11institutetext: University of São Paulo, Institute of Science and Technology
11email<EMAIL_ADDRESS>
# Optimizing Diffusion Rate and Label Reliability in a Graph-based Semi-
supervised Classifier

Bruno Klaus de Aquino Afonso 11 0000-0003-2086-1054 Lilian Berton 11
0000-0003-1397-6005

###### Abstract

Semi-supervised learning has received attention from researchers, as it allows
one to exploit the structure of unlabeled data to achieve competitive
classification results with much fewer labels than supervised approaches. The
Local and Global Consistency (LGC) algorithm is one of the most well-known
graph-based semi-supervised (GSSL) classifiers. Notably, its solution can be
written as a linear combination of the known labels. The coefficients of this
linear combination depend on a parameter $\alpha$, determining the decay of
the reward over time when reaching labeled vertices in a random walk. In this
work, we discuss how removing the self-influence of a labeled instance may be
beneficial, and how it relates to leave-one-out error. Moreover, we propose to
minimize this leave-one-out loss with automatic differentiation. Within this
framework, we propose methods to estimate label reliability and diffusion
rate. Optimizing the diffusion rate is more efficiently accomplished with a
spectral representation. Results show that the label reliability approach
competes with robust $\ell_{1}$-norm methods and that removing diagonal
entries reduces the risk of overfitting and leads to suitable criteria for
parameter selection.††This study was financed in part by the Coordenação de
Aperfeiçoamento de Nível Superior - Brasil (CAPES) - Finance Code 001, and São
Paulo Research Foundation (FAPESP) grant #18/01722-3

###### Keywords:

Machine learning leave-one-out semi-supervised learning graph-based approaches
label propagation eigendecomposition.

##  1 Introduction

Machine learning (ML) is the subfield of Computer Science that aims to make a
computer learn from data [11]. The task we’ll be considering is the problem of
classification, which requires the prediction of a discrete label
corresponding to a class.

Before the data can be presented to an ML model, it must be represented in
some way. Our input is nothing more than a collection of $n$ examples. Each
object of this collection is called an instance. For most practical
applications, we may consider an instance to be a vector of $d$ dimensions.
Graph-based semi-supervised learning (GSSL) relies a lot on matrix
representations, so we will be representing the observed instances as an input
matrix

$\mathbf{X}\in\mathbb{R}^{n{\times}d}.$ (1)

A ML classifier should learn to map an input instance to the desired output.
To do this, it needs to know the labels (i.e. the output) associated with
instances contained in the training set. In semi-supervised learning, only
some of the labels are known in advance [3]. Once again, it is quite
convenient to use a matrix representation, referred to as the (true) label
matrix

$\displaystyle\mathbf{Y}_{ij}=\begin{cases}1&\text{ if the $i$-th instance is
associated with the $j$-th class}\\\ 0&\text{otherwise}\end{cases}$ (2)

Most approaches are inductive in nature so that we can predict the labels of
instances not seen before deployment. However, most GSSL methods are
transductive, which simply means that we are only interested in the fixed but
unknown set of labels corresponding to unlabeled instances [17]. Accordingly,
we may represent this with a classification matrix:

$\displaystyle\mathbf{F}\in\mathbb{R}^{n{\times}c}$ (3)

In order to separate the labeled data $\mathcal{L}$ from the unlabeled data
$\mathcal{U}$, we divide our matrices as following:

$\displaystyle\mathbf{X}$
$\displaystyle=\left[\mathbf{X_{\mathcal{L}}}^{\top},\mathbf{X_{\mathcal{U}}}^{\top}\right]^{\top}$
(4) $\displaystyle\mathbf{Y}$
$\displaystyle=\left[\mathbf{Y_{\mathcal{L}}}^{\top},\mathbf{Y_{\mathcal{U}}}^{\top}\right]^{\top}$
(5) $\displaystyle\mathbf{F}$
$\displaystyle=\left[\mathbf{F_{\mathcal{L}}}^{\top},\mathbf{F_{\mathcal{U}}}^{\top}\right]^{\top}$
(6)

The idea of SSL is appealing for many reasons. One of them is the possibility
to integrate the toolset developed for unsupervised learning. Namely, we may
use unlabeled data to measure the density $P(\mathbf{x})$ within our
$d$-dimensional input space. Once that is achieved, the only thing left is to
take advantage of this information. To do this, we have to make use of
assumptions about the relationship between the input density $P(\mathbf{x})$
and the conditional class distribution $P(y\\!\mid\\!\mathbf{x})$. If we are
not assuming that our datasets satisfy any kind of assumption, SSL can
potentially cause a significant decrease compared to baseline performance
[14]. This is currently an active area of research: safe semi-supervised
learning is said to be attained when SSL never performs worse than the
baseline, for any choice of labels for the unlabeled data. This is indeed
possible in some limited circumstances, but also provably impossible for
others, such as for a specific class of margin-based classifiers [9].

In Figure 1, we illustrate which kind of dataset is suitable for GSSL. There
are two clear spirals, one corresponding to each class. In a bad dataset for
SSL, we can imagine the spiral structure to be a red herring, i.e. something
misguiding. A very common assumption for SSL is the smoothness assumption,
which is one of the cornerstones for GSSL classifiers. It states that “If two
instances $\mathbf{x}_{1}$, $\mathbf{x}_{2}$ in a high-density region are
close, then so should be the corresponding outputs $y_{1}$, $y_{2}$” [3].
Another important assumption is that the (high-dimensional) data lie (roughly)
on a low-dimensional manifold, also known as the manifold assumption [3]. If
the data lie on a low-dimensional manifold, the local similarities will
approximate the manifold well. As a result, we can, for example, increase the
resolution of our image data without impacting performance.

Graph-based semi-supervised learning has been used extensively for many
different applications across different domains. In computer vision, these
include car plate character recognition [catunda2019car] and hyperspectral
image classification [13]. GSSL is particularly appealing if the underlying
data has a natural representation as a graph. As such, it has been a promising
approach for drug property prediction from the structure of molecules [8].
Moreover, it has had much application in knowledge graphs, such as the
development of web-scale recommendation systems [16].

(a) Ground truth

(b) Observed labels

Figure 1: An ideal scenario for semi-supervised learning

GSSL methods put a greater emphasis on using geodesics by expressing
connectivity between instances through the creation of a graph. Many
successful deep semi-supervised approaches use a similar yet slightly weaker
assumption, namely that small perturbations in input space should cause little
corresponding perturbation on the output space [12].

It turns out that we can best express our concepts by defining a measure of
similarity, instead of distance. In particular, we search for an affinity
matrix $\mathbf{W}\in\mathbb{R}^{n{\times}n}$, such that

$\mathbf{W}_{ij}=\begin{cases}w(\mathbf{x}_{i},\mathbf{x}_{j})\in\mathbb{R}&\text{if
$\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ are considered neighbors}\\\
0&\text{otherwise}\end{cases}$ (7)

where $w$ is some function determining the similarity between any two
instances $\mathbf{x}_{i},\mathbf{x}_{j}$. When constructing an affinity
matrix in practice, instances are not considered neighbors of themselves, i.e.
we have $\forall i\in\\{1..n\\}:\mathbf{W}_{ii}=0$.

The specification of an affinity matrix is a necessary step for any GSSL
classifier, and its sparsity is often crucial for reducing computational
costs. There are many ways to choose a neighborhood. Most frequently, it is
constructed by looking at the K-Nearest neighbors (KNN) of a given instance.

One last important concept to GSSL is that of the graph Laplacian operator.
This operator is analogue to the Laplace-Beltrami operator on manifolds. There
are a few graph Laplacian variants, such as the combinatorial Laplacian

$\mathbf{L}_{\mathbb{C}}=\mathbf{D}-\mathbf{W}$ (8)

where $\mathbf{D}$, called the degree matrix, is a diagonal matrix whose
entries are the sum of each row of $\mathbf{W}$. There is also the normalized
Laplacian, whose diagonal is the identity matrix $\mathbf{I}$:

${\mathbf{L}_{\mathbb{N}}=\mathbf{I}-\mathbf{D}^{-\frac{1}{2}}\mathbf{W}\mathbf{D}^{-\frac{1}{2}}=\mathbf{D}^{-\frac{1}{2}}\mathbf{L}_{\mathbb{C}}\mathbf{D}^{-\frac{1}{2}}}$
(9)

Each graph Laplacian $\mathbf{L}$ induces a measure of smoothness with respect
to the graph on a given classification matrix $\mathbf{F}$, namely

${\widetilde{S}_{\mathbf{L}}}(\mathbf{F})=\frac{1}{2}\sum_{k=1}^{c}(\mathbf{F}_{[:,k]})^{\top}\mathbf{L}(\mathbf{F}_{[:,k]})=\frac{1}{2}tr(\mathbf{F}^{\top}\mathbf{L}\mathbf{F})$
(10)

where $tr$ is the trace of the matrix and $c$ the number of classes. If we
consider each column $\mathbf{f}$ of $\mathbf{F}$ individually, then we can
express graph smoothness of each graph Laplacian as

$\displaystyle\mathbf{f}^{\top}\mathbf{L}_{\mathbb{C}}\mathbf{f}$
$\displaystyle=\sum_{1\leq i,j\leq
n}\mathbf{W}_{ij}(\mathbf{f}_{i}-\mathbf{f}_{j})^{2}$ (11)
$\displaystyle\mathbf{f}^{\top}\mathbf{L}_{\mathbb{N}}\mathbf{f}$
$\displaystyle=\sum_{1\leq i,j\leq
n}\left(\frac{\mathbf{W}_{ij}}{\sqrt{\mathbf{D}}_{ii}}\mathbf{f}_{i}-\frac{\mathbf{W}_{ij}}{\sqrt{\mathbf{D}}_{jj}}\mathbf{f}_{j}\right)^{2}$
(12)

Each graph Laplacian also has an eigendecomposition:

$\mathbf{L}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{\top}$ (13)

where the set of columns of $\mathbf{U}$ is an orthonormal basis of
eigenfunctions, with $\mathbf{\Lambda}$ a diagonal matrix with the
eigenvalues. As $\mathbf{U}$ is a unitary matrix, any real-valued function on
the graph may be expressed as a linear combination of eigenfunctions. We map a
function to the spectral domain by pre-multiplying it by $\mathbf{U}^{\top}$
(also called graph Fourier transform, also known as GFT). Additionally, pre-
multiplying by $\mathbf{U}$ gives us the inverse transform. This spectral
representation is very useful, as eigenfunctions that are smooth with respect
to the graph have smaller eigenvalues. Outright restricting the amount of
eigenfunctions is known as smooth eigenbasis pursuit[6], a valid strategy for
semi-supervised regularization.

In this work, we explore the problem of parameter selection for the Local and
Global Consistency model. This model yields a propagation matrix, which itself
depends on a fixed parameter $\alpha$ determining the diffusion rate. We show
that, by removing the diagonal in the propagation matrix used by our baseline,
a leave-one-out criterion can be easily computed. Our first proposed algorithm
attempts to calculate label reliability by optimizing the label matrix,
subject to constraints. This approach is shown to be competitive with robust
$\ell_{1}$-norm classifiers. Then, we consider the problem of optimizing the
diffusion rate. Doing this in the usual formulation of LGC is impractically
expensive. However, we show that the spectral representation of the problem
can be exploited to easily solve an approximate version of the problem.
Experimental results show that minimizing leave-one-out error leads to good
generalization.

The remaining of this paper is organized as follows. Section 2 summarizes the
key ideas and concepts that are related to our work. Section 3 presents our
two proposed algorithms: LGC_LVO_AutoL for determining label reliability, and
LGC_LVO_AutoD for determining the optimal diffusion rate parameter. Our
methodology is detailed in Section 4, going over the basic framework and
baselines. The results are presented in Section 5. Lastly, concluding remarks
are found in Section 6.

##  2 Related work

In this section, we present some of the algorithms and concepts that are
central to our approach. We describe the inner workings of our baseline
algorithm, and how eliminating diagonal entries of its propagation matrix may
lead to better generalization.

###  2.1 Local and Global Consistency

The Local and Global Consistency (LGC) [17] algorithm is one of the most
widely known graph-based semi-supervised algorithms. It minimizes the
following cost:

$\mathcal{Q}(\mathbf{F})=\frac{1}{2}\left(tr(\mathbf{F}^{\top}\mathbf{L}_{\mathbb{N}}\mathbf{F})+\mu\big{\lVert}\mathbf{F}-\mathbf{Y}\big{\rVert}^{2}\right)$
(14)

LGC addresses the issue of label reliability by introducing the parameter
$\mu\in(0,\infty)$. This parameter controls the trade-off between fitting
labels, and achieving high graph smoothness.

