# Multi-Label Continual Learning for the Medical Domain: A Novel Benchmark

Marina Ceccon
Università degli studi di Padova
Padova
<EMAIL_ADDRESS>
&Davide Dalle Pezze
Università degli studi di Padova
Padova
<EMAIL_ADDRESS>
Alessandro Fabris
Max Planck Institute
for Security and Privacy
Bochum
<EMAIL_ADDRESS>
&Gian Antonio Susto
Università degli studi di Padova
Padova
<EMAIL_ADDRESS>

###### Abstract

Multi-label image classification in dynamic environments is a problem that
poses significant challenges. Previous studies have primarily focused on
scenarios such as Domain Incremental Learning and Class Incremental Learning,
which do not fully capture the complexity of real-world applications. In this
paper, we study the problem of classification of medical imaging in the
scenario termed New Instances & New Classes, which combines the challenges of
both new class arrivals and domain shifts in a single framework. Unlike
traditional scenarios, it reflects the realistic nature of CL in domains such
as medical imaging, where updates may introduce both new classes and changes
in domain characteristics. To address the unique challenges posed by this
complex scenario, we introduce a novel approach called Pseudo-Label Replay.
This method aims to mitigate forgetting while adapting to new classes and
domain shifts by combining the advantages of the Replay and Pseudo-Label
methods and solving their limitations in the proposed scenario. We evaluate
our proposed approach on a challenging benchmark consisting of two datasets,
seven tasks, and nineteen classes, modeling a realistic Continual Learning
scenario. Our experimental findings demonstrate the effectiveness of Pseudo-
Label Replay in addressing the challenges posed by the complex scenario
proposed. Our method surpasses existing approaches, exhibiting superior
performance while showing minimal forgetting.

_K_ eywords Continual Learning $\cdot$ Medical Imaging $\cdot$ Multi-Label

##  1 Introduction

Chest X-ray images are commonly used to detect a wide range of diseases,
including pulmonary and cardiac conditions. Early detection and timely
intervention are critical for improving patient outcomes and reducing
mortality rates [1, 2]. In recent years, several studies have proven the
efficacy of using Deep Learning models to detect diseases from chest X-ray
images [3, 4]. While these models allowed for improved state-of-the-art in the
field, many significant challenges remain before considering their use to
support decision-making in realistic scenarios. One such challenge is ensuring
the model’s adaptability to dynamic environments, where shifts in input data
distribution may occur over time [5, 6].

For example, each hospital may employ machines with different image
acquisition techniques, resulting in subtle yet consequential shifts in data
distribution. These discrepancies can perturb the performance of previously
trained models, underscoring the need for ongoing recalibration and
refinement.[3]. Another consideration is the potential need to detect diseases
not initially incorporated into the trained model. For instance, the labeling
of additional classes may have been provided at a later point in time,
reflecting evolving medical understanding and diagnostic criteria [7].

These observations highlight the importance of model flexibility and
scalability in accommodating new data and expanding diagnostic capabilities as
medical knowledge progresses. Failure to address this can severely limit the
model’s utility and constrain its ability to effectively serve patient needs
in the face of emerging health challenges. Therefore, maintaining an adaptable
framework capable of incorporating new disease classifications is crucial for
ensuring the continued relevance and efficacy of diagnostic algorithms [5]. A
paradigm called Continual Learning (CL) has emerged in the literature to solve
this problem. CL enables the model to adapt to new data while retaining
knowledge from the old data. For these reasons, our work considers applying CL
strategies to the problem of pathology classification of chest X-ray images.

Previous studies in this domain focused on solving the problem in two
different settings: Domain Incremental Learning (DIL) and Class Incremental
Learning (CIL). In DIL, the system faces new tasks with shifts in the domain
(i.e., shifts in the input data distribution), whereas in CIL, new classes are
continuously added to existing tasks [8]. However, these scenarios fail to
accurately represent real-world challenges where both new classes and domain
shifts can happen. Thus, a more realistic scenario termed New Instances & New
Classes (NIC) is studied and analyzed in this work. NIC combines the
challenges of both new class arrivals and domain shifts within a single
framework, mirroring the complexities often encountered in realistic
applications like medical imaging [9]. Therefore, we propose a novel benchmark
for the medical imaging field based on the NIC scenario, considering nineteen
classes, seven tasks, and two domains.

In this scenario, we evaluate the performance of state-of-the-art CL methods.
In particular, replay-based and distillation-based approaches are often used
to solve the CIL scenario in previous works [7, 5]. However, when applied to
the NIC scenario, these approaches face limitations in a multi-label setting.
Replay-based approaches may suffer from task interference between the samples
from the current task and the replayed samples[10]. Moreover, distillation-
based methods deteriorate when old classes do not reappear in future tasks
[11]. These considerations outline challenges that previously proposed methods
cannot handle properly.

This work introduces a novel approach called Pseudo-Label Replay to address
such challenges. This approach leverages Pseudo-Labeling [12] and Replay [13]
techniques to integrate information from previous tasks while adapting to new
data streams. Experimental results demonstrate the effectiveness of Pseudo-
Label Replay in handling a complex NIC scenario. The proposed approach
outperforms existing methods, achieving significant performance improvements.
This highlights the proposed method’s potential to address the complexities of
real-world CL scenarios and improve the overall performance of multi-label
image classification systems.

Overall, we make the following contributions:

  * •

We devise and motivate a novel benchmark for CL in medical imaging, based on
the challenging NIC scenario, to study flexible collaboration across hospitals
and research centers.

  * •

We propose Pseudo-Label Replay, a novel method designed specifically for the
NIC scenario, which integrates Pseudo-Labeling and Replay techniques to handle
new classes and domain shifts effectively.

  * •

We demonstrate the superiority of Pseudo-Label Replay over existing methods
through experimental validation.

In addition, to promote further research in the domain and facilitate the
advancement of novel methodologies along with comparisons with state-of-the-
art approaches, we make the code available.111
https://anonymous.4open.science/r/pseduolabel_replay-6720/README.md
Our work is structured as follows: Sec. 2 provides an overview of the CL
literature, focusing on medical applications, particularly the classification
of chest X-rays. Sec. 3 presents a dynamical benchmark for multi-label medical
imaging, while Sec. 4 proposes a novel strategy. Sec. 5 presents our
experimental setting, focusing on the methods and metrics utilized, and Sec. 5
describes the results of our experiments. Lastly, Sec. 7 concludes this work
by discussing limitations and future research directions.

Figure 1: Scheme of the multi-label CL setting in the context of
classification of chest X-rays. Diagnostic capabilities are expanded over time
with new tasks.

##  2 Related Work

In this section, we provide an overview of the most popular CL scenarios and
methods in Sec. 2.1. Following that, we focus on the medical domain and chest
X-ray image classification for the CL setting in Sec. 2.2.

###  2.1 Continual Learning

CL has emerged as a framework to adapt models to new data distributions
without forgetting what they have already learned [14]. In particular, the
main problem tackled by the CL methods is that of _catastrophic forgetting_ :
by fine-tuning the new distributions of data, the knowledge of old data gets
completely corrupted [15, 13].

####  2.1.1 Scenarios

In the CL literature, most works refer to three CL scenarios based on the type
of shift in the data stream and the availability of the task ID [8]. In the
first scenario, called Task-Incremental, the model receives assistance during
testing by receiving both the input sample and the corresponding task
identification, making the prediction process easier [14]. On the other hand,
in the Domain-Incremental setting, it is assumed to have a sequence of tasks
where each task differs because of a shift in the input data distribution
while the set of classes remains common for all tasks. In this case, only the
input is provided without additional information during testing, which makes
the problems more challenging. Similarly, the Class-Incremental scenario
forbids access to the task provenience during the test. The difference with
the previous scenario is that, in this case, each task contains new classes;
in other words, the set of classes is not constant among the tasks. Hence, in
this scenario, during the testing phase, the model must be able to internally
deduce to which task the given sample belongs and then perform the prediction
on the sample [8].

