# Few-shot Domain-Adaptative Visually-fused Event Detection from Text

Fatemeh Shiri, Farhad Moghimifar,
Reza Haffari, Yuan-Fang Li  Department of Data Science and AI
Faculty of Information Technology
Monash University, Melbourne, Australia
<EMAIL_ADDRESS>Van Nguyen  Information Sciences Division
Defence Science and Technology Group
Adelaide, Australia
<EMAIL_ADDRESS>
###### Abstract

Incorporating auxiliary modalities such as images into event detection models
has attracted increasing interest over the last few years. The complexity of
natural language in describing situations has motivated researchers to
leverage the related visual context to improve event detection performance.
However, current approaches in this area suffer from data scarcity, where a
large amount of labelled text-image pairs are required for model training.
Furthermore, limited access to the visual context at inference time negatively
impacts the performance of such models, which makes them practically
ineffective in real-world scenarios. In this paper, we present a novel domain-
adaptive visually-fused event detection approach that can be trained on a few
labelled image-text paired data points. Specifically, we introduce a visual
imaginator method that synthesises images from text in the absence of visual
context. Moreover, the imaginator can be customised to a specific domain. In
doing so, our model can leverage the capabilities of pre-trained vision-
language models and can be trained in a few-shot setting. This also allows for
effective inference where only single-modality data (i.e. text) is available.
The experimental evaluation on the benchmark M2E2 dataset shows that our model
outperforms existing state-of-the-art models, by up to 11 points.

###### Index Terms:

event detection, multimodal event detection, few-shot learning, image
generation

##  I Introduction

Event detection is a sub-task of event extraction in Natural Language
Processing (NLP) that aims at extracting information about events from text. A
main challenge of this task is rooted in the complexity of the natural
language. Over the years, a wide variety of approaches, from rule-based
methods [1, 2, 3, 4] to end-to-end neural network based models [5, 6, 7, 8]
have been proposed to address this task. While these techniques have shown
promising results on the existing standard benchmarks, they are heavily
reliant on a single modality, i.e. text (Figure 2), and fail to account for
the real-world grounding in other related modalities.

Utilising other modalities such as images to introduce salient information
that complements what is contained in text has shown to be effective in other
areas such as Machine Translation [9, 10, 11]. However, developing
_multimodal_ approaches for event detection is still an under-explored area.
Consider Figure 1 as an example from the M2E2 dataset [12]. The term “custody”
in the input text could indicate the occurrence of an _Arrest:Jail_ event
type, as it implies the action of arresting someone. This interpretation is
complemented by the visual features embodied in the image. Generally,
incorporating the underlying visual representation of a situation can improve
the data efficiency of event detection methods and help to uncover
linguistically indescribable or missed events.

One of the main challenges in developing such multimodal event detection
models is the lack of large amounts of labelled data, which is especially
critical for fine-tuning techniques based on large language models [12]
(Figure 2). This issue hinders the real-world practicality where domain-
specific multi-modal data (i.e. paired text and image) may be scarce.
Additionally, accessing visual context in real-world scenarios might be
limited, which can also affect the performance of a multimodal event detection
model at the inference time.

Figure 1: An example of Multimedia VOA news from the M2E2 dataset [12]. The
textual input in this article reports an event of the type Arrest:Jail, which
is complemented by the visual context. The trigger word for this event is
”custody”, and the bounding box in the accompanying image highlights the man
being taken into custody.

To overcome these limitations, we propose a few-shot visually-fused model for
event detection that can effectively handle data scarcity during both training
and inference. During training, we address the problem of having access to
only a limited number of labeled image-text pairs with a few-shot transfer
learning model. We initialize the parameters of our model by leveraging a pre-
trained large vision-language model and update them to accommodate the
distribution of the target data, thus addressing the targeted task more
effectively.

Figure 2: The comparison among (a) conventional text-only event detection, (b)
multimodal supervised event detection, and (c) our proposed end-to-end few-
shot visually fused event detection model.

In addition, we introduce a customised _Visual Imaginator_ model that can
learn to synthesise visual representations of the textual input using only a
few examples during training. This approach reduces the domain shift between
the data that the pre-trained model was developed on and our target dataset,
enabling our proposed Visual Imaginator to generate customised domain-specific
images that complement the textual input. This approach helps us address the
challenge of having access to single-modality data, i.e., text. We contemplate
that our model can leverage synthetically generated images in lieu of missing
actual images during inference to perform multimodal event detection. As a
result, our proposed approach for event detection can achieve accuracy
comparable to that of multimodal models where both text and images are fully
available.

In summary, our contributions are as follows:

  * •

We formulate the multimodal event detection task as a few-shot learning
problem and propose a novel visually-fused event detection framework,
alleviating the issue of data scarcity.

  * •

Our proposed customised Visual Imaginator module can synthesise domain-
specific images using textual input to address the challenge of missing text-
image pairs at inference time.

  * •

Experiments on a standard benchmark dataset for event detection confirm the
superiority of our model over state-of-the-art models in low-resource and few-
shot scenarios.

##  II Background

Our approach is related to three lines of research.

###  II-A Multimodal Event Detection

Most event extraction methods extract event information from single-modal
data, mainly text, and ignore the contribution of visual media [13, 14, 15,
16]. However, in real-world scenarios, data is often multimodal. For instance,
news articles, policy/regulation documents, intelligence reports and social
media posts typically contain text and accompanying images. A number of event
extraction methods have brought forth encouraging results by retrieving
additional related visual knowledge [17, 18].