LGC has an analytic solution. To see this, we take the partial derivative of
the cost with respect to $\mathbf{F}$:

$\displaystyle\frac{\partial\mathcal{Q}}{\partial\mathbf{F}}$
$\displaystyle=\frac{1}{2}\frac{\partial
tr(\mathbf{F}^{\top}\mathbf{L}_{\mathbb{N}}\mathbf{F})}{\partial\mathbf{F}}+\frac{1}{2}\mu\frac{\partial\big{\lVert}\mathbf{F}-\mathbf{Y}\big{\rVert}^{2}}{\partial\mathbf{F}}$
(15) $\displaystyle=((1+\mu)\mathbf{I}-\mathbf{S})\mathbf{F}-\mu\mathbf{Y}$
(16)

where
$\mathbf{S}=\mathbf{D}^{-\frac{1}{2}}\mathbf{W}\mathbf{D}^{-\frac{1}{2}}=\mathbf{I}-\mathbf{L}_{\mathbb{N}}$.
By dividing the above by ($1+\mu$), we observe that this derivative is zero
exactly when

$(I-\alpha\mathbf{S})\mathbf{F}=\beta\mathbf{Y}$ (17)

with

$\alpha=\frac{1}{1+\mu}\in(0,1)$ (18)

and

$\beta=1-\alpha$ (19)

The matrix $(\mathbf{I}-\alpha\mathbf{S})$ can be shown to be positive-
definite and therefore invertible, so the optimal $\mathbf{F}$ can be obtained
as

$\mathbf{F}=\beta(I-\alpha\mathbf{S})^{-1}\mathbf{Y}$ (20)

We hereafter refer to $(I-\alpha S)^{-1}$ as the propagation matrix
$\mathbf{P}$. Each entry $\mathbf{P}_{ij}$ represents the amount of label
information from $X_{j}$ that $X_{i}$ inherits. It can be shown that the
inverse is a result of a diffusion process, which is calculated via iteration:

$\displaystyle F(0)=\mathbf{Y}$ (21) $\displaystyle
F(t\\!+\\!1)=\alpha\mathbf{S}F(t)+(1\\!-\\!\alpha)\mathbf{Y}$ (22)

Moreover, it can be shown that the closed expression for $F$ at any iteration
is

$F(t)=(\alpha\mathbf{S})^{t-1}\mathbf{Y}+(1-\alpha)\sum_{i=0}^{t-1}(\alpha\mathbf{S})^{i}\mathbf{Y}$
(23)

S is similar to $D^{-1}W$, whose eigenvalues are always in the range $[-1,1]$
[17]. This ensures the first term vanishes as $t$ grows larger, whereas the
second term converges to $\mathbf{P}Y$. Consequently, $\mathbf{P}$ can be
characterized as

$\displaystyle\mathbf{P}$
$\displaystyle=(1-\alpha)\lim_{t\rightarrow\infty}\sum_{i=0}^{t}(\alpha\mathbf{S})^{i}$
(24)
$\displaystyle=(1-\alpha)\lim_{t\rightarrow\infty}\sum_{i=0}^{t}\alpha^{i}\mathbf{D}^{\frac{1}{2}}(\mathbf{D}^{-1}\mathbf{W})^{i}\mathbf{D}^{-\frac{1}{2}}$
(25)

The transition probability matrix $\widetilde{W}=\mathbf{D^{-1}W}$ makes it so
we can interpret the process as a random walk. Let us imagine a particle
walking through the graph according to the transition matrix. Assume it began
at a labeled vertex $v_{a}$, and at step $i$ it reaches a labeled vertex
$v_{b}$, initially labeled with class $c_{b}$. When this happens, $v_{a}$
receives a confidence boost to class $c_{b}$. Alternatively, one can say that
this boost goes to the entry which corresponds to the contribution from
$v_{b}$ to $v_{a}$, i.e. $\mathbf{P}_{ab}$. This boost is proportional to
$\alpha^{i}$. This gives us a good intuition as to the role of $\alpha$. More
precisely, the contribution of vertices found later in the random walk decays
exponentially according to $\alpha^{i}$.

###  2.2 Self-influence and leave-one-out-error

There is one major problem with LGC’s solution: the diagonal of $\mathbf{P}$.
At first glance, we would think that “fitting the labels ” means looking for a
model that explains our data very well. In reality, this translates to
memorizing the labeled set. The main problem resides within the diagonal of
the propagation matrix. Any entry $\mathbf{P}_{ii}$ stores the self-influence
of a vertex, which is calculated according to the expected reward obtained by
looping around and visiting itself. The optimal solution w.r.t. label fitting
occurs when $\alpha$ tends to zero. For labeled instances, an initial reward
is given for the starting vertex itself, and the remaining are essentially
ignored.

We argue that the diagonal is directly related to overfitting. It essentially
tells the model to rely on the label information it knows. There are a few
analogies to be made: say that we are optimizing the number of neighbors $k$
for a KNN classifier. The analog of “LGC-style optimal label fitting” would be
to include each labeled instance as a neighbor to itself, and set $k=1$. This
is obviously not a good criterion. The answer that maximizes a proper “label
fitting” criteria, in this case, is selecting $k$ that minimizes
classification error with the extremely important caveat: directly using each
instance’s own label is prohibited.

In spite of the problems we have presented, the family of LGC solutions
remains very interesting to consider. We will have to eliminate diagonals,
however. Let

$\mathbf{H}(\alpha):=\left(\mathbf{(\mathbf{I}-\alpha\mathbf{S})^{-1}}-diag((\mathbf{I}-\alpha\mathbf{S})^{-1})\right)_{\mathcal{L}}\mathbf{Y}_{\mathcal{L}}$
(26)

By eliminating the diagonal, we obtain, for each label, its classification if
it were not included in the label propagation process. As such, it can be
argued that minimizing

$\big{\lVert}\mathbf{H}(\alpha)-\mathbf{Y}_{\mathcal{L}}\big{\rVert}^{2}$ (27)

also minimizes the leave-one-out (LOO) error. There is an asterisk: each
instance is still used as unlabeled data, but this effect should be
insignificant. It is also interesting to row-normalize (a small constant
$\epsilon$ may be added for stabilization) the rows, so that we end up with
classification probabilities.

Previously, we developed a semi-supervised leave-one-out filter [4, 2]. We
also managed to reduce the amount of storage used by only calculating a
propagation submatrix. The LOO-inspired criterion encourages label information
to be redundant, so labels that are incoherent with the implicit model are
removed.

The major drawback of our proposal is that it needs an extra parameter $r$,
which is the number of labels to remove. The optimal $r$ is usually around the
number of noisy labels, which is unknown to us.This was somewhat addressed in
[4]: we can instead use a threshold, which tells us how much labels can
deviate from the original model. Nonetheless, it is desirable to solve this
problem in a way that removes such a parameter. We will do this by introducing
a new optimization problem.

##  3 Proposal

In this work, we consider the optimal value of ${\alpha}$ for our LOO-inspired
loss [4]. The objective is to develop a method to minimize surrogate losses
based on LOO error for the LGC GSSL classifier and evaluate the generalization
of its solutions.

We use the term “surrogate loss”[9] to denote losses such as squared error,
cross-entropy and so on. We use this term to purposefully remind that the
solution $\mathbf{H}(\alpha)$ that minimizes the loss on the test set does not
also necessarily be the one that maximizes accuracy. We use automatic
differentiation as our optimization procedure. As such, we call our approach
LGC_LVO_AutoL and LGC_LVO_AutoD when learning label reliability and diffusion
rate, respectively. We exploit the fact that we can compute the gradients of
our loss.

###  3.1 LGC_LVO_AutoL: Automatic Correction of Noisy Labels

Let $\mathbf{P}$ be the propagation matrix, whose submatrix
$\mathbf{P}_{\mathcal{L}\mathcal{L}}$ corresponds to kernel values between
labeled data only. Then, the modified LGC solution is given by
$\widetilde{\mathbf{P}}\mathbf{Y}_{\mathcal{L}}$, where

$\widetilde{\mathbf{P}}:=(\mathbf{P}_{\mathcal{L}\mathcal{L}}-diag(\mathbf{P}_{\mathcal{L}\mathcal{L}}))$
(28)

Our optimization problem is to optimize a diagonal matrix $\mathbf{\Omega}$
indicating the updated reliability of each label:

$\displaystyle\min_{\mathbf{\Omega}}\texttt{LOSS}({deg(\widetilde{\mathbf{P}}\mathbf{\Omega}\mathbf{Y}_{\mathcal{L}})^{-1}\widetilde{\mathbf{P}}\mathbf{\Omega}\mathbf{Y}_{\mathcal{L}},\;\;\mathbf{Y}_{\mathcal{L}}})\;\;\;\;\;\;\;\text{such
that }\;\;\forall i:\mathbf{\Omega}_{ii}\geq 0$ (29)

where LOSS denotes some loss function such as squared error or cross-entropy.

###  3.2 LGC_LVO_AutoD: Automatic choice of diffusion rate

In [4], we use a modified version of the power method to calculate the
submatrix $\mathbf{P}_{\mathcal{L}}$. This is enough to give us the answer in
a few seconds for a fixed $\alpha$, but would quickly turn into a huge
bottleneck if we were to constantly update $\alpha$. Our new approach uses the
graph Fourier transform. In other words, we adapt the idea of smooth
eigenbasis pursuit to this particular problem. Let us write the propagation
matrix using eigenfunctions:

$\displaystyle\mathbf{P}$
$\displaystyle=\sum_{t=0}^{\infty}\alpha^{t}\mathbf{S}^{t}$ (30)
$\displaystyle=\sum_{t=0}^{\infty}\alpha^{t}(\mathbf{I}-\mathbf{\mathbf{L}_{\mathbb{N}}})^{t}$
(31)

Using the graph Fourier transform, we have that

$\displaystyle\mathbf{\mathbf{L}_{\mathbb{N}}}$
$\displaystyle=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{\top}$ (32)
$\displaystyle\mathbf{I}$
$\displaystyle=\mathbf{U}\mathbf{I}\mathbf{U}^{\top}$ (33)

So it follows that

$\displaystyle\mathbf{P}$
$\displaystyle=\mathbf{U}\widetilde{\Lambda}\mathbf{U}^{\top}$ (34)

where $\forall i\in\\{1..N\\}$:

$\displaystyle\widetilde{\Lambda}_{ii}$
$\displaystyle=\sum_{t=0}^{\infty}(\alpha(1-\Lambda_{ii}))^{t}$ (35)
$\displaystyle\\{\big{\lVert}\alpha(1-\Lambda_{ii})\big{\rVert}<1\\}$
$\displaystyle=\frac{1}{1-(\alpha(1-\Lambda_{ii}))}$ (36)
$\displaystyle=\frac{1}{(1-\alpha)\times 1+(\alpha)\times\Lambda_{ii}}$ (37)

In practice, we can assume that $\mathbf{U}$ is an $l{\times}p$ matrix, with
$l$ the number of labeled instances and $p$ the chosen amount of
eigenfunctions. The diagonal entries are given by

$\displaystyle\mathbf{P}_{ii}$
$\displaystyle=(\mathbf{U}\widetilde{\Lambda}\mathbf{U}^{\top})_{ii}$ (38)
$\displaystyle=(\mathbf{U}^{\top})_{[:,i]}\widetilde{\Lambda}(\mathbf{U}^{\top})_{[:,i]}$
(39)
$\displaystyle=\Sigma_{i=1}^{p}\mathbf{U}_{ik}^{2}\widetilde{\Lambda}_{kk}$
(40)
$\displaystyle=(\mathbf{U}_{[i,:]})(\mathbf{U}_{[i,:]}\widetilde{\Lambda})^{\top}$
(41)

Next, we’ll analyze the complexity of calculating the leave-one-out error for
a given diffusion rate $\alpha$, given that we have stored the first $p$
eigenfunctions in the matrix $\mathbf{U}$. Let
$\mathbf{U}_{\mathcal{L}}\in\mathbb{R}^{p{\times}l}$ be the matrix of
eigenfunctions with domain restricted to labeled instances. Exploiting the
fact that $\mathbf{Y}_{\mathcal{U}}=0$, it can be shown that

$\mathbf{P}_{\mathcal{L}\mathcal{L}}=\mathbf{U}_{\mathcal{L}}\mathbf{\widetilde{\Lambda}}(\mathbf{U}_{\mathcal{L}}^{\top}\mathbf{Y}_{\mathcal{L}})$
(42)

We can precompute
$(\mathbf{U}_{\mathcal{L}}^{\top}\mathbf{Y}_{\mathcal{L}})\in\mathbb{R}^{p{\times}c}$
with $\mathcal{O}(plc)$ multiplications. For each diffusion rate candidate
$\alpha$, we obtain
$\mathbf{U}_{\mathcal{L}}\mathbf{\widetilde{\Lambda}}\in\mathbb{R}^{l{\times}p}$
by multiplying each restricted eigenfunction with its new eigenvalue. The only
thing left is to post-multiply it by the pre-computed matrix. As a result, we
can compute the leave-one-out error for arbitrary $\alpha$ with
$\mathcal{O}(plc)$ operations.

