# Multitask Learning for Low Resource Spoken Language Understanding

###### Abstract

We explore the benefits that multitask learning offer to speech processing as
we train models on dual objectives with automatic speech recognition and
intent classification or sentiment classification. Our models, although being
of modest size, show improvements over models trained end-to-end on intent
classification. We compare different settings to find the optimal disposition
of each task module compared to one another. Finally, we study the performance
of the models in low-resource scenario by training the models with as few as
one example per class. We show that multitask learning in these scenarios
compete with a baseline model trained on text features and performs
considerably better than a pipeline model. On sentiment classification, we
match the performance of an end-to-end model with ten times as many
parameters. We consider 4 tasks and 4 datasets in Dutch and English.

Index Terms—  Spoken Language Understanding, Transformers, Speech Processing,
Multitask Learning, Pretraining

##  1 Introduction

Spoken language understanding (SLU) generally follows a pipeline approach,
where an automatic speech recognition (ASR) system is followed by a natural
language understanding (NLU) module. These systems know multiple limitations,
including cascading errors from one module to the next or discarding relevant
information related to speech not captured by text. This motivated research
directed towards end-to-end SLU that removed the explicit speech transcription
required by older systems [1, 2]. End-to-end approaches concerned only with
SLU tasks, however, struggle to match the performance of pipeline systems.
Indeed, these specific datasets are considerably smaller and neural networks
notably require large amounts of data to perform well. To circumvent this data
scarcity issue, pretraining and multitask learning offer an elegant solution.
Pretraining allows applying what a neural network learned from a large dataset
to another domain by partly or entirely initializing the weights in the model
of interest with the weights from the pretrained model. If the tasks share
certain characteristics with the ultimate objective and the domain shift
between the datasets is not too big, we can decrease considerably the amount
of training examples and computation resources necessary to train the final
model. During the actual training or finetuning of the model, a trade-off
appears, as further training will inevitably lead to a specialization in favor
of the new task and catastrophic forgetting of what was learned during the
pretraining phase. In contrast, with multitask learning, the tasks are trained
in parallel. In this case, a similar trade-off occurs, caused by competing
optimization objectives. This approach relies strongly on the assumption that
the multiple objectives share a common ground and that the ability to solve
one helps on solving others. It is easy to imagine many examples for which
this assumption might hold or not, depending on the tasks and datasets
considered. In this work, we investigate synergies between spoken language
understanding and speech recognition. One task of interest is intent
classification (IC), where a system must identify an intent with any number of
arguments. A second task investigated here is sentiment classification, where
the goal is to determine whether a spoken utterance expresses a positive,
negative or neutral sentiment. Having a system able to transcribe speech to
text is likely to be helpful for such tasks and requires fewer data and fewer
parameters to reach a good performance [3, 4]. These are two aspects that we
will focus on in this work: We compare a multitask model optimized for both
speech transcription and intent recognition or sentiment classification with a
model pretrained on ASR then finetuned on SLU. We also look into edge cases in
IC by simulating training sets with few examples. Further, we limit our model
sizes considerably for the advantages in terms of speed and energy consumption
that it brings. Experiments are performed in English and in Dutch.

##  2 Architecture

The model is composed of a generic transformer encoder on which we stack
different modules, each specialized for one task. We experiment with different
arrangements in order to find the optimal architecture that allows each task
to benefit from - rather than compete with - each other. We have selected four
tasks of interest: automatic speech recognition (ASR), intent classification
(IC) and sentiment classification (SC). In this work, we explore the synergies
between each task and ASR and leave other combinations for future works.
Additionally, we explicitly want our models to be relatively small. Smaller
models require less energy both during training and inference, which is good
from a cost and environmental perspective.

###  2.1 Encoder

The encoder is the only component that is entirely shared with all the tasks.
It is composed of a VGG-like feature extractor followed by transformer encoder
layers. The role of the convolutional frontend is two-fold: decrease the
dimension along the frequency and time axes and learn the local relationships
in the sequence. This has been successfully used in previous works to replace
the positional encoding traditionally required by transformers [5]. A sequence
of speech features of length $T$ is reduced to a length $T^{\prime}<T$ by the
CNN, where $T^{\prime}\approx T/4$. The resulting features are further
processed by a series of 12 self-attention and feedforward layers. The output
of the encoder is a sequence of features of 256 units. It is used as input to
the other modules and is never optimized in a standalone fashion. The encoder
has under 18 million parameters. Wav2Vec 2.0 [6] has more than 300 million.

###  2.2 Automatic Speech Recognition

For the ASR module, we use a hybrid setting with a transformer decoder coupled
with a CTC classification layer [7]. The prediction units are 5,000 subword
units. The CTC layer assigns to each acoustic unit a symbol that is either a
token from the vocabulary or the blank symbol. A sentence can be derived
either using a greedy algorithm that first removes duplicated symbols then
blank tokens, or with a more fine-grained approach such as beam search. The
transformer decoder generates one token at a time by comparing previously
generated tokens with the acoustic evidence brought by the encoder. Following
[7], the loss corresponding to this task combines a CTC loss [8] and a cross-
entropy loss. We also use label smoothing to avoid overconfident predictions.
The total number of parameters for the ASR head is 13.3 million.

###  2.3 Intent Classification

Intent is composed of an action (e.g. move) and a number of arguments (e.g.
where: forward, how: slowly). We encode the intent as a multi-hot vector where
the first $k$ bits correspond to the $k$ actions, the following $l$ bits to
the $l$ possible values for the first argument and so on. Inspired by [9], we
use multihead class attention to summarize the sequence of embeddings into one
prediction. We introduce a special learned token, $x_{\text{CLS}}$, that is
attending to each element in the sequence. In the original notation from [10],
we substitute the embedded queries, $Q$, with $x_{\text{CLS}}$. However,
contrarily to [9], we exclude the CLS token from the keys and values. Class
attention compares the class embeddings, $x_{\text{CLS}}$, to the sequence of
representations, $X$, to identify and extract relevant information from the
embedded sequence. The task loss is binary cross-entropy. The intent
classification module has only 825 thousand parameters.

###  2.4 Sentiment Classification

For sentiment classification, we use the same architecture as in Section 2.3.
The loss for this task is cross-entropy. We apply label smoothing to avoid
overconfident predictions.

##  3 Methodology

In this section, we describe how we train and evaluate the models. For each
language, we have selected one ASR dataset to pretrain the encoder and ASR
decoder and two datasets to perform multitask learning (MTL). Pretraining has
shown its benefits in numerous applications and serves here as a shortcut to
save considerable computing resources by initializing parts of our model. We
pretrain our model with ESPnet’s recipe for ASR. At this stage, we freeze the
encoder’s parameters, as our experiments did not show improvements when the
encoder was updated. The decoder is trained with teacher forcing, meaning that
the true tokens are used as input rather than the decoder’s own prediction.
Although this introduces a difference between training and inference, it also
makes the training procedure faster by allowing parallel training, and more
stable by avoiding introducing errors while the encoder is still learning.

###  3.1 Training

All downstream datasets have at least two targets: transcriptions for ASR and
labels for IC and SC. Given a speech utterance, the model predicts both
targets. Then, the weighted sum of the errors relative to each task is
backpropagated through the network and the parameters are updated accordingly.
The decoder is thus trained jointly on ASR and either IC or SC. We set the
loss weights experimentally by measuring the performance on the validation
set.

###  3.2 Evaluation

We evaluate the models by measuring the task performance on a left-out subset.
For intent classification, we report the accuracy. Note that for calculating
the accuracy, we consider that a prediction is correct if all the elements are
correct – intent and arguments. For sentiment classification, we report the
macro averaged f1-score.
In addition, we explore the ability of our model to learn in data shortage
scenarios, using as few training examples as possible. We follow [11]’s
direction and train our multitask models with varying amount of training data
for one speaker. In practice, for each speaker dataset, we define several
training sets of increasing sizes to train and evaluate the models. This
allows us to build learning curves that show the average performance of the
same model on increasing training sizes. We compare these results with two
types of baseline features: one is obtained by encoding the gold transcription
with a pretrained BERT (NLP in Figure 2), and the second is obtained by
transcribing speech utterances with an ASR model and encoding them with a
pretrained BERT (Pipeline in Figure 2).