The problem of event extraction from multimodal data has not been investigated
until recently [12], and it remains an under-explored area. [12] introduced a
multimedia structured common space construction method to take advantage of
the existing image-caption pairs and single-modal annotated data for weakly
supervised training. [18] showed that including the visual context results in
better extraction performance. [17] proposed a dual recurrent multimodal model
to conduct deep interactions between images and sentences for modality
features aggregation. [19] proposed an approach based on pre-trained vision-
language models [20] for addressing multimodal event extraction. However,
these approaches require a complete set of text-image pairs for training.
Furthermore, these models suffer from the data scarcity issue, which is
especially critical in low-resource situations. Unlike these approaches, we
propose a model capable of synthesising the visual context to alleviate the
problem of _low-resource_ data. To tackle ambiguities, one may draw on visual
clues to provide clarity. We propose an automatic event detection approach,
which leverages visual information.

###  II-B Vision-language Pretraining models

Recent years, vision-language pretraining models have been widely studied [21,
22, 23, 24, 25, 26]. It has been shown that visual, structural information
such as scene graphs [27] is useful to pretraining these models. However,
event structural knowledge is not well captured in pretraning models,
resulting in deficiencies in tasks related to verb comprehension [28].
Following the vision-language pretrained model CLIP [20], [19] encoded
structural event semantics and knowledge to enhance vision-language
pretraining.

###  II-C Text-to-Image Generation

Generating images from text has been extensively studied [29, 30, 31].
Representative works use GANs [31, 32, 33] to synthesize photo-realistic
scenes with high semantic fidelity to their conditioned text descriptions.
DALL-E [29] proposes an autoregressive Transformer with discrete VAEs [34] to
create images from text for a wide range of concepts expressible in natural
language. [35] proposed to generate high-quality images using diffusion models
by establishing connections between diffusion models and variational
inference. While our approach is inspired by these works, the goal of the
present work is to synthesise discrete visual representations for improving
event detection instead of generating high-quality photo-realistic images.

##  III Methodology

###  III-A Architecture

Figure 3 illustrates the architecture of our Visually-Fused Event Detection
(VF-Event) framework, which consists of two key components: (1) the visual
imagination module and (2) the pre-trained vision-language module. The
customized imagination module can address the challenges of multimodal event
detection without requiring images at inference time.

---
Figure 3: An illustration of our VF-Event framework. Top: Customizing the
imagination module with samples from support set. Middle: Fine-tuning of the
vision-language module with the support set. Bottom: Inference process of VF-
Event in the absence of visual inputs.

###  III-B Task Formulation

Our model learns to extract the event type of a query instance $s$ given a
support set $\mathbb{S}$ and a set of pre-defined event types $\mathcal{E}$,
which appears in the support set $\mathbb{S}$. We employ a $N$+1-way, $K$-shot
setting, in which there are $N$ clusters representing $N$ event types and an
additional cluster representing the none event type, and each cluster contains
$K$ data points. In addition, there are corresponding visual contexts
$\mathbb{V}$ that are paired with instances in $\mathbb{S}$. The support set
is denoted as follows:

$\displaystyle\mathbb{S}=\\{$
$\displaystyle(s_{1}^{1},v_{1}^{1},e_{1}),\dots,(s_{1}^{K},v_{1}^{K},e_{1}),$
(1) $\displaystyle\dots$
$\displaystyle(s_{N}^{1},v_{N}^{1},e_{N}),\dots,(s_{N}^{K},v_{N}^{K},e_{N}),$
$\displaystyle(s_{N+1}^{1},v_{N+1}^{1},e_{none}),\dots,(s_{N+1}^{K},v_{N+1}^{K},e_{none})\\}$

where:

  * •

each $s_{i}^{j}=(w_{1},w_{2},\dots)$ represents the $j$-th instance of the
support set of event type $e_{i}$, and $w_{m}$ is a token from the vocabulary
$\mathcal{W}$.

  * •

$\mathbb{V}=\\{v_{1}^{1},\dots,v_{1}^{K},\dots,v_{N}^{K}\\}$ are visual
contexts. $v_{i}^{j}\in\mathbb{V}$ is the image paired with the $j$-th
instance of even type $e_{i}$ in the support set.

  * •

$\mathcal{E}=\\{e_{1},\dots,e_{N},e_{none}\\}$ s the set of labels,
representing the pool of all possible event types, and $e_{none}$ is a special
label for non-event.

###  III-C Event Detection

We formulate the task of event detection as a multimodal classification task,
where our proposed model outputs the event type $e$ with the highest
probability among all possible event types $\mathcal{E}$.

The encoder $\textbf{ENC}(\cdot)$ consists of a language encoder
($\textbf{ENC}_{L}$) and a visual encoder ($\textbf{ENC}_{V}$).
$\textbf{ENC}_{L}$ computes the hidden representations (${H}_{s}$) of the
input instance $s$ and a $\textbf{ENC}_{V}$ compute the visual hidden
representations (${H}_{v}$) of the input image $v$.

Hence, the hidden representation (${H}$) of the input $(s,v)$ is formulated
as:

${H}=\textbf{ENC}(s,v)=[\textbf{ENC}_{L}(s);\textbf{ENC}_{V}(v)]$ (2)

The $\textbf{FNN}(\cdot)$ is a feedforward neural network with a soft-max
layer which computes the probability distribution over the possible event
types:

$P_{\theta_{class}}~{}(e|s,v)=\textbf{FNN}_{\theta_{FNN}}(w_{class}.\textbf{ENC}_{\theta_{ENC}}(s,v)+b_{class}),$
(3)

where
$\textbf{ENC}_{\theta_{\textbf{ENC}}}=(\textbf{ENC}_{L},\textbf{ENC}_{V})$
refers to the encoder structure and $\textbf{FNN}_{\theta_{\textbf{FNN}}}$
refers to the feedforward neural network.
$\theta_{\textbf{class}}:=\\{\theta_{\textbf{ENC}},\theta_{\textbf{FNN}}\\}$
denotes the parameters of our classification model. Finally, we compute the
predicted event type by:

$\widehat{e}=argmax_{e\in\mathcal{E}}P_{\theta_{class}}(e|s,v),$ (4)

Given the support set $\mathbb{S}$, the VF-Event architecture can be trained
by minimising the cross-entropy loss:

$\mathcal{L}_{class}=\mathbb{E}_{s,v,e}[-\log P(e|s,v;\theta_{class})]$ (5)

###  III-D Training in a few-shot setting

During training, a support set of instances paired with images with K examples
per event type is available, the VF-Event model outputs the event type with
the highest probability (Figure 3, middle).

