diff --git a/main.tex b/main.tex index e73065d2e8a7ef4efc795d922abee8a002f4b04d..78718754ea2f13b71b37d17717893de25c266ff7 100644 --- a/main.tex +++ b/main.tex @@ -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 Ref.\,\onlinecite{beyer2022a}. -\subsection{Analysis of Results} - -\subsubsection{Particle-particle instabilities} +\subsection{Singlet-Triplet decomposition} To determine the nature of particle-particle instabilities we calculate the P-channel truncated unity susceptibility as previously described in Ref.~\cite{beyer2022a,klebl2022a}. @@ -295,9 +293,8 @@ the respective spin's irrep. A careful symmetry analysis in conjunction with the possible combinations is presented in the following. -\todo{Helical state as limit} -\subsubsection{Irreducible Representations under Rashba-SOC} +\subsection{Irreducible Representations under Rashba-SOC} \begin{table} \centering @@ -312,20 +309,23 @@ presented in the following. \sym A 2 & --- & $d_z$ & $\Delta_{p_y} d_x + \Delta_{p_x} d_y$\\ \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 & + \hspace{0.15cm} % Fix spacing in the table $\begin{bmatrix} \numberedHexagon{$\Delta_{p_x}$}{1}{-1}{-2}{-1}{1}{2} \\ \numberedHexagon{$\Delta_{p_y}$}{1}{1}{0}{-1}{-1}{0} \end{bmatrix} $ - & $\begin{bmatrix} + \hspace{0.15cm} % Fix spacing in the table + & + $\begin{bmatrix} d_x \\ d_y \end{bmatrix}$ & $\begin{bmatrix} \Delta_{p_x} \\ \Delta_{p_y} - \end{bmatrix} d_z $ \\[3em] + \end{bmatrix} d_z $ \\[6em] \sym E 2 & $\left[ \begin{aligned} @@ -352,7 +352,7 @@ presented in the following. \end{bmatrix} \end{gathered}$ \hspace{0.5cm} - \\[3em] \hline + \\[6em] \hline \end{tabular} \caption{ \captiontitle{Basis functions of irreducible representations} @@ -362,59 +362,50 @@ presented in the following. two-spin basis function, in terms of the typical superconducting psuedo- vector formulation $d$. The fourth column is the total spatial and two-spin basis function, calculated as decribed in the text.} - \label{tab:irrep-basis-fns} + \label{tab:irrep_basis_fns} \end{table} -As mentioned above, when analysing the results of the FRG flow, we classify the -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 second coloumn of Table \ref{tab:irrep-basis-fns}. - -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 -third column of Table \ref{tab:irrep-basis-fns} . - -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. +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 +in \cref{tab:irrep_basis_fns}, second column. + +If the \su 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 +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. +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. +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 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: +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?}. +Either results in: \begin{equation} \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 -fourth column of Table \ref{tab:irrep-basis-fns}. +We show all possible basis functions for each irrep in +\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} @@ -613,7 +604,7 @@ effect of the Rashba SOC. \appendix \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 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 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 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} \bibliography{references}