Subsequently, a more challenging scenario termed New Instances & New Classes
(NIC) has emerged. NIC combines the challenges of both new class arrivals and
domain shifts within a single framework, mirroring the complexities often
encountered in realistic applications like medical imaging. This combination
of different types of shifts between tasks makes this scenario more
challenging and suitable for representing intricate settings, such as the
medical domain. The concept of the NIC scenario was initially introduced and
investigated by Lomonaco et al. (2017) [9]. It was examined within the context
of a single-label classification problem, where it is assumed there are no
intersections among tasks regarding labels or samples.

To the best of our knowledge, this scenario was never analyzed in multi-label
settings. Indeed, the multi-label image classification problem has been
examined by previous works under the DIL setting in Dalle Pezze et al. (2023)
[16] and under CIL in Kim et al. (2020) [17] and Liang et al. (2021)[18].
However, these studies operate under the strong assumption that there are no
intersections among tasks regarding data and labels, thus simplifying the
scenario and making it more easily solvable.

####  2.1.2 Approaches

The methods in the CL literature can be grouped into three big families of
strategies known as rehearsal-based, regularization-based, and architecture-
based approaches.

Rehearsal-based techniques assume storing and reusing past data samples during
training. Within this category, various approaches exist; a prominent method
is Experience Replay [19], simply referred to as Replay. This method stores a
portion of the old data in memory. While training on a new task, the new data
is combined with the data from the memory buffer to maintain knowledge of old
tasks. This method has demonstrated remarkable performance in multi-class
classification problems, consistently demonstrating its effectiveness in all
three CL scenarios (DIL, CIL, TIL) [20]. However, it was proven to fail in
multi-label CIL settings [11]. Indeed, interference between the current task
samples and the replayed samples occurs since they only contain the
information on the labels of the relative task.

Regularization-based approaches consider additional penalties or constraints
during training to maintain the memory of old tasks. A well-known method in
this category is Learning without Forgetting (LwF) [21]. LwF is based on the
idea of _knowledge distillation_ proposed originally by Hinton et al. (2015)
[22]. In the process of distillation, the knowledge acquired by a model from
prior tasks is condensed into a compact form, such as soft target
probabilities, which are then utilized during the training of subsequent
tasks. In particular, during training, the loss is computed as the combination
of the current loss, computed on the model outputs and on the current task
targets, and the distillation loss, computed on the current model outputs and
the old model outputs. This additional KD loss encourages the model to produce
outputs similar to the old model’s.

When evaluated on the classic problem of multi-class image classification,
this model is proven not optimal [8]. On the other hand, this method
demonstrated good performance in multi-label settings such as Object
Detection, where objects from old tasks are present in the new images but not
labeled [23].

Pseudo-Label has emerged as a successful approach in Continual Object
Detection [12]. As in LwF, the model trained on the previous task is saved in
memory and used while training on a new task. While visiting the data of a new
task, the new task samples are passed through the old models, and the
predictions on the old labels are added to the new ground truth targets.

Architecture-based approaches, as the name suggests, are methods that focus on
changing the original model’s architecture to help maintain the old knowledge;
methods in this family follow different approaches [24, 25, 26].

Moreover, in CL literature, some methods propose combining different
strategies families to obtain better and more robust performance. For example,
in Mi et al. (2020) [27], samples from previous tasks are periodically
replayed to the current model, and an adaptive distillation loss is used.
Similarly, Hou et al. (2018) [28] combines a distillation-based approach with
Replay. Kalb et al. (2021) [11] combine a distillation-based approach with
Replay for the Semantic Segmentation problem in the CL setting.

To the best of our knowledge, our work is the first to combine the Pseudo-
Label strategy with the Replay approach.

###  2.2 CL in the Medical Domain

Previous works have explored both DIL and CIL scenarios within the context of
CL methods for classifying chest X-ray images. For the DIL scenario, Lenga et
al. (2020) [3] conduct a study considering two tasks based on chest X-ray
images; Srivastava et al. (2021) [6] address a similar setting through a
replay-based approach.

Singh et al. (2023)[7] consider a CIL scenario with three tasks encompassing
12 classes, gradually introducing new classes over time. Similarly, Akundi et
al. (2022) [29] study a CIL scenario with five tasks, each with only one
class, for a total of 5 classes considered. One notable limitation of existing
studies is the relatively short streams with few classes, hindering a
comprehensive evaluation of the efficacy of the studied CL methods. Another
limitation is the choice of scenarios, such as DIL or CIL, which fail to
describe the challenges of the medical domain fully.

The challenges posed by multi-label settings are commonly encountered in CL
when solving problems such as Object Detection [23] and Semantic Segmentation
[11]. In these studies, the authors usually opt for distillation-based
approaches instead of replay-based ones since they were proven to alleviate
the concealed old classes problem in multi-label settings [12].

In the same way, Akundi et al. (2022) [29] propose using distillation-based
approaches in the medical imaging domain. However, previous works found a
significant degradation in the performance of these approaches when assuming
that certain classes encountered at the start of the data stream rarely
reappear in subsequent tasks [11]. These considerations bring additional
challenges that previously proposed methods could not handle correctly.

Figure 2:  Benchmark proposed for the medical imaging field. It represents a
New Instances & new Classes scenario, where some tasks have new classes while
other tasks have already seen classes but with a shift in the input data
distribution. Our proposed stream consists of a sequence of seven tasks
totaling nineteen classes across two domains.

##  3 Proposed Scenario

We consider the problem of flexible collaboration across medical institutions,
enabling effective data pooling and model sharing. The nature of healthcare
systems and processes leads to three challenges. First, medical data is highly
sensitive and subject to stringent data protection regulations [30]. Sharing
and processing medical data across multiple institutions is especially
difficult as hospitals are independent organizations with distinct data
controllers [31]. Informed consent can enable broader processing of medical
data, but it is challenging to obtain for a large and representative group of
patients [32].

Furthermore, medical diagnosis encompasses numerous disease labels,
necessitating the curation and annotation of a comprehensive dataset across
multiple pathologies, which can take several years [33]. It is crucial for
models to evolve incrementally and incorporate information about new diseases
as they become available [34]. New diseases may surface, as was the case with
COVID-19. Adapting existing Deep Learning models trained to classify other
pathologies also to classify the new disease is desirable. Moreover, the
classification of rare pathologies benefits is often carried out by extending
models for related yet different conditions [35]. Typically, access to the
original data used for training the model is not feasible due to data
protection regulations. Consequently, a scenario where the model is updated
with only a small subset of historical data, like the CIL setting, becomes
crucial.

Third, domain shift is inevitable, e.g., due to differences in medical
equipment, acquisition protocols, and evolving populations [3, 36, 37]. This
is especially true for datasets spanning multiple sites, which may serve areas
with different demographics or provide specialized care, e.g., to different
age ranges. In general, medical processes adopted at one hospital influence
data acquisition and cause artifacts in the data [38].

Overall, both DIL and CIL scenarios are common in the medical domain. Clearly,
they do not represent mutually exclusive settings. Hence, a situation in which
both new domains and new classes appear is not only possible but also very
plausible, and it needs to be addressed.