##  4 Experiments

We explore datasets in both English and Dutch. We use the Corpus Gesproken
Nederlands (CGN) [12] and Librispeech [13] for pretraining. The 80-dimensional
MEL-filterbanks are pre-computed for all the datasets. We modified ESPnet [7]
to fit our specific requirements. For the actual training of the multitask
models, we use 2 datasets in Dutch and 2 datasets in English.

###  4.1 Datasets

#### Corpus Gesproken Nederlands

[12] is a collection of recordings in Dutch and Flemish collected from various
sources totaling more than 900 hours. They are divided into 15 components (‘a’
to ‘o’) based on their nature. After removing short and overlapping
utterances, we divide each component into three subsets that will serve as
training, validation and test sets. We leave out three components (‘a’, ‘c’
and ‘d’) because they differ considerably from the other ones. The subsets
from the remaining components, totaling 415 hours of speech, are concatenated
together to form the training, validation and test sets.

#### Librispeech

[13] contains about a thousand hours of read English speech derived from
audiobooks. The authors provide official splits for training, validating and
testing the models and this dataset serves as reference in ASR. We use all 960
hours for training.

#### Grabo

[11] is composed of spoken commands, mostly in Dutch (one speaker speaks in
English) intended to control a robot. There are 36 commands that were repeated
15 times by each of the 11 speakers. More precisely, the robot can perform
eight actions, each being further defined by some attributes (e.g., the robot
can move forward or backward, rapidly or slowly). We follow the same
methodology as [14].

#### Patience Corpus

[15] is derived from a card game where the player is giving vocal instructions
to move cards between different stacks on the table. The dataset contains
recordings from 8 different Flemish speakers. The intent of the player is
represented as a multi-hot vector. Again, we use the same splitting
methodology as [14].

#### Fluent Speech Commands

[2] contains 30,043 utterances from 97 English speakers. Each utterance is a
command to control smart-home appliances or virtual assistant. We use the
challenge splits proposed in [16]. They propose two test sets, one with
unknown speakers (FSC-S) and one with unknown utterances (FSC-U)

#### SLUE-VoxCeleb

[17] is a benchmark dataset for sentiment classification. The task is to
classify each utterance as being positive, negative or neutral.

###  4.2 Model Selection

(a) Multitask Learning

(b) Single task (SLU)

Fig. 1: Comparison of different settings on Grabo for the multitask model
where the SLU module is connected to the output of the encoder (encoder.11) or
the first (ASR.0), third (ASR.2) or last layer (ASR.5) of the ASR decoder.
Each box represents the distribution of the score across speakers and folds
where each model was trained on a dataset with 2 examples per class.
Performance of a multitask model trained on ASR+SLU (left) and a model trained
solely on SLU (right) is shown.

In this experiment, we compare the model performance on the SLU task depending
on the features it receives, namely intermediary representations from the
first, third or last layer of the ASR decoder or from the encoder’s output. We
perform the experiment first on Grabo dataset where we build small training
sets with 2 examples per class to simulate a data shortage scenario. All the
models are initialized with the same pretrained weights to ensure comparable
results. We perform this operation three times and report the accuracy in
Figure 1. The model where the SLU module connects to the encoder’s output
shows the largest variability and the lowest average performance. The
encoder’s output corresponds to acoustic units that do not have much to do
with language. The decoder, however, produces a sequence of subword units
given the acoustic evidence and the previous predicted tokens. When the SLU
module is connected to the encoder’s output, it must summarize a longer
sequence where each element conveys acoustic rather than linguistic
information. In contrast, aside from the obvious advantage that the SLU head
must cope with shorter sentences, the implicit language model in the decoder
gives another meaning to the tokens that is dictated by the masked language
modelling objective. These representations consider the acoustics through the
cross-attention but also linguistic elements with the self-attention. The
model trained on features from the last layer of the decoder shows an
increased number of outliers, depicted by the diamonds in Figure 1. This
indicates that there might be a trade-off between using better-defined or
lower-level features as we change the capacity of the ASR module dedicated to
only one task. In other words, when the SLU head is connected to the last
decoder layer, the decoder has fewer parameters to learn elements necessary to
ASR but detrimental, or unnecessary to SLU. Consequently, competition arises
between both objectives.
On Figure 1, the performance of the models trained with a unique objective is
lower than when we optimize for both ASR and SLU. This confirms that
performing both tasks helps the performance on SLU. The difference between the
models’ performance in Figure 1(b) is not statistically significant, even when
the encoder’s output is used. This indicates that the improved performance
between the models is due to the dual objective rather than the increased
capacity given by the additional layers.

###  4.3 Data Shortage Scenario

(a) Grabo

(b) Patience Corpus

Fig. 2:  Comparison of different models trained with increasing train set
sizes on Grabo (left) and Patcor (right). Each curve represents the accuracy
on the test set as a function of the size of the training set. The bands
around the curves represent a 68% confidence interval.

When data is scarce, multitask learning can do a lot, as showed in Figure 2.
We compare our multitask model (blue) with two baselines: we either use ASR
transcripts or gold transcription to generate features with BERTje [18]. These
sequences of representations are used as input to train a SLU module. As the
Grabo dataset contains the exact same set of sentences repeated multiple
times, we do not finetune BERT in order not to make the task trivial. We
observe remarkable results on both datasets, Grabo and Patience, especially in
low resource scenario where our model shows impressive performance compared to
the baselines. The Patience Corpus is considerably more complicated than Grabo
because of the diversity of training examples and the larger number of
possible outputs. Nonetheless, our model shows a better performance for small
training sizes although for larger training sets, the NLP model’s accuracy
score rises above the results of the multitask model.

###  4.4 Results

Model | Grabo | FSC-S | FSC-U
---|---|---|---
MTL | 100 | 98.2 | 88.1
ESPnet-SLU [19] | 97.2 | 97.5 | 78.5
Table 1: Accuracy of the models on the test set of our model compared to the baseline proposed in [19]. FSC-S and FSC-U correspond to the Fluent Challenge splits, respectively with unknown speakers and with unknown utterances. Model | # params | macro-f1
---|---|---
MTL (Ours) | 31 M | 49.82
NLP | 390 M | 64.3
Pipeline | 707 M | 63.3
E2E | 317 M | 48.5
Table 2: Performance on the test set of our model on sentiment classification
(VoxCeleb) compared to the baseline [17].

Table 1 compares our results with the baseline models proposed in [19] on
Fluent and Grabo. As our models are pretrained on a large variety of speakers,
it is not surprising that they are robust to unseen speakers. However, it
seems that the combination of pretraining and multitask learning also
considerably helps the model to generalize to unseen output patterns, as
evidenced by the strong score on FSC-U. At this stage, we did not observe
strong evidence that SLU positively impacts ASR scores. In Table 2, we compare
the performance of our model on the sentiment classification task. The model
labelled NLP corresponds to the performance of DeBERTa large [20] on the gold
transcriptions. Pipeline is composed of Wav2Vec 2.0 large [6] followed by
DeBERTa large. E2E correspond to Wav2Vec 2.0 without a dedicated language
model for decoding. The results shown here are reported in [17]. Although our
model only slightly improves over the end-to-end baseline, it does so with
considerably fewer parameters.

##  5 Conclusion

In this work, we proposed a multitask-learning model to do both automatic
speech recognition and either intent classification or sentiment
classification in parallel. We see an improvement over doing IC independently,
even more so in low-resource scenario where it is beneficial to access
knowledge from written as well as spoken language. Further, we reach a better
performance than the end-to-end baseline on sentiment classification, although
we are using a model that is ten times smaller. In the future, we will keep
working on this topic by including more tasks to train in parallel.
Additionally, we did not mention the impact of MTL on ASR results, which we
will explore in more details. Finally, we used in this work class attention,
defined in Section 2.3. It has proved to be a very efficient addition to our
model but requires an article of its own to explore all its benefits.