We have shown that, by using the propagation submatrix, we can re-compute the
propagation submatrix $\mathbf{P}_{\mathcal{L}\mathcal{L}}$ in
$\mathcal{O}(plc)$ time. In comparison, the previous approach [4] requires,
for each diffusion rate, a total of $O(tknlc)$ operations, assuming $t$
iterations of the power method and a sparse affinity matrix with average node
degree equal to $k$. Even if we use the full eigendecomposition ($p=n$), this
new approach is significantly more viable for different learning rates.
Moreover, we can lower the choice of $p$ to use a faster, less accurate
approximation.

##  4 Methodology

####  4.0.1 Basic Framework

We have a configuration dispatcher which enables us to vary a set of
parameters (for example, the chosen dataset and the parameters for affinity
matrix generation).

We start out by reading our dataset, including features and labels. We use the
random seed to select the sampled labels, and create the affinity matrix
$\mathbf{W}$ necessary for LGC. For automatic label correction, we assume that
$\alpha$ is given as a parameter. For choosing the automatic diffusion rate,
we need to extract the $p$ smoothest eigenfunctions as a pre-processing step.
Next, we repeatedly calculate the gradient and update either $\alpha$ or
$\mathbf{\Omega}$ to minimize LOO error. After a set amount of iterations, we
return the final classification and perform an evaluation on unlabeled
examples.

The programming language of choice is Python 3, for its versatility and
support. We also make use of the tensorflow-gpu [1] package, which massively
speeds up our calculations and also enables automatic differentiation of loss
functions. We use a Geforce GTX 1070 GPU to speed up inference, and also for
calculating the k-nearest neighbors of each instance with the faiss-gpu
package [7].

####  4.0.2 Evaluation and baselines

In this work, our datasets have a roughly equal number of labels for each
class. As such, we will report the mean accuracy, as well as its standard
deviation. However, one distinction is that we calculate the accuracy
independently on labeled and unlabeled data. This is done to better assess
whether our algorithms are improving classification on instances outside the
labeled set, or if it outperforms its LGC baseline only when performing
diagnosis of labels.

Our approach, LGC_LVO_Auto, is compatible with any differentiable loss
function, such as mean squared error (MSE) or cross-entropy (xent). The chosen
optimizer was Adam, with a learning rate of 0.7 and 5000 iterations. For the
approximation of the propagation matrix in LGC_LVO_AutoL, we used $t=1000$
iterations throughout.

Perhaps the most interesting classifier to compare our approach to is the LGC
algorithm, as that is the starting point and the backbone of our own approach.
We will also be comparing our results with the ones reported by [6]. These
include: Gaussian Fields and Harmonic Functions (GFHF) [18], Graph Trend
Filtering [15], Large-Scale Sparse Coding (LSSC) [10] and Eigenfunction [5].

##  5 Results

This section presents the results of employing our two approaches
LGC_LVO_AutoL and LGC_LVO_AutoD on ISOLET and MNIST datasets, compared to
other graph-based SSL algorithms from the literature.

###  5.1 Experiment 1: LGC_LVO_AutoL on ISOLET

####  5.1.1 Experiment setting

In this experiment, we compared LGC_LVO_AutoL to the baselines reported in
[6], specifically for the ISOLET dataset. Unlike the authors, our 20 different
seeds also control both the label selection and noise processes. The graph
construction was performed exactly as in [6], a symmetric 10-nearest neighbors
graph with the width $\sigma$ of the RBF kernel set to 100. We emphasize that
the reported results by the authors correspond to the best-performing
parameters, divided for each individual noise level. In [6], $\lambda_{1}$ is
set to $10^{5}$, $10^{2}$, $10^{2}$ and $10^{2}$ for the respective noise
rates of 0%, 20%, 40% and 60%; $\lambda_{2}$ is kept to 10, and the number of
eigenfunctions is $m=30$. We could not find any implementation code for SIIS,
so we had to manually reproduce it ourselves. As for parameter selection for
LGC_LVO_AutoL, we simply set $\alpha=0.9$ (equivalently, $\mu=0.1111$). We
reiterate that having a single parameter is a strength of our approach. In
future work, we will try to combine LGC_LVO_AutoL with LGC_LVO_AutoD to fully
eliminate the need for parameter selection.

####  5.1.2 Experiment results

The results are contained in Tables 1(a) and 1(b). With respect to the
accuracy on unlabeled examples, we observed that:

  * •

SIIS appeared to have a slight edge in the noiseless scenario.

  * •

LGC’s own inherent robustness was evident. When 60% noise was injected, it
went from $84.72$ to $70.69$, a decrease of $16.55\%$. In comparison, SIIS had
a decrease of $14.39\%$; GFHF a decrease of $22.08\%$; GTF a decrease of
$21.82\%$.

  * •

With $60\%$ noise, LGC_LVO_AutoL(xent) decreased its accuracy by $11.52\%$, so
LGC_LVO_AutoL(xent) had the lowest percentual decrease.

  * •

LGC_LVO_AutoL(MSE) disappointed for both labeled and unlabeled instances.

  * •

LGC_LVO_AutoL was not noticeably superior to LGC when there was less than 60%
noise.

With respect to the accuracy on unlabeled examples, we observed that:

  * •

LGC was unable to correct noisy labels.

  * •

LGC_LVO_AutoL$(xent)$ discarded only 5% of the labels for the noiseless
scenario, which is better than SIIS and LSSC.

  * •

Moreover, LGC_LVO_AutoL$(xent)$ had the highest average accuracy on labeled
instances for $20\%,40\%,60\%$ label noise.

(a) Accuracy on unlabeled examples only Dataset | Noise Level | LSSC | GTF | GFHF | SIIS
---|---|---|---|---|---
| 0% |  84.8 $\pm$ 0.0 |  70.1 $\pm$ 0.0 |  86.5 $\pm$ 0.0 | 85.4 $\pm$ 0.0
ISOLET | 20% |  82.8 $\pm$ 0.3 |  69.9 $\pm$ 0.2 |  81.6 $\pm$ 0.4 | 84.9 $\pm$ 0.6
(1040/7797 labels) | 40% |  78.5 $\pm$ 0.6 |  59.8 $\pm$ 0.3 |  79.7 $\pm$ 1.0 | 80.2 $\pm$ 1.3
reported results | 60 % |  67.5 $\pm$ 1.8 |  54.8 $\pm$ 0.5 |  67.4 $\pm$ 1.5 | 74.9 $\pm$ 1.4
Dataset | Noise Level | LGC |  LGC_LVO_AutoL (MSE) |  LGC_LVO_AutoL (XENT) | SIIS
| 0% |  84.71 $\pm$ 0.56 |  84.21$\pm$0.4 |  84.22$\pm$0.45 | 85.24 $\pm$ 0.32
ISOLET | 20% |  82.89 $\pm$ 0.59 |  81.6$\pm$0.63 |  82.56$\pm$0.62 | 83.69 $\pm$ 0.33
(1040/7797 labels) | 40% |  79.33 $\pm$ 0.92 |  77.73$\pm$0.96 |  80.23$\pm$0.74 | 80.88 $\pm$ 0.77
our results | 60% |  70.69 $\pm$ 1.01 |  68.98$\pm$1.81 | 74.51$\pm$1.75 |  72.97 $\pm$ 1.16

(b) Accuracy on labeled examples after label correction Dataset | Noise Level | LSSC | GTF | GFHF | SIIS
---|---|---|---|---|---
| 0% |  89.9 $\pm$ 0.0 |  95.8 $\pm$ 0.0 | 100.00 $\pm$ 0.0 |  91.1 $\pm$ 0.0
ISOLET | 20% |  87.7 $\pm$ 0.3 |  79.8 $\pm$ 0.7 |  80.00 $\pm$ 0.00 | 90.5 $\pm$ 0.8
(1040/7797 labels) | 40% |  82.9 $\pm$ 0.9 |  63.3 $\pm$ 0.4 |  60.00 $\pm$ 0.00 | 83.6 $\pm$ 1.0
reported results | 60 % |  71.8 $\pm$ 1.7 |  55.3 $\pm$ 0.6 |  40.00 $\pm$ 0.00 | 77.4 $\pm$ 1.0
Dataset | Noise Level | LGC |  LGC_LVO_AutoL (MSE) |  LGC_LVO_AutoL (XENT) | SIIS
| 0% | 99.9 $\pm$ 0.02 |  97.36$\pm$0.52 |  95.01$\pm$0.58 |  90.24 $\pm$ 0.69
ISOLET | 20% |  80.84 $\pm$ 0.27 |  90.93$\pm$1.28 | 91.52$\pm$0.79 |  88.5 $\pm$ 1.07
(1040/7797 labels) | 40% |  60.24 $\pm$ 0.20 |  82.54 $\pm$ 0.92 | 87.19$\pm$0.89 |  85.25 $\pm$ 0.96
our results | 60% |  40.00 $\pm$ 0.04 |  71.16 $\pm$ 1.79 | 79.14$\pm$1.72 |  76.34 $\pm$ 1.53

Table 1: Accuracy on ISOLET dataset

###  5.2 Experiment 2: LGC_LVO_AutoL on MNIST

####  5.2.1 Experiment setting

This experiment was based on [10], where a few classifiers were tested on the
MNIST dataset subject to label noise. In that paper, the parameters for the
graph were tuned to minimize cross-validation errors. Moreover, an anchor
graph was used, which is a large-scale solution. We did not use such a graph,
as our TensorFlow iterative implementation of LGC_LVO_AutoL was efficient
enough to perform classification on MNIST in just a few seconds. As we also
included the results for LGC (without anchor graph), it is interesting to
observe that its accuracy decreases similarly to the previously reported
results: the main difference is better performance for the noiseless scenario,
which is to be expected (the anchor graph is an approximation).

Once again, we simply set $\alpha=0.9$ for LGC_LVO_AutoL. We used a symKNN
matrix with $k=15$ neighbors, and a heuristic sigma $\sigma=423.57$ obtained
by taking one-third of the mean distance to the 10th neighbor (as in [3]).

(a) Accuracy on unlabeled examples only Dataset | Noise Level | LSSC* | Eigenfunction* | LGC* (anchor graph) |
---|---|---|---|---|---
MNIST | 0% | 93.1 $\pm$ 0.7 |  73.8 $\pm$ 1.6 |  90.4 $\pm$ 0.7 |
(100/70000 labels) | 15% | 91.1 $\pm$ 2.0 |  68.6 $\pm$ 2.8 |  83.5 $\pm$ 1.6 |
reported results | 30% | 89.0 $\pm$ 3.6 |  61.9 $\pm$ 4.0 |  74.4 $\pm$ 2.8 |
Dataset | Noise Level |  LGC_LVO_AutoL (MSE) |  LGC_LVO_AutoL (XENT) | LGC |
MNIST | 0% |  91.7 $\pm$ 0.7 |  92.69 $\pm$ 1.19 | 93.09 $\pm$ 0.92 |
(100/70000 labels) | 15% |  86.48 $\pm$ 2.59 | 90.45 $\pm$ 2.13 |  85.40 $\pm$ 1.66 |
our results | 30% |  81.33 $\pm$ 4.43 | 84.46 $\pm$ 3.89 |  74.58 $\pm$ 2.6 |

(b) Accuracy on labeled examples after label correction Dataset | Noise Level |  LGC_LVO_AutoL (MSE) |  LGC_LVO_AutoL (XENT) | LGC |
---|---|---|---|---|---
MNIST | 0% |  99.5 $\pm$ 0.59 |  98.05 $\pm$ 0.59 | 100.00 $\pm$ 0.0 |
(100/70000 labels) | 15% |  95.05 $\pm$ 2.01 | 96.10 $\pm$ 1.3 |  85.00 $\pm$ 0.0 |
our results | 30% |  85.55$\pm$4.88 | 89.75 $\pm$ 4.06 |  70.00 $\pm$ 0.0 |

Table 2: Accuracy on MNIST dataset

####  5.2.2 Experiment results

The results are found in Tables 2(a) and 2(b). With respect to the accuracy on
unlabeled examples, we observed that:

  * •

LGC_LVO_AutoL with cross-entropy improved the LGC baseline significantly on
unlabeled instances. For 30% label noise, mean accuracy increases from 74.58%
to 84.46%.

  * •

The mean squared error loss is once again consistently inferior to cross-
entropy when there is noise.

  * •

LGC_LVO_AutoL was not able to obtain better results than LSSC.

With respect to the accuracy on labeled examples, we observed that:

  * •

LGC was not able to correct the labeled instances.