During inference, when visual inputs are not available, VF-Event relies on the
Visual Imaginator to synthesize an image given query text $x$. Then we use an
image encoder to compute visual hidden presentations ${H}_{\widehat{v}}$ of
the synthesized image. We then adapt the visual imagination based on the few-
shot data.

Our visual imagination module is based on Denoising Diffusion Probabilistic
Models [35] (Figure 3, top). This generative model consists of a pre-trained
autoencoder that maps images to a spatial latent code, a corresponding decoder
that learns to map the latent representation back to the image, and a
diffusion model that is conditioned on the textual input ($s$). Inspired by
[36], during inference, we use the textual input of $(s,v)$ from the support
set to condition the model to regenerate $v$. Furthermore, we use the input
text $s$ to synthesise visual context $\widehat{v}$. We then resort to the
following reconstruction-based loss function to train the visual imagination
module on the synthesised image $\widehat{v}$:

$\displaystyle\mathcal{L}_{visual}$
$\displaystyle=\mathbb{E}_{s,v,\widehat{v},\epsilon}[\omega||F_{\theta_{v}}(\alpha
v+\sigma\epsilon,s)-\widehat{v}||_{2}^{2}$ (6)

where $F_{\theta_{v}}$ denotes the conditional visual imagination. $\alpha$
and $\sigma$ are terms that control the noise schedule. $v$ is the ground
truth image from the support set. $\epsilon\sim\mathcal{N}(0,I)$ is a noise
term. The only trainable parameters of the visual imagination module are those
of the textual encoder, and we keep the other parameters frozen.

TABLE I: Experimental results of few-shot multimodal event detection on M2E2 dataset. The results are reported on considering 5, 10, 15, and 20 samples of training examples. Methods | 5-shot | 10-shot | 15-shot | 20-shot
---|---|---|---|---
F1 | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall
Text2Event [13] | 6.59 | 27.27 | 3.75 | 30.38 | 30.77 | 30.00 | 43.04 | 43.59 | 42.5 | 56.25 | 54.22 | 58.44
Valhalla [37] | 16.36 | 20.79 | 15.55 | 17.52 | 18.63 | 17.77 | 26.89 | 37.39 | 28.88 | 24.44 | 29.74 | 25.28
Ferozen [38] | 40.36 | 43.26 | 41.11 | 42.01 | 43.21 | 42.22 | 43.19 | 44.33 | 43.33 | 46.11 | 47.11 | 45.97
Clip [20] | 47.93 | 50.15 | 50 | 50.38 | 55.7 | 50 | 52.06 | 54.78 | 52.22 | 60.06 | 62.65 | 60.91
VF-Event | 50.00 | 50.94 | 50 | 50.58 | 60.95 | 48.88 | 63.79 | 70.14 | 62.22 | 65.23 | 67.6 | 64.36

In this work, we train our model using a combination of the loss functions in
equations 5 and 6. We control the contribution of the additional visual loss
by a hyperparameter $\beta$.

$\mathcal{L}=\mathcal{L}_{class}+\beta\mathcal{L}_{visual}.$ (7)

##  IV Experiments

In this section, we describe the experimental setup used to evaluate the
performance of our model, VF-Event, and present the results of the experiments
comparing VF-Event with the baselines.

###  IV-A Experimental Setup

Dataset. We performed our experiments on the M2E2 dataset [12], a multimodal
news dataset that expands upon ACE [39]. The dataset consists of 6,167
sentences, with 1,297 event mentions across 8 event types: Movement.Transport,
Conflict.Attack, Conflict.Demonstrate, Justice.ArrestJail, Contact.PhoneWrite,
Contact.Meet, Life.Die, and Transaction.TransferMoney. To evaluate the
performance of our model with limited training data, we considered a few-shot
setting where only _K_ (5, 10, 15, and 20) data points with both text and
image pairs were used during the training session. Furthermore, during
inference, only text was provided to the model, and paired images were
discarded.

Evaluation Metrics. Following previous works in addressing this task [12], the
results are reported using macro-averaged Precision (P), Recall (R) and
F-measure (F-1) score metrics.

Baselines. In order to validate the proposed model through experimental
comparison, we selected the following multimodal event detection models as the
baselines:

  * •

Text2Event [13]: A text-only baseline that trains a pre-train language model
(T5) without any visual information.

  * •

Valhalla [37]: A multimodal sequence-to-sequence model for machine translation
by leveraging visual hallucination at test time. We train Valhalla by training
the model to generate only the event type.

  * •

Frozen [38]: A method for transforming large language models into multimodal
few-shot learning systems by extending the soft-prompting philosophy of prefix
tuning.

  * •

Clip [20]: A recent model, based on Contrastive Language-Image Pre-training,
learns a multi-modal embedding space, which can be used to estimate the
semantic similarity between a given text and an image.

Experimental Settings. We initialized our Visual Imaginator using the
parameters of the pre-trained stable diffusion model [40]. For text and image
encoding, we employed CLIP ViT-L/14 [20] with an embedding size of 768 and a
dropout of 0.3 over the textual encoder. We use the Huggingface transformer
library 111https://github.com/huggingface/transformers to implement the Visual
Imaginator and the encoder of our model. Optimization was performed using Adam
[41] with a learning rate of $2e-5$. We used a 1-layer feedforward neural
network, and set $\beta$ to 0.01. Our model was trained for 50 epochs with a
batch size of 4. For the baselines, we used the source code and implementation
details provided by the authors 222As we were unable to access the source code
of the Frozen model, we implemented the model by closely following the
descriptions provided in the paper. This ensured that our implementation was
as close as possible to the original model.. We prepared the data for all
models in the same way.

