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Commit 04abfa68 authored by Jacob Beyer's avatar Jacob Beyer
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Stephan now really should look at it first

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......@@ -225,14 +225,14 @@ neighbor density-density interaction according to
where $\bar \sigma$ is the opposing spin to $\sigma$.
\subsection{Truncated Unity Functional Renormalization Group}
In order to properly resolve the different possible particle-particle and
particle-hole instabilities of the triangular lattice Rashba-Hubbard model it
is imperative to use an unbiased method.
From the small selection of unbiased methods (DMFRG, Monte-Carlo,...\citen),
we choose the functional renormalization group (FRG) for its wide applicability
and advantageous numerical scaling.
Furthermore, the results of FRG can trivially exhibit singlet-triplet mixing,
which we expect to find in SOC systems.
To ensure predictions of triplet superconductivity are not overridden by
particle-hole instabilities, it is imperative to allow their occurrence within
the theory.
From the small selection of methods providing this unbiased view (DMFRG,
Monte-Carlo,...\citen), we choose the functional renormalization group (FRG)
for its wide applicability and advantageous numerical scaling.
Importantly, the results of FRG allow for singlet-triplet mixing, expected in
SOC systems.
\begin{figure}
\centering
......@@ -251,15 +251,14 @@ which we expect to find in SOC systems.
\end{figure}
Thorough derivations of the FRG equations can be found in
Refs.\,\onlinecite{metzner2012,dupuis2021,platt2013,beyer2022a}
among others, here we only give the briefest overview of our chosen
approximations:
Refs.\,\onlinecite{metzner2012,dupuis2021,platt2013,beyer2022a} among others,
we only give the briefest overview of our chosen flavor of FRG:
The diagrammatic representation of our single-loop, non-\su{} flow-equation
is given in \cref{fig:diags}.
During the calculation we neglect higher loop orders, the flow equations for
the self-energy $\frac{\dd}{\dd\Lambda}\Gamma^{(2)}$ as well as the
three-particle interactions $\frac{\dd}{\dd\Lambda}\Gamma^{(6)}$, focusing on
the effective two-particle interaction $V$.
three-particle interactions $\frac{\dd}{\dd\Lambda}\Gamma^{(6)}$, focusing
solely on the effective two-particle interaction $V$.
As regulator for the two-point Greens function we choose the
$\Omega$-cutoff\,\cite{husemann2009}.
Moreover, we restrict ourselves to zero temperature $T = 0$ and static
......@@ -267,11 +266,11 @@ vertices.
To reduce computational complexity we employ the truncated unity extension to
the FRG\,\cite{lichtenstein2017,husemann2009,wang2012}.
Importantly, this retains momentum conservation at the vertices, broken in
$N$-patch schemes\,\cite{honerkamp2001}, but required to accurately
capture spin-momentum locking. For a more complete technical discussion of the
approximations and numerical implementations we refer the reader to
Ref.\,\onlinecite{beyer2022a}.
Importantly, this retains momentum conservation at the vertices -- broken in
$N$-patch schemes\,\cite{honerkamp2001} -- required to accurately
capture spin-momentum locking.
We compared and validated our calculations according to the procedures
presented in Ref.\,\onlinecite{beyer2022a}.
\subsection{Singlet-Triplet decomposition}
To determine the nature of particle-particle instabilities we calculate the
......@@ -288,6 +287,9 @@ While in the absence of Rashba-SOC we expect the weight to reside fully in
either the singlet or the triplet subspace, at finite SOC they are both
expected to be finite, with the exact fractions being the first characteristic
of the superconducting state.
In fact, \todo{finish} in the limit of strong SOC, we expect the state to be
confined to the helical basis, which is fully singlet-triplet mixed \citen.
The second important property is the state's total irreducible representation
(irrep), which must be equal in both spin spaces\,\cite{sigrist1987}.
This can be obtained from the simulated momentum's irrep in conjunction with
......@@ -608,7 +610,7 @@ effect of the Rashba SOC.
\begin{acknowledgements}
The German Research Foundation (DFG) is acknowledged for support through
RTG 1995, within the Priority Program SPP 2244 \enquote{2DMP}/ S.R.
RTG 1995, within the Priority Program SPP 2244 \enquote{2DMP}. S.R.
acknowledges support from the Australian Research Council (FT180100211 and
DP200101118).
The authors gratefully acknowledge the scientific support
......
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