# TSInterpret: A unified framework for time series interpretability

Jacqueline Höllig<EMAIL_ADDRESS>
Information Process Engineering
FZI Forschungszentrum Informatik
76131 Karlsruhe, Germany Cedric Kulbach<EMAIL_ADDRESS>
Information Process Engineering
FZI Forschungszentrum Informatik
76131 Karlsruhe, Germany Steffen Thoma<EMAIL_ADDRESS>
Information Process Engineering
FZI Forschungszentrum Informatik
76131 Karlsruhe, Germany

###### Abstract

With the increasing application of deep learning algorithms to time series
classification, especially in high-stake scenarios, the relevance of
interpreting those algorithms becomes key. Although research in time series
interpretability has grown, accessibility for practitioners is still an
obstacle. Interpretability approaches and their visualizations are diverse in
use without a unified api or framework. To close this gap, we introduce
TSInterpret111https://github.com/fzi-forschungszentrum-informatik/TSInterpret,
an easily extensible open-source Python library for interpreting predictions
of time series classifiers that combines existing interpretation approaches
into one unified framework. The library features (i) state-of-the-art
interpretability algorithms, exposes a (ii) unified API enabling users to work
with explanations in a consistent way, and provides (iii) suitable
visualizations for each explanation.

Keywords: Time Series; Interpretability; Feature Attribution; Counterfactual
Explanation

##  1 Introduction

Although time series data are omnipresent in industry and daily life, deep
learning methods have only been applied to time series data in the last
decade. Before, time series classification (TSC) primarily focused on
classifications based on discriminatory features (e.g., learning based on the
whole time series, intervals, or shapelets (Bagnall et al. (2017))). Compared
to deep methods, they do not have the ”black-box”- character as a significant
drawback. However, with the increasing accuracy of deep learning models and
their ability to cope with vast amounts of data, their application on time
series classification increased, leading to a need for interpretability,
especially in high-risk settings. Various interpretability methods for time
series classification are available (Rojat et al. (2021)). However, the usage
of those methods is not yet standardized: The proposed methods often lack a)
open code (e.g., Siddiqui et al. (2021)), b) an easy-to-use interface (e.g.,
Ismail et al. (2020)), or c) visualization (e.g., Guilleme et al. (2019)),
making the application of those methods inconvenient and thereby hindering the
usage of deep learning methods on safety-critical scenarios.

Although unifying frameworks (e.g., tf-explain (Meudec (2021)), Alibi (Klaise
et al. (2021)), or captum (Kokhlikyan et al. (2020))) have already been
developed for various data types (tabular, image, text) and algorithms (e.g.,
GradCam (Selvaraju et al. (2020)), Counterfactuals (Mothilal et al. (2020)),
SHAP (Lundberg and Lee (2017))), an interpretability framework for time-series
data is still missing. Due to the different structures and properties of time-
ordered data, most approaches and frameworks are not directly applicable to
time series (Ismail et al. (2020)). Further, time series are not intuitive for
human understanding (Siddiqui et al. (2019)) and need additional
visualizations. Therefore, we propose TSInterpret, a framework implementing
algorithms for time series classification. In this work, we provide

  * •

a review of existing interpretability libraries,

  * •

a unified framework for time series interpretability,

  * •

and unified visualizations for the implemented algorithms.

##  2 Need for Interpretability in Time Series Classification

Temporal data is ubiquitous and encountered in many real-world applications
ranging from electronic health records (Rajkomar et al. (2018)) to cyber
security (Susto et al. (2018)). Although almost omnipresent, time series
classification has been considered one of the most challenging problems in
data mining for the last two decades (Yang and Wu (2006); Esling and Agon
(2012)). With the rising data availability and accessibility (e.g., provided
by the UCR / UEA archive (Bagnall et al. (2017); Dau et al. (2019))), hundreds
of time series classification algorithms have been proposed. Although deep
learning methods have been successful in the field of Computer Vision (CV) and
Natural Language Processing (NLP) for almost a decade, application on time
series has only occurred in the past few years (e.g., Fawaz et al. (2019);
Rajkomar et al. (2018); Susto et al. (2018); Ruiz et al. (2021)). Deep
learning models have been shown to achieve state-of-art results on time series
classification (e.g., Fawaz et al. (2019)). However, those methods are black-
boxes due to their complexity which limits their application to high-stake
scenarios (e.g., in medicine or autonomous driving), where user trust and
understandability of the decision process are crucial.

Although much work has been done on interpretability in CV and NLP, most
developed approaches are not directly applicable to time series data. The time
component impedes the usage of existing methods (Ismail et al. (2020)). Thus,
increasing effort is put into adapting existing methods to time series (e.g.,
LEFTIST based on SHAP / Lime (Guilleme et al. (2019)), Temporal Saliency
Rescaling for Saliency Methods (Ismail et al. (2020)), Counterfactuals (Ates
et al. (2021); Delaney et al. (2021))), and developing new methods
specifically for time series interpretability (e.g., TSInsight based on
autoencoders (Siddiqui et al. (2021)), TSViz for interpreting CNN (Siddiqui et
al. (2019))). For a survey of time series interpretability, please refer to
Rojat et al. (2021).

Compared to images or textual data, humans cannot intuitively and
instinctively understand the underlying information contained in time series
data. Therefore, time series data, both uni- and multivariate, have an
unintuitive nature, lacking an understanding at first sight (Siddiqui et al.
(2019)). Hence, providing suitable visualizations of time series
interpretability becomes crucial.

For instance, health data tasks like cardiac disease detection from
electrocardiogram data (ECG5000, Dau et al. (2019)) or seizure prediction from
the data of a tri-axial accelerometer (Epilepsy, Bagnall et al. (2018)) are
examples for the need of interpretability within time series data. Those
classification tasks are in a sensitive field regarding people’s lives.
Therefore, decisions need to be carefully taken based on solid evidence. Wrong
medications or unidentified diseases can have long-lasting effects on a
patient’s health. To allow physicians to make such data-driven decisions with
the help of machine learning, interpretations of black-box models are crucial.
It is not only relevant if a patient has epilepsy or a cardiac disease but why
or which indications are there for such. Further, implementing a non-
interpretable machine learning model in medicine raises legal and ethical
issues (Amann et al. (2020)).

##  3 Related Work

Name | Backend | Data | Task | Scope | Output
---|---|---|---|---|---
Tabular |  Text |  Image |  Time Series |  Classification |  Regression |  Model Agnostic |  Model Specific |  Local |  Global |  FA |  IB |  RB
AIX360 (Arya et al. (2019)) | PYT,TF, BB | X | X | X |  | X | X | X | X | X | X | X | X | X
ALIBI (Klaise et al. (2021)) | BB | X | X | X |  | X | X | X | X | X | X | X | X |
Anchor (Ribeiro et al. (2018)) | BB | X | X |  |  |  |  | X |  | X |  |  | X |
Captum (Kokhlikyan et al. (2020)) | PYT | X | X | X |  | X | X | X | X | X |  | X |  |
carla (Pawelczyk et al. (2021)) | PYT, TF, SK | X |  |  |  | X | X | X |  | X |  |  | X |
DALEX (Baniecki et al. (2021)) | SK,TF | X |  |  |  | X | X | X |  | X | X | X |  |
DeepExplain (Ancona et al. (2018)) | TF |  | X | X |  | X | X |  |  | X | X | X |  |
Dice (Mothilal et al. (2020)) | PYT, TF, BB | X |  |  |  | X | X | X |  | X |  |  | X |
ELI5 (eli5 org (2017)) | BB | X | X | X |  | X |  | X |  | X |  | X |  |
H2O (Hall et al. (2019)) | - | X |  |  | X | X | X | X | X | X | X | X |  |
iNNvestigate (Alber et al. (2019)) | TF |  |  | X |  | X |  |  | X | X | X | X |  |
InterpretMl (Nori et al. (2019)) | BB, WB | X |  |  |  | X | X | X | X | X | X | X |  |
Lucid (tensorflow (2018)) | TF |  |  | X |  | X | X |  | X | X | X | X |  |
OmniXAI (Yang et al. (2022)) | PYT, TF, SK | X | X | X | X | X | X | X | X | X | X | X | X |
pytorch-cnn-visualizations (Ozbulak (2019)) | PYT |  |  | X |  | X |  |  | X | X |  | X |  |
SHAP (Lundberg and Lee (2017)) | PYT, TF, BB | X | X | X |  | X | X | X | X | X | X | X |  |
Skater (Oracle (2018)) | BB | X | X | X |  | X | X |  |  | X | X | X | X |
Tf-Explain (Meudec (2021)) | TF |  |  | X |  | X |  |  | X | X |  | X |  |
TorchRay (Fong et al. (2019)) | PYT |  |  | X |  | X |  |  | X |  |  | X |  |
What-if (Wexler et al. (2019)) | TF | X | X | X |  | X | X | X |  |  |  |  | X |
wildboar (Samsten (2020)) | - |  |  |  | X | X | X |  | X | X |  |  | X |

Table 1: Overview over recent explanation libraries and their time series
capabilities. BB: Black-Box, TF: Tensorflow, PYT: PyTorch, SK: scikit-learn

Numerous open-source tools for machine learning evolved in the last few years.
In the following, we focus on libraries implementing various post-hoc
interpretability methods installable via PyPI. For the summary in Table 2,
only active libraries, i.e., libraries with some development activity in the
past 12 months on GitHub, are included. The comparison is conducted according
to the scope of included explanations (Model Agnostic vs. Model-specific,
Global vs. Local) and library properties (supported model libraries, data
types, and tasks). Model Agnostic methods apply to any classification
algorithms, whereas model-specific algorithms only work on a subset of
classification algorithms (e.g., Cam on Convolutional Neural Networks). Global
interpretability methods interpret the model’s overall decision-making
process, while local interpretability focuses on the interpretation of a
single instance and the prediction of that instance222More information on the
taxonomy are available in Rojat et al. (2021).. Further, the return type of
the explanation methods are taken into account. Feature Attribution methods
(FA) return a per-feature attribution score based on the feature’s
contribution to the model’s output (e.g., GradCam (Selvaraju et al. (2020)) or
SHAP (Lundberg and Lee (2017))), instance-based methods (IB) calculate a
subset of relevant features that must be present to retain or remove a change
in the prediction of a given model (e.g., counterfactuals (Mothilal et al.
(2020)) or anchors (Ribeiro et al. (2018))), while rule-based methods (RB)
derive rules.

Except for the libraries wildboar (Samsten (2020)), OmniXAI (Yang et al.
(2022)), and H2O (Hall et al. (2019)), no library provides interpretability
methods on time series. However, the support of OmniXAI (Yang et al. (2022))
is restricted to anomaly detection, univariate time series and includes only
non-time-series-specific interpretability methods. Wildboar (Samsten (2020))
focuses on temporal machine learning (classification, regression, and anomaly
detection) and only provides a limited number of (counterfactual)
interpretability methods as additional features. A unified framework for time
series interpretability like captum or ALIBI enabling easy-to-use
interpretability is still missing.

##  4 Library Design

The API Design is orientated on the scikit design paradigms consistency,
sensible defaults, composition, nonprofilarity of classes, and inspection
(Buitinck et al. (2013)).

Figure 1: Structure of TSInterpret.

LABEL:lst:InterpretabilityExample shows the workflow of the library in a
coding sample. Given a trained machine learning model and an instance to be
classified: First, the desired interpretability method is imported (line 1)
and instantiated (line 2), followed by explaining the instance (line 3), and
finally, the generation of the plot (line 4).

Listing 1: API Usage Example.

⬇

from TSInterpret.InterpretabilityModels.Saliency.TSR import TSR

int_mod = TSR(model, item.shape[-1], item.shape[-2])

exp = int_mod.explain(item, labels = label, TSR = True)

int_mod.plot(item,exp)

Consistency

All implemented objects share a consistent interface. Every interpretability
method inherits from the interface InterpretabilityBase to ensure that all
methods contain a method explain and a plot function. The plot function is
implemented on the level below based on the output structure provided by the
interpretability algorithm to provide a unified visualization experience
(e.g., in the case of Feature Attribution, the plot function visualizes a
heatmap on the original sample). If necessary, those plots are refined by the
Mechanism layer. This is necessary to ensure suitable representation as the
default visualization can sometimes be misinterpreted (e.g., the heatmap used
in the plot function of InterpretabilityBase allows positive and negative
values, while TSR is scaled to $[0,1]$. Using the same color pattern for both
scales would lead to a high risk of misinterpreting results while comparing
TSR with LEFTIST.). The explain function is implemented on the method level.

Sensible Defaults

TSInterpret provides reasonable defaults for most parameters by providing the
default parameterizations for each method from the designated papers. Those
parameters can easily be changed if needed by providing alternative values
during model instantiation.

Composition

Many interpretability methods for time series classification are based on
already existing methods for tabular, image, or text data (e.g., Ismail et al.
(2020) is based amongst others on Lundberg and Lee (2017)). Whenever feasible,
existing implementation of such algorithms are used (e.g., SHAP (Lundberg and
Lee (2017)), captum (Kokhlikyan et al. (2020)), or tf-explain (Meudec (2021)).

Nonprofilarity of classes

TSInterpret implements the interpretability algorithms as custom classes.
Datasets, instances, and results are represented as NumPy arrays, Lists, or
Tuples, instead of classes. For instance, the counterfactual method returns a
Tuple of the counterfactual time series and the label (list, int).
Hyperparameters are regular strings or numbers.

Inspection

TSInterpret stores and exposes the parameter of the interpretability
algorithms as public attributes. In some methods, parameters have a
significant impact on the obtained results. Making those parameters publicly
available through attributes facilitates experimenting with hyperparameters.

##  5 Algorithm Overview

As depicted in section 3, our framework provides support for various time
series interpretability types, resulting in interpretations and visualization
for a variety of use cases. The current version of the library includes the
interpretability algorithms listed in Table 2. Their implementation in
TSInterpret is based on code provided by the authors of the algorithms, which
got adapted to the framework and extended to ensure the availability on
multiple model backends (e.g., PyTorch or TensorFlow).

Method | Model | Explanations | Type | Dataset
---|---|---|---|---
NUN-CF (Delaney et al. (2021)) | TF, PYT, SK | IB | uni | y
CoMTE (Ates et al. (2021)) | TF, PYT,SK | IB | multi | y
LEFTIST (Guilleme et al. (2019)) | TF, PYT, SK | FA | uni | y
TSR (Ismail et al. (2020)) | TF, PYT | FA | multi | n
Table 2: Interpretability Methods implemented in TSInterpret. BB: Black-Box,
TF: Tensorflow, PYT: PyTorch, SK: scikit-learn

NUN