Figure 4: A qualitative comparison of the ground-truth images to synthesised
images. Left: the original image. Middle: the image generated by the pre-
trained stable diffusion model [35]. Right: the image synthesized by our
customized imagination module VF-Event.

###  IV-B Observations and Discussions

Table I shows the performance of the models on the M2E2 dataset, where at the
training time only _K_ data points with both modalities are provided. The
superiority of our model can be observed from the F1, precision and recall
scores associated with event type detection. On F1 score, our model achieves a
substantial 43% improvement on 5-shot and a 9% improvement on 20-shot over the
text-only model Text2Event [13] on event type detection. Our VF-Event model
significantly outperforms the text-only baselines on all four few-shot
settings, which demonstrates the effectiveness of visual imagination for text-
only. This result also indicates that when the number of training datasets are
less (i.e., 5), unimodal, text-only models fail to extract information from
events and fusing visual information during inference, results in poor
detection of event types. Our proposed approach, as shown in Table I, improves
on all compared methods, achieving best precision, recall and F1 scores under
different few-shot settings in all but one metric (Recall in 10-shot).

In summary, these consistent improvements clearly show that visual imagination
can effectively leverage visual representation from the source sentences at
test time for improved event detection.

###  IV-C Qualitative Analysis of the Visual Imaginator

In this section, we present two samples of images generated by our model. We
randomly selected two data points from the test set of M2E2 and used the
textual information to condition our visual imagination to generate the visual
representation of the text (Figure 4). As can be seen in these examples, while
the ground-truth images fail to fully portray the textual information, our
proposed model incorporates this information in the visual representation. For
instance, in the first example, the original image fails to provide any visual
clues about _threw stone_ or _fire/smoke_. In contrast, our proposed Visual
Imaginator captures this information (Figure 4, first row). Similarly, in the
second example, _air war_ is properly portrayed in the image generated by our
model, whereas it is missed in the ground truth image as well as the image
generated by stable diffusion (Figure 4, second row). It can be observed that
sometimes the synthesized images are more informative compared to their
original counterparts. This indicates the ability of our VF-Event model to
generate context-relevant images during inference.

TABLE II: Ablation study on M2E2 dataset: Effects of the visual and textual modality and the Visual Imaginator on the performance of our model in few-shot event detection. VF-Event (L): no images. VF-Event (V): no text. VF-Event (RET): image-retrieval. Methods | 5-shot | 10-shot | 15-shot | 20-shot
---|---|---|---|---
F1 | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall | F1 | Precision | Recall
VF-Event (L) | 46.19 | 49.83 | 45.55 | 50.26 | 60.86 | 47.77 | 64.09 | 70.02 | 62.13 | 62.65 | 64.85 | 63.21
VF-Event (V) | 17.40 | 16.88 | 20.0 | 16.57 | 22.39 | 18.88 | 14.70 | 18.23 | 14.44 | 17.23 | 29.07 | 17.24
VF-Event (RET) | 39.32 | 40.97 | 41.11 | 50.17 | 57.67 | 50.0 | 51.0 | 59.44 | 48.88 | 54.76 | 54.44 | 56.32
VF-Event | 50.00 | 50.94 | 50 | 50.58 | 60.95 | 48.88 | 63.79 | 70.14 | 62.22 | 65.23 | 67.6 | 64.36

###  IV-D Ablation study

We conducted an ablation study to investigate the impact of each modality in
few-shot event detection. We evaluated the performance of our model under two
ablation settings: (1) when the visual context was fully disregarded during
training (VF-Event $(\texttt{L})$), and (2) when the textual context was
disregarded during training (VF-Event $(\texttt{V})$). Moreover, we replaced
our proposed Visual Imaginator with an image retrieval method [42] and
evaluated the performance of our model under this setting (VF-Event
$(\texttt{RET})$).

Table II summarises the results of our ablation study. As shown, VF-Event
$(\texttt{L})$ outperforms VF-Event $(\texttt{V})$ in all settings, indicating
that textual modality carries richer information than visual modality for
event detection. However, using both modalities outperforms models trained on
only one modality (i.e. text), across all metrics for different support sets
(with the exception of the F1-score for the 15-shot setting). This observation
demonstrates the complementary effect of visual context for event detection.
Moreover, VF-Event $(\texttt{RET})$ performs worse than, but close to VF-Event
$(\texttt{L})$, demonstrating the effectiveness of our model when an image
retrieval model is used instead of the Visual Imaginator. Finally, our full
model with Visual Imaginator outperforms VF-Event $(\texttt{RET})$ in all
settings, with a margin of up to 12 points in F1-score. This observation
emphasizes the importance of our proposed Visual Imaginator in synthesizing
context-related information.

##  V Conclusion

This work addresses the challenges of event detection when the model faces
low-resource multimodal data, during training and inference. We propose a few-
shot visually-fused model which combines a pre-trained multimodal module with
a customised Visual Imaginator module. Using the few-shot learning approach
and leveraging only a few labelled paired image-text samples, our model adapts
Visual Imaginator to a scenario of interest, thereby reducing the domain shift
between the original domain of the pre-trained models and the target domain.
The customised Visual Imaginator module also allows for effective event
detection when the visual modality is absent by providing relevant visual
context to the model during inference. Our experiments demonstrate that our
proposed approach effectively exploits the low-resource training data, and
outperforms state-of-the-art techniques for event detection.

