{equation}
+For the case of $\alpha=0$ --- no SOC --- \su 2 spin rotation symmetry enforces
+the degeneracy of the the spin triplet states, which are decoupled from the point
+symmetry group of the crystal field\,\cite{sigrist1987}.
+\todo{the state we get is...}
+
+When we now add SOC, the overall symmetry of the Hamiltonian is reduced and the
+spin-triplet state degeneracy is broken. The resulting superconducting state is
+doubly degenerate, fixed in the $d_x, d_y$ spin subspaces with $p$-wave spatial
+pairing, as shown on the right hand side of \cref{fig:sing}.
+This is however not enough to identify the irrep of the state, as we are left
+with a total space of possibilities in \cref{eqn:irrep-breakdown}.
 In our case, this is disambiguated by the degeneracy of the eigenstates, i.e.,
 because we find a pair of eigenstates we must be in the \sym E 2 irreducible
-representation's subspace.
-For other disambiguations one could have considered the transformation behavior
-of the eigenstates of the system as shown in the right side of \cref{fig:sing}.
-Their transformation behavior clearly does not follow \sym A 1 or \sym B 2
-representations, we therefore conclude that we remain in the \sym E 2
-irreducible representation.
+representation's subspace. This can be confirmed by identifying the state
+plotted on the right hand side of figure \cref{fig:sing} with the basis
+functions as listed in Table\,\ref{tab:irrep_combinations}.
 
 \begin{figure*}
     \includegraphics{plots/sc_inst.pdf}