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Commit 28a5a1d0 authored by Jacob Beyer's avatar Jacob Beyer
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...@@ -271,9 +271,7 @@ capture spin-momentum locking. For a more complete technical discussion of the ...@@ -271,9 +271,7 @@ capture spin-momentum locking. For a more complete technical discussion of the
approximations and numerical implementations we refer the reader to approximations and numerical implementations we refer the reader to
Ref.\,\onlinecite{beyer2022a}. Ref.\,\onlinecite{beyer2022a}.
\subsection{Analysis of Results} \subsection{Singlet-Triplet decomposition}
\subsubsection{Particle-particle instabilities}
To determine the nature of particle-particle instabilities we calculate the To determine the nature of particle-particle instabilities we calculate the
P-channel truncated unity susceptibility as previously described in P-channel truncated unity susceptibility as previously described in
Ref.~\cite{beyer2022a,klebl2022a}. Ref.~\cite{beyer2022a,klebl2022a}.
...@@ -295,9 +293,8 @@ the respective spin's irrep. ...@@ -295,9 +293,8 @@ the respective spin's irrep.
A careful symmetry analysis in conjunction with the possible combinations is A careful symmetry analysis in conjunction with the possible combinations is
presented in the following. presented in the following.
\todo{Helical state as limit}
\subsubsection{Irreducible Representations under Rashba-SOC} \subsection{Irreducible Representations under Rashba-SOC}
\begin{table} \begin{table}
\centering \centering
...@@ -312,20 +309,23 @@ presented in the following. ...@@ -312,20 +309,23 @@ presented in the following.
\sym A 2 & --- & $d_z$ \sym A 2 & --- & $d_z$
& $\Delta_{p_y} d_x + \Delta_{p_x} d_y$\\ & $\Delta_{p_y} d_x + \Delta_{p_x} d_y$\\
\sym B 1 & \numberedHexagon{$\Delta_f$}{-1}{1}{-1}{1}{-1}{1} & --- & --- \\ \sym B 1 & \numberedHexagon{$\Delta_f$}{-1}{1}{-1}{1}{-1}{1} & --- & --- \\
\sym B 2 & --- & --- & $\Delta_{f} d_z$ \\ \hline \sym B 2 & --- & --- & $\Delta_{f} d_z$ \\[0.5em] \hline \\
\sym E 1 & \sym E 1 &
\hspace{0.15cm} % Fix spacing in the table
$\begin{bmatrix} $\begin{bmatrix}
\numberedHexagon{$\Delta_{p_x}$}{1}{-1}{-2}{-1}{1}{2} \\ \numberedHexagon{$\Delta_{p_x}$}{1}{-1}{-2}{-1}{1}{2} \\
\numberedHexagon{$\Delta_{p_y}$}{1}{1}{0}{-1}{-1}{0} \numberedHexagon{$\Delta_{p_y}$}{1}{1}{0}{-1}{-1}{0}
\end{bmatrix} $ \end{bmatrix} $
& $\begin{bmatrix} \hspace{0.15cm} % Fix spacing in the table
&
$\begin{bmatrix}
d_x \\ d_x \\
d_y d_y
\end{bmatrix}$ \end{bmatrix}$
& $\begin{bmatrix} & $\begin{bmatrix}
\Delta_{p_x} \\ \Delta_{p_x} \\
\Delta_{p_y} \Delta_{p_y}
\end{bmatrix} d_z $ \\[3em] \end{bmatrix} d_z $ \\[6em]
\sym E 2 & \sym E 2 &
$\left[ $\left[
\begin{aligned} \begin{aligned}
...@@ -352,7 +352,7 @@ presented in the following. ...@@ -352,7 +352,7 @@ presented in the following.
\end{bmatrix} \end{bmatrix}
\end{gathered}$ \end{gathered}$
\hspace{0.5cm} \hspace{0.5cm}
\\[3em] \hline \\[6em] \hline
\end{tabular} \end{tabular}
\caption{ \caption{
\captiontitle{Basis functions of irreducible representations} \captiontitle{Basis functions of irreducible representations}
...@@ -362,59 +362,50 @@ presented in the following. ...@@ -362,59 +362,50 @@ presented in the following.
two-spin basis function, in terms of the typical superconducting psuedo- two-spin basis function, in terms of the typical superconducting psuedo-
vector formulation $d$. 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.} spatial and two-spin basis function, calculated as decribed in the text.}
\label{tab:irrep-basis-fns} \label{tab:irrep_basis_fns}
\end{table} \end{table}
As mentioned above, when analysing the results of the FRG flow, we classify the The divergent eigenstate will belong to one of the irreps of the point symmetry
divergent superconducting instability by its symmetry group of the lattice (\textit{i.e.} \sym C {6v}).
behaviour\,\cite{sigrist1987}. \todo{better citation for symmetry analysis in When the state is a spin singlet or \su, this can be identified by matching the
FRG?} The divergent eigenstate will belong to one of the irreducible spacial symmetry behavior with one of the spacial irrep basis functions shown
representations, or irreps, of the point symmetry group of the lattice. This in \cref{tab:irrep_basis_fns}, second column.