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# The Isotope Effect and Critical Magnetic Fields of Superconducting YH6: A
Migdal-Eliashberg Theory Approach

S. Villa-Cortés<EMAIL_ADDRESS><EMAIL_ADDRESS>Instituto de
Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48,
72570, Puebla, Puebla, México  Nevada Extreme Conditions Laboratory,
University of Nevada Las Vegas, Las Vegas, Nevada 89154, USA  O. De la Peña-
Seaman  Instituto de Física, Benemérita Universidad Autónoma de Puebla,
Apartado Postal J-48, 72570, Puebla, Puebla, México  Keith V. Lawler  Nevada
Extreme Conditions Laboratory, University of Nevada Las Vegas, Las Vegas,
Nevada 89154, USA  Ashkan Salamat  Nevada Extreme Conditions Laboratory,
University of Nevada Las Vegas, Las Vegas, Nevada 89154, USA  Department of
Physics & Astronomy, University of Nevada Las Vegas, Las Vegas, Nevada 89154,
USA

###### Abstract

The emergence of near-ambient temperature superconductivity under pressure in
the metal hydride systems has motivated a desire to further understand such
remarkable properties, specifically critical magnetic fields. YH6 is suggested
to be a departure from conventional superconductivity, due to apparent
anomalous behavior. Using density functional calculations in conjunction with
Migdal-Eliashberg theory we show that in YH6 the critical temperature and the
isotope effect under pressure, as well as the high critical fields, are
consistent with strong-coupling conventional superconductivity; a property
anticipated to extend to other related systems. Furthermore, the strong-
coupling corrections occur to the expected BCS values for the Ginzburg-Landau
parameter ($\kappa_{1}(T)$), London penetration depth ($\lambda_{L}(T)$),
electromagnetic coherence length ($\xi(T)$), and the energy gap
($\Delta_{0}$).

superconductivity, isotope effect, Eliashberg theory, superhydride, high
pressure

Over the last decade, a new class of materials of stoichiometric to hydrogen-
rich metal hydrides under pressure has emerged with theoretical predictions
being made about their crystal structures, electronic, dynamic, and coupling
properties [1, 2, 3, 4]. As a result of those periodic table spanning
predictions [5], several conventional-superconductor candidates have been
proposed with critical temperatures ($T_{c}$) approaching room temperature [6,
7, 8]. The experimental breakthrough for these materials came with the
discovery of phonon-mediated superconductivity in H3S, with a maximum $T_{c}$
of $203$ K measured at 155 GPa [9, 10, 11]. Subsequently, high-$T_{c}$
superconductivity measurements were reported in other compounds such as LaH10
with a $250$–$260$ K $T_{c}$ at $170$ GPa [12, 13]; YH9 with $T_{c}=262$ K at
$182$ GPa [14]; YH6 with $T_{c}=220$ K at approx. $160$ GPa [15, 16]; more
recently, CaH6 with $T_{c}=215$ K at $172$ GPa [17]; as well as a reported
carbonaceous sulfur hydride with $T_{c}=287$ K at $267$ GPa [18, 19]. Most of
the theoretical works on the superconducting state of these novel metal
hydrides have shown that a strong electron-phonon coupling and high-energy
hydrogen phonon modes play a key role in the high $T_{c}$ calculated values,
concluding that they are phonon-mediated strong-coupling superconductors [3,
20, 21, 22, 23, 24, 25, 2, 26, 27, 28, 29, 30, 31, 1, 4, 32].

Beyond the $T_{c}$, there are several other important properties of a
superconducting material such as the isotope effect coefficient, the upper,
lower, and thermodynamic critical magnetic fields, the penetration depth, and
the coherence length. The first has been crucial in elucidating the mechanism
responsible for Cooper-pair formation in conventional superconductors. While
the last gives us the response of the materials to an external magnetic field.
The lower and upper critical magnetic fields ($H_{c1}$ and $H_{c2}$) are
specially interesting, since they give a measure of the Meissner effect and
the magnitude of the external magnetic field at which superconductivity is
totally suppressed. Due to experimental difficulties [33], measurements of the
critical magnetic fields of the metal hydrides at high pressures have been
done only at temperatures near $T_{c}$. From that data the slope of
$H_{c2}(T)$ is fitted and $H_{c2}(0)$ is extrapolated using the Ginzburg-
Landau (GL) [34] or the Werthamer–Helfand-Hohenberg (WHH) [35] models, giving
rise to different values of $H_{c2}(0)$. For example, the reported $H_{c2}(0)$
values for YH6, the aim of this work, are 107(157) T and 76(102) T at 160 GPa
and 200 GPa, respectively, for GL(WHH) [15, 36]. Similar differences are
reported in other metal hydrides [9, 12, 33]. Furthermore, these extrapolated
values of $H_{c2}(0)$, in conjunction with GL model, Bardeen-Cooper-Schrieffer
(BCS) theory [37], and empirical relations [38, 39, 40, 41], are used to get a
complete description of other relevant quantities, like the critical magnetic
fields, penetration depth ($\lambda$), coherence length ($\xi$), as well as
the GL parameter ($\kappa=\lambda/\xi$).

While the $T_{c}$ of the metal hydrides has been widely studied theoretically
within a strong-coupling formalism (Migdal-Eliashberg (ME) theory [42]), there
are no reports where these other important properties, like the critical
magnetic fields and related lengths, are calculated from first principles
within a strong-coupling formalism, where corrections to the empirical and
weak-coupling values are expected. This, in conjunction with the lack of
experimental measurements of the critical fields at low temperatures, has
generated confusion and some doubts about the nature of the kind of
superconductors they belong to [38, 39, 40, 41]. For YH6, this has even
suggested a possible deviation from conventional superconductivity [36]. The
aim of this communication is to show that the critical magnetic fields, the
London penetration depth, and the electromagnetic coherence length can be
calculated from first principles using density functional theory (DFT) in
conjunction with ME theory, showing the conventional nature of YH6 and
providing a general prescription to describe the superconducting state in the
high-$T_{c}$ metal hydrides.