  * •

LGC_LVO_AutoL with cross-entropy improved the LGC baseline significantly on
labeled instances. With 30% noise, accuracy goes from 70.00% to 89.75%.

###  5.3 Experiment 3: LGC_LVO_AutoD on MNIST

Experiment settings were kept the same as Experiment 2 (without noise), and
$p=300$ eigenfunctions were extracted. In Figure 2, we show accuracy on the
labeled set as red, and on the unlabeled set as purple. The $x$ values on the
horizontal axis relate to $\alpha$ as following: $\alpha=2^{-1/x}$. Looking at
Figure 2(b), we can see that, if we do not remove the diagonal, there is much
overfitting and the losses reach their minimum value much earlier than does
the accuracy on unlabeled examples. When the diagonal is removed, the loss-
minimizing estimates for $\alpha$ get much closer to the optimal one (Figure
2(a)). Moreover, by removing the diagonal we obtain a much better estimation
of the accuracy on unlabeled data. Therefore, we can simply select the
$\alpha$ corresponding to the best accuracy on known labels.

(a) Diagonal removed

(b) Diagonal NOT removed

Figure 2: LGC_LVO_AutoD on MNIST. The $x$ coordinate on the horizontal axis
determines diffusion rate $\alpha=2^{-1/x}$ for $x\geq 1$. The orange, blue
and green curves represent the surrogate losses: mean absolute error, mean
squared error, cross-entropy. Moreover, the accuracy on the known labeled data
is shown in red, and the accuracy on unlabeled data in purple.

##  6 Concluding remarks

We have proposed the LGC_LVO_Auto framework, based on leave-one-out validation
of the LGC algorithm. This encompasses two methods, LGC_LVO_AutoL and
LGC_LVO_AutoD, for estimating label reliability and diffusion rates. We use
automatic differentiation for parameter estimation, and the eigenfunction
approximation is used to derive a faster solution for LGC_LVO_AutoD, in
particular when having to recalculate the propagation matrix for different
diffusion rates.

Overall, LGC_LVO_AutoL produced interesting results. For the ISOLET dataset,
it was very successful at the task of label diagnosis, being able to detect
and remove labels with overall better performance than every $\ell_{1}$-norm
method. On the other hand, did not translate too well for unlabeled instance
classification. For the MNIST dataset, performance is massively boosted for
unlabeled instances as well. In spite of outperforming its LGC baseline by a
wide margin, it could not match the reported results of LSSC on MNIST.
Preliminary results showed that LGC_LVO_AutoD is a viable way to get a good
estimate of the optimal diffusion rate, and removing the diagonal entries
proved to be the crucial step for avoiding overfitting.

For future work, we will be further evaluating LGC_LVO_AutoD. We will also try
to integrate LGC_LVO_AutoD and LGC_LVO_AutoL together into one single
algorithm. Lastly, we will aim to extend LGC_LVO_AutoD to a broader class of
graph-based kernels, in addition to the one resulting from the LGC baseline.

## References

  * [1] Abadi, M., et al.: TensorFlow: Large-scale machine learning on heterogeneous systems (2015), http://tensorflow.org/, software available from tensorflow.org
  * [2] de Aquino Afonso, B.K.: Analysis of Label Noise in Graph-Based Semi-Supervised Learning. Master’s thesis (2020)
  * [3] Chapelle, O., Schölkopf, B., Zien, A. (eds.): Semi-Supervised Learning. MIT Press, Cambridge, MA (2006), http://www.kyb.tuebingen.mpg.de/ssl-book
  * [4] de Aquino Afonso, B.K., Berton, L.: Identifying noisy labels with a transductive semi-supervised leave-one-out filter. Pattern Recognition Letters (2020). https://doi.org/https://doi.org/10.1016/j.patrec.2020.09.024, http://www.sciencedirect.com/science/article/pii/S0167865520303603
  * [5] Fergus, R., Weiss, Y., Torralba, A.: Semi-supervised learning in gigantic image collections. In: Advances in neural information processing systems. pp. 522–530 (2009)
  * [6] Gong, C., Zhang, H., Yang, J., Tao, D.: Learning with inadequate and incorrect supervision. In: Data Mining (ICDM), 2017 IEEE International Conference on. pp. 889–894. IEEE (2017)
  * [7] Johnson, J., Douze, M., Jégou, H.: Billion-scale similarity search with gpus. IEEE Transactions on Big Data (2019)
  * [8] Kearnes, S., McCloskey, K., Berndl, M., Pande, V., Riley, P.: Molecular graph convolutions: moving beyond fingerprints. Journal of computer-aided molecular design 30(8), 595–608 (2016)
  * [9] Krijthe, J.H.: Robust semi-supervised learning: projections, limits & constraints. Ph.D. thesis, Leiden University (2018)
  * [10] Lu, Z., Gao, X., Wang, L., Wen, J.R., Huang, S.: Noise-robust semi-supervised learning by large-scale sparse coding. In: AAAI. pp. 2828–2834 (2015)
  * [11] Mitchell, T.: Machine Learning. McGraw-Hill, New York (1997)
  * [12] Miyato, T., Maeda, S.i., Ishii, S., Koyama, M.: Virtual adversarial training: a regularization method for supervised and semi-supervised learning. IEEE transactions on pattern analysis and machine intelligence (2018)
  * [13] Shao, Y., Sang, N., Gao, C., Ma, L.: Probabilistic class structure regularized sparse representation graph for semi-supervised hyperspectral image classification. Pattern recognition 63, 102–114 (2017)
  * [14] Van Engelen, J.E., Hoos, H.H.: A survey on semi-supervised learning. Machine Learning 109(2), 373–440 (2020)
  * [15] Wang, Y.X., Sharpnack, J., Smola, A.J., Tibshirani, R.J.: Trend filtering on graphs. The Journal of Machine Learning Research 17(1), 3651–3691 (2016)
  * [16] Ying, R., He, R., Chen, K., Eksombatchai, P., Hamilton, W.L., Leskovec, J.: Graph convolutional neural networks for web-scale recommender systems. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. pp. 974–983 (2018)
  * [17] Zhou, D., Bousquet, O., Lal, T.N., Weston, J., Schölkopf, B.: Learning with local and global consistency. In: Advances in neural information processing systems. pp. 321–328 (2004)
  * [18] Zhu, X., Ghahramani, Z., Lafferty, J.: Semi-supervised learning using gaussian fields and harmonic functions. In: Proceedings of the Twentieth International Conference on International Conference on Machine Learning. pp. 912–919. AAAI Press (2003)


††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Optimizing one dimensional superconducting diodes: Interplay of Rashba spin-
orbit coupling and magnetic fields

Sayak Bhowmik  Institute of Physics, Sachivalaya Marg, Bhubaneswar, Orissa
751005, India  Homi Bhabha National Institute, Training School Complex,
Anushakti Nagar, Mumbai 400094, India  Dibyendu Samanta  Department of
Physics, Indian Institute of Technology, Kanpur 208016, India  Ashis K. Nandy
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of Science
Education Research,An OCC of Homi Bhabha National Institute, Jatni 752050,
India  Arijit Saha<EMAIL_ADDRESS>Institute of Physics, Sachivalaya Marg,
Bhubaneswar, Orissa 751005, India  Homi Bhabha National Institute, Training
School Complex, Anushakti Nagar, Mumbai 400094, India  Sudeep Kumar Ghosh
0000-0002-3646-0629<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Technology, Kanpur 208016, India

###### Abstract

The superconducting diode effect (SDE) refers to the non-reciprocal nature of
the critical current (maximum current that a superconductor can withstand
before turning into a normal metal) of a superconducting device. Here, we
investigate SDE in helical superconductors with broken inversion and time-
reversal symmetry, focusing on a prototypical Rashba nanowire device
proximitized by an $s$-wave superconductor and subjected to external magnetic
fields. Using a self-consistent Bogoliubov-de Gennes mean-field formalism, we
analyze the interplay between linear and higher-order spin-orbit coupling
(SOC), bulk supercurrents, and external magnetic fields. Our results
demonstrate that Rashba nanowires with only linear SOC can achieve incredibly
large diode efficiency $\gtrsim 45\%$ through the interplay of longitudinal
and transverse magnetic fields. Notably, higher-order SOC enables finite diode
efficiency even without a longitudinal Zeeman field, which can be utilized to
reveal its presence and strength in nanowires. We present a comprehensive
phase diagram of the device elucidating the emergent Fulde-Ferrell-Larkin-
Ovchinnikov (FFLO) superconducting state and demonstrate that proximitized
Rashba nanowires offer a versatile, practical platform for SDE, with potential
realizations in existing material systems. These results provide crucial
insights for optimizing SDE in nanoscale superconducting devices, paving the
way for next-generation dissipationless quantum electronics.

##  I Introduction

The invention of diodes marked a pivotal breakthrough in solid-state
electronics, with their distinctive non-reciprocal current flow making them
fundamental to semiconductor devices Scaff and Ohl (1947); Shockley (1949). In
quantum materials, inversion and time-reversal symmetry crucially determine
charge current nonreciprocity. Magnetochiral anisotropy (MCA), arising in
systems with broken inversion symmetry Tokura and Nagaosa (2018), is a
dominant mechanism for this effect. MCA leads to non-reciprocity in the
magnetoresistance with respect to the current direction and can be greatly
enhanced by superconducting correlations Qin  _et al._ (2017); Wakatsuki  _et
al._ (2017). In superconductors, MCA can lead to an extreme manifestation of
nonreciprocity known as the superconducting diode effect (SDE), where
resistance is zero in one current direction and non-zero in the other Nadeem
_et al._ (2023); Nagaosa and Yanase (2024); Narita and Ono (2024). SDE thus
realizes an ideal diode with perfect rectification. Such a scenario arises in
the regime where the critical current that causes superconductor to normal
metal transition differ in the opposite directions
($|J_{c}^{+}|\neq|J_{c}^{-}|$) Nadeem  _et al._ (2023); Nagaosa and Yanase
(2024). Recent experiments in engineered superlattice systems Ando  _et al._
(2020); Sundaresh  _et al._ (2023); Narita and Ono (2024) and bulk materials/
thin films Itahashi  _et al._ (2020); Schumann  _et al._ (2020); Wakatsuki
_et al._ (2017); Nadeem  _et al._ (2023) have spurred research interests into
more efficient superconducting diode devices, including in twisted multilayer
graphene Lin  _et al._ (2022); Díez-Mérida  _et al._ (2023) and transition
metal dichalcogenides Bauriedl  _et al._ (2022); Yun  _et al._ (2023).

The inversion symmetry breaking required for critical current non-reciprocity
can be achieved through extrinsic mechanisms (e.g., asymmetric geometry or
artificial superlattice potential Sundaresh  _et al._ (2023); Burlakov  _et
al._ (2014); Narita and Ono (2024)) or intrinsic mechanisms like Rashba spin-
orbit coupling (SOC) Ando  _et al._ (2020); Yuan and Fu (2022); Ilić and
Bergeret (2022); Turini  _et al._ (2022); Roig  _et al._ (2024); Halterman
_et al._ (2022); Daido and Yanase (2022); Karabassov  _et al._ (2022); Legg
_et al._ (2022). Correspondingly, SDE can be broadly classified into two
types: extrinsic and intrinsic. Although predicted decades ago Levitov  _et
al._ (1985); Edelstein (1996), the mechanism of SDE is beginning to be
understood only recently, leading to extensive investigations into its
potential for dissipationless electronic circuits Yuan and Fu (2022); Daido
_et al._ (2022); Ilić and Bergeret (2022); He  _et al._ (2022); Roig  _et al._
(2024); Pal  _et al._ (2022); de Picoli  _et al._ (2023); Banerjee and
Scheurer (2024); Nadeem  _et al._ (2023). Recent theoretical studies Yuan and
Fu (2022); Daido  _et al._ (2022); Ilić and Bergeret (2022); He  _et al._
(2022), using phenomenological Ginzburg-Landau theory and mean-field analysis
of the two-dimensional Rashba-Zeeman Hubbard model, have shown that intrinsic
SDE can arise in systems with broken inversion and time-reversal symmetry. In
the helical superconducting ground state, this occurs due to asymmetry in
Cooper pair momentum in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state
Fulde and Ferrell (1964); Larkin and Ovchinnikov (1965), resulting in Kramers
non-degenerate $s$-wave Cooper pairs Daido  _et al._ (2022); Yuan and Fu
(2022); Nadeem  _et al._ (2023).