## References

  * [1] D. Serdyuk, Y. Wang, C. Fuegen, A. Kumar, B. Liu, and Y. Bengio, “Towards end-to-end spoken language understanding,” in _ICASSP_ , 2018.
  * [2] L. Lugosch, M. Ravanelli, P. Ignoto, V. S. Tomar, and Y. Bengio, “Speech Model Pre-Training for End-to-End Spoken Language Understanding,” in _Interspeech_ , 2019.
  * [3] P. Haghani, A. Narayanan, M. A. U. Bacchiani, G. Chuang, N. Gaur, P. J. M. Mengibar, D. Qu, R. Prabhavalkar, and A. Waters, “From audio to semantics: Approaches to end-to-end spoken language understanding,” in _Spoken Language Technology Workshop (SLT), 2018 IEEE_ , 2018.
  * [4] M. Rao, A. Raju, P. Dheram, B. Bui, and A. Rastrow, “Speech to semantics: Improve ASR and NLU jointly via all-neural interfaces,” in _Interspeech 2020, 21st Annual Conference of the International Speech Communication Association, Virtual Event, Shanghai, China, 25-29 October 2020_ , H. Meng, B. Xu, and T. F. Zheng, Eds. ISCA, 2020, pp. 876–880. [Online]. Available: https://doi.org/10.21437/Interspeech.2020-2976
  * [5] A. Mohamed, D. Okhonko, and L. Zettlemoyer, “Transformers with convolutional context for ASR,” _CoRR_ , 2019.
  * [6] A. Baevski, Y. Zhou, A. Mohamed, and M. Auli, “wav2vec 2.0: A framework for self-supervised learning of speech representations,” in _NIPS_ , 2020.
  * [7] S. Watanabe, T. Hori, S. Karita, T. Hayashi, J. Nishitoba, Y. Unno, N. E. Y. Soplin, J. Heymann, M. Wiesner, N. Chen, A. Renduchintala, and T. Ochiai, “Espnet: End-to-end speech processing toolkit,” _CoRR_ , 2018.
  * [8] A. Graves, S. Fernández, and F. Gomez, “Connectionist temporal classification: Labelling unsegmented sequence data with recurrent neural networks,” in _ICML_ , 2006.
  * [9] H. Touvron, M. Cord, A. Sablayrolles, G. Synnaeve, and H. Jégou, “Going deeper with image transformers,” _CoRR_ , vol. abs/2103.17239, 2021. [Online]. Available: https://arxiv.org/abs/2103.17239
  * [10] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, u. Kaiser, and I. Polosukhin, “Attention is all you need,” in _NIPS_ , 2017\.
  * [11] V. Renkens, S. Janssens, B. Ons, J. F. Gemmeke, and H. Van hamme, “Acquisition of ordinal words using weakly supervised NMF,” in _IEEE, SLT_ , 2014\.
  * [12] N. Oostdijk, “Het corpus gesproken nederlands,” _Nederlandse Taalkunde_ , 2000\.
  * [13] V. Panayotov, G. Chen, D. Povey, and S. Khudanpur, “Librispeech: An asr corpus based on public domain audio books,” in _ICASSP_ , 2015.
  * [14] V. Renkens and H. Van hamme, “Capsule networks for low resource spoken language understanding,” in _Interspeech_ , 2018.
  * [15] N. Tessema, B. Ons, J. van de Loo, J. F. Gemmeke, G. De Pauw, W. Daelemans, and H. Van hamme, “Metadata for corpora patcor and domotica-2,” _ESAT, KU Leuven, Belgium_ , 2013.
  * [16] S. Arora, A. Ostapenko, V. Viswanathan, S. Dalmia, F. Metze, S. Watanabe, and A. W. Black, “Rethinking end-to-end evaluation of decomposable tasks: A case study on spoken language understanding,” in _22nd Annual Conference of the International Speech Communication Association (Interspeech 2021)_ , Brno, Czech Republic, August 2021.
  * [17] S. Shon, A. Pasad, F. Wu, P. Brusco, Y. Artzi, K. Livescu, and K. J. Han, “SLUE: new benchmark tasks for spoken language understanding evaluation on natural speech,” _CoRR_ , vol. abs/2111.10367, 2021. [Online]. Available: https://arxiv.org/abs/2111.10367
  * [18] W. de Vries, A. van Cranenburgh, A. Bisazza, T. Caselli, G. van Noord, and M. Nissim, “Bertje: A dutch BERT model,” _CoRR_ , 2019.
  * [19] S. Arora, S. Dalmia, P. Denisov, X. Chang, Y. Ueda, Y. Peng, Y. Zhang, S. Kumar, K. Ganesan, B. Yan, N. T. Vu, A. W. Black, and S. Watanabe, “Espnet-slu: Advancing spoken language understanding through espnet,” _CoRR_ , vol. abs/2111.14706, 2021. [Online]. Available: https://arxiv.org/abs/2111.14706
  * [20] P. He, X. Liu, J. Gao, and W. Chen, “Deberta: Decoding-enhanced bert with disentangled attention,” _CoRR_ , vol. abs/2006.03654, 2020.


# Colossal orbital-Edelstein effect in non-centrosymmetric superconductors

Luca Chirolli  Department of Physics, University of California, Berkeley,
CA-94720  NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza
San Silvestro 12, I-56127 Pisa, Italy  Maria Teresa Mercaldo  Dipartimento di
Fisica “E. R. Caianiello”, Università di Salerno, IT-84084 Fisciano (SA),
Italy  Claudio Guarcello  Dipartimento di Fisica “E. R. Caianiello”,
Università di Salerno, IT-84084 Fisciano (SA), Italy  Francesco Giazotto
NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza San
Silvestro 12, I-56127 Pisa, Italy  Mario Cuoco  SPIN-CNR, IT-84084 Fisciano
(SA), Italy  Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno,
IT-84084 Fisciano (SA), Italy

###### Abstract

In superconductors that lack inversion symmetry, the flow of supercurrent can
induce a non-vanishing magnetization, a phenomenon which is at the heart of
non-dissipative magneto-electric effects, also known as Edelstein effects. For
electrons carrying spin and orbital moments a question of fundamental
relevance deals with the orbital nature of magneto-electric effects in
conventional spin-singlet superconductors with Rashba coupling. Remarkably, we
find that the supercurrent-induced orbital magnetization is more than one
order of magnitude greater than that due to the spin, giving rise to a
colossal magneto-electric effect. The induced orbital magnetization is shown
to be sign tunable, with the sign change occurring for the Fermi level lying
in proximity of avoiding crossing points in the Brillouin zone and in the
presence of superconducting phase inhomogeneities, yielding domains with
opposite orbital moment orientation. The orbital-dominated magneto-electric
phenomena, hence, have clear-cut marks for detection both in the bulk and at
the edge of the system and are expected to be a general feature of multi-
orbital superconductors without inversion symmetry breaking.

Introduction.— The success in designing spintronic and spinorbitronic devices
relies on effects to generate, manipulate and detect spin-polarized currents.
The Rashba spin-orbit coupling [1, 2] has a special role in this context
because it allows to directly tune the electron spin orientation through its
propagation and, viceversa, to achieve a spin control of the electron
trajectories [3], thus giving rise to a large variety of magneto-electric
effects. The most notable manifestations are the spin Hall effect [4, 5, 6, 7]
as well as the direct and inverse Edelstein effects (EE and IEE) [8, 9, 10],
with a magnetization being induced by an electric current or, conversely, a
non-equilibrium magnetization leading to charge current, respectively.