To this end, we start by showing that the behavior of the critical temperature
under pressure can be reproduced within the ME formalism, using as input the
Eliashgberg function ($\alpha^{2}F\left(\omega\right)$) calculated from first
principles in the optimized $lm\bar{3}m$ crystal structure (see Supplemental
Material (SM) [43]). As a first step, the Linearized Migdal-Eliashberg
Equations (LMEE, Eq. S8) are solved at a fixed pressure, where $T_{c}$ is
known from experiment. Here, the experimental $T_{c}\,=\,$220 K at 160 GPa
[15] for YH6 and $T_{c}\,=\,$165 K at 200 GPa [36] for YD6 are employed as the
initial data. From this solution, the functional derivative of $T_{c}$ with
respect to $\alpha^{2}F\left(\omega\right)$ is calculated within the Bergmann
and Rainer [44, 45] formalism. These functional derivatives allow a link
between the observed changes in $\alpha^{2}F\left(\omega\right)$, by a
pressure variation (from $P_{0}$ to $P_{i}$), to changes in the critical
temperature $\Delta T_{c}(P_{0},P_{i})$ (Eq. S17). Fig. 1a shows the behavior
of $T_{c}$ under pressure for both, YH6 and YD6. It can be observed that our
calculated $T_{c}$ is in excellent agreement with the experimental values in
the whole pressure interval (160 – 330 GPa), and even better than the
calculations reported previously in the literature, showing that the behavior
of $T_{c}(P)$ is driven mainly by the changes in the electron-phonon
interaction (Fig. S4). The temperature isotope effect coefficient, $\alpha$,
is calculated taking into account the changes in the electron-electron and the
electron-phonon interaction (Eq. S19), coming from phonon changes due to the
isotope mass substitution, within the Rainer and Culetto [46, 47, 48]
formalism. Fig. 1b presents $\alpha$ as a function of pressure, also in
excellent agreement with experiment. This first step confirms the conventional
nature of superconductivity in YH6 and show us that the solutions of the gap
function, $\tilde{\Delta}_{n}$, correctly reproduce the superconducting state
using the $\alpha^{2}F\left(\omega\right)$ determined from first principles
calculations as an input for the LMEEs.

Figure 1: Calculated $T_{c}(P)$ for both compounds, YH6 and YD6 (a), and the
isotope effect coefficient $\alpha$ (b) within the ME formalism (Eqs. S18 and
S19 in SM [43]). For comparison, it is shown the experimental data, as well as
previously reported $T_{c}$ calculations.

We now focus on the critical magnetic fields. In the dirty limit for a strong-
coupling superconductor, $H_{c2}(T)$ has to be evaluated from the LMEEs (Eqs.
S13 and S14) in the presence of a homogeneous magnetic field [49] where the
pair-breaking parameter $\rho(T)$ and $H_{c2}(T)$ are related by

$H_{c2}(T)=\rho(T)/eD,$ (1) $D=l_{tr}v_{F}/3,$ (2)

where $D$ is the diffusion constant, $v_{F}$ the Fermi velocity, $l_{tr}$ the
mean free path, and $e$ is the electron charge. Figure 2 shows the temperature
dependence of $H_{c2}(T)$ calculated within the ME (Eq. 1, solid lines), GL
(dashed lines), and WHH (doted lines) formalism for YH6 at 160 and 200 GPa,
and YD6 at 173 GPa. These pressures were selected to make a direct comparison
with available experimental data (symbols) [15, 36]. $v_{F}$ was calculated
from the dispersion of the electronic band structure (see Table 1), and
$l_{tr}=1.655(1.645)$ Å for YH6(YD6) was fitted using Eq. 2 to get the
experimental $H_{c2}$ data close to $T_{c}$. Near $T_{c}$, the calculated
values of $H_{c2}(T)$ are similar for the three models, and in very good
agreement with experiment. As the temperature starts to decrease, the
difference between the models’ results increases, with the ME results being in
between the higher WHH and lower GL values. The largest differences between ME
and the GL and WHH models ($\Delta{H_{c2}}$) are at $T=0$ K, with values that
go from $\Delta{H_{c2}}(\mbox{YH${}_{6}$})=7(18)$ T at 200 GPa to as high as
$\Delta{H_{c2}}(\mbox{YH${}_{6}$})=17(33)$ T at 160 GPa, in respect to
WHH(GL). Although the GL theory is applicable to practically all
superconductors, it is a phenomenological theory restricted to temperatures
close to $T_{c}$. Therefore, GL theory is not expected to give accurate
results at lower temperatures, giving rise to the huge differences at T=0 K
between itself and the other theories. The WHH formalism is based on weak-
coupling BCS theory, and thus should be valid for the whole temperature range;
whereas the ME formalism covers weak- and strong-coupling. Thus, the ME
results show a distinct strong-coupling correction to the WHH results.

Figure 2: Calculated upper critical field for both compounds, YH6 and YD6,
within the Migdal-Eliashberg formalism. For comparison, it is shown the
experimental data (symbols) [15, 36] and the extrapolated behavior within the
GL and WHH models taken from [15, 36].

To calculate the thermodynamic critical field, $H_{c}$, of an isotropic
strong-coupling superconductor, the Non-linear MEEs (NLMEE) have to be solved
(Eqs. S5 and S6). With the knowledge of $\tilde{\Delta}_{n}$ and
$\tilde{\omega}_{n}$, the difference in free energy between the normal and the
superconducting states of the metal, $\Delta F(T)=F_{n}-F_{s}$, can be
calculated directly. By definition, $H_{c}$ is given by the relation
$H_{c}(T)=[8\pi(\Delta F(T))]^{1/2}$. Then, from the calculated fields, the GL
parameter and the lower critical field can be evaluated from the relations
$\kappa_{1}=(1/\sqrt{2})H_{c2}/H_{c}$ and
$H_{c1}H_{c2}=H_{c}^{2}\text{ln}(\kappa)$, respectively. Expanding Figure 2 to
higher pressures to span the whole pressure range studied, Figure 3 shows the
calculated behavior of $H_{c}(T)$, $H_{c1}(T)$, and $H_{c2}(T)$ for YH6 (solid
lines) and YD6 (dashed lines) at 160, 200, and 250 GPa. As a function of
pressure, there is a steady shift of the critical fields to lower values as
the pressure is incremented. In particular, at $T$=0 K there is a considerable
suppression in $H_{c2}$ from 141.5(110.8) T to 111.8(91.0) T for YH6(YD6), at
160 and 250 GPa respectively (see Table 1). As it can be seen, the three
fields show an isotopic shift to lower values due to the replacement of
hydrogen by deuterium.