A Rashba nanowire device proximitized to an s-wave superconductor and
subjected to external magnetic fields serves as a prototypical platform for
realizing helical superconductivity Oreg  _et al._ (2010); Sau  _et al._
(2010); Lutchyn  _et al._ (2010); Stanescu  _et al._ (2011); Legg  _et al._
(2022); de Picoli  _et al._ (2023). While previous phenomenological/ non-self-
consistent studies Legg  _et al._ (2022) demonstrated the potential for SDE in
such systems, they reported low diode efficiencies ($\sim 2\%$). To enhance
the relatively low diode efficiency of this device and better understand its
properties, it’s crucial to systematically investigate the superconducting
order parameter using self-consistent methods Daido  _et al._ (2022); Ilić and
Bergeret (2022); de Picoli  _et al._ (2023). Recent studies Daido  _et al._
(2022); Ilić and Bergeret (2022); de Picoli  _et al._ (2023) have shown that
this approach can potentially uncover pathways for improvement across various
parameters. In this paper, we address this gap through a comprehensive, self-
consistent mean-field investigation of the interplay between various Rashba
SOC terms and external magnetic fields in determining emergent FFLO pairing
and highly efficient superconducting diodes in a Rashba nanowire device. We
show that even with only linear Rashba SOC we can achieve a very large diode
efficiency ($\gtrsim 45\%$) by suitably tuning the parameters in this system.
We extend beyond linear Rashba SOC to include higher-order (cubic) Rashba SOC
terms, which are not only naturally present in semiconducting nanowires Campos
_et al._ (2018); Kammermeier  _et al._ (2018); Sonehara  _et al._ (2022) but
also can dominate the normal state properties Vajna  _et al._ (2012); Moriya
_et al._ (2014); Nakamura  _et al._ (2012); Michiardi  _et al._ (2015);
Shanavas (2016); Liu  _et al._ (2018); Peng  _et al._ (2023); Amundsen  _et
al._ (2024). Notably, we demonstrate that higher-order Rashba SOC can induce
finite diode efficiency ($\gtrsim 15\%$) even without longitudinal magnetic
fields. This behavior differs qualitatively from the linear SOC case and can
be used to detect the presence of higher-order SOC terms in the nanowire. Our
study establishes the Rashba nanowire device as a promising platform for
highly efficient superconducting diodes, crucial for advancing nanoscale
superconducting electronics.

The remainder of this article is structured as follows. Sec. II introduces the
proximity-induced Rashba nanowire device and outlines the self-consistent
mean-field formalism used to investigate properties of the superconducting
ground state in the nanowire. Sec. III presents our main findings on
nonreciprocal charge transport in this device. We examine cases with both
linear and higher-order Rashba SOC terms, exploring their interplay with
Zeeman fields to optimize the SDE. Finally, Sec. IV summarizes our results,
discusses potential material realizations, and proposes future research
directions.

Figure 1: Schematic of the proximity-induced Rashba nanowire device: (a) The
schematic of our proposed device consisting of a Rashba nanowire in close
proximity to an $s$-wave superconductor and in presence of external Zeeman
fields: $B_{x}$ and $B_{z}$ to investigate SDE. The strength of the linear
Rashba SOC is denoted by $\alpha$, while the cubic Rashba SOC strengths are
represented by $\delta$ and $\gamma$ as illustrated in Eq. (2). (b) and (c):
The dispersions of the normal state and the Bogoliubov quasi particles in the
superconducting state are qualitatively shown in (b) and (c) respectively
considering both linear and higher order SOC.

##  II Model and Self-consistent mean-field formalism

We consider a single channel one-dimensional semiconducting nanowire along the
$z$-direction with both linear and higher order Rashba SOC terms Campos  _et
al._ (2018); Kammermeier  _et al._ (2018); Sonehara  _et al._ (2022); hso
placed in close proximity to a three dimensional bulk $s$-wave superconductor
as schematically shown in Fig. 1(a). In the presence of externally applied
magnetic fields, the normal state physics of the nanowire is governed by the
one-dimensional Rashba-Zeeman Hamiltonian (Alicea, 2012)

$\mathcal{H}_{0}=\sum_{s,s^{\prime}}\int
dz\,\psi_{s}^{\dagger}(z)h_{ss^{\prime}}(z)\psi_{s^{\prime}}(z),$
$\\!\\!\hat{h}(z)=\frac{p_{z}^{2}}{2m}-\mu+\left(\frac{\alpha}{\hbar}p_{z}+\frac{\delta}{\hbar^{3}}p_{z}^{3}\right)\sigma_{x}+\frac{\gamma}{\hbar^{3}}p_{z}^{3}\sigma_{y}+\mathbf{B}\cdot\boldsymbol{\sigma}\,.$
(1)

Here, the spin degree of freedom is labeled by
$s,s^{\prime}\in\\{\uparrow,\downarrow\\}$. $\psi_{s}^{\dagger}(z)$ creates a
fermion at the position $z$ with spin $s$, while
$\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the vector of the
Pauli matrices in spin space and $\mu$ is the chemical potential.
$p_{z}=\frac{\hbar}{i}\partial_{z}$ is the momentum operator in the direction
of current flow ($z$-direction) within the nanowire. The strength of the usual
linear Rashba SOC is denoted by $\alpha$ and the cubic Rashba SOC of strengths
are $\delta$ and $\gamma$. The presence of Rashba SOC breaks the inversion
symmetry of the system. The externally applied magnetic fields lead to the
Zeeman term characterized by $\mathbf{B}=(B_{x},B_{y},B_{z})$ that breaks
time-reversal symmetry. We assume the radius of the nanowire to be small such
that there is only one occupied mode, i.e., single channel. This nanowire
Hamiltonian [Eq. (1)] in momentum space can be written as

$\mathcal{H}_{0}=\sum_{s,s^{\prime}}\sum_{k}\,c_{k,s}^{\dagger}h_{ss^{\prime}}(k)c_{k,s^{\prime}},$
$\hat{h}(k)=\xi_{k}+\left[\left(\alpha k+\delta k^{3}\right)\sigma_{x}+\gamma
k^{3}\sigma_{y}\right]+\mathbf{B}\cdot\boldsymbol{\sigma}\,.$ (2)

Here, $k$ represents the momentum along the nanowire ($z$-direction) and
$\xi_{k}=\frac{\hbar^{2}k^{2}}{2m}-\mu$. The bulk energy spectrum of the
nanowire obtained by diagonalizing $\hat{h}_{k}$ is given by

$\epsilon_{\pm}(k)=\xi_{k}\pm\sqrt{(\alpha k+\delta k^{3}+B_{x})^{2}+(\gamma
k^{3}+B_{y})^{2}+B_{z}^{2}}$

where, $\pm$ label the two helicty bands. Note that, the Zeeman field $B_{z}$
(longitudinal field) opens a Zeeman gap while $B_{x}$ (transverse field)
creates asymmetry between the two bands as schematically shown in Fig. 1(b).
On the other hand, $B_{y}$ opens up a gap as well as creates band asymmetry
(not shown). Throughout the majority of this paper, we will discuss the
results with $B_{y}=0$ for simplicity and will separately discuss the effects
of nonzero $B_{y}$ for completeness.

We seek to elucidate the nature of the superconducting ground state in the
Rashba nanowire in the presence of $s$-wave superconducting correlations that
can be induced via the proximity effect in presence of a bulk $s$-wave
superconductor, as well as supercurrents (see Fig. 1(a)). In general, the
presence of supercurrents can lead to a superconducting state with finite
momentum Cooper pairs termed as the FFLO state. To establish that, we consider
an attractive Hubbard type interaction in the bulk $s$-wave superconductor
described by the Hamiltonian

$\mathcal{H}_{I}=-\frac{U}{2}\sum_{s,s^{\prime}}\int
d^{3}{\bf{r}}\,\psi_{s}^{\dagger}({\bf{r}})\psi_{s^{\prime}}^{\dagger}({\bf{r}})\psi_{s^{\prime}}({\bf{r}})\psi_{s}({\bf{r}})\,,$
(3)

where, $U$ is the attractive Hubbard interaction strength. We now decouple the
Hubbard interaction term ($\mathcal{H}_{I}$) in the finite momentum $s$-wave
channel within the mean-field approximation Daido  _et al._ (2022); Daido and
Yanase (2022); de Picoli  _et al._ (2023) and consider these correlations to
be proximity induced in the nanowire. Then, using the Bogoliubov de-Gennes
(BdG) mean-field formalism the effective Hamiltonian of the nanowire takes the
form

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\,dz\,\Psi^{\dagger}(z)\mathcal{H}_{BdG}(z)\Psi(z)+\mathcal{E}_{0}\,,$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\mathcal{H}_{BdG}(z)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\hat{h}(z)&\hat{\Delta}(z)\\\
-\hat{\Delta}^{*}(z)&-\hat{h}^{*}(z)\end{bmatrix};\,\hat{\Delta}(z)=-i\sigma_{y}\Delta
e^{iqz}\,,$ (4)

where,
$\Psi(z)=[\psi_{\uparrow}(z),\psi_{\downarrow}(z),\psi_{\uparrow}^{\dagger}(z),\psi_{\downarrow}^{\dagger}(z)]^{T}$
is the Nambu spinor inside the nanowire,
$\mathcal{E}_{0}=\frac{L}{U}|\Delta|^{2}$ with $L$ denoting the length of the
nanowire. The $s$-wave FFLO order parameter is given by
$\Delta(z)=\Delta(-i\sigma_{y})e^{iqz}$, where $q$ is the Cooper pair
momentum. The Hamiltonian in Eq. (4) effectively describes the physics of the
Rashba nanowire device schematically shown in Fig. 1(a). To proceed further,
we write the mean-field Hamiltonian [Eq. (4)] in the momentum space as

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k}\Psi_{k}^{\dagger}\mathcal{H}_{BdG}(k)\Psi_{k}+\mathcal{E}_{0}\,,$
$\displaystyle=$
$\displaystyle\sum_{n,k}\left(E_{n,k}\gamma^{\dagger}_{n,k}\gamma_{n,k}-\frac{E_{n,k}}{2}\right)+\mathcal{E}_{0}\,,$
$\text{\rm
with,}\;\;\mathcal{H}_{BdG}(k)=\begin{bmatrix}\hat{h}(k+\frac{q}{2})&-i\sigma_{y}\Delta\\\
i\sigma_{y}\Delta&-\hat{h}(-k+\frac{q}{2})^{*}\end{bmatrix}\,.$ (5)

Here,
$\Psi_{\mathbf{k}}\equiv[c_{k+q/2,\uparrow},c_{k+q/2,\downarrow},c_{-k+q/2,\uparrow}^{\dagger},c_{-k+q/2,\downarrow}^{\dagger}]^{T}$
and $E_{n,k}(q)$ denotes the Bololiubov quasiparticle energies. The behavior
of the Bogoliubov quasiparticle spectrum is schematically shown in Fig. 1(c).
This depicts that the introduction of the pairing potential $\Delta$ opens a
gap in the spectrum. As in the case of the nanowire spectrum, the asymmetry in
BdG bands occurs due to the application of the Zeeman field $B_{x}$, and the
Zeeman field $B_{z}$ makes the bands distinctly gapped.

Figure 2: FFLO ground state with linear SOC: (a) The self-consistent
superconducting gap $\Delta(q)$ is shown as a function of the Cooper-pair
momentum $q~{}$(nm-1) for $B_{x}/\Delta_{0}=0.3$ with only linear Rashba SOC
($\alpha=100$ meV-nm). $\Delta_{0}$ is the BCS gap in the absence of any
external magnetic field ($\mathbf{B}=0$). (b) The Cooper pair momentum of the
FFLO ground state $q_{0}$ as a function of $B_{x}$. The presence of the FFLO
ground state ($q_{0}\neq 0$) is established only when both $B_{x}$ and $B_{z}$
are finite. Other model parameters are chosen as:
$(B_{y},\mu/\Delta_{0},U,\beta^{-1})$=($0$, $0.5$, $25.55$ meV, $0.1$ meV).
Figure 3: FFLO ground state with both linear and higher order SOC: (a) The
self-consistent superconducting gap $\Delta(q)$ is depicted as a function of
Cooper-pair momentum $q~{}$ for $B_{z}=0$ with both the linear Rashba SOC term
($\alpha=100$ meV-nm) and the cubic Rashba SOC terms ($\delta=\gamma=10$ eV-
nm3). (b) The Cooper pair momentum of the FFLO ground state $q_{0}~{}$ as a
function of $B_{x}$ is shown. Note that the FFLO state is stable even when
$B_{z}=0$ in the presence of a finite $B_{x}$. Other system parameters are
chosen as: $(B_{y},\mu/\Delta_{0},U,\beta^{-1})$=($0$, $0.5$, $16.45$ meV,
$0.1$ meV).

The condensation energy $\Omega(q)$ of the system, defined as the difference
of free energy per unit length between the superconducting and normal state
Kinnunen  _et al._ (2018); Daido  _et al._ (2022), is

$\Omega(q,\Delta)=F(q,\Delta)-F(q,0)\,,$ (6)

where, $F(q,\Delta)$ is the free energy density of the nanowire given by

$F(q,\Delta)=-\frac{1}{L\beta}\sum_{n,k}\ln\left[1+e^{-\beta
E_{n,k}(q)}\right]+\frac{|\Delta(q)|^{2}}{U}\,,$ (7)

where, $\beta=(k_{B}T)^{-1}$ with $k_{B}$ being the Boltzmann constant and $T$
is the temperature.