Recently, it has been recognized that an effect akin to the spin-Rashba (SR)
in two-dimensional electron systems (2DES) arises due to the coupling between
the atomic orbital angular momentum ${\bf L}$ and the crystal wave-vector
${\bf k}$ [11, 12, 13, 14]. For materials having electronic states with non-
vanishing ${\bf L}$ close to the Fermi level, an intrinsic crystalline
potential or an applied electric field that break spatial inversion can yield
non-local odd-parity matrix elements among distinct atomic orbitals. The
resulting orbital Rashba effect (ORE) [11, 12, 13, 14], in analogy with the
spin Rashba effect (SRE), confers chirality to the electronic states at the
Fermi level, that acquire a momentum-dependent orbital texture [15, 13, 17,
16, 14].

Orbital physics can then give rise to orbital analogue phenomena of the spin
Hall and spin Edelstein effects, namely orbital Hall [18, 19, 20] and orbital
Edelstein effects [21, 22, 23, 24], with the latter leading to a current-
induced orbital magnetization. Brought into the context of non-dissipative
superconducting magnetoelectric effects, so far the focus has been mostly on
the spin degrees of freedom for spin-Rashba type superconductors [25, 26, 27,
28, 29, 30] including the generalization to all gyrotropic crystal point
groups [31]. However, since electrons carry both spin and orbital moments,
assessing the role of electrons’ orbital moment in setting out the magneto-
electric effects is of fundamental relevance.

In this manuscript, we study the emergence of magneto-electric effects induced
by a supercurrent flow in 2D non-centrosymmetric superconductors featuring an
orbital Rashba (OR) coupling. We show that the supercurrent-induced orbital
moments can exhibit striking marks with an extraordinary large amplitude and
sign tunability. We find that the strength of the orbital polarization is more
than one order of magnitude greater than that due to the spin, for nominally
equal Rashba coupling strength. This colossal effect is then general,
irrespective of the intensity of the OR coupling compared to the Fermi energy.
The orbital Edelstein effect becomes particularly enhanced for the Fermi level
lying in proximity of avoiding crossing points in the band structure induced
by the orbital Rashba coupling, where it also features a sign change. The
orientation of the induced orbital magnetization can be suitably tuned by
sweeping the Fermi level across the avoided crossing, providing a knob to
master the response. The sign change occurs not only for uniform
configurations but also in the presence of superconducting phase
inhomogeneities thus providing a route to devise domains with opposite
magnetization. We discuss direct experimental probes of the effect and the
impact that an applied electric field can have in guiding the suppression of
the superconducting state by orbital magneto-electric effects.

Figure 1: Schematic setup showing the Edelstein effect in (a) spin- and (b)
orbital- Rashba 2D systems with a supercurrent $J_{x}$ flowing along the
$x$-axis and the corresponding induced magnetization in the transverse
$y$-direction. Induced (c) spin- and (d) orbital-polarization integrated in
the whole Brillouin zone for a 2D Rashba system as a function of the chemical
potential for $\alpha=0.1t$ and $q=0.02$ in unit of the inverse atomic
distance. The maximum of the induced orbital moment $\langle L_{y}\rangle$ is
about 60 times larger than the corresponding spin polarization $\langle
S_{y}\rangle$ for the same amplitude of the Rashba coupling. In (d) the
employed parameters are $t^{\prime}=0.4t$ and $\delta_{z}=0.5t$.  Figure 2:
(a) Spin-split Rashba bands for $k_{y}=0$ (plotted for positive $k_{x}$) and
$k_{y}=\pi/2$ (for negative $k_{x}$) for $\alpha=0.1t$. The color map
indicates the amplitude and sign of the $k$-resolved spin polarization in the
normal state. (b) Induced spin-polarization for the spin-Rashba model in the
superconducting state, as a function of the chemical potential $\mu$ for two
representative cuts in the Brillouin zone as shown in (a). (c) Cuts of the
orbital-resolved band structure for $\alpha=0.1t$ at $k_{y}=0$ (positive
$k_{x}$) and at $k_{y}=0.9\pi$ (negative $k_{y}$), respectively. The color map
indicates the amplitude and sign of the $k$-resolved angular momentum
polarization in the normal state. (d) Energy-resolved induced orbital moment
in the superconducting state for the two cuts reported in (c). The sign change
of the $\langle L_{y}\rangle_{SC}$ occurs in proximity of the band touching at
$\mu/t\sim-1.67(2.24)$ for $k_{y}=0$ ($k_{y}=0.9\pi$), respectively. The
employed parameters are $t^{\prime}=0.4t$ and $\delta_{z}=0.5t$.

Model.— In order to directly evaluate the contribution due to the electrons’
orbital moment and compare it with the spin one, we consider a 2D electronic
system that explicitly includes the spin or orbital Rashba couplings. For the
OR case, the minimal description is based on three bands arising from atomic
orbitals spanning the $L=1$ angular momentum subspace. This model applies for
$p$ orbitals or $d$ orbitals, e.g. $\\{d_{yz}$, $d_{xz}$, $d_{xy}\\}$. As
several materials can be effectively described with similar multi-orbital
degrees of freedom close to the Fermi level, we keep the analysis general and
refer for simplicity to $d_{a}$ orbitals, with $a=(yz,xz,xy)$, at each site
assuming a square lattice characterized by $C_{4v}$ point group symmetry. The
breaking of mirror symmetry, due to intrinsic crystalline potential or
externally applied electric field, sets out a polar axis ($z$) resulting into
an orbital Rashba interaction that couples the atomic angular momentum ${\bf
L}$ with the crystal wave-vector ${\bf k}$. The Hamiltonian in momentum space
is generally written as

$h_{0}({\bf k})=\sum_{a}\epsilon_{a}({\bf k})\mathbb{P}_{a}+\alpha\left({\bf
g}_{\bf k}\wedge\hat{\bf L}\right)\cdot\hat{z}-\mu,$ (1)

where $\epsilon_{a}({\bf
k})=-2t^{x}_{a}\cos(k_{x})-2t^{y}_{a}\cos(k_{y})-\delta_{a}$ are dispersion
relations resulting from symmetry-allowed nearest-neighbor hopping amplitudes,
where $t_{yz}^{y}=t_{xz}^{x}=t_{xy}^{x}=t_{xy}^{y}=t$,
$t_{yz}^{x}=t_{xz}^{y}=t^{\prime}$, and
$\mathbb{P}_{a}=(\hat{L}^{2}-\hat{L}_{a}^{2})/2$ is a projector on the orbital
$a$ subspace. At the $\Gamma$ point a degeneracy occurs between, say, the $xz$
and the $yz$ orbitals, thus for convenience we set
$\delta_{xz}=\delta_{yz}=0$. In turn, the orbital $xy$ can show a finite
crystal field splitting $\delta_{xy}\equiv\delta_{z}$ that depends on the
specific crystalline distortions.

The three bands are coupled at finite momentum by the orbital Rashba term
described by a coupling constant $\alpha$ and involving the components of the
angular momentum that in a matrix form are given by
$[\hat{L}_{k}]_{lm}=i\epsilon_{klm}$ with $\epsilon_{klm}$ the Levi-Civita
tensor. Here, the OR coupling is specified by a vector ${\bf g}({\bf
k})=(\sin(k_{x}),\sin(k_{y}),0)$ in momentum space that sets out the chirality
of the orbital texture.

We now assume a finite chemical potential and a purely local ($s-$wave)
attractive interaction of strength $-U<0$ leading to pairing of electrons with
same orbital character. This gives rise to spin-singlet superconductivity
described by a uniform gap $\Delta=-U\langle\hat{d}_{a,\downarrow}({\bf
r})\hat{d}_{a,\uparrow}({\bf r})\rangle$ on the Fermi surface, with
$\hat{d}_{a,s}({\bf r})$ fermionic annihilation operators associated to
$d_{a}$ orbitals at position ${\bf r}$ with spin $s$.