The strong-coupling behavior of the calculated $H_{c2}(T)$ and $H_{c}(T)$ by
the ME-model is confirmed by the deviation function, $D(t)$ (Eqs. S27 and
S28), which shows the standard behavior (positive values) for strong-coupling
materials, in contrast to the intermediate-coupling behavior (change in sign)
that is observed in the WHH model (Fig. S6). Fig. 3d shows the parameter
$\kappa_{1}(T)$, which has values between approximately 24 and 27 for both
compounds at $T$=0 K (Table 1), while at $T_{c}$ (where $\kappa_{1}$ is
similar to the GL $\kappa$ [50]) this parameter has its minimum values and
vary from 17.5 to 17.7. The ratio $\kappa_{1}(0)/\kappa_{1}(T_{c})$ give us an
estimation of the strong-coupling correction to BCS values. While in the weak-
coupling formalism there is a universal ratio
$\kappa_{1}(0)/\kappa_{1}(T_{c})$=1.12 [49], the values for YH6(YD6) vary from
1.49(1.5) to 1.38(1.4) at 160 and 250 GPa, respectively. Such enhanced values
clearly show that these systems are within the strong coupling regime across
this noted pressure range, and the slight decrease with respect to pressure
shows the tendency towards a less strong-coupling (intermediate) regime, as
expected from the general trend under pressure of the coupling parameter (Fig.
S4).

Figure 3:  Calculated thermodynamic (a), lower (b), and upper (c) critical
magnetic fields within the Migdal-Eliashberg formalism for YH6 (solid lines)
and YD6(dashed lines) at 160 (blue), 173 (green), 200 (red), and 250 (black)
GPa. The parameter $\kappa_{1}(T)=(1/\sqrt{2})H_{c2}(T)/H_{c}(T)$ (d) is used
in conjunction with the relation $H_{c1}H_{c2}=H_{c}^{2}\text{ln}(\kappa)$ to
get the lower critical magnetic field (b). Also shown are the electromagnetic
coherence length (e) and the London penetration depth (f) in the clean limit.

Despite expressions which are valid within Migdal-Eliashberg theory being
derived several years ago [51, 52], there are very few strong-coupling
superconductor systems with reported numerical results of their
electromagnetic properties. Here we are interested in the magnetic field
penetration depth (in the London limit) $\lambda_{L}(T)$, and the
electromagnetic coherence length $\xi(T)$. Both quantities are studied in the
clean limit, due to the very large energy scale associated with the phonon
frequencies and superconducting gap for the metal hydrides, as was previously
suggested for H3S by Nicol and Carbotte [20]. The London-limit penetration
depth, which applies when $\lambda_{L}(T)\gg\xi(0)$, is given by

$\lambda_{L}(T)=\left[\frac{4}{3}\pi
N(0)e^{2}v_{F}^{2}T\mu_{0}\sum_{n=1}^{\infty}\frac{\tilde{\Delta}_{n}^{2}}{\left(\tilde{\omega}_{n}^{2}+\tilde{\Delta}_{n}^{2}\right)^{3/2}}\right]^{-1/2},$
(3)

where $\mu_{0}$ is the permeability and $N(0)$ is the single spin density of
electronic states at the Fermi energy. The electromagnetic coherence length,
which describes the nonlocality in the electromagnetic response of a
superconductor, is given by

$\xi(T)=\frac{v_{F}\hslash}{2}\frac{\left[\sum_{n=1}^{\infty}\frac{\tilde{\Delta}_{n}^{2}}{\left(\tilde{\omega}_{n}^{2}+\tilde{\Delta}_{n}^{2}\right)^{3/2}}\right]}{\left[\sum_{n=1}^{\infty}\frac{\tilde{\Delta}_{n}^{2}}{\tilde{\omega}_{n}^{2}+\tilde{\Delta}_{n}^{2}}\right]}.$
(4)

The solutions of the NLMEEs (Eqs. S5 and S6), $\tilde{\Delta}$ and
$\tilde{\omega}$, are required to get both $\lambda_{L}(T)$ and $\xi(T)$.
Numerical results for the temperature variation of $\xi(T)$ are given in Fig.
3e. In superconductors, $\xi(T)$ is the range of the perturbation of the
current density caused by an applied electromagnetic field, which is different
from the GL coherence length $\xi$ that describes the perturbation of the
superconducting pair density. In the weak-coupling theory, both quantities are
related at $T=0$ K through the relation $\xi=0.739\xi(0)$. From these strong-
coupling formalism calculations, $\xi(0)$ is found to be 1.71(2.18) nm and
2.11(2.58) nm for YH6(YD6) at 160 and 250 GPa, respectively. Using the GL
relation, $\xi=\sqrt{\frac{\phi_{0}}{2\pi H_{c2}}}$, the GL coherence length
$\xi$ is found to be 1.53(1.72) nm and 1.73(1.94) nm for YH6(YD6), at the same
applied pressures which are in agreement with GL experimental estimations of
1.4–1.8 nm at 160 GPa for YH6 [15, 36]. These values yield ratios $\xi/\xi(0)$
0.895(0.787) and 0.82(0.75) for YH6(YD6) at 160 and 250 GPa, respectively,
which clearly deviate from the weak-coupling limit of 0.739. As the
temperature increases, $\xi(T)$ falls quickly near $T_{c}$, where the ratio
$\xi(T_{c})/\xi(0)$ drops to a value of about 0.82(0.83) for YH6(YD6) at 160
GPa, which is larger than the BCS value of 0.752 [53].

