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Jacob Beyer
Triangular Rashba
Commits
f574b67a
Commit
f574b67a
authored
1 year ago
by
Jacob Beyer
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f574b67a
...
@@ -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
_{
6
v
}
)
$
.
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
_{
6
v
}
)
$
.
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
_{
6
v
}
)
$
, the
Given the symmetry group representation acting on a single spin
$
S
(
C
_{
6
v
}
)
$
, the
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