Magnetization.— The Edelstein effect in spin Rashba coupled superconductors
manifests as a finite spin polarization in response to an applied supercurrent
(Fig. 1(a)). The wedge product of polar vectors associated to the direction of
the mirror symmetry breaking, ${\bf z}$, and the supercurrent ${\bf J}$,
allows the construction of an axial vector

${\bf M}\sim{\bf z}\wedge{\bf J},$ (2)

and an in-plane spin magnetization, orthogonal to the applied bias current, is
expected (Fig. 1(a)). As in the SRE, an orbital moment develops on the Fermi
surface and an analogous orbital Edelstein effect for the ORE is expected as
in Fig. 1(c), with a finite orbital magnetization ${\bf M}=\langle{\bf
L}\rangle$ (in unit of the Bohr magneton $\mu_{B}$).

In the presence of a current bias the order parameter acquires a position-
dependent phase, $\hat{\Delta}({\bf r})=\hat{\Delta}e^{i{\bf q}\cdot{\bf r}}$
for a finite ${\bf q}$ sustained by the bias current. The phase bias can be
absorbed in the fermionic operators $\hat{d}_{a,s}\to\hat{d}_{a,s}e^{-i{\bf
q}\cdot{\bf r}/2}$ through a gauge transformation and appears as a ${\bf q}/2$
shift in momentum with opposite sign for particles and holes. For weak values
of the applied current we can expand the magnetization at first order in the
momentum ${\bf q}$ and write the magnetization in linear response theory as

$\langle\hat{L}_{\mu}\rangle=-\frac{1}{2}\chi_{\mu\nu}q_{\nu},$ (3)

with $\chi_{\mu\nu}$ a static susceptibility between the angular momentum
$\hat{L}_{\mu}$ and the velocity operator defined as $\hat{v}_{\mu}=\partial
h_{0}/\partial k_{\mu}$, that is composed by a normal and an anomalous term
and acquires no structure in particle-hole space. Thanks to the presence of a
finite superconducting gap in the spectrum and the spontaneous breaking of
U(1) symmetry, the static susceptibility can be obtained by energy correction
at second-order perturbation theory and is essentially determined by the
matrix elements
$v_{x}^{n,n^{\prime}}L_{y}^{n^{\prime}n}\equiv\langle\phi_{n,{\bf
k}}|\hat{v}_{x}|\phi_{n^{\prime},{\bf k}}\rangle\langle\phi_{n^{\prime},{\bf
k}}|\hat{L}_{y}|\phi_{n,{\bf k}}\rangle$, with $|\phi_{n,{\bf k}}\rangle$
eigenstates of $h_{0}({\bf k})$, Eq. (1), with eigenvalues $\epsilon_{n,k}$.
Since we will consider separately the case of spin and orbital Rashba
coupling, we use a unique coupling constant $\alpha$ for both cases.

The comparison of the outcome of the total spin and orbital moments for the
spin- and orbital-Rashba 2DES is reported in Fig. 1(b),(d). There are two
striking features that can be immediately observed. The first one concerns
with the amplitude of the effect. The evolution of the magnetization as a
function of the electron filling, via a change in the chemical potential,
shows that the induced orbital moment is generally much larger than the spin
moment, with a maximum in the orbital channel that can be up to about 60 times
bigger than the spin one, even for nominally equal coupling constant. The
second relevant aspect is the sign tunability for values of the chemical
potential that are in proximity of avoiding crossings at small transverse to
the current momenta, $k_{y}$, in the electronic structure. The sign change of
the total response for the ORE case (occurring at $\mu/t\sim 1.8$ in Fig. 1
(d)) has its origin in the avoided crossings appearing in the band structure
(Fig. 2 (c) and (d)). The latter occurs in generic points in the Brillouin
zone because orbitals, differently from spin, are not forced to be degenerate
at high symmetry point by Kramers’ theorem, that guarantees spin degeneracy.
In particular, the symmetry of the orbitals and the crystal point group of the
material may allow a crystal field splitting to appear. The latter can then
shift the avoided crossing at arbitrary energy, resulting in orbital tunable
magneto-electric effects not only in semi-metallic but also in metallic
states. In turn, the sign change of the spin moment for the SRE is due to the
sublattice symmetry of the tight-binding model at specific points in parameter
space and manifests at $\mu=0$.

Figure 3: Schematics of the evolution of the bands around the $\Gamma$ point
in the $C_{\infty v}$ symmetry by varying a) the orbital mass imbalance
$m_{z}/m$ for $\delta_{z}=0$ and b) the crystal field splitting $\delta_{z}$
for $m_{z}=5m$. Edelstein susceptibility versus the chemical potential $\mu$
in units of $E_{\rm OR}=m\alpha^{2}/2$ by varying c) the orbital mass ratio
$m_{z}/m$ for $\delta_{z}=0$ and d) the crystal field splitting
$\delta_{z}/E_{\rm OR}$ for $m_{z}=5m$.

A simple way to understand the different spin and orbital response of the
system is to consider 1D cuts of the band structure by fixing the transverse
momentum $k_{y}$ (Fig. 2). In particular, for $k_{y}=0,\pi$, one band in the
orbital-Rashba case completely decouples. In this specific case, the matrix
element appearing in the susceptibility (see Eq. (8) of Methods) reads

$v_{x}^{+,-}L^{-,+}_{y}=\alpha(\epsilon_{1}-\epsilon_{2})\frac{(\epsilon_{1}-\epsilon_{2})\partial_{x}g_{x}-g_{x}(v^{1}_{x}-v^{2}_{x})}{4\alpha^{2}g_{x}^{2}+(\epsilon_{1}-\epsilon_{2})^{2}},$
(4)

with $\partial_{x}\equiv\partial/\partial k_{x}$,
$\epsilon_{1(2)}=\epsilon_{yz(xz)}(k_{x})$, and
$v_{1(2)}=\partial_{x}\epsilon_{yz(xz)}(k_{x})$. Clearly, in the case of spin-
Rashba, the unperturbed spin degenerate bands satisfy
$\epsilon_{1}=\epsilon_{2}$ and the Edelstein susceptibility is exactly zero
(Fig. 2(b)). In turn, the susceptibility for the orbital-Rashba case is non
zero due to the difference in effective masses and crystal field splitting
that characterize the dispersions originating from different orbitals. At
finite momentum $k_{y}$ the susceptibility in the spin-Rashba case becomes
finite and allows a direct comparison of the response between the two cases.
In Fig. 2(a) we show the bands at $k_{y}=0(\pi/2)$ for positive (negative)
$k_{x}$ of the spin-Rashba model for $\alpha=0.1t$ and in Fig. 2(b) the
induced spin magnetization evaluated for $q=0.02$ (in units of the inverse
atomic lattice constant). The linear response condition is satisfied and the
magnetization is proportional to the applied momentum via the susceptibility.
A finite response is found for non-zero $k_{y}$, peaked at the band splitting
nearby the $\Gamma$ point.

The Edelstein magnetization for the case of the ORE is more interesting. In
Fig. 2(c) we show the bands at $k_{y}=0(0.9\pi)$ for positive (negative)
$k_{x}$ of the orbital-Rashba model for $t^{\prime}=0.4t$ and $\alpha=0.1t$. A
light effective mass band, originating from the $d_{xy}$ orbital, detaches and
shifts down in energy with respect to the other two bands originating from the
$d_{yz}$ and $d_{xz}$ orbitals, that remain degenerate at the $\Gamma$ point.
As a result the light and heavy bands cross at finite momentum and an avoided
crossing is generated at finite $\alpha$. This is generally expected to occur
for 2DES. An additional crystal field splitting $\delta_{z}=0.5t$ is added to
tune the avoided crossing position.

In Fig. 2(d) the induced orbital moment for $q=0.02$ is shown. The response is
finite for every $k_{y}$ and a sign change appears that is pinned to the
avoided crossing. This is understood at $k_{y}=0$ by inspection of Eq. (4). At
the avoided crossing, $\epsilon_{1}=\epsilon_{2}$ and the sign of the matrix
element is determined by the velocity difference $v^{1}_{x}-v^{2}_{x}$. As the
two velocities interchange at the avoided crossing the magnetization features
a sign change. Furthermore, as shown by the color code in Fig. 2(c), the
expectation value of $\hat{L}_{y}$ is maximal at the avoided crossing,
yielding peaks above and below the sign change. By increasing $k_{y}$,
$\hat{L}_{y}$ decreases in amplitude according to the structure of the orbital
Rashba, and, due to the admixture with the third band a shift of the avoided
crossing in energy occurs. The full response (Fig. 1(d)), integrated on all
$k_{y}$’s, features a vanishing point in proximity of the highest in energy
avoided crossing and its magnitude is about sixty times larger than in the
spin-Rashba case.

