Skip to content
GitLab
Explore
Sign in
Primary navigation
Search or go to…
Project
T
Triangular Rashba
Manage
Activity
Members
Labels
Plan
Issues
Issue boards
Milestones
Wiki
Requirements
Code
Merge requests
Repository
Branches
Commits
Tags
Repository graph
Compare revisions
Snippets
Locked files
Build
Pipelines
Jobs
Pipeline schedules
Test cases
Artifacts
Deploy
Releases
Package registry
Container registry
Model registry
Operate
Environments
Terraform modules
Monitor
Incidents
Service Desk
Analyze
Value stream analytics
Contributor analytics
CI/CD analytics
Repository analytics
Code review analytics
Issue analytics
Insights
Model experiments
Help
Help
Support
GitLab documentation
Compare GitLab plans
GitLab community forum
Contribute to GitLab
Provide feedback
Terms and privacy
Keyboard shortcuts
?
Snippets
Groups
Projects
Show more breadcrumbs
Jacob Beyer
Triangular Rashba
Commits
28a5a1d0
Commit
28a5a1d0
authored
1 year ago
by
Jacob Beyer
Browse files
Options
Downloads
Patches
Plain Diff
Reworked Matts Text
parent
8f7bd452
No related branches found
No related tags found
No related merge requests found
Changes
1
Show whitespace changes
Inline
Side-by-side
Showing
1 changed file
main.tex
+48
-57
48 additions, 57 deletions
main.tex
with
48 additions
and
57 deletions
main.tex
+
48
−
57
View file @
28a5a1d0
...
...
@@ -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
}
\subs
ubs
ection
{
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
$
\\
[
3
em]
\end
{
bmatrix
}
d
_
z
$
\\
[
6
em]
\sym
E 2
&
$
\left
[
\begin
{
aligned
}
...
...
@@ -352,7 +352,7 @@ presented in the following.
\end
{
bmatrix
}
\end
{
gathered
}$
\hspace
{
0.5cm
}
\\
[
3
em]
\hline
\\
[
6
em]
\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
_{
6
v
}$
, 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
_{
6
v
}$
. 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
$
E
2
\otimes
E
2
$
?
}
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
\
c
ref
{
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
_{
6
v
}$
), 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
_
fn
s
}
.
\bibliographystyle
{
apsrev4-2
}
\bibliography
{
references
}
...
...
This diff is collapsed.
Click to expand it.
Preview
0%
Loading
Try again
or
attach a new file
.
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Save comment
Cancel
Please
register
or
sign in
to comment