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Commit f574b67a authored by Jacob Beyer's avatar Jacob Beyer
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...@@ -5,6 +5,7 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2} ...@@ -5,6 +5,7 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2}
\usepackage[version=4]{mhchem} \usepackage[version=4]{mhchem}
\usepackage[dvipsnames]{xcolor} \usepackage[dvipsnames]{xcolor}
\usepackage{bbold}
\definecolor{first}{RGB}{0,153,230} \definecolor{first}{RGB}{0,153,230}
\definecolor{second}{RGB}{40,37,110} \definecolor{second}{RGB}{40,37,110}
...@@ -150,6 +151,13 @@ studies\,\cite{wolf2018,wolf2022a}, we treat the interacting part ...@@ -150,6 +151,13 @@ studies\,\cite{wolf2018,wolf2022a}, we treat the interacting part
$H_\mathrm{int}$ with the functional renormalization group (FRG), a short $H_\mathrm{int}$ with the functional renormalization group (FRG), a short
introduction is given below. introduction is given below.
Previous FRG studies including Rashba-SOC have been conducted on the
square-lattice Rashba Hubbard model in Ref.~\onlinecite{beyer2023}, on the
triangular lattice with attractive electron electron interactions in
Ref.~\onlinecite{schober2016}.
Studies for SOC of non-Rashba type have been conducted on twisted bilayer
\ce{PtSe2} in Ref.~\onlinecite{klebl2022a}.
\section{Model and Method} \section{Model and Method}
\label{sec:modelmethod} \label{sec:modelmethod}
...@@ -214,7 +222,6 @@ neighbor density-density interaction according to ...@@ -214,7 +222,6 @@ neighbor density-density interaction according to
\end{equation} \end{equation}
where $\bar \sigma$ is the opposing spin to $\sigma$. where $\bar \sigma$ is the opposing spin to $\sigma$.
\subsection{Truncated Unity Functional Renormalization Group} \subsection{Truncated Unity Functional Renormalization Group}
In order to properly resolve the different possible particle-particle and In order to properly resolve the different possible particle-particle and
particle-hole instabilities of the triangular lattice Rashba-Hubbard model it particle-hole instabilities of the triangular lattice Rashba-Hubbard model it
...@@ -224,12 +231,6 @@ we choose the functional renormalization group (FRG) for its wide applicability ...@@ -224,12 +231,6 @@ we choose the functional renormalization group (FRG) for its wide applicability
and advantageous numerical scaling. and advantageous numerical scaling.
Furthermore, the results of FRG can trivially exhibit singlet-triplet mixing, Furthermore, the results of FRG can trivially exhibit singlet-triplet mixing,
which we expect to find in SOC systems. which we expect to find in SOC systems.
Previous FRG studies including Rashba-SOC have been conducted on the
square-lattice Rashba Hubbard model in Ref.~\onlinecite{beyer2023}, on the
triangular lattice with attractive electron electron interactions in
Ref.~\onlinecite{schober2016}.
Studies for SOC of non-Rashba type have been conducted on twisted bilayer
\ce{PtSe2} in Ref.~\onlinecite{klebl2022a}.
\begin{figure} \begin{figure}
\centering \centering
...@@ -273,14 +274,16 @@ Ref.\,\onlinecite{beyer2022a}. ...@@ -273,14 +274,16 @@ Ref.\,\onlinecite{beyer2022a}.
\subsection{Analysis of Results} \subsection{Analysis of Results}
\subsubsection{Particle-particle instabilities} \subsubsection{Particle-particle instabilities}
To accurately determine the nature of particle-particle instabilities we To determine the nature of particle-particle instabilities we calculate the
calculate the truncated unity susceptibility as previously described in P-channel truncated unity susceptibility as previously described in
Ref.~\cite{beyer2022a,klebl2022a}. Ref.~\cite{beyer2022a,klebl2022a}.
The obtained eigenstate we decompose into singlet-like ($\Psi$) and The obtained eigenstate we decompose into singlet-like ($\Psi$) and
triplet-like ($d_x, d_y, d_z$) components, calculating the magnitude the triplet-like ($d_x, d_y, d_z$) components, calculating the magnitude the
respective subspaces. respective subspaces according to
\todo{I think the d basis should be defined explicitly here, \begin{equation}
heavily used after this point.} v = [\Psi(\bvec k) \mathbb{1} + \bvec d(\bvec k)
\cdot \hat{\bvec{\sigma}}] i \hat \sigma_y \, .
\end{equation}
Notably, at finite Rashba-SOC the eigenstate is expected to have finite weight Notably, at finite Rashba-SOC the eigenstate is expected to have finite weight
in both subspaces, due to the singlet-triplet mixing. in both subspaces, due to the singlet-triplet mixing.
...@@ -624,21 +627,28 @@ effect of the Rashba SOC. ...@@ -624,21 +627,28 @@ effect of the Rashba SOC.
\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 constructing To get the irreducible representations of the two spins, we begin by
the symmetry group representation of for a single spin, which we denote as $S(C_{6v})$. constructing the symmetry group representation of for a single spin, which we
The representation of the whole group can be constructed from the two generators, denote as $S(C_{6v})$. The representation of the whole group can be
which are one of the $C_6$ rotations: constructed from the two generators, which are one of the $C_6$ rotations:
\begin{equation} \begin{equation}
S(C_6) = \exp \bigg( \frac{-i \pi \hat{\tau}_z}{6} \bigg) = S(C_6) = \exp \bigg( \frac{-i \pi \hat{\sigma}_z}{6} \bigg)
=
\begin{pmatrix} \begin{pmatrix}
\sqrt{3} - i & 0 \\ \sqrt{3} - i & 0 \\
0 & \sqrt{3} + i 0 & \sqrt{3} + i
\end{pmatrix} , \end{pmatrix} ,
\end{equation} \end{equation}
where $\tau_z$ is the third Pauli matrix, as well as a reflection $\sigma_1$, where $\hat\sigma_z$ is the third Pauli matrix, as well as a reflection $m_y$,
which we will take to be the reflection in the $y$-axis: which we will take to be the mirror w.r.t. the $y$-axis:
\begin{equation} \begin{equation}
S(\sigma_1) = i \hat{\tau}_x \, . S(m_y) = i \hat{\sigma}_x
=
\begin{pmatrix}
0 & i \\
i & 0
\end{pmatrix}
\, .
\end{equation} \end{equation}
Given the symmetry group representation acting on a single spin $S(C_{6v})$, the Given the symmetry group representation acting on a single spin $S(C_{6v})$, the
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