Orbital splitting and mass imbalance.— As previously anticipated, one of the
main difference between SR and OR cases is the absence of a Kramers’s theorem
that forces an orbital degeneracy at high symmetry points in the Brillouin
zone. It follows that the Edelstein effect in the OR case shows a strong
dependence on the effective masses and crystal field splittings in amplitude
and sign. To simplify the analysis we expand the Hamiltonian around the
$\Gamma$ point, where the orbital Rashba term becomes invariant for any
rotation about the $z$ axis, and extend the $C_{\infty v}$ symmetry to the
effective masses. This way, the avoided crossing observed in Fig. 1 are now at
the same energy for all $k_{x}$ and $k_{y}$ and the dependence of the
Edelstein effect on effective masses and crystal field splitting becomes
manifest. In turn, the Edelstein response for the SR case around the $\Gamma$
point is featureless, with no sign change and a constant susceptibility for
positive $\mu$ that resembles the curve $m=m_{z}$ in Fig. 3 (c).

Figure 4: (a) Sketch of the superconducting phase across the superconductor
along a given direction in real space with a change of the phase gradient by
$\Delta\phi$ around the position $i_{x}=0$. (b) Real space dependence of the
orbital polarization renormalized to the bulk value $\langle L_{y,b}\rangle$
reported in (c) for several values of the chemical potential. In proximity of
the domain boundary ($i_{x}=0$) the orbital moment has an enhancement of the
amplitude of about 10 times with respect to the bulk, and a sign change
occurring at a distance of about the superconducting coherence length. (d)
Plot of the spatial resolved orbital moment for a representative value of
$\mu$ and $\alpha$ for different values of the superconducting order parameter
$\Delta_{0}$. The increase of the gap amplitude induces a sign change of the
orbital polarization at a shorter distance from the phase domain boundary thus
scaling with the coherence length amplitude. The employed parameters are
$t^{\prime}=0.4t$, $\delta_{z}=0.5t$, while the hopping amplitude at the
domain boundary is $t_{b}=0.75t$.

In Fig. 3(a) we show the evolution of the bands by increasing or decreasing
the ratio $m_{z}/m$. For $m_{z}/m\neq 1$ two of the three bands tend to
collapse on top of the other, so that their relative contribution vanishes and
a partial cancellation of the contributions from different bands takes place.
The susceptibility, shown in Fig. 3(c) is dominated by the residual
contribution, that is proportional to the difference in the Fermi momenta and
in the case $m_{z}/m>1$ the signal increases.

In Fig. 3(b) we show the evolution of the bands by varying the value of the
crystal field splitting $\delta_{z}$ and we choose a large difference in the
effective masses, $m=5m_{z}$. This way, the band edge of the light mass band
shifts down in energy for positive $\delta_{z}$ and up in energy for negative
$\delta_{z}$. An avoided crossing is then produced for positive $\delta_{z}$,
as in the case illustrated in Fig. 2(c), and an interchange of the velocity
above and below the avoided crossing produces a sign change in the
susceptibility. The evolution from negative to positive $\delta_{z}$ is shown
in Fig. 3(d), and a clear sign change appears. We point out that the SRE case
around the $\Gamma$ point is featureless, showing no sign change and a
constant susceptibility [30] resembling the $m=m_{z}$ line in Fig. 3 (c).

Figure 5: Sketch of an experimental setup for the detection of the orbital
moments generated by the supercurrent flow. We consider non-centrosymmetric or
ultra-thin superconductors with $z$ being the polar direction associated with
inversion symmetry breaking. In the case of a uniform magnetization we expect
that a magnetic flux along the direction of the orbital moment can be measured
by SQUID along the lateral profile of the superconductor. (b) In presence of
small phase inhomogeneities there is a reorientation of the orbital moments in
real space that results into opposite fluxes along the lateral edge. (c) The
application of an electric field with lateral split gates leads to an out-of-
plane orbital polarization and in turn a magnetic flux that is oriented
perpendicular to the surface of the superconductor.

Spatial-dependent orbital Edelstein effects. — The amplitude and sign of the
ORE has also distinctive marks when considering its spatial dependence in the
case of inhomogeneous superconducting phase profile. We have solved the
orbital Rashba model (Eq. (1)) for a 2D system with size $L_{x}\times L_{y}$,
$L_{x}=200$, in unit of the inter-atomic distance, and assuming translational
invariance for $L_{y}$. The supercurrent profile includes a phase gradient in
real space along a given direction (Fig. 4(a)) and a phase defect in the
spatial profile of the supercurrent. Thus, we have a domain boundary with a
weak modification of the phase gradient expressed by the variation
$\Delta\phi$ across the domain boundary at a given position. The aim here is
to assess the spatial dependence of the induced orbital moment. The presence
of a weak phase gradient change leads to two striking features. We find a
general enhancement of the induced orbital moment (Fig. 4(b)) compared to the
value in the bulk (Fig. 4(c)). Moreover, the orbital Edelstein effect exhibits
a sign variation of the orbital moment in real space thus yielding domains
with opposite orientation. The spatial extension of the domain with an
amplification of the orbital polarization is of the order of the
superconducting coherence length $\xi_{SC}$. We verify that for a ballistic
superconductor, for which $\xi_{SC}\sim v_{F}/\Delta$ with $v_{F}$ the Fermi
velocity, an increase of the gap amplitude leads to a reduction of the size of
the region with enhanced orbital polarization (Fig. 4(d)).

Discussion.— The amplitude of the orbital contribution is extraordinary large
if compared to that of the spin for nominally equal coupling constant $\alpha$
and a colossal Edelstein response is demonstrated. Although similar in
structure to the SRE, the origin of the ORE does not rely on the atomic spin-
orbit interaction, instead it is solely due to a finite momentum-dependent
overlap between Wannier orbitals enabled by the broken mirror symmetry. For
this reason the magnitude of the OR coupling is typically much larger [32, 33,
34, 35, 36] than the corresponding one in the spin channel. These features
open the possibility to both a wide amplitude tunability of the Edelstein
effect and sign variation through a gate that changes the charge density and
allows us to sweep the Fermi level around avoided crossings. An ideal platform
is represented by 2D superconductors realized at LAO/STO (LaAlO3/SrTiO3)
interfaces [37], where an electric field has been shown to be effective in
varying the electron density and tuning the chemical potential close to
orbitally driven avoiding crossings [38]. Furthermore, since the disclosed
orbital Edelstein effect requires solely s-wave superconductivity, its
application can be extended to ultrathin elemental superconductors as well as
hybrid implementations through proximized superconductivity.

Detection of the Edelstein effect can employ a SQUID (or other magnetic
spectroscopic probes) since it is able to probe the spatial distribution of
the magnetic field generated by the local magnetic moments, as depicted in
Fig. 5 (a). The SQUID technique can directly give access to the dipolar field
distribution with direction change occurring around phase defects or similar
inhomogeneities, as depicted in Fig. 5 (b). Additionally, for inversion
symmetry breaking generated via a side gate, as shown in Fig. 5 (c), an out-
of-plane magnetic flux with the amplitude controlled by the electric field, is
expected to be observed at the edge of the superconductor. The size of the
induced local orbital moment can reach about $10^{-4}\mu_{\text{B}}$, and even
higher by varying the OR coupling, which convert into dipolar fields of about
hundred Gauss in close proximity of the 2DES. The dipolar field of the induced
magnetization has a different spatial distribution compared to that due to the
supercurrent especially in the presence of domains with opposite orbital
moments, thus allowing a separate detection of the two contributions.