In this work, we model an NIC scenario in chest X-ray imaging across two
sites. We consider the ChestX-ray14 dataset, compiled from patients of the NIH
Clinical Center (NIH) [39] and the CheXpert dataset, with radiographies from
patients treated at the Stanford Hospital (CXP) [1]. Both datasets contain
information on 14 diseases, 7 of which are in common. We removed non-pathology
labels from CXP, leaving us with 19 classes. We divide CXP into three tasks
and the NIH dataset into four tasks, ordering them sequentially to create the
NIC stream of tasks represented in Figure 2. Each task models the availability
of new data at one hospital. Between tasks $i$ and $i+1$, new classes or data
from known classes are added, but with a shift in the input data distribution,
it becomes available for a new hospital.

In each task, all and only the images containing the associated pathologies in
the corresponding dataset are present.222As described in Section 5.1, we allow
for 3% Replay, i.e., a small percentage of (randomly selected) data from old
tasks $[0,\dots,i-1]$ is available in task $i$ from patients who provide
informed consent.

Since several images contain labels from different tasks, there exists an
intersection among tasks, presenting a realistic scenario where task overlaps
challenge traditional replay-based methods. Moreover, we include rare diseases
in the initial tasks to test the stability of models, i.e., their ability to
keep classifying old classes correctly despite a low probability of the
respective disease appearing (without a label) in future tasks.

##  4 Proposed approach: Pseudo-Label Replay

As previously discussed, in the considered scenario, rehearsal-based
approaches suffer from task interference since the same (or similar) data seen
in previous tasks with old labels can reappear again in the future with a
different set of labels, as is shown in Fig. 1. This problem is partially
solved by knowledge distillation from the model learned during the previous
task. However, the efficacy of distillation-based approaches is guaranteed
only if, over time, images with old labels reappear, enabling the model to
remember how to predict the old classes correctly. In more detail, the old
model is used to predict the old classes in the new images. However, if the
prediction of the old model always predicts the absence of such labels, then
the information provided is limited, reducing the method’s efficacy.

In other words, the main challenges of our scenario are represented by the
non-null intersection between tasks, which is an obstacle for replay-based
approaches, and by the presence of rare pathologies with low prevalence in
future tasks, which obstructs distillation-based approaches. To tackle these
challenges, we propose Pseudo-Label Replay, described in Pseudocode 1.

Input: Model $f_{\theta}$, Replay Memory $(X_{m},Y_{m}^{L_{o}})$ of previous
old tasks, Dataset of Task n $(X_{n},Y_{n}^{L_{n}})$, new classes set $L_{n}$,
old classes set $L_{o}$

Output: $f_{\theta}$, $X_{m}$, $Y_{m}$

(1) Apply Pseudo-labeling on new data;

$Y_{n}^{L_{o}}\leftarrow f_{\theta}(X_{n})$;

$Y_{n}\leftarrow Y_{n}^{L_{o}}\cup Y_{n}^{L_{n}}$;

(2) Train with replay;

$X_{train}\leftarrow X_{m}\cup X_{n}$;

$Y_{train}\leftarrow Y_{m}\cup Y_{n}$;

Train $f_{\theta}$ using $(X_{train},Y_{train})$;

(3) Apply Pseudo-labeling on old data;

$Y_{m}^{L_{n}}\leftarrow f_{\theta}(X_{m})$;

$Y_{m}\leftarrow Y_{m}^{L_{o}}\cup Y_{m}^{L_{n}}$;

(4) Creation of Replay Memory for the new task;

$X^{\prime}_{n},Y^{\prime}_{n}\leftarrow$ subset$(X_{n},Y_{n})$;

$X_{m}\leftarrow X_{m}\cup X^{\prime}_{n}$;

$Y_{m}\leftarrow Y_{m}\cup Y^{\prime}_{n}$;

return _$f_{\theta}$ , $X_{m}$, $Y_{m}$_;

Algorithm 1 Pseudo-Label Replay

As described in step (1) of the Pseudocode and represented in point (1) of
Fig. 3, before training on a new task, the targets of the current task are
modified to add information on previous classes, as done in Pseudo-Label. This
information is given by the teacher model saved in memory, whose outputs on
the new samples are binarized using appropriate thresholds. The samples with
modified targets are then passed to the model for training, together with some
samples replayed from the memory buffer, as represented in step (2). The loss
on the current task samples is computed considering all labels seen up to now,
while the loss on the Replayed samples is computed considering only the old
classes. At step (3), after training on the new task, the samples already
present in the buffer are modified to include the information on the new
classes, using the newly trained model and the pseudo-labels derived from it.
Lastly, in step (4), a small portion of the current samples, with the targets
containing the pseudo-labels added before training, is saved in the memory
buffer.

Figure 3: Visual representation of Pseudo-Label Replay.

The advantages of Pseudo-Label Replay are threefold: (1) the Replay is
optimized, since the targets give information not only on the tasks they were
taken from but on all tasks up to the current one; (2) at the same time, task
interference is reduced; (3) finally, by Replaying samples that contain old
classes, the limitations of distillation are overcome.

##  5 Experimental Setting

We describe the implementation details of the experiments regarding the
utilized CL methods in Sec. 5.1. Then, we describe the considered evaluation
metrics in Sec. 5.2.

###  5.1 Considered methods

We evaluated multiple CL methods and compared their performance with our
approach. In particular, we considered:
Joint Training: this method corresponds to training on all tasks
simultaneously. As previously explained, in real-world scenarios, this
approach is often infeasible; we consider it to be an upper bound.
Fine-Tuning: this method involves sequential training on each task without
employing additional techniques to prevent catastrophic forgetting; we
consider it a lower bound for CL approaches.
Replay: for this method, after training on each task, we store in a memory
buffer a portion of samples from said task. We choose as size of the memory
buffer 3% of the original dataset size. If $M$ is the size of the buffer,
after training on each task $i$, there are a total of $\frac{M}{i}$ samples
from each task in the buffer.
Learning without Forgetting (LwF): after training on each task, we save in
memory the optimal model. While training on task $i$, the loss is computed as
$L=L_{1}+\tau L_{2}$, where $L_{1}$ is the loss for the current task, while
$L_{2}$ is the distillation loss, computed on the current model outputs and
the old model outputs, considering only the old labels. The hyper-parameter
$\tau$ handles the trade-off between stability on the old tasks and plasticity
on the new one; we set $\tau$ = 2, following prior work [21].
Pseudo-Label: we use the model saved in memory to modify the ground truth
targets, adding the old model predictions for the old labels. As thresholds
for the prediction, we use the thresholds found to optimize the F1 score on
the validation sets relative to the old tasks.
LwF Replay: we combine the two methods by computing the loss $L=L_{1}+\tau
L_{2}$ on both the new samples and the Replayed ones. The hyper-parameters are
chosen as in LwF; the Replay buffer size is 3%.
Pseudo-Label Replay: for our approach, we combine the Replay and Pseudo-Label
strategies with the same buffer size and threshold setting strategy.

###  5.2 Considered metrics

When evaluating models trained to classify medical images like chest X-rays, a
common and frequently used metric is the AUC ROC [40, 3, 6]. The ROC curve is
a graphical representation of the true positive rate (TPR) against the false
positive rate (FPR) at various threshold settings; the AUC is defined as a
scalar value representing the area under the ROC curve. The AUC is a valuable
metric for evaluating the overall discriminative power of a classification
model. However, it is not considered sufficient for datasets with skewed class
balance. Indeed, the resulting value may be misleading, hiding the poor
performance of a model [41]. Multi-label datasets are often imbalanced; this
also holds for the considered datasets, as several labels are rarely present.