Figure 4: Superconducting diode effect with linear SOC: (a) The supercurrent
density ($J$) is shown as a function of $q~{}$(nm-1) considering solely linear
Rashba SOC and $B_{x}/\Delta_{0}=0.3$. The nonreciprocity in the supercurrent,
$J(q)\neq-J(-q)$, is achieved in this case only when both $B_{x}$ and $B_{z}$
are finite. It then leads to nonreciprocity in the critical supercurrents,
$J_{c}^{+}\neq|J_{c}^{-}|$, hence exhibiting a finite diode efficiency
($\eta\neq 0$). Here, $J_{0}\equiv
J_{c}(B_{z}=0)=|J_{c}^{+}(B_{z}=0)|=|J_{c}^{-}(B_{z}=0)|$. (b) and (c): The
corresponding diode efficiency ($\eta$) is shown as a function of $B_{x}$
(with fixed $B_{z}$) and $B_{z}$ (with fixed $B_{x}$) with only linear Rashba
SOC in the panels (b) and (c) respectively. The system parameters are chosen
as: $(B_{y},\alpha,\mu/\Delta_{0},U,\beta^{-1})$=($0$, $100$ meV-nm, $0.5$,
$25.55$ meV, $0.1$ meV).

We determine the order parameter $\Delta(q)$ for a given value of $q$ by self-
consistently solving the gap equation, obtained via minimizing the free energy
$F(q,\Delta)$ in Eq. (7), given by

$\Delta(q)=-\frac{U}{L}\sum_{n,k}\frac{\partial
E_{n,k}}{\partial\Delta^{*}}\,n_{F}(E_{n,k})$ (8)

where $n_{F}(E_{n,k})=1/(1+e^{\beta E_{n,k}})$ is the Fermi function. Using
the self-consistent solution of $\Delta(q)$ in Eq. (6), we optimize the
condensation energy with respect to $q$ to obtain the Cooper pair momentum
($q_{0}$) of the true FFLO superconducting ground state. The BCS gap in the
absence of any external magnetic fields ($\mathbf{B}=0$) is denoted by
$\Delta_{0}$ and we choose the value of $U$ in such a way that we are in the
weak coupling BCS regime with $\Delta_{0}\sim 1$ meV. In the rest of the
paper, we present results in the low temperature limit choosing a fixed value
of $\beta^{-1}=0.1$ meV much smaller than $\Delta_{0}$.

The variation of the self-consistent solutions of the superconducting order
parameter $\Delta(q)$ and the momentum of the Cooper pairs $q_{0}$ in the FFLO
ground state with the external Zeeman fields are shown in Fig. 2 and Fig. 3
with only linear SOC case and, with both the linear and higher order SOC case
respectively. We note from Fig 2(a) and Fig 3(a) that for both the cases the
superconducting gap vanishes for a critical value of the Cooper pair momentum
either above a positive value of $q=q_{c}^{+}$ or below a negative value of
$q=q_{c}^{-}$. Fig 2(b) and Fig. 3(b) show that $q_{0}$ varies linearly with
$B_{x}$ for $B_{x}\ll\Delta_{0}$ for a given value of $B_{z}$. We note Fig
2(b) that in the case of only linear SOC $q_{0}$ is finite only when
$B_{z}\neq 0$ and $B_{x}\neq 0$. This is because when only linear SOC is
present the effect of the SOC can be gauged away by using a spin-dependent
gauge transformation Braunecker  _et al._ (2010) and then when only $B_{x}\neq
0$ the band structure is symmetric leading to zero momentum Cooper pairs.
However, when both $B_{x}$, $B_{z}\neq 0$, then in the transformed basis a
spiral magnetic field texture is realized leading to momentum dependent
asymmetry in the band structure and hence leading to finite momentum Cooper
pairs. In contrast, when both the linear and cubic SOC terms are present, the
effect of SOC can not be gauged away and as a result only $B_{x}\neq 0$ is
sufficient to give rise to $q_{0}\neq 0$ in this case. Consequently, this
feature also gets manifested in realizing finite SDE in this Rashba nanowire
device, as extensively discussed in the next section.

Figure 5: Superconducting diode effect with both linear and higher order SOC:
(a) The supercurrent density ($J$) as a function of $q~{}$(nm-1) with both
cubic Rashba SOC ($\delta$ = $\gamma$ = $10$ eV-nm3) and linear Rashba SOC
($\alpha=100$ meV-nm) choosing $B_{z}=0$. The nonreciprocity in the critical
supercurrents: $|J_{c}^{+}|\neq|J_{c}^{-}|$ (also implying a finite diode
efficiency $\eta\neq 0$) is achieved for any finite value of $B_{x}$. Here,
$J^{\prime}_{0}\equiv
J_{c}(B_{x}=B_{z}=0)=|J_{c}^{+}(B_{x}=B_{z}=0)|=|J_{c}^{-}(B_{x}=B_{z}=0)|$.
(b) and (c): The corresponding diode efficiency ($\eta$) is depicted as a
function of $B_{x}$ (with fixed $B_{z}$) and $B_{z}$ (with fixed $B_{x}$) in
the panels (b) and (c) respectively. We have chosen the other model parameters
as: $(B_{y},\mu/\Delta_{0},U,\beta^{-1})$=($0$, $0.5$, $16.45$ meV, $0.1$
meV).

##  III Non-reciprocal transport

The supercurrent density $J(q)$ is computed from the condensation energy
$\Omega(q,\Delta)$ in Eq. (6) as Yuan and Fu (2022); Daido  _et al._ (2022);
Legg  _et al._ (2022),

$J(q)=-2e\,\frac{\partial\Omega(q,\Delta)}{\partial q}.$ (9)

The sign of $J(q)$ indicates the direction of the supercurrent flow. The
critical (optimal) values of the supercurrent along and opposite to the flow
direction within the range $q_{c}^{-}\leq q\leq q_{c}^{+}$ for which the
superconducting phase is stable, are denoted as $J_{c}^{+}$ and $J_{c}^{-}$
respectively. When $|J_{c}^{+}|\neq|J_{c}^{-}|$, we have a finite SDE in the
system and the efficiency or the quality factor of the SDE is defined as

$\eta=\frac{|J_{c}^{+}|-|J_{c}^{-}|}{|J_{c}^{+}|+|J_{c}^{-}|}\ .$ (10)

We first compute $J(q)$ over the range $q_{c}^{-}\leq q\leq q_{c}^{+}$ using
the self-consistent solutions $\Delta(q)$ for each value of $q$ (shown in Fig
2(a) and Fig 3(a) for example). Then finding its optimal values $J_{c}^{+}$
and $J_{c}^{-}$, we evaluate the diode efficiency ($\eta$) numerically. We
systematically analyze the SDE in our Rashba nanowire device considering two
broadly distinct scenarios: with only linear SOC and, with both linear and
higher order SOC for various system parameters as discussed in the following
two subsections.

###  III.1 Case-I: Rashba nanowire with only linear SOC

In the presence of only linear Rashba SOC, we uncover the SDE by investigating
the behavior of the supercurrent density ($J(q)$) and the diode efficiency
($\eta$) for different system parameters as presented in Fig. 4. Fig. 4(a)
shows that the supercurrent density is symmetric with respect to $q$ for
$B_{z}=0$ even with $B_{x}\neq 0$ Legg  _et al._ (2022). We note that as we
start increasing the magnitude of $B_{z}$ with $B_{x}\neq 0$, $J(q)$ becomes
more and more asymmetric, leading to nonreciprocal critical supercurrents,
$|J_{c}^{+}|\neq|J_{c}^{-}|$ and hence to finite SDE. This is a direct
consequence of the fact that the presence of FFLO state and the asymmetry in
critical momenta: $|q_{c}^{+}|\neq|q_{c}^{-}|$ is distinctly realized only in
presence of both the fields $B_{x},B_{z}$ as noted in Fig. 2(a).

The diode efficiency as a function of $B_{x}$ for fixed values of $B_{z}$ and
as a function of $B_{z}$ for fixed values of $B_{x}$ are shown in Fig. 4(b)
and Fig. 4(c) respectively. In both cases, $\eta$ increases linearly at low
field strengths, reaches a maximum at moderate values, and then decreases to
zero as the field strengths continue to intensify. This behavior can be
understood as a consequence of a competition between the asymmetry in the band
structure of the system and changes in the superconducting pairing
susceptibility as the magnetic field strengths are varied, consistent with
previous studies Yuan and Fu (2022); Turini  _et al._ (2022). Note that, the
self-consistent BCS gap vanishes for very large values of the Zeeman fields
$B_{x}$ and $B_{z}$ consequently leading to zero efficiency, however the
resultant nanowire spectrum might still be gapped like an ordinary insulator.

Preceding investigations on a Rashba nanowire featuring only linear Rashba SOC
Legg  _et al._ (2022) reported maximum diode efficiency of $\sim 2\%$, where
the superconducting order parameter has been fixed to a particular value.
Contrastingly, our study reveals that the self-consistently derived
superconducting order parameter within the full parameters space has
significant influence on the performance of the diode where the highest $\eta$
value is found to be $\gtrsim 45\%$. This is evident from calculation
presented in Fig. 4(b,c). The vanishing of the superconducting gap within our
self-consistent treatment necessitates restricting the maximum values of the
magnetic fields used in our model. We note that in this case $B_{y}$ also
plays a similar role as $B_{z}$ due to the geometry of the setup (see Fig.
1(a)) under consideration and hence further analysis considering $B_{y}\neq 0$
is not presented. However, $B_{y}$ plays a significant and distinct role when
we consider the effect of cubic Rashba SOC terms as discussed in the next
subsection.

###  III.2 Case-II: Rashba nanowire with both linear and higher order SOC

The variation of $J(q)$ and $\eta$ in the case when both the linear
($\alpha\neq 0$) and cubic Rashba SOC terms ($\delta,\gamma\neq 0$) are
present in the nanowire are shown in Fig. 5. First of all, comparing Fig. 5(a)
with Fig. 4(a), we note that the cubic Rashba SOC terms interestingly lead to
asymmetry in $J(q)$ even with $B_{z}=0$ as long as $B_{x}\neq 0$. This leads
to nonreciprocal critical supercurrents, $|J_{c}^{+}|\neq|J_{c}^{-}|$ with a
single magnetic field $B_{x}\neq 0$ in contrast to only linear SOC case
discussed earlier. The variation of $\eta$ with the magnetic fields $B_{x}$
and $B_{z}$ with $B_{y}=0$ are shown in Fig. 5(b, c). We note from Fig. 5(b)
that the diode efficiency at a fixed $B_{z}$ exhibits a linear increase as a
function of $B_{x}$ for small values of $B_{x}\ll\Delta_{0}$, eventually
attaining a maxima around $B_{x}/\Delta_{0}=0.4$ similar to the case with only
linear Rashba SOC described earlier. This behavior, as in the earlier linear
SOC case Yuan and Fu (2022); Turini  _et al._ (2022), results from two
competing effects: the system’s asymmetric band structure versus changes in
superconducting pairing susceptibility as magnetic field strength varies. The
maximum efficiency increases as one increases the magnitude of $B_{z}$.
However, $\eta$ shows only marginal enhancement at a fixed value of $B_{x}$
when $B_{z}$ is varied as depicted in Fig. 5(c).

Figure 6: Diode efficiency phase diagram with both linear and higher order
SOC: The variation of the diode efficiency $\eta$ is shown in the plane of
cubic Rashba SOC parameters $\delta-\gamma$ choosing “optimal” values of the
Zeeman fields $(B_{x}/\Delta_{0},B_{z}/\Delta_{0})=(0.4,0.0)$ in panel (a) and
$(B_{x}/\Delta_{0},B_{z}/\Delta_{0})=(0.4,0.5)$ in panel (b), respectively. We
note a small enhancement in the maximum value of $\eta$ for finite $B_{z}$.
The other system parameters are chosen as:
$(B_{y},\alpha,\mu/\Delta_{0},\beta^{-1})$=$(0$, $100$ meV-nm, $0.5$, $0.1$
meV). Figure 7: The diode efficiency $\eta$ as a function of $B_{y}$ with both
linear and cubic Rashba SOC terms for $B_{x}=0$ and finite
$B_{x}/\Delta_{0}=0.3$ are shown in (a) and (b) respectively. The chosen model
parameters are $(\alpha,\delta,\gamma,\mu/\Delta_{0},\beta^{-1})$=($100$ meV-
nm, $10$ eV-nm3, $10$ eV-nm3, $0.5$, $0.1$ meV).