The Edelstein magnetization represents an explicitly time-reversal symmetry
breaking perturbation, that, in turn, can have a non-trivial detrimental back
action on the superconductor itself. A reduction of the critical current can
result from the induced dipolar fields, especially when the size of the
induced moment locally overcomes the critical magnetic fields. The
supercurrent can thus be an optimal indicator in systems where an applied
electric field varies the intensity of the Rashba coupling. We argue that the
orbital Edelstein effect could have a non-trivial role in the suppression of
critical current recently reported in experiments where an electric field is
applied to metallic superconductors [39].

Acknowledgments.— L.C. acknowledges the European Commission for funding
through the MSCA Global Fellowship grant TOPOCIRCUS-841894. M.C., M.T.M and
F.G. acknowledge support by the EU’s Horizon 2020 research and innovation
program under Grant Agreement nr. 964398 (SUPERGATE). F.G. acknowledges the
European Research Council under Grant Agreement No. 899315-TERASEC, and the
EU’s Horizon 2020 research and innovation program under Grant Agreement No.
800923 (SUPERTED) for partial financial support.

##  .1 Methods.

BCS superconductivity is described by introducing the Nambu basis,
$\Psi^{\dagger}({\bf r})=(\hat{\bf d}^{\dagger}_{\uparrow}({\bf r}),\hat{\bf
d}^{T}_{\downarrow}({\bf r}))$, with $\hat{\bf d}_{s}({\bf r})$ a column
vector of annihilation Fermionic operators $\hat{d}_{a,s}$ associated to the
different orbitals. The mean-field Hamiltonian is then written as $H=\int
d{\bf r}\Psi^{\dagger}({\bf r}){\cal H}({\bf r})\Psi({\bf r})$, with ${\cal
H}({\bf r})$ the Bogoliubov deGennes Hamiltonian featuring an explicit
position dependence via the phase of the superconducting gap $\Delta({\bf
r})=\Delta e^{i{\bf q}\cdot{\bf r}}$, with ${\bf q}$ sustained by a finite
bias current. Performing a gauge transformation on particle and hole states,
$\Psi({\bf r})\to e^{i{\bf q}\cdot{\bf r}\tau_{z}/2}\Psi({\bf r})$, we can map
the position dependence of the gap in a momentum shift of opposite sign for
particle and holes, so that the BdG Hamiltonian in momentum space takes the
form

${\cal H}({\bf k},{\bf q})=\left(\begin{array}[]{cc}h_{0}({\bf k}+{\bf
q}/2)&\hat{\Delta}({\bf q})\\\ \hat{\Delta}^{\dagger}({\bf
q})&-h^{*}_{0}(-{\bf k}+{\bf q}/2)\end{array}\right),$ (5)

where $\hat{\Delta}({\bf q})$ is a gap matrix proportional to the identity and
we neglect the momentum dependence of the gap amplitude.

Following the same approach employed for the spin Rashba coupled
superconductor, in the OR case the magnetization in response to an applied
supercurrent is given by $\langle\hat{L}_{\mu}\rangle=T\sum_{i\omega_{n},{\bf
k}}{\rm Tr}[G_{i\omega_{n}}({\bf k},{\bf q})\hat{L}_{\mu}]$, where the Green’s
function in the Nambu space is given by $-G^{-1}_{i\omega_{n}}({\bf k},{\bf
q})=-i\omega_{n}+{\cal H}({\bf k},{\bf q})$. Expansion of the full Green’s
function at first order in ${\bf q}$ yields the expression Eq. (3) in terms of
a static susceptibility

$\chi_{\mu\nu}=-T\sum_{i\omega_{n},{\bf k}}{\rm Tr}\left[G_{i\omega_{n}}({\bf
k})\hat{L}_{\mu}G_{i\omega_{n}}({\bf k})\hat{v}_{\nu}({\bf k})\right],$ (6)

with the velocity operator is explicitly given by

$\displaystyle\hat{v}_{\mu}({\bf k})$ $\displaystyle=$
$\displaystyle\sum_{a}\frac{\partial\epsilon_{a}({\bf k})}{\partial
k_{\mu}}\mathbb{P}_{a}+\alpha\hat{\bf L}\wedge\hat{z}\cdot\frac{\partial{\bf
g}({\bf k})}{\partial k_{\mu}}.$ (7)

Assuming to apply the current bias along the $x$ direction, the susceptibility
can be written as

$\displaystyle\chi_{yx}$ $\displaystyle=$ $\displaystyle 2\sum_{{\bf
k},n>n^{\prime}}\frac{1}{E_{n,k}+E_{n^{\prime},k}}\left(1-\frac{\epsilon_{n,k}\epsilon_{n^{\prime},k}+\Delta^{2}}{E_{n,k}E_{n^{\prime},k}}\right)$
(8) $\displaystyle\times$ $\displaystyle{\rm Re}\langle\phi_{n,{\bf
k}}|\hat{v}_{x}|\phi_{n^{\prime},{\bf k}}\rangle\langle\phi_{n^{\prime},{\bf
k}}|\hat{L}_{y}|\phi_{n,{\bf k}}\rangle.$

This expression is easily adapted to describe the Edelstein response in the
spin-Rashba case by replacing the dispersion $\epsilon_{a}({\bf k})$ with
degenerate bands, the angular momentum $\hat{L}_{\mu}$ with spin Pauli
matrices, and an overall factor 1/2 to avoid double counting spin-degeneracy.
This way, a direct comparison between the two models can be made.

In the $C_{v\infty}$ case around the $\Gamma$ point the eigenvalues of the
band Hamiltonian $h_{0}$ have the general expression
$\epsilon_{0,k}=\epsilon_{k}$ and
$\epsilon_{\pm,k}=\epsilon_{k}+\delta\epsilon_{\pm,k}$, with
$\delta\epsilon_{\pm,k}=(\epsilon^{z}_{k}-\epsilon_{k}\pm\sqrt{4k^{2}\alpha^{2}+(\epsilon^{z}_{k}-\epsilon_{k})^{2}})/2$,
$\epsilon^{z}_{k}=k^{2}/2m_{z}-\mu-\delta_{z}$, and
$\epsilon_{k}=k^{2}/2m-\mu$. The matrix elements that enter the susceptibility
are then given by

$\displaystyle\left\langle v_{x}^{0,\pm}L^{\pm,0}_{y}\right\rangle_{\phi}$
$\displaystyle=$
$\displaystyle\pm\frac{\alpha\delta\epsilon_{\pm}}{2(\epsilon_{k}^{+}-\epsilon_{k}^{-})},$
(9) $\displaystyle\left\langle v_{x}^{+,-}L^{-,+}_{y}\right\rangle_{\phi}$
$\displaystyle=$
$\displaystyle\frac{\alpha(\epsilon_{k}-\epsilon^{z}_{k})}{(\epsilon^{+}_{k}-\epsilon^{-}_{k})^{2}}\left(\epsilon_{k}-\epsilon^{z}_{k}+\frac{\alpha
k^{2}}{m_{z}}-\frac{\alpha k^{2}}{m}\right),$

where $\langle\ldots\rangle_{\phi}$ denotes average over the azimuthal angle
$\phi=\tan^{-1}(k_{y}/k_{x})$. We see that $\left\langle
v_{x}^{+,-}L^{-,+}_{y}\right\rangle_{\phi}$ essentially agrees with Eq. (4).