Therefore, we evaluate the model performance also considering the macro F1
score since it is a common metric when dealing with multi-label classification
problems [42, 43]. The macro F1 is the average F1 score calculated for each
label. The F1 score quantifies the balance between precision and recall, hence
it is an appropriate choice for the evaluation of imbalanced datasets. Its
focus on positives, i.e., images where a given pathology is present, makes F1
especially relevant in the medical domain.

For the evaluation over a stream of tasks, we consider the Average AUC or
Average F1 over all the pathologies of all tasks up to $i$, as typical in CL
papers [29, 3]. Similarly, we measure the percentual forgetting, proposed in
previous papers in CL literature [16, 17], hence the percentage difference
between the final performance of the model on each task and the performance
just after training on said task. Lastly, we consider the Relative gap between
each method’s final performance and the upper baseline’s performance, i.e.,
Joint Training.

##  6 Results

(a) AUC

(b) F1

Figure 4: Comparison between the average AUC and F1 score on all the task seen
so far between Pseudo-Label Replay and prior methods.

(a) Learning without Forgetting (LwF)

(b) Learning without Forgetting (LwF) Replay

(c) Pseudo-Label

(d) Pseudo-Label Replay

Figure 5: Comparison between the average F1 score on each task between Pseudo-
Label Replay and prior methods.

In the next part, we discuss the outcomes produced by each method when applied
to the proposed challenging benchmark. Specifically, we will elucidate the
achieved results concerning the F1 performance in Sec. 6.1. Subsequently, we
delve deeper by examining the forgetting behavior exhibited by each method in
Sec. 6.2.

###  6.1 Performance results

In the following analysis, we discuss the final values of AUC and F1 score,
obtained after the model is trained on all tasks, and averaged on all runs.
These values are depicted in Fig. 4(a) and Fig. 4(b), respectively, and they
are reported in Tab. 1. Concerning the CL methods, we observe no sizable
changes in the average AUC scores among different CL methods, except for Fine-
Tuning and Replay, which exhibit a lower performance.

Table 1: The F1 and AUC metrics are reported as defined in the Sec. 5.2. In bold is reported the best value for each metric among the compared CL approaches. The forgetting (low is better) is also reported to comprehensively evaluate the approaches, while the relative gap represents the discrepancy between the final value of F1 of each method and the value of F1 of the Joint Training. | Metrics
---|---
Strategy |  Avg. F1 $\uparrow$ |  Avg. AUC $\uparrow$ |  Forgetting F1  $\downarrow$ | Relative gap $\downarrow$
Joint Training | 0.38 | 0.79 | - | -
Fine-Tuning | 0.02 | 0.55 | 100% | 95%
Replay | 0.15 | 0.65 | 59% | 60%
LwF | 0.22 | 0.68 | 41% | 42%
LwF Replay | 0.18 | 0.68 | 48% | 52%
Pseudo-Label | 0.23 | 0.69 | 19% | 40%
Pseudo-Label Replay | 0.28 | 0.70 | 1.6% | 26%

Focusing on F1, despite its drawback in task interference, Replay demonstrates
a slight capacity to mitigate forgetting, yielding an average F1 score of
0.15. Although not particularly high, this outperforms the Fine-Tuning (lower-
bound) strategy, which only achieves an average F1 score of 0.02. This
discrepancy in performance between the two methods can be observed from the
relative gap as well: 95% for Fine-Tuning and 60% for Replay. LwF outperforms
Replay, as expected from the literature [23]. In detail, LwF exhibits a
significant improvement over Replay, achieving an average F1 score of 0.22.
Pseudo-Label performs similarly to LwF, with a marginally higher F1 score of
0.23. The two methods perform similarly in terms of the relative gap as well:
LwF achieves a value of 42%, while Pseudo-Label is 40%.

We observe that the combination of Replay and LwF does not solve the
forgetting issue. Indeed, the value achieved by the LwF Replay method
($\text{average F1}=0.18$) is significantly lower than both LwF and Pseudo-
Label methods taken singularly, and its relative gap is higher (52%).This is
expected due to the interference caused by Replay, which is not overcome by
combining the method with LwF. This suggests that combining any distillation
method with Replay is not sufficient to overcome the challenges of our
scenario.

Pseudo-Label Replay achieves an F1 score of 0.28 and a relative gap of 26%,
with a large margin over the abovementioned methods.

###  6.2 Forgetting Analysis

We provide an analysis of the forgetting factor based on the results reported
in Fig. 5 and the forgetting metric reported in Tab. 1. In Fig. 5, four plots
are shown, corresponding to the best-performing methods from the previous
section: LwF, LwF Replay, Pseudo-Label, and Pseudo-Label Replay.

These plots illustrate the performance of each method for every task from its
initial learning phase and in all subsequently seen tasks.

For LwF and the Pseudo-Label, we can observe that the F1 score tends to
decrease slightly when the model is trained on new classes. In particular, the
decline is steeper for LwF: indeed, as reported in Tab. 1, the percentual
forgetting is 42%, while for Pseudo-Label, it is 19%. On the other hand, in
the previous section, it was found that the final values of the F1 score are
similar for the two results. This discrepancy suggests that subtle differences
in approach and implementation between these strategies significantly
influence the forgetting metric.

As mentioned in the previous section, combining the LwF strategy with Replay
does not improve the performance of LwF; on the contrary, the forgetting is
stronger. This can also be observed by the plot in Fig. 5(b). Indeed, we can
observe that the performance of some tasks decreases significantly when
training on new tasks due to the interference between the new and the replayed
data. In particular, from Tab. 1, we observe that the forgetting value in this
case is 48.3%.

When examining our proposed approach, Pseudo-Label Replay, we note a
remarkably low forgetting rate of 1.6%. This performance surpasses that of its
closest competitor, Pseudo-Label, by a significant margin.

Overall, Pseudo-Label Replay displays excellent stability with nearly flat F1
curves.

##  7 Conclusions and Future Work

Our work proposes a novel benchmark specifically designed for evaluating CL
methods in the context of multi-label medical image classification. This
benchmark includes different medical imaging datasets, pathologies, and
imaging conditions to provide a comprehensive evaluation platform for
continual learning algorithms. Specifically, we design the new scenario by
considering realistic conditions in the medical domain, combining new classes
and domains into a challenging benchmark.

To tackle these challenges, we propose a novel method combining Pseudo-
Labeling and Replay to integrate information from previous tasks while
adapting to new data streams. Our experimental results from the proposed
benchmark demonstrate Pseudo-Label Replay’s effectiveness in handling this
complex scenario. Our method surpasses existing approaches, exhibiting
superior performance by a large margin while showing minimal forgetting. While
our approach has demonstrated high effectiveness in handling multi-label
classification problems within the CL scenario, its applicability to other
settings still needs to be explored. In future work, we plan to evaluate the
effectiveness of Pseudo-Label Replay in other settings, such as object
detection and semantic segmentation problems.

Finally, while our approach significantly improves performance compared to the
other CL approaches, a noticeable gap with the upper bound achieved by Joint
Training remains. Future work should explore new approaches to enhance model
performance in the CL setting and close this gap.

