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Jacob Beyer
Triangular Rashba
Commits
ea1bfd27
Commit
ea1bfd27
authored
1 year ago
by
Jacob Beyer
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Merge branch 'master' of rwth_git:jacob.beyer/triangular_rashba
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ea1bfd27
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@@ -79,13 +79,13 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2}
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...
@@ -298,20 +298,49 @@ The allowed combinations can be obtained by a careful symmetry analysis.
...
@@ -298,20 +298,49 @@ The allowed combinations can be obtained by a careful symmetry analysis.
\begin{table}
\begin{table}
\centering
\centering
\begin{tabular}
{
ccc
}
\toprule
\begin{tabular}
{
cccc
}
\toprule
{$
\mu
$}
&
Spatial
&
Spin
\\
\hline
{$
\mu
$}
&
Spatial
&
Spin
&
Total
\\
\hline
\sym
A 1
&
\numberedHexagon
{$
\Delta
_
s
$}{
1
}{
1
}{
1
}{
1
}{
1
}{
1
}
&
$
\hat
{
\mathbf
{
d
}}_
0
$
\\
\sym
A 1
&
\numberedHexagon
{$
\Delta
_
s
$}{
1
}{
1
}{
1
}{
1
}{
1
}{
1
}
\sym
A 2
&
-
&
$
\hat
{
\mathbf
{
d
}}_
z
$
\\
&
$
\hat
{
\mathbf
{
d
}}_
0
$
\sym
B 1
&
\numberedHexagon
{$
\Delta
_
f
$}{
-1
}{
1
}{
-1
}{
1
}{
-1
}{
1
}
&
\\
\hline
\\
&
$
\begin
{
gathered
}
\Delta
_{
s
}
\hat
{
\mathbf
{
d
}}_
0
\\
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
x
-
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
y
\end
{
gathered
}$
\\
\sym
A 2
&
-
&
$
\hat
{
\mathbf
{
d
}}_
z
$
&
$
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
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}}_
x
+
\Delta
_{
p
_
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}
\hat
{
\mathbf
{
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$
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B 1
&
\numberedHexagon
{$
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_
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$}{
-1
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-1
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1
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-1
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-
&
-
&
$
\Delta
_{
f
}
\hat
{
\mathbf
{
d
}}_
z
$
\\
\hline
\\
\sym
E 1
&
\sym
E 1
&
$
\left
[
\numberedHexagon
{$
\Delta
_{
p
_
x
}$}{
1
}{
-
1
}{
-
2
}{
-
1
}{
1
}{
2
}
$
\begin
{
bmatrix
}
\numberedHexagon
{$
\Delta
_{
p
_
y
}$}{
1
}{
1
}{
0
}{
-
1
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-
1
}{
0
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\right
]
$
\numberedHexagon
{$
\Delta
_{
p
_
x
}$}{
1
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-
1
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-
2
}{
-
1
}{
1
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2
}
\\
&
$
[
\hat
{
\mathbf
{
d
}}_
x,
\hat
{
\mathbf
{
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y
]
$
\\
[3em]
\numberedHexagon
{$
\Delta
_{
p
_
y
}$}{
1
}{
1
}{
0
}{
-
1
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-
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\end
{
bmatrix
}
$
&
$
\begin
{
bmatrix
}
\hat
{
\mathbf
{
d
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x
\\
\hat
{
\mathbf
{
d
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y
\end
{
bmatrix
}$
&
$
\begin
{
bmatrix
}
\Delta
_{
p
_
x
}
\\
\Delta
_{
p
_
y
}
\end
{
bmatrix
}$
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[3em]
\sym
E 2
&
\sym
E 2
&
$
\left
[
\numberedHexagon
{$
\Delta
_{
d
_{
x
^
2-y
^
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-
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[
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\numberedHexagon
{$
\Delta
_{
d
_{
x
^
2-y
^
2
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-
1
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-
1
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2
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-
1
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-
1
}{
2
}
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{$
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_{
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xy
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1
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{$
\Delta
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\right
]
$
\right
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$
&
\\
[3em]
\hline
&
-
&
$
\begin
{
gathered
}
[
\Delta
_{
d
_{
xy
}}
,
\Delta
_{
d
_{
x
^
-
y
^
2
}}
]
\hat
{
\mathbf
{
d
}}_
0
\\
\Delta
_
f
[
\hat
{
\mathbf
{
d
}}_
x,
\hat
{
\mathbf
{
d
}}_
y
]
\\
\begin
{
aligned
}
[
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
x
-
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
y,
\\
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
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x
+
\Delta
_{
p
_
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[3em]
\hline
\end{tabular}
\end{tabular}
\caption
{
\caption
{
\captiontitle
{
Basis functions of irreducible representations
}
\captiontitle
{
Basis functions of irreducible representations
}
...