## References

  * [1] E. I. Rashba, Properties of semiconductors with an extremum loop. I. Cyclotron and combinational Resonance in a magnetic field perpendicular to the plane of the loop, Sov. Phys. Solid State 2, 1109 (1960).
  * [2] G. Dresselhaus, Spin-Orbit Coupling Effects in Zinc Blende Structures, Phys. Rev. 100, 580 (1955).
  * [3] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, New perspectives for Rashba spin-orbit coupling, Nat. Mater. 14, 871 (2015).
  * [4] M. Dyakonov and V. Perel, Current-induced spin orientation of electrons in semiconductors, Phys. Lett. A 35, 459 (1971).
  * [5] J. E. Hirsch, Spin Hall Effect, Phys. Rev. Lett. 83, 1834 (1999).
  * [6] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Observation of the spin Hall effect in semiconductors, Science 306, 1910 (2004).
  * [7] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall Effects, Rev. Mod. Phys. 87, 1213 (2015).
  * [8] A. G. Aronov and Y. B. Lyanda-Geller, Nuclear electric resonance and orientation of carrier spins by an electric field, JETP Lett. 50, 431 (1989).
  * [9] V. M. Edelstein, Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems, Solid State Commun. 73, 233 (1990).
  * [10] K. Shen, G. Vignale, and R. Raimondi, Microscopic Theory of the Inverse Edelstein Effect, Phys. Rev. Lett. 112, 096601 (2014).
  * [11] S. R. Park, C. H. Kim, J. Yu, J. H. Han, and C. Kim, Orbital-angular-momentum based origin of Rashba-type surface band splitting, Phys. Rev. Lett. 107, 156803 (2011).
  * [12] J.-H. Park, C. H. Kim, J.-W. Rhim, and J. H. Han, Orbital Rashba effect and its detection by circular dichroism angle-resolved photoemission spectroscopy, Phys. Rev. B 85, 195401 (2012).
  * [13] B. Kim, P. Kim, W. Jung, Y. Kim, Y. Koh, W. Kyung, J. Park, M. Matsunami, Shin-ichi Kimura, J. S. Kim, J. H. Han, and C. Kim, Microscopic mechanism for asymmetric charge distribution in Rashba-type surface states and the origin of the energy splitting scale, Phys. Rev. B 88, 205408 (2013).
  * [14] M. T. Mercaldo, P. Solinas, F. Giazotto, and M. Cuoco, Electrically tunable superconductivity through surface orbital polarization, Phys. Rev. Applied 14, 034041 (2020).
  * [15] J.-H. Park, C. H. Kim, H.-W. Lee, and J. H. Han, Orbital chirality and Rashba interaction in magnetic bands, Phys. Rev. B 87, 041301(R) (2013).
  * [16] B. Kim, C. H. Kim, P. Kim, W. Jung, Y. Kim, Y. Koh, M. Arita, K. Shimada, H. Namatame, M. Taniguchi, J. Yu, and C. Kim, Spin and orbital angular momentum structure of Cu(111) and Au(111) surface states, Phys. Rev. B 85, 195402 (2012).
  * [17] S. R. Park, J. Han, C. Kim, Y. Y. Koh, C. Kim, H. Lee, H. J. Choi, J. H. Han, K. D. Lee, N. J. Hur, M. Arita, K. Shimada, H. Namatame, and M. Taniguchi, Chiral Orbital-Angular Momentum in the Surface States of Bi2Se3, Phys. Rev. Lett. 108, 046805 (2012).
  * [18] S. Zhang and Z. Yang, Intrinsic Spin and Orbital Angular Momentum Hall Effect, Phys. Rev. Lett. 94, 066602 (2005).
  * [19] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. Inoue, Intrinsic spin Hall effect and orbital Hall effect in 4d and 5d transition metals, Phys. Rev. B 77, 165117 (2008).
  * [20] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Giant Intrinsic Spin and Orbital Hall Effects in Sr2MO4 (M = Ru, Rh, Mo), Phys. Rev. Lett. 100, 096601 (2008).
  * [21] L. S. Levitov, Y. V. Nazarov, and G. M. Éliashberg, Magneto-electric effects in conductors with mirror isomer symmetry, Sov. Phys. JETP 61, 133 (1985).
  * [22] S. Zhong, J. E. Moore, I. Souza, Gyrotropic Magnetic Effect and the Magnetic Moment on the Fermi Surface Phys. Rev. Lett. 116, 077201 (2-16).
  * [23] L. Salemi, M. Berritta, A. K. Nandy, and P. M. Oppeneer, Orbitally dominated Rashba-Edelstein effect in noncentrosymmetric antiferromagnets, Nat. Comm. 10, 5381 (2019).
  * [24] A. Johannsson, B. Göbel, J. Henk, M. Bibes, and I. Mertig Spin and orbital Edelstein effects in a two-dimensional electron gas: Theory and application to SrTiO3 interfaces, Phys. Rev. Research 3, 013275 (2021).
  * [25] V. M. Edelstein, Magnetoelectric effect in polar superconductors, Phys. Rev. Lett. 75, 2004 (1995).
  * [26] V. M. Edelstein, Magnetoelectric effect in dirty superconductors with broken mirror symmetry, Phys. Rev. B 72, 172501 (2005).
  * [27] S. Yip, Noncentrosymmetric superconductors, Annu. Rev. Condens. Matter Phys., 5, 15 (2014).
  * [28] F. Konschelle, I. V. Tokatly, and F. S. Bergeret, Theory of the spin-galvanic effect and the anomalous phase shift $\phi_{0}$ in superconductors and Josephson junctions with intrinsic spin-orbit coupling, Phys. Rev. B 92, 125443 (2015).
  * [29] I. V. Tokatly, Usadel equation in the presence of intrinsic spin-orbit coupling: A unified theory of magnetoelectric effects in normal and superconducting systems, Phys. Rev. B 96, 060502(R) (2017).
  * [30] J. J. He, K. Hiroki, K. Hamamoto, and N. Nagaosa, Spin supercurrent in two-dimensional superconductors with Rashba spin-orbit interaction, Commun. Physics 2, 128 (2019).
  * [31] W.-Y. He and K.T. Law, Novel magnetoelectric effects in gyrotropic superconductors and a case study of transition metal dichalcodenides, Phys. Rev. Research 2, 012073 (2020).
  * [32] D. Go, J.-P. Hanke, P. M. Buhl, F. Freimuth, G. Bihlmayer, H.-W. Lee, Y. Mokrousov, and S. Blügel, Toward surface orbitronics: giant orbital magnetism from the orbital Rashba effect at the surface of sp-metals, Sci. Rep. 7, 46742 (2017).
  * [33] T. Park, J. Hong, and J. H. Shim, Enhancement of Giant Rashba Splitting in BiTeI under Asymmetric Interlayer Interaction, Journal of the Phys. Soc. of Japan 89, 044701 (2020).
  * [34] D. Go, D. Jo, T. Gao, K. Ando, S. Blügel, H.-W. Lee, and Y. Mokrousov, Orbital Rashba effect in a surface-oxidized Cu film, Phys. Rev. B 103, L121113 (2021)
  * [35] V. Sunko, H. Rosner, P. Kushwaha, S. Khim, F. Mazzola, L. Bawden, O. J. Clark, J. M. Riley, D. Kasinathan, M. W. Haverkort, T. K. Kim, M. Hoesch, J. Fujii, I. Vobornik, A. P. Mackenzie, and P. D. C. King, Maximal Rashba-like spin splitting via kinetic-energy-coupled inversion-symmetry breaking, Nature 549, 492 (2017).
  * [36] M. Ünzelmann, H. Bentmann, P. Eck, T. Kißlinger, B. Geldiyev, J. Rieger, S. Moser, R. C. Vidal, K. Kißner, L. Hammer, M. A. Schneider, T. Fauster, G. Sangiovanni, D. Di Sante, and F. Reinert, Orbital-Driven Rashba Effect in a Binary Honeycomb Monolayer AgTe, Phys. Rev. Lett. 124, 176401 (2020).
  * [37] N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, J. Mannhart, Superconducting Interfaces Between Insulating Oxides, Science 317, 1196 (2007).
  * [38] A. D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, J.-M. Triscone, Electric field control of the LaAlO3/SrTiO3 interface ground state, Nature 456, 624 (2008).
  * [39] G. De Simoni, F. Paolucci, P. Solinas, E. Strambini, and F. Giazotto, Metallic supercurrent field-effect transistor. Nat. Nano. 13, 802 (2018).