Fig. 3f shows the temperature dependence of $\lambda_{L}(T)$. $\lambda_{L}(0)$
is found to be approximately 159 nm for both, YH6 and YD6 at 160 GPa, which is
close to the reported BCS values of 164 and 147 nm for H3S and LaH10,
respectively [41]. The strong-coupling deviation function of
$[\lambda_{L}(0)/\lambda_{L}(T)]^{1/2}$ with respect to the two fluid model
for YH6 is shown in Fig. S6. The deviation function (Eq. S29), gives a minimum
value of -0.05 GPa, which is in contrast to the -0.22 GPa result from BCS
weak-coupling theory [54]. By means of Padé approximants [55], the energy gap,
$\Delta_{0}$, can be found from an analytic continuation to the real axis of
$\tilde{\Delta}_{n}$ (computed from the NLMEEs). Here, $\Delta_{0}$ is
46.68(37.6) meV and 37.7(30.9) meV for YH6(YD6) at 160 and 250 GPa,
respectively. From these values, the BCS ratios $2\Delta_{0}/k_{B}T_{c}$ are
4.94(5.13) and 4.50(4.74) which shows a strong-coupling correction to the 3.52
BCS value.

Table 1: Zero-temperature calculated parameters of the superconducting state at selected pressures for YH6 and YD6 using the solutions of the NLMEEs and LMEEs, in conjunction with Eqs. 1, 3 and 4 for the upper critical field, London penetration depth ($\lambda_{L}$), and coherence length ($\xi$), respectively. The critical fields are in units of Tesla, the Fermi velocity ($v_{F}$) in $\times 10^{5}m/s$ (Eq. S26), $\lambda_{L}$ and $\xi$ in nm. |  |  | YH6 |  |  | YD6
---|---|---|---|---|---|---
P |  |  | $v_{F}$ | $H_{c}\left(0\right)$ | $H_{c1}\left(0\right)$ | $H_{c2}\left(0\right)$ | $\kappa_{1}\left(0\right)$ | $\lambda_{L}\left(0\right)$ | $\xi\left(0\right)$ |  |  | $v_{F}$ | $H_{c}\left(0\right)$ | $H_{c1}\left(0\right)$ | $H_{c2}\left(0\right)$ | $\kappa_{1}\left(0\right)$ | $\lambda_{L}\left(0\right)$ | $\xi\left(0\right)$
160 |  |  | 8.76 | 3.79 | 0.33 | 141.5 | 26.37 | 159.15 | 1.712 |  |  | 8.75 | 2.96 | 0.26 | 110.8 | 26.48 | 158.38 | 2.18
180 |  |  | 8.68 | 3.59 | 0.33 | 129.0 | 25.57 | 157.98 | 1.884 |  |  | 8.68 | 2.83 | 0.25 | 103.0 | 25.76 | 157.38 | 2.34
200 |  |  | 8.61 | 3.45 | 0.31 | 122.3 | 25.04 | 157.40 | 1.954 |  |  | 8.59 | 2.74 | 0.25 | 97.7 | 25.27 | 157.22 | 2.43
220 |  |  | 8.53 | 3.35 | 0.31 | 116.7 | 25.64 | 158.12 | 2.164 |  |  | 8.51 | 2.68 | 0.25 | 91.9 | 24.93 | 157.93 | 2.54
250 |  |  | 8.39 | 3.25 | 0.30 | 111.6 | 24.27 | 159.04 | 2.110 |  |  | 8.38 | 2.61 | 0.24 | 90.9 | 24.60 | 159.28 | 2.58

To summarize, the superconducting properties and electromagnetic field
response of the yttrium hydride YH6 are computed here from first principles,
using DFT in conjunction with ME theory. From these, the experimental behavior
of $T_{c}$ as a function of applied pressure and the isotopic effect can be
perfectly reproduced within the harmonic approximation. Similarly, the
calculated $H_{c2}(T)$ with the ME formalism shows excellent agreement with
the available experimental data at temperatures near $T_{c}$. As $T$ goes to
zero, it shows intermediate values between the GL and WHH models, providing an
important strong-coupling correction to these currently used phenomenological
models. The implemented formalism even allows a full description of the $H-T$
phase diagram by calculating $H_{c}(T)$, $H_{c1}(T)$, and $\kappa_{1}(T)$.
Finally, the description of the YH(D)6 superconducting state is completed by
calculating $\xi(T)$ and $\lambda_{L}(T)$ within the clean limit, as well as
the energy gap $\Delta_{0}$. From our results, we found that any deviation
from BCS behavior is well explained as a strong-coupling correction, and in
this basis we are able to discard any possible anomalous behavior or departure
from conventional superconductivity as was previously suggested [36]. Even
more, we consider that ME theory, in conjunction with Rainer-Bergmann [49] and
Nam’s [51, 52] formalism, can provide a general prescription to describe the
superconducting state in the high-$T_{c}$ metal hydrides.

###### Acknowledgements.

This research is funded in part by the Gordon and Betty Moore Foundation’s
EPiQS Initiative, Grant GBMF10731, partially supported by the Consejo Nacional
de Ciencia y Tecnología (CONACyT, México) under Grant No.
FOP16-2021-01-320399, as well as by the U.S. Department of Energy, Office of
Basic Energy Sciences under Award Number DE-SC0020303. The authors thankfully
acknowledge computer resources, technical advice, and support provided by
Laboratorio Nacional de Supercómputo del Sureste de México (LNS), a member of
the CONACYT national laboratories.

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