@@ -321,6 +350,33 @@ The allowed combinations can be obtained by a careful symmetry analysis.
...
@@ -321,6 +350,33 @@ The allowed combinations can be obtained by a careful symmetry analysis.
\label
{
tab:mom-irreps
}
\label
{
tab:mom-irreps
}
\end{table}
\end{table}
\begin{table}
\centering
\begin{tabular}
{
cccc
}
\toprule
{$
\mu
$}
&
\multicolumn
{
3
}{
c
}{
Basis functions
}
\\
\hline
\sym
A 1
&
$
\Delta
_{
s
}
\hat
{
\mathbf
{
d
}}_
0
$
&
$
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
x
-
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
y
$
&
\\
\sym
A 2
&
&
$
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
x
+
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
y
$
&
\\
\sym
B 1
&
&
&
\\
\sym
B 2
&
$
\Delta
_{
f
}
\hat
{
\mathbf
{
d
}}_
z
$
&
&
\\
\hline
\sym
E 1
&
$
[
\Delta
_{
p
_
x
}
,
\Delta
_{
p
_
y
}
]
\hat
{
\mathbf
{
d
}}_
z
$
&
&
\\
\sym
E 2
&
$
[
\Delta
_{
d
_{
xy
}}
,
\Delta
_{
d
_{
x
^
-
y
^
2
}}
]
\hat
{
\mathbf
{
d
}}_
0
$
&
$
\Delta
_
f
[
\hat
{
\mathbf
{
d
}}_
x,
\hat
{
\mathbf
{
d
}}_
y
]
$
&
$
\begin
{
aligned
}
[
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
x
-
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
y,
\\
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
x
+
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
y
]
\end
{
aligned
}$
\\
\hline
\end{tabular}
\caption
{
\captiontitle
{
Basis functions of irreducible representation
}
We show the possible basis functions for two spins that are
antisymmetric under exchange of quantum indices.
}
\label
{
tab:total-irreps
}
\end{table}
% \begin{table}
% \begin{table}
% \centering
% \centering
% \begin{tabular}{ccc} \toprule
% \begin{tabular}{ccc} \toprule
...
@@ -402,31 +458,6 @@ The results is the following decomponsition:
...
@@ -402,31 +458,6 @@ The results is the following decomponsition:
All of the basis functions of the combined spin-momentum system are given in Table
All of the basis functions of the combined spin-momentum system are given in Table
\ref
{
tab:total-irreps
}
.
\ref
{
tab:total-irreps
}
.
\begin{table}
\centering
\begin{tabular}
{
cccc
}
\toprule
{$
\mu
$}
&
\multicolumn
{
3
}{
c
}{
Basis functions
}
\\
\hline
\sym
A 1
&
$
\Delta
_{
s
}
\hat
{
\mathbf
{
d
}}_
0
$
&
$
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
x
-
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
y
$
&
\\
\sym
A 2
&
&
$
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
x
+
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
y
$
&
\\
\sym
B 1
&
&
&
\\
\sym
B 2
&
$
\Delta
_{
f
}
\hat
{
\mathbf
{
d
}}_
z
$
&
&
\\
\hline
\sym
E 1
&
$
[
\Delta
_{
p
_
x
}
,
\Delta
_{
p
_
y
}
]
\hat
{
\mathbf
{
d
}}_
z
$
&
&
\\
\sym
E 2
&
$
[
\Delta
_{
d
_{
xy
}}
,
\Delta
_{
d
_{
x
^
-
y
^
2
}}
]
\hat
{
\mathbf
{
d
}}_
0
$
&
$
\Delta
_
f
[
\hat
{
\mathbf
{
d
}}_
x,
\hat
{
\mathbf
{
d
}}_
y
]
$
&
$
\begin
{
aligned
}
[
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
x
-
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
y,
\\
\Delta
_{
p
_
x
}
\hat
{
\mathbf
{
d
}}_
x
+
\Delta
_{
p
_
y
}
\hat
{
\mathbf
{
d
}}_
y
]
\end
{
aligned
}$
\\
\hline
\end{tabular}
\caption
{
\captiontitle
{
Basis functions of irreducible representation
}
We show the possible basis functions for two spins that are
antisymmetric under exchange of quantum indices.
}
\label
{
tab:total-irreps
}
\end{table}
\section
{
Results for the nearest-neighbor model
}
\section
{
Results for the nearest-neighbor model
}
...
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