Modifying the values of $\delta$ and $\gamma$ evidently changes the band
dispersions in the normal and superconducting state of the system and
consequently the efficiency of the superconducting diode will also change. We
depict the diode efficiency phase diagrams in the $\delta-\gamma$ plane in
Fig. 6. We obtain the phase diagrams by varying $U$ for every configuration of
($\delta,\gamma$) such that the self-consistent BCS gap $\Delta_{0}$ remains
fixed. For the $B_{z}=0$ and $B_{x}\neq 0$ case, we can achieve a peak diode
efficiency of approximately $15\%$ for an optimal choice of higher-order
Rashba SOC parameters as shown in Fig. 6(a). We emphasize that this behavior
is qualitatively different from the linear SOC case, where the efficiency is
zero with only the $B_{x}$ field. This distinctive response, arising solely
from higher-order SOC terms, provides a method to detect and quantify the
strength of these terms in a nanowire device.

The application of an additional magnetic field $B_{z}$ readily elevates
$\eta$ to approximately $20\%$, as illustrated in Fig. 6(b) since effects of
both the linear and cubic SOC terms are relevant. Note that the presence of
highly asymmetric behavior in the supercurrent, given a specific set of system
parameters, does not necessarily guarantee a corresponding high level of non-
reciprocity in the super-currents. As is evident from Fig. 6(a,b), the system
might exhibit nearly negligible diode effect ($\eta\sim 0$) even though the
parameters are set such that there is asymmetry in supercurrent (top right
corners of Fig. 6(a,b)).

Note that the results discussed so far were obtained by fixing $B_{y}=0$. The
effect of finite $B_{y}$ in $\eta$ is shown in the Fig. 7. In this case,
$B_{y}$ serves the role of both $B_{x}$ and $B_{z}$ in realizing a finite SDE
by creating both gap and asymmetry in the band structure. Fig. 7(a) and Fig.
7(b) show that the application of $B_{y}$ produces systematic increase in
$\eta$ with increasing $B_{y}$ for different values of $B_{z}$ with $B_{x}=0$
and $B_{x}\neq 0$ respectively. However, $B_{y}$ can not increase the
efficiency beyond a certain limit because the superconducting state becomes
unstable after some value of $B_{y}$ when the other magnetic fields are
present as well.

Figure 8: The diode efficiency ($\eta$) with linear SOC and with cubic SOC of
different type where $\delta=10$ eV-nm3 but $\gamma=0$ as a function of
$B_{x}$ (with fixed $B_{z}$) and $B_{z}$ (with fixed $B_{x}$) are shown in (a)
and (b) respectively. Other parameters are:
$(\alpha,\mu/\Delta_{0},B_{y},\beta^{-1})$=($100$ meV-nm, $0.5$, $0$, $0.1$
meV).

For completeness, we consider a special case with a different type of the
higher order Rashba SOC terms hso  when only $\delta\neq 0$ and $\gamma=0$
(with $\alpha\neq 0$ as usual). The variation of the corresponding diode
efficiencies with the magnetic fields are shown in Fig. 8. A finite diode
efficiency with a maximum of $\sim 12\%$ even in presence of a finite $B_{x}$
only is obtained as shown in Fig. 8(a) which is also in contrast to the linear
SOC case. Thus the presence of even a single cubic Rashba parameter $\delta$
is sufficient to manifest SDE. By further application of $B_{z}$ the diode
efficiency $\eta$ is readily enhanced upto $\sim 21\%$ as a function of
$B_{x}$ shown in Fig. 8(a). There is only a marginal increase in efficiency by
increasing $B_{z}$ for fixed $B_{x}$ as shown in the Fig. 8(b).

##  IV Summary and conclusions

This article analyzes how the interplay between linear and cubic Rashba SOC
terms, combined with external Zeeman fields, can optimize the efficiency of
SDE in a proximitized one-dimensional nanowire device. Using self-consistent
mean-field analysis, we demonstrate that SDE, characterized by nonreciprocal
critical currents ($|J_{c}^{+}|\neq|J_{c}^{-}|$), arises from Cooper pair
momentum asymmetry in the helical superconducting ground state of this time-
reversal and inversion symmetry broken system. Remarkably, we achieve a diode
efficiency $\eta\gtrsim 45\%$ with only linear Rashba SOC significantly higher
than the previously reported efficiency of $\eta\sim 2\%$ from non-self-
consistent analyses Legg  _et al._ (2022). In this case with only linear SOC,
at least two magnetic fields are required to obtain a finite diode efficiency:
one along the SOC direction and one perpendicular to it. This requirement
stems from the fact that linear SOC effects can be eliminated through a spin-
dependent gauge transformation Braunecker  _et al._ (2010), resulting in a
conventional BCS superconducting state with zero Cooper pair momentum when
only one Zeeman field is present. However, the presence of two orthogonal
Zeeman fields creates a spiral magnetic field texture via the gauge
transformation, leading to momentum-dependent band asymmetry and asymmetric
FFLO Cooper pairs Braunecker  _et al._ (2010). In contrast, higher-order SOC
effects cannot be eliminated via gauge transformations, allowing for finite
diode efficiency even with only one Zeeman field unlike the linear SOC case
and this behavior can be used to detect the presence of higher-order SOC in
nanowires. However, including higher-order SOC doesn’t improve the maximum
diode efficiency (only $\eta\sim 20\%$), this value remains significant
compared to previously reported diode efficiencies Legg  _et al._ (2022);
Nadeem  _et al._ (2023). Our results underscore the importance of self-
consistently determining the order parameter of the superconducting ground
state to fully understand the capability and optimize Rashba nanowire devices
as superconducting diodes.

Our findings can be readily experimentally tested in spin orbit coupled Rashba
nanowires (Alicea, 2012; Liang and Gao, 2012; Manchon  _et al._ , 2015;
Bercioux and Lucignano, 2015) such as zinc-blende InSb and wurtzite InAs, with
lengths of approximately 100 nm, where cubic Rashba SOC interactions are
significant depending on the growth direction Campos  _et al._ (2018).
Superconductivity can be proximity induced in these nanowires by common bulk
$s$-wave superconductors like Nb or Al. Future research directions include
investigating SDE in proximity-induced multichannel Rashba nanowires with
overlapping channels Potter and Lee (2010); Lutchyn  _et al._ (2011); Law and
Lee (2011); Alicea (2012), extending beyond the single-channel case examined
here. Additionally, the influence of topological superconductivity realized in
the nanowire and the nanowire device geometry on SDE warrants further
exploration Liu  _et al._ (2023); Kutlin and Mel’nikov (2020); Kopasov  _et
al._ (2021); Legg  _et al._ (2023); Nesterov  _et al._ (2016); Spånslätt
(2018). An intriguing prospect is the potential replication of magnetic field
effects using electric fields via the Rashba-Edelstein effect Vignale and
Tokatly (2016); Smirnov and Golub (2017); Piatti (2021); Nadeem  _et al._
(2023), which could lead to the realization of field-free SDE.

##  V Acknowledgments

SKG acknowledges financial support from SERB, Government of India via the
Startup Research Grant: SRG/2023/000934 and IIT Kanpur via the Initiation
Grant (IITK/PHY/2022116). A.K.N. acknowledges the financial support from
Department of Atomic Energy (DAE), Govt. of India, through the project Basic
Research in Physical and Multidisciplinary Sciences via RIN4001. The authors
acknowledge discussions with Koushik Mandal, Krishnendu Sengupta and Amit
Agarwal. DS and SKG utilized the Andromeda server at IIT Kanpur for numerical
calculations. SB and AS acknowledge SAMKHYA: High-Performance Computing
Facility provided by Institute of Physics, Bhubaneswar, for numerical
computations.