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#  Giant anomalous thermal Hall effect in tilted type-I magnetic Weyl
semimetal Co3Sn2S2

Abhirup Roy Karmakar<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Kharagpur, W.B. 721302, India  S. Nandy  Department of
Physics, University of Virginia, Charlottesville, VA 22904, USA  A. Taraphder
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Kharagpur, W.B. 721302, India  G. P. Das<EMAIL_ADDRESS>Research
Institute for Sustainable Energy (RISE), TCG Centres for Research and
Education in Science and Technology, Sector V, Salt Lake, Kolkata 700091,
India

###### Abstract

The recent discovery of magnetic Weyl semimetal Co3Sn2S2 opens up new avenues
for research into the interactions between topological orders, magnetism, and
electronic correlations. Motivated by the observations of large anomalous Hall
effect because of large Berry curvature, we investigate another Berry
curvature-induced phenomenon, the anomalous thermal Hall effect in Co3Sn2S2.
We study it with and without strain, using a Wannier tight-binding Hamiltonian
derived from first principles density functional theory calculations. We first
identify this material as a tilted type-I Weyl semimetal based on the band
structure calculation. Within the quasi-classical framework of Boltzmann
transport theory, a giant anomalous thermal Hall signal appears due to the
presence of large Berry curvature. Surprisingly, the thermal Hall current
changes and even undergoes a sign-reversal upon varying the chemical
potential. Furthermore, applying about 13 GPa stress, an enhancement as large
as 33% in the conductivity is observed; however, the tilt vanishes along the
path connecting the Weyl nodes. In addition, we have confirmed the validity of
the Wiedemann-Franz law in this system for anomalous transports. We propose
specific observable signatures that can be directly tested in experiments.

DOI: 10.1103/PhysRevB.106.245133

##  I Introduction

Topologically protected gapped and gapless materials have attracted immense
attention lately due to their unique properties [1, 2]. In line with band
structure terminology, they are known as topological insulators [3, 4] and
topological semimetals respectively [2, 5]. The three-dimensional (3D)
realizations of topological semimetals, which include Dirac and Weyl
semimetals (DSM, WSM), have been classified in accordance with the symmetries
they possess as well as the nature of band dispersion near the Fermi level [2,
5].

In the case of a WSM, the Weyl nodes, which are earmarked by their locations
in momentum and energy space where the non-degenerate bands linearly touch
each other, always appear in pairs with well-defined but opposite chiralities
[6, 7, 8, 9, 10, 11]. Each node of the pairs individually acts as the source
(+ve chirality) or sink (-ve chirality) of the Berry curvature, which can be
viewed as an effective magnetic field in momentum space [12]. In order to have
a topological charge associated with the Weyl node, WSM has to break either
the time-reversal symmetry (TRS) or the space inversion symmetry (IS) which
leads to the classification of Weyl semimetals: (i) IS broken but TR-symmetric
WSM and (ii) TRS broken but inversion-symmetric WSM or magnetic WSM [9, 10,
11, 13]. Compared to the IS broken WSM, the magnetic WSM gives a platform to
study the interplay between topological orders, magnetism and electronic
correlations, leading to exotic quantum states. In addition, it can also
generate a true nodal WSM phase when the Fermi energy coincides with the Weyl
nodes since the pair of Weyl nodes remains at the same energy to preserve IS.

Ever since their discovery, Weyl semimetals have been surprising us with their
unique and fascinating transport properties, such as nontrivial Berry
curvature induced anomalous Hall effect (AHE) and anomalous Nernst effect
(ANE) [14, 15, 16, 17, 18, 19, 20, 21], chiral anomaly induced negative
longitudinal magnetoresistance [22, 23, 24, 25, 26, 27], planar Hall effect
[28, 29, 30, 31, 32, 33, 34, 35, 36], thermoelectric phenomena [37, 38, 39],
gyrotropic birefringence [40], Magnus Hall effect [41], and chiral magnetic
effect [42].

These topological transport properties in WSM have been studied extensively
both theoretically and experimentally. However, in the context of real
materials, most of these studies are based on the IS-broken but TR-symmetric
WSMs. After the first experimental discovery of WSM in TaAs in 2015 [43, 44],
the other WSM candidates discovered so far are non-magnetic. On the other
hand, despite many proposed candidates, a direct experimental verification of
magnetic WSM is challenging. In 2019, a Shandite material Co3Sn2S2, in the
ferromagnetic phase, has been identified as a Weyl semimetal with three pairs
of Weyl points in the Brillouin zone [45, 46, 47]. Due to its intrinsic
magnetism, as well as a large Berry curvature, Co3Sn2S2 shows large anomalous
Hall effect, anomalous Nernst effect, planar Hall effect and chiral anomaly-
induced magnetotransport phenomena, all of which have been studied extensively
[48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. However, another important
effect that is gaining attention lately, viz. the anomalous thermal Hall
effect (ATHE), has not been explored in this particular class of material as
yet. ATHE, in contrast to conventional heat conduction, causes the generation
of heat current in the direction transverse to the applied temperature
gradient in the absence of any external magnetic field. It is also known as
the anomalous Righi-Leduc effect and can serve as an additional source of
dissipationless heat current [59, 60, 61]. Similar to other anomalous
transports, this one is caused by the non-trivial topology of the bands, with
Berry curvature playing the role of the magnetic field. The finite results of
the thermal Hall effect represent both the system’s topology and the anomalous
scattering of quasiparticle excitations. ATHE has been utilized as a potent
instrument for investigating charge-neutral quasiparticles in insulating
quantum materials [62, 63, 64, 65]. This particular transport phenomenon may
also be effective in the directional control of heat currents.

In this paper, we investigate the anomalous thermal Hall effect of the
magnetic WSM Co3Sn2S2 in detail. We start by studying the electronic structure
of this material and obtain a Wannier tight-binding model from the first
principles calculation. We identify this material as a type-I Weyl semimetal
with tilted cones for the first time using DFT calculations which agrees with
very recent experimental findings [66, 67]. Then we calculate the Berry
curvature which exhibits very high values at some specific points in the
Brillouin zone, namely the Weyl nodes. Interestingly, the anomalous thermal
Hall conductivity (ATHC) obtained by integrating the Berry curvature comes out
to be very large without the presence of any external field when compared with
previous observations [68, 69, 65, 70]. The result is validated by confirming
that it satisfies the Wiedemann-Franz law quite accurately at low temperature.
However, at higher temperatures the conductivity loses its linear relationship
with temperature. By plotting ATHC over a range of chemical potentials we find
that the heat current can be tuned as well as reversed by changing the
chemical potential, although the maximum value is attained at its original
Fermi level. Furthermore, we study the effect of external pressure on ATHC by
applying uniaxial compressive stress on the crystal along different
directions. We find that the application of stress along the direction
perpendicular to the kagome-plane drags the Weyl points closer to the Fermi
level which in turn enhances the transverse heat current. We also observe that
the tilt along the connecting line between the Weyl nodes decreases to zero
upon applying stress. However, stresses along lateral directions did not
improve the conductivity and, therefore, we only focus our attention to the
z-axis compression.

The remainder of this paper is organized as follows: in Sec. II we discuss the
crystal structure, underlying symmetries and magnetic moments of Co3Sn2S2.
Then we elaborate the computational methods employed throughout the work.
After that we study its electronic band structure, density of states, as well
as Fermi surface in Sec. III. In Sec. IV we investigate ATHE and study its
behavior with chemical potential and temperature. Then in Sec. V we discuss
the effect of stress on the system and show how it can be used to enhance the
ATHC. And finally, we summarize our work, the results obtained and possible
future directions in Sec. VI.

##  II Material and Method

Figure 1: Crystal structure of Co3Sn2S2. The left part of the figure shows the
rhombohedral primitive unit cell of space group R-3m. The cobalt and tin atoms
form a ferromagnetic kagome lattice which is shown at the right.