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 If the \su symmetry is broken by the introduction of SOC, the transformation
eigenstate, and the irrep of the entire state can be fully understood by behaviour of the spins under symmetry group operations must be
symmetry of the spatial pairing, \textit{i.e.} by analysing the symmetry of the considered\,\cite{kaba2019}.
real space bond order or the momentum space pairing. For example, the Here, the total irrep can be understood only as a combination of
triangular lattice has a point symmetry group $C_{6v}$, and the irrep breakdown the spatial symmetry and the symmetry behavior of the two spins, \textit{i.e.}
of superconducting states can be classified by matching the spatial pairing the total symmetry transformation behavior.
with one of the spatial irrep basis functions. A breakdown of these functions To construct appropriate basis functions, we first consider the transformation
shown in the second coloumn of Table \ref{tab:irrep-basis-fns}. 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
When the $SU(2)$ symmetry of the spins is broken by the introduction of \cref{tab:irrep_basis_fns}, third column.
spin-oribtal coupling, the transformation behaviour of
the spins under symmetry group actions can no longer be ignored\,\cite{kaba2019}. The total irreps is constructed as the tensor product of the spin-pair and
Instead, we aim to build an irrep basis by understanding the symmetry of the spatial irreps.
spatial component of our state in combination with the two spins of our We retain only the products that yield physical states, \textit{i.e.} those
superconducting states, \textit{i.e.} how the whole state transforms under which are antisymmetric under exchange of all quantum indices ($E_2 \otimes
group symmetry actions. E_1$ would yield spacially even spin-triplet pairing and is discarded).
Except for the case $\sym E 1 \otimes \sym E 1$, \todo{what about $E2
To construct the irrep basis functions for the overall states, we first consider \otimes E2$?} either the spacial or the spin-pair space is one-dimensional.
how just the two spins transform under the point symmetry group $C_{6v}$. The details on how In those cases, multiplication of the irreps' group characters yields the
the two-spin irrep basis functions were calculated can be found in total state's characters, and thus the total irrep.
Appendix\,\ref{app:spin-irreps}, and the functions themselves are listed in the
third column of Table \ref{tab:irrep-basis-fns} . The $\sym E 1 \otimes \sym E 1$ case requires a further decomposition.
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.
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 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 representation, as described in \cref{app:spin_irreps} \todo{Is this actually
theoretic calculation exploiting the orthogonality of the group characters \todo{source?}. in here?}, or by a group theory calculation exploiting the orthogonality of
The results is the following decomponsition: the group characters \todo{source?}.
Either results in:
\begin{equation} \begin{equation}
\sym E 1 \otimes \sym E 1 = \sym A 1 \oplus \sym B 2 \oplus \sym E 2 \, . \sym E 1 \otimes \sym E 1 = \sym A 1 \oplus \sym B 2 \oplus \sym E 2 \, .
\end{equation} \end{equation}
All of the basis functions of the combined spin-momentum system are given in We show all possible basis functions for each irrep in
fourth column of Table \ref{tab:irrep-basis-fns}. \cref{tab:irrep_basis_fns}, fourth column.
\subsection{Limit of strong spin-orbit coupling}
\todo{Helical state as limit}
\section{Results for the nearest-neighbor model} \section{Results for the nearest-neighbor model}
...@@ -613,7 +604,7 @@ effect of the Rashba SOC. ...@@ -613,7 +604,7 @@ effect of the Rashba SOC.
\appendix \appendix
\section{Irreducible representation basis functions in the spin representation} \section{Irreducible representation basis functions in the spin representation}
\label{app:spin-irreps} \label{app:spin_irreps}
To get the irreducible representations of the two spins, we begin by To get the irreducible representations of the two spins, we begin by
constructing the symmetry group representation of for a single spin, which we constructing the symmetry group representation of for a single spin, which we
...@@ -651,7 +642,7 @@ where $\chi(g)$ is the character of group element $g$, $|G|$ is the order of the ...@@ -651,7 +642,7 @@ 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 $\mu$. 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 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 operators. For the two spins, the calculated basis functions are
listed in Table\,\ref{tab:spin-irreps}. listed in Table\,\ref{tab:irrep_basis_fns}.
\bibliographystyle{apsrev4-2} \bibliographystyle{apsrev4-2}
\bibliography{references} \bibliography{references}
......
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