## References

  * Scaff and Ohl (1947) J. H. Scaff and R. S. Ohl, “Development of silicon crystal rectifiers for microwave radar receivers,” The Bell System Technical Journal 26, 1–30 (1947).
  * Shockley (1949) W. Shockley, “The theory of p-n junctions in semiconductors and p-n junction transistors,” The Bell System Technical Journal 28, 435–489 (1949).
  * Tokura and Nagaosa (2018) Yoshinori Tokura and Naoto Nagaosa, “Nonreciprocal responses from non-centrosymmetric quantum materials,” Nature communications 9, 3740 (2018).
  * Qin  _et al._ (2017) F Qin, W Shi, T Ideue, M Yoshida, A Zak, Reshef Tenne, T Kikitsu, D Inoue, D Hashizume, and Y Iwasa, “Superconductivity in a chiral nanotube,” Nature communications 8, 14465 (2017).
  * Wakatsuki  _et al._ (2017) Ryohei Wakatsuki, Yu Saito, Shintaro Hoshino, Yuki M Itahashi, Toshiya Ideue, Motohiko Ezawa, Yoshihiro Iwasa, and Naoto Nagaosa, “Nonreciprocal charge transport in noncentrosymmetric superconductors,” Science advances 3, e1602390 (2017).
  * Nadeem  _et al._ (2023) Muhammad Nadeem, Michael S Fuhrer, and Xiaolin Wang, “The superconducting diode effect,” Nature Reviews Physics 5, 558–577 (2023).
  * Nagaosa and Yanase (2024) Naoto Nagaosa and Youichi Yanase, “Nonreciprocal transport and optical phenomena in quantum materials,” Annual Review of Condensed Matter Physics 15, 63–83 (2024).
  * Narita and Ono (2024) Hideki Narita and Teruo Ono, “Superconducting diode effect in artificial superlattices,” JSAP Review 2024, 240206 (2024).
  * Ando  _et al._ (2020) Fuyuki Ando, Yuta Miyasaka, Tian Li, Jun Ishizuka, Tomonori Arakawa, Yoichi Shiota, Takahiro Moriyama, Youichi Yanase, and Teruo Ono, “Observation of superconducting diode effect,” Nature 584, 373–376 (2020).
  * Sundaresh  _et al._ (2023) Ananthesh Sundaresh, Jukka I Väyrynen, Yuli Lyanda-Geller, and Leonid P Rokhinson, “Diamagnetic mechanism of critical current non-reciprocity in multilayered superconductors,” Nature Communications 14, 1628 (2023).
  * Itahashi  _et al._ (2020) Yuki M Itahashi, Toshiya Ideue, Yu Saito, Sunao Shimizu, Takumi Ouchi, Tsutomu Nojima, and Yoshihiro Iwasa, “Nonreciprocal transport in gate-induced polar superconductor SrTiO3,” Science advances 6, eaay9120 (2020).
  * Schumann  _et al._ (2020) Timo Schumann, Luca Galletti, Hanbyeol Jeong, Kaveh Ahadi, William M. Strickland, Salva Salmani-Rezaie, and Susanne Stemmer, “Possible signatures of mixed-parity superconductivity in doped polar $\mathrm{SrTi}{\mathrm{O}}_{3}$ films,” Phys. Rev. B 101, 100503 (2020).
  * Lin  _et al._ (2022) Jiang-Xiazi Lin, Phum Siriviboon, Harley D Scammell, Song Liu, Daniel Rhodes, K Watanabe, T Taniguchi, James Hone, Mathias S Scheurer, and JIA Li, “Zero-field superconducting diode effect in small-twist-angle trilayer graphene,” Nature Physics 18, 1221–1227 (2022).
  * Díez-Mérida  _et al._ (2023) J Díez-Mérida, A Díez-Carlón, SY Yang, Y-M Xie, X-J Gao, J Senior, K Watanabe, T Taniguchi, X Lu, Andrew P Higginbotham, _et al._ , “Symmetry-broken josephson junctions and superconducting diodes in magic-angle twisted bilayer graphene,” Nature Communications 14, 2396 (2023).
  * Bauriedl  _et al._ (2022) Lorenz Bauriedl, Christian Bäuml, Lorenz Fuchs, Christian Baumgartner, Nicolas Paulik, Jonas M Bauer, Kai-Qiang Lin, John M Lupton, Takashi Taniguchi, Kenji Watanabe, _et al._ , “Supercurrent diode effect and magnetochiral anisotropy in few-layer NbSe2,” Nature communications 13, 4266 (2022).
  * Yun  _et al._ (2023) Jonginn Yun, Suhan Son, Jeacheol Shin, Giung Park, Kaixuan Zhang, Young Jae Shin, Je-Geun Park, and Dohun Kim, “Magnetic proximity-induced superconducting diode effect and infinite magnetoresistance in a van der Waals heterostructure,” Phys. Rev. Res. 5, L022064 (2023).
  * Burlakov  _et al._ (2014) AA Burlakov, VL Gurtovoi, AI Il’in, Alexey V Nikulov, and VA Tulin, “Superconducting quantum interference device without josephson junctions,” JETP letters 99, 169–173 (2014).
  * Yuan and Fu (2022) Noah F. Q. Yuan and Liang Fu, “Supercurrent diode effect and finite-momentum superconductors,” Proceedings of the National Academy of Sciences 119, e2119548119 (2022).
  * Ilić and Bergeret (2022) S. Ilić and F. S. Bergeret, “Theory of the supercurrent diode effect in rashba superconductors with arbitrary disorder,” Phys. Rev. Lett. 128, 177001 (2022).
  * Turini  _et al._ (2022) Bianca Turini, Sedighe Salimian, Matteo Carrega, Andrea Iorio, Elia Strambini, Francesco Giazotto, Valentina Zannier, Lucia Sorba, and Stefan Heun, “Josephson diode effect in high-mobility insb nanoflags,” Nano Letters 22, 8502–8508 (2022).
  * Roig  _et al._ (2024) Mercè Roig, Panagiotis Kotetes, and Brian M. Andersen, “Superconducting diodes from magnetization gradients,” Phys. Rev. B 109, 144503 (2024).
  * Halterman  _et al._ (2022) Klaus Halterman, Mohammad Alidoust, Ross Smith, and Spencer Starr, “Supercurrent diode effect, spin torques, and robust zero-energy peak in planar half-metallic trilayers,” Phys. Rev. B 105, 104508 (2022).
  * Daido and Yanase (2022) Akito Daido and Youichi Yanase, “Superconducting diode effect and nonreciprocal transition lines,” Phys. Rev. B 106, 205206 (2022).
  * Karabassov  _et al._ (2022) T. Karabassov, I. V. Bobkova, A. A. Golubov, and A. S. Vasenko, “Hybrid helical state and superconducting diode effect in superconductor/ferromagnet/topological insulator heterostructures,” Phys. Rev. B 106, 224509 (2022).
  * Legg  _et al._ (2022) Henry F. Legg, Daniel Loss, and Jelena Klinovaja, “Superconducting diode effect due to magnetochiral anisotropy in topological insulators and Rashba nanowires,” Phys. Rev. B 106, 104501 (2022).
  * Levitov  _et al._ (1985) LS Levitov, Yu V Nazarov, and GM Eliashberg, “Magnetostatics of superconductors without an inversion center,” JETP Lett 41, 365 (1985).
  * Edelstein (1996) Victor M Edelstein, “The ginzburg-landau equation for superconductors of polar symmetry,” Journal of Physics: Condensed Matter 8, 339 (1996).
  * Daido  _et al._ (2022) Akito Daido, Yuhei Ikeda, and Youichi Yanase, “Intrinsic superconducting diode effect,” Phys. Rev. Lett. 128, 037001 (2022).
  * He  _et al._ (2022) James Jun He, Yukio Tanaka, and Naoto Nagaosa, “A phenomenological theory of superconductor diodes,” New Journal of Physics 24, 053014 (2022).
  * Pal  _et al._ (2022) Banabir Pal, Anirban Chakraborty, Pranava K Sivakumar, Margarita Davydova, Ajesh K Gopi, Avanindra K Pandeya, Jonas A Krieger, Yang Zhang, Mihir Date, Sailong Ju, _et al._ , “Josephson diode effect from cooper pair momentum in a topological semimetal,” Nature physics 18, 1228–1233 (2022).
  * de Picoli  _et al._ (2023) Tatiana de Picoli, Zane Blood, Yuli Lyanda-Geller, and Jukka I. Väyrynen, “Superconducting diode effect in quasi-one-dimensional systems,” Phys. Rev. B 107, 224518 (2023).
  * Banerjee and Scheurer (2024) Sayan Banerjee and Mathias S. Scheurer, “Enhanced superconducting diode effect due to coexisting phases,” Phys. Rev. Lett. 132, 046003 (2024).
  * Fulde and Ferrell (1964) Peter Fulde and Richard A. Ferrell, “Superconductivity in a strong spin-exchange field,” Phys. Rev. 135, A550–A563 (1964).
  * Larkin and Ovchinnikov (1965) AI Larkin and Yu N Ovchinnikov, “Nonuniform state of superconductors,” Soviet Physics-JETP 20, 762–762 (1965).
  * Oreg  _et al._ (2010) Yuval Oreg, Gil Refael, and Felix von Oppen, “Helical liquids and majorana bound states in quantum wires,” Phys. Rev. Lett. 105, 177002 (2010).
  * Sau  _et al._ (2010) Jay D. Sau, Sumanta Tewari, Roman M. Lutchyn, Tudor D. Stanescu, and S. Das Sarma, “Non-abelian quantum order in spin-orbit-coupled semiconductors: Search for topological majorana particles in solid-state systems,” Phys. Rev. B 82, 214509 (2010).
  * Lutchyn  _et al._ (2010) Roman M. Lutchyn, Jay D. Sau, and S. Das Sarma, “Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
  * Stanescu  _et al._ (2011) Tudor D. Stanescu, Roman M. Lutchyn, and S. Das Sarma, “Majorana fermions in semiconductor nanowires,” Phys. Rev. B 84, 144522 (2011).
  * Campos  _et al._ (2018) Tiago Campos, Paulo E. Faria Junior, Martin Gmitra, Guilherme M. Sipahi, and Jaroslav Fabian, “Spin-orbit coupling effects in zinc-blende insb and wurtzite inas nanowires: Realistic calculations with multiband $\mathbf{k}\cdot{}\mathbf{p}$ method,” Phys. Rev. B 97, 245402 (2018).
  * Kammermeier  _et al._ (2018) Michael Kammermeier, Paul Wenk, Florian Dirnberger, Dominique Bougeard, and John Schliemann, “Spin relaxation in wurtzite nanowires,” Phys. Rev. B 98, 035407 (2018).
  * Sonehara  _et al._ (2022) Junpei Sonehara, Michael Kammermeier, Dai Sato, Daisuke Iizasa, Ulrich Zülicke, Shutaro Karube, Junsaku Nitta, and Makoto Kohda, “Anisotropic spin dynamics in semiconductor narrow wires from the interplay between spin-orbit interaction and planar magnetic field,” Phys. Rev. B 105, 094434 (2022).
  * Vajna  _et al._ (2012) Sz. Vajna, E. Simon, A. Szilva, K. Palotas, B. Ujfalussy, and L. Szunyogh, “Higher-order contributions to the Rashba-Bychkov effect with application to the Bi/Ag(111) surface alloy,” Phys. Rev. B 85, 075404 (2012).
  * Moriya  _et al._ (2014) Rai Moriya, Kentarou Sawano, Yusuke Hoshi, Satoru Masubuchi, Yasuhiro Shiraki, Andreas Wild, Christian Neumann, Gerhard Abstreiter, Dominique Bougeard, Takaaki Koga, and Tomoki Machida, “Cubic rashba spin-orbit interaction of a two-dimensional hole gas in a strained-$\mathrm{Ge}/\mathrm{SiGe}$ quantum well,” Phys. Rev. Lett. 113, 086601 (2014).
  * Nakamura  _et al._ (2012) H. Nakamura, T. Koga, and T. Kimura, “Experimental evidence of cubic Rashba effect in an inversion-symmetric oxide,” Phys. Rev. Lett. 108, 206601 (2012).
  * Michiardi  _et al._ (2015) Matteo Michiardi, Marco Bianchi, Maciej Dendzik, Jill A. Miwa, Moritz Hoesch, Timur K. Kim, Peter Matzen, Jianli Mi, Martin Bremholm, Bo Brummerstedt Iversen, and Philip Hofmann, “Strongly anisotropic spin-orbit splitting in a two-dimensional electron gas,” Phys. Rev. B 91, 035445 (2015).
  * Shanavas (2016) K. V. Shanavas, “Theoretical study of the cubic Rashba effect at the ${\mathrm{SrTiO}}_{3}$ (001) surfaces,” Phys. Rev. B 93, 045108 (2016).
  * Liu  _et al._ (2018) Hong Liu, E. Marcellina, A. R. Hamilton, and Dimitrie Culcer, “Strong spin-orbit contribution to the hall coefficient of two-dimensional hole systems,” Phys. Rev. Lett. 121, 087701 (2018).
  * Peng  _et al._ (2023) Xiong-Tao Peng, Fang Lin, Liang Chen, Lin Li, Dong-Hui Xu, and Jin-Hua Sun, “Tunable correlation effects of magnetic impurities by cubic Rashba spin-orbit coupling,” Phys. Rev. B 107, 165148 (2023).
  * Amundsen  _et al._ (2024) Morten Amundsen, Jacob Linder, Jason W. A. Robinson, Igor Žutić, and Niladri Banerjee, “Colloquium: Spin-orbit effects in superconducting hybrid structures,” Rev. Mod. Phys. 96, 021003 (2024).
  * (50) The form of the Higher order Rashba SOC terms depends on the particular symmetry directions in which the nanowire is grown Campos  _et al._ (2018).
  * Alicea (2012) Jason Alicea, “New directions in the pursuit of majorana fermions in solid state systems,” Reports on progress in physics 75, 076501 (2012).
  * Kinnunen  _et al._ (2018) Jami J Kinnunen, Jildou E Baarsma, Jani-Petri Martikainen, and Päivi Törmä, “The Fulde–Ferrell–Larkin–Ovchinnikov state for ultracold fermions in lattice and harmonic potentials: a review,” Reports on Progress in Physics 81, 046401 (2018).
  * Braunecker  _et al._ (2010) Bernd Braunecker, George I. Japaridze, Jelena Klinovaja, and Daniel Loss, “Spin-selective peierls transition in interacting one-dimensional conductors with spin-orbit interaction,” Phys. Rev. B 82, 045127 (2010).
  * Liang and Gao (2012) Dong Liang and Xuan PA Gao, “Strong tuning of rashba spin–orbit interaction in single InAs nanowires,” Nano Letters 12, 3263–3267 (2012).
  * Manchon  _et al._ (2015) Aurelien Manchon, Hyun Cheol Koo, Junsaku Nitta, Sergey M Frolov, and Rembert A Duine, “New perspectives for Rashba spin–orbit coupling,” Nature materials 14, 871–882 (2015).
  * Bercioux and Lucignano (2015) Dario Bercioux and Procolo Lucignano, “Quantum transport in Rashba spin–orbit materials: a review,” Reports on Progress in Physics 78, 106001 (2015).
  * Potter and Lee (2010) Andrew C. Potter and Patrick A. Lee, “Multichannel generalization of kitaev’s majorana end states and a practical route to realize them in thin films,” Phys. Rev. Lett. 105, 227003 (2010).
  * Lutchyn  _et al._ (2011) Roman M. Lutchyn, Tudor D. Stanescu, and S. Das Sarma, “Search for majorana fermions in multiband semiconducting nanowires,” Phys. Rev. Lett. 106, 127001 (2011).
  * Law and Lee (2011) K. T. Law and Patrick A. Lee, “Robustness of majorana fermion induced fractional josephson effect in multichannel superconducting wires,” Phys. Rev. B 84, 081304 (2011).
  * Liu  _et al._ (2023) Zhaochen Liu, Linghao Huang, and Jing Wang, “Josephson diode effect in topological superconductor,” arXiv preprint arXiv:2311.09009 (2023).
  * Kutlin and Mel’nikov (2020) A. G. Kutlin and A. S. Mel’nikov, “Geometry-dependent effects in majorana nanowires,” Phys. Rev. B 101, 045418 (2020).
  * Kopasov  _et al._ (2021) A. A. Kopasov, A. G. Kutlin, and A. S. Mel’nikov, “Geometry controlled superconducting diode and anomalous josephson effect triggered by the topological phase transition in curved proximitized nanowires,” Phys. Rev. B 103, 144520 (2021).
  * Legg  _et al._ (2023) Henry F. Legg, Katharina Laubscher, Daniel Loss, and Jelena Klinovaja, “Parity-protected superconducting diode effect in topological josephson junctions,” Phys. Rev. B 108, 214520 (2023).
  * Nesterov  _et al._ (2016) Konstantin N. Nesterov, Manuel Houzet, and Julia S. Meyer, “Anomalous josephson effect in semiconducting nanowires as a signature of the topologically nontrivial phase,” Phys. Rev. B 93, 174502 (2016).
  * Spånslätt (2018) Christian Spånslätt, “Geometric josephson effects in chiral topological nanowires,” Phys. Rev. B 98, 054508 (2018).
  * Vignale and Tokatly (2016) Giovanni Vignale and I. V. Tokatly, “Theory of the nonlinear Rashba-Edelstein effect: The clean electron gas limit,” Phys. Rev. B 93, 035310 (2016).
  * Smirnov and Golub (2017) D. S. Smirnov and L. E. Golub, “Electrical spin orientation, spin-galvanic, and spin-hall effects in disordered two-dimensional systems,” Phys. Rev. Lett. 118, 116801 (2017).
  * Piatti (2021) Erik Piatti, “Ionic gating in metallic superconductors: A brief review,” Nano Express 2, 024003 (2021).