The shandite material Co3Sn2S2 crystallizes in a rhombohedral structure with a
layer of quasi-2D Co3Sn sandwiched between S and Sn atoms [71, 72]. The unit
cell falls under the space group R-3m (no. 166) with a lattice parameter of
5.38 Å and consists of a mirror plane ($M_{010}$), inversion symmetry and
$C_{3z}$ rotational symmetry. The material possesses a strong ferromagnetism
originating from the Cobalt atoms which sit on a kagome lattice in the ab
plane in the hexagonal representation as shown in Fig. 1. The magnetic moment
is 0.29 and 0.31 $\mu_{B}$/Co from neutron-diffraction [73] and magnetization
measurement [74] respectively. The measurements further show that the magnetic
moments are directed along the c-axis with a Curie temperature ($T_{c}$) 177K.
Electronic structure of also reveals that it has a half-metallic nature with a
gap in the minority spin channel [75, 76].

We have performed first principles density functional theory (DFT)
calculations using Vienna Ab initio Simulation Package (VASP) [77] in the
projector augmented wave (PAW) approximation. Generalized gradient
approximation (GGA) was considered for exchange-correlation functional in the
Perdew-Burke-Ernzerhof (PBE) scheme [78]. A 15$\times$15$\times$15 Monk-horst
grid was taken to fill the 3D Brillouin zone. From the basic DFT calculation
we obtained the value of the magnetic moment of around 0.33 $\mu_{B}$/Co,
which agrees well with the experiments and then studied the electronic
structure as described in Sec. III. We projected the Bloch wave functions into
the maximally localized Wannier functions (MLWFs) and obtained Hamiltonian
matrix elements between the MLWFs with the help of Wannier90 package [79]. As
we shall see later, the d-orbital of Co and p-orbitals of Sn and S have the
largest contribution near Fermi level and, therefore, they were chosen as the
projections for the Wannierization. In order for a thorough investigation of
the band structure and calculation of ATHC, we derived a Hamiltonian using the
Wannier tight-binding model with 54 bands from the matrix elements [79, 80].
All other numerical calculations were performed in Python programming
language.

##  III Electronic structure

(a)

(b)

(c)

(d)

(e)

Figure 2: Electronic structure of Co3Sn2S2. (a) The spin-polarized (red for
spin-up and blue for spin-down channel) and SOC-included (dotted green curves)
band structure plotted along the high symmetry direction
W$-$T$-$U$-$L$-$$\Gamma$. The spin-down channel has a gap, whereas the spin-up
channel is gapless and has two band-crossings along U$-$L$-$$\Gamma$ which are
parts of a nodal ring. (b) The orbital-projected density of states. The large
area under red curve implies that the Co-d orbital dominates near Fermi level
with negligible contributions from other orbitals. The $+$ve and $-$ve values
correspond to spin-up and spin-down channels respectively. (c) The Brillouin
zone of the system with a truncated octahedral shape. The green line shows
high-symmetry direction for the band structure plot. (d)-(e) Fermi surface for
the valence and conduction bands respectively. Both the bands cross Fermi
level.

In this section, we investigate the electronic band structure of Co3Sn2S2 in
detail, both in the absence and in the presence of spin-orbit coupling (SOC).
In the absence of SOC, the calculated electronic band structure, along the
path W$-$T$-$U$-$L$-$$\Gamma$, is shown in Fig. 2(a). The corresponding high-
symmetry points in the truncated octahedron-shaped Brillouin zone is shown in
Fig. 2(c) with the green line indicating the high-symmetry path for the band
structure. By calculating the spin-polarized band structure along the
aforementioned path, we find (Fig. 2(a)) that the spin-down channel (blue) is
insulating with a gap of 0.44 eV, whereas the spin-up channel (red) is gapless
confirming its half-metallic nature. One of the main interest in the band
structure lies in the two crossings along the path U$-$L$-$$\Gamma$.
Energetically they are located slightly above and below the Fermi level. As
the interaction between two spin-channels is avoided in this particular case,
the mirror-symmetry of the Hamiltonian remains intact and leads to a nodal
ring situated in the mirror plane. It turns out that these two crossings are
parts of that particular nodal ring itself. Interestingly, due to the other
two symmetries, namely the inversion symmetry and $C_{3z}$, six such nodal
rings in total are present in the entire Brillouin zone.

$k_{1}$ |  $k_{2}$ |  $k_{3}$ | $E-E_{f}(meV)$ | $Chirality$
---|---|---|---|---
$\ \ \ 0.000\ $ | $\ \ \ 0.425\ $ | $\ \ \ 0.063\ $ | $62.9$ | $+1$
$\ \ \ 0.000\ $ | $-0.425\ $ | $-0.063\ $ | $62.5$ | $-1$
$\ \ \ 0.425\ $ | $\ \ \ 0.000\ $ | $\ \ \ 0.063\ $ | $62.3$ | $+1$
$-0.425\ $ | $\ \ \ 0.000\ $ | $-0.063\ $ | $62.6$ | $-1$
$-0.425\ $ | $-0.425\ $ | $-0.365\ $ | $62.7$ | $+1$
$\ \ \ 0.425\ $ | $\ \ \ 0.425\ $ | $\ \ \ 0.365\ $ | $62.5$ | $-1$
Table 1: The locations (fractional coordinates) of the Weyl nodes in the
Brillouin zone. Their energies above Fermi level and the chiralities are also
shown in last two columns. Figure 3: Tilted Weyl cones. Band structure showing
Weyl nodes of opposite chiralities is plotted along their connecting line.
Their locations being $(0,0.425,0.063)$ and $(0,-0.425,-0.063)$ respectively
in fractional coordinates. One of them has been zoomed-in to show the tilting
of the cone with respect to vertical line via the red line.

As the compound consists of elements with relatively large atomic numbers, it
is desirable to incorporate spin-orbit interactions (which, typically, vary as
the fourth power of the effective nuclear charge $Z$). The mirror-symmetry of
the Hamiltonian gets broken with the introduction of SOC and hence the nodal
ring, that was present earlier, collapses into a pair of Weyl points with
opposite chiralities. The electronic band structure, including SOC, is shown
with the dotted green curves in Fig. 2(a). The aforementioned band crossings
around the Fermi level are gapped out, which essentially indicates that the
Weyl nodes are not located on the high-symmetry line. Upon scanning the first
Brillouin zone, we found six such Weyl nodes in total of alternate chiralities
which happens due to the extant inversion and $C_{3z}$ crystal symmetries. All
the nodes happen to be at around 63 meV above the Fermi level owing to the
presence of TRS. Their locations in the Brillouin zone as well as
corresponding energies and chiralities are listed in Table 1. They are very
close to Fermi level and hence we expect a significant contribution of the
Weyl nodes to the transport.

We have plotted the first two Weyl nodes of Tab. 1 in Fig. 3. Upon zooming in,
it is found that the cones are tilted with respect to the vertical. To
investigate this quantitatively, we consider an effective two-band Hamiltonian
for a tilted WSM which is written as,

$H=v\ \bm{k.\sigma}+\bm{t.k}+\epsilon_{0}$ (1)

where $\bm{k}$ is the momentum relative to the Weyl node. $v$, $\bm{t}$, and
$\epsilon_{0}$ being the Fermi velocity, tilt parameter, and Weyl node energy
respectively. The cones shown in Fig. 3 are tilted primarily along the $k_{y}$
direction. From our DFT calculation, we obtain the value of $t_{y}$ and $v$ as
0.19 and 0.65 eV$\cdot\AA$. The ratio $|\frac{t_{y}}{v}|$ turns out to be
around 0.29 ($<$ 1) which indicates that the WSM is of type-I. In a similar
fashion, the tilt in other two pairs can also be shown, and they are along
different directions. It is important to note that our calculations were done
in lattice model [14, 39] with 54 bands rather than this two-band linearized
model.

The Co atoms consist of partially filled d-orbital and, therefore, the
electronic correlations are expected to play an important role in this system.
Recently, Xu et al. [81], have estimated the Coulomb interaction to be around
4 eV in this material by combining optical spectroscopy measurements with
many-body theoretical calculations. We have tried to incorporate the Coulomb
interaction in our calculations following the simplified (rotationally
invariant) approach, given by Dudarev et al. [82]. However, it made the Weyl
nodes disappear by gapping out the band crossings. This implies that the
single-electron picture is not sufficient to reproduce the whole physics.
Since we are only interested in features caused by the Weyl nodes (coming from
the crossing of valence and conduction bands), we ignore the effects of
correlation in the foregoing. We have also plotted the Fermi surfaces for the
valence and conduction bands responsible for the Weyl nodes in Fig. 2(d) and
2(e) respectively. It is no surprise that both the bands have contributions to
the Fermi surface, which is also visible in the band structure. To get an
insight into the orbital contributions, we have computed the orbital-
projected, spin-polarized density of states around $E_{f}$. It is clearly
visible from Fig. 2(b) that the major contribution comes from the d-orbital of
Co. As we move away from $E_{f}$, contributions from the p-orbitals of Sn and
S atoms begin to arise. We can also see that the DOS profile of spin-up
(positive values) and spin-down (negative values) channels are not exactly
same. And that inequality is owed to the ferromagnetism, coming predominantly
from the $3d$-orbital of Co.

##  IV Anomalous Thermal Hall Effect

In this section, we will investigate the anomalous thermal Hall effect using
the structural and electronic properties of Co3Sn2S2. The ATHE refers to the
appearance of a transverse heat current as a first-order response to an
applied longitudinal thermal gradient. In linear response regime, the equation
for the heat current in the presence of applied thermal gradient
($\bm{\nabla}T$) and electric field ($\bm{E}$) is given by,

$\bm{J}_{Q}=T\alpha.\bm{E}-\kappa.\bm{\nabla}{T}$ (2)

where $\alpha$ and $\kappa$ in Eq. 2 are the Peltier and thermal conductivity
tensors corresponding to Nernst-Ettinghausen and Leduc-Righi effects
respectively. The transverse component of these tensors can be expressed as
[83, 84, 85] $\alpha_{xy}=\frac{k_{B}e}{h}c_{1},\
\kappa_{xy}=-\frac{k_{B}^{2}T}{h}c_{2}$ where,

$c_{i}=\int\frac{d\bm{k}}{(2\pi)^{3}}\sum_{n}\Omega_{\textbf{k}n}\int_{\epsilon_{\bm{k}n}-\mu}^{\infty}d\epsilon_{\bm{k}n}(\beta\epsilon_{\bm{k}n})^{i}\dfrac{\partial
f_{eq}(\epsilon_{\bm{k}n})}{\partial\epsilon_{\bm{k}n}}$ (3)

with $\Omega_{\textbf{k}n}$, $\epsilon_{\bm{k}n}$, $\mu$ and $f_{eq}$ being
the Berry curvature, energy eigenvalue of the n-th band, chemical potential
and the equilibrium Fermi distribution function respectively. $\beta$ is
written in place of $1/(k_{B}T)$, where $k_{B}$ and $T$ are the Boltzmann
constant and temperature respectively. Upon simplifying the above equation, we
can write the expression for anomalous thermal Hall coefficient as

$\kappa_{xy}=\frac{k_{B}^{2}T}{h}\int\frac{d\bm{k}}{(2\pi)^{3}}\sum_{n}\Omega_{\textbf{k}n}[\frac{\pi^{2}}{3}+\beta^{2}(\epsilon_{\bm{k}n}-\mu)^{2}f(\epsilon_{\bm{k}n}-\mu)\\\
-[ln(1+e^{-\beta(\epsilon_{\bm{k}n}-\mu)})]^{2}-2Li_{2}[1-f(\epsilon_{\bm{k}n}-\mu)]]$
(4)

where $Li_{2}(z)$ is the polylogarithm function of order 2.

(a)

(b)

(c)

(d)

Figure 4:  Berry curvature and anomalous thermal Hall conductivity. (a) The
strengths of the Berry curvature (z-component) corresponding to the bands
forming Weyl nodes are shown by color variation of the bands along high-
symmetry path W$-$T$-$U$-$L$-$$\Gamma$. (b) The total Berry curvature is shown
on the mirror-symmetric plane ($k_{y}=0$). The sharp peaks correspond to the
Weyl nodes present on this plane. (c) Anomalous thermal Hall conductivity
(ATHC) plotted with respect to chemical potential $\mu$ at $T=50$ K. The heat
current can be controlled as well as reversed by changing the chemical
potential of the system. However, the maximum value is achieved at $\mu=$ Ef.
(d) The relationship between ATHC and temperature. Note that $\kappa_{xy}$
varies linearly with $T$ at low temperatures up to 80 K after which it
gradually deviates from linearity with increasing T. The chemical potential
was set equal to the Fermi level.

It is important to note that we are considering only the transverse response
to the applied thermal gradient $\nabla T$ in presence of nontrivial Berry
curvature without any external magnetic field. In order to calculate ATHE, we
must discuss first the Berry curvature profile of the system as we are
considering only the Berry curvature-induced contribution. Since the TRS is
broken in Co3Sn2S2 due to its inherent ferromagnetism, it possesses finite
values of Berry curvature throughout the Brillouin zone and sharp peaks at the
locations of the Weyl nodes (Tab. 1). The z-component of the Berry curvature
($\Omega_{z}$) along the high symmetry path (W$-$T$-$U$-$L$-$$\Gamma$) as well
as on the mirror-plane (kx-kz) are depicted in Figures 4(a) and 4(b)
respectively. The values in the former are significantly lower than those in
the latter because the Weyl nodes have dislocated from the high-symmetry line
after activating the spin-orbit coupling.

The plane plot, however, could capture the exact locations of the nodes as
shown in Table 1 because the Berry curvature attains highest value at the Weyl
points. The values at the momentum locations (0.425, 0, 0.0625) and (-0.425,
0, -0.0625) turn out to be in the range of 105. After obtaining $\kappa_{xy}$
using Eq. 4, we plot it as a function of chemical potential $\mu$ in Fig.
4(c). The magnitude of the ATHC is significantly large compared to other
materials even in the absence of any external magnetic field [68, 69, 65, 70]
and it reaches the maximum value of 0.146 W/K-m at its original Fermi level.
In addition, we find that the direction of thermal Hall current can be
reversed by tuning the chemical potential by, for example, doping the
material. It is worth mentioning at this point that we have also performed a
spin-polarized calculation of ATHC after switching off SOC where the material
is not in Weyl semimetal phase. However, both spin channels produced null
results at the original Fermi level.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5:  Strain-induced band dispersion. (a)-(c) SOC-included band
structures for compressive stresses of 0, 5.6, and 12.9 GPa along z-axis. The
valence band starts to flatten as the stress is increased. The local band gap
at L point shows an expanding nature throughout the process. (d)-(f) The band
dispersion around two of the Weyl nodes along their connecting line in the
Brillouin zone. The nodes move towards the Fermi level with increasing stress
and finally at 12.9 GPa they fall directly on the Fermi level. The tilting of
the cones decreases with stress and almost vanish at 12.9 GPa. The red line in
the inset indicates tilting with respect to the vertical.

To understand the main source of this large effect we analyze all the terms in
Eq. 4. As the system possesses inversion symmetry, the full integration of the
Berry curvature over the Brillouin zone is zero. Hence, the first constant
term of the kernel written inside the square bracket has no contribution. At
low temperatures, the integration of second and third terms inside the square
bracket turns out to be exactly equal and opposite. We would like to point out
that these terms give a very large contribution at the high temperature limit.
Hence, the only contribution to ATHC comes from the polylogarithm function
multiplying with Berry curvature at $T\rightarrow 0$. It is important to note
that the $x$ and $y$ components of the Berry curvature lead to a very small
Hall conductivity (less than 5% of that due to the z-component) and hence only
the xy-component of ATHC has been shown.

At low temperature, Eq. 4 is reduced to

$\kappa_{xy}=\frac{\pi^{2}}{3}\frac{k_{B}^{2}T}{h}\int\frac{d\bm{k}}{(2\pi)^{3}}\sum_{n}\Omega_{n}\theta(\mu-\epsilon_{\bm{k}n})=L_{0}T\sigma_{xy}$
(5)

where $L_{0}$ is known as the Lorenz number and has a value of $2.44\times
10^{-8}\ W\Omega K^{-2}$. The Eq. 5 satisfies the Wiedemann-Franz law which
states that the ratio of the thermal conductivity and the electrical
conductivity of the material is directly proportional to temperature. To
validate this from our calculation, we have computed the anomalous Hall
conductivity ($\sigma_{xy}$) at low temperature and computed the ratio of
$\sigma_{xy}$ and $\kappa_{xy}$. The proportionality constant comes out to be
very close to $L_{0}$ over the range of chemical potential ($\mu$) at 50 K.
However, as T is increased, Eq. 5 no longer remains valid.

We plot $\kappa_{xy}$ over a large range of temperature $T$ in Fig. 4(d). When
$T$ is low, $\kappa_{xy}$ increases linearly with T up to 80 K according to
the Wiedemann-Franz law. However, as the temperature is further increased, it
slowly loses its linear character and after a certain temperature ($\sim$
400K), it starts to decrease. This indicates that the Wiedemann-Franz law for
anomalous transports is only valid at low temperatures. In the next section we
see what happens to the transport properties when one applies external
pressure on this system.

##  V Effects of strain

In this section we shall investigate the effect of strain on the anomalous
transport properties in Co3Sn2S2. Here we restrict ourselves to only uniaxial
compressive stress which is achieved by introducing strain in the lattice
vector. For this purpose, we reduce the unit cell length along z-direction
gradually by up to 5% which is equivalent to a stress of around 13 GPa along
the z-axis. At every step of increasing stress, the structure has been
geometry optimized by keeping the cell volume fixed while relaxing the atomic
positions. And it is important to note that no structural transition has taken
place during the process.

Figure 6:  The anomalous thermal Hall conductivity plotted against chemical
potential for different values of compressive stress along z-axis. At $\mu=$
Ef, ATHC increases by about 33% for a stress of 12.9 GPa.

To investigate what happens to the electronic properties, we plot the SOC-
included band structure in Fig. 5 (upper row) for three different stresses.
First thing we observe is that the Fermi level gets shifted away from the
valence band (gapped nodal ring region) as the pressure is increased. In
addition, the valence band flattens out in some regions of the k-path. As a
consequence, the local band gap at $L$ point also gets expanded. The band
dispersion around the first pair of the Weyl nodes are also shown in the lower
part of Fig. 5. It is quite remarkable to observe that the Weyl nodes are
being dragged towards the Fermi level as the stress is increased and at a
stress of 12.9 GPa the nodes fall directly on the Fermi level which initially
were at 63 meV above $E_{f}$ in normal condition. Another interesting thing to
note here is, the tilting of the Weyl nodes decreases with increasing stress.
This can be observed in the insets in figures 5(d)-(f) where the tilt-axis
(red line) becomes almost vertical at 12.9 GPa. Quantitatively, the tilt ratio
$|\frac{t}{v}|$ becomes $\sim 0.02$ in Fig. 5(f).

As topological effects come from the bands nearest to the Fermi level, we
expect that this movement of the Weyl points towards $E_{f}$ should show up as
an enhancement in the topological transport phenomena. In order to see that
quantitatively, we plot anomalous thermal Hall conductivity for different
values of external stress in Fig. 6. Despite having almost same pattern, the
values of ATHC do vary at some regions along $\mu$ axis. At their original
Fermi levels ATHC increases by about 33% with just 5% compressive strain,
which is huge. But, in the left part of the figure ATHC is seen to decrease
with increasing stress. However, since experimentally one measures the outcome
from the original Fermi level, the conductivity will increase. We had also
performed calculations with strain along lateral directions, but did not
observe any enhancement. Hence, we conclude from this section that the effect
of uniaxial compressive strain in this magnetic Weyl semimetal is quite
significant which can enhance the anomalous thermal Hall conductivity
drastically.

##  VI Summary

To summarize, we have investigated the Berry curvature-induced anomalous
thermal Hall effect in magnetic Weyl semimetal Co3Sn2S2 using a combination of
the first principles DFT calculation and quasi-classical Boltzmann transport
theory. From DFT calculation, we first obtain the electronic band structure of
the material and observe that the system transits from a nodal-line semimetal
to a Weyl semimetal by turning on the spin-orbit coupling. We show that the
first Brillouin zone contains three pairs of Weyl nodes in total, created by
the $C_{3z}$ rotational symmetry and they are located at the same energy due
to the presence of inversion symmetry. We have shown from DFT calculations
that the Weyl cones are tilted and also calculated the tilt parameter which
indicates that the system is a type-I WSM. From a tight-binding Hamiltonian,
derived from Wannier functions, we have calculated the Berry curvature which
shows large enhancement especially at the positions of the Weyl nodes.

Next, using the semiclassical Boltzmann transport theory, we have computed the
Berry curvature-induced anomalous thermal Hall conductivity of Co3Sn2S2 which
came out to be very large. By varying the chemical potential the thermal Hall
current can be tuned and even the direction can be reversed for some
particular values. We have also checked the Wiedemann-Franz law and found that
this law is well satisfied at low temperatures. In addition, to further
enhance the thermal conductivity in Co3Sn2S2, we apply uniaxial compressive
stress. This essentially moves the Weyl nodes towards the Fermi level which in
turn enhances ATHC as the anomalous transport properties are directly related
to the Weyl nodes. To be quantitative, the conductivity increased by about 33%
for a compressive strain of just 5% along the z-axis of the unit cell. It is
worthwhile to mention that the tilt of the Weyl cones vanishes along the
connecting line on application of stress.

We would like to point out that since the d-orbitals of Co atom contribute the
most to the bands near Fermi level which create the Weyl nodes and these
orbitals are partially filled, one can expect a non-trivial role of electronic
correlation in this material. However, the incorporation of Coulomb
interaction in DFT (i.e., DFT+U) gaps out the Weyl node crossing which is
unphysical and, therefore, one has to go beyond the single-electron Hartree-
Fock picture to fully understand the effects of correlation in this material.

## Acknowledgments

A.R.K. acknowledges National Supercomputing Mission (NSM) for providing
computing resources of ‘PARAM Shakti’ at IIT Kharagpur, which is implemented
by C-DAC and supported by the Ministry of Electronics and Information
Technology (MeitY) and Department of Science and Technology (DST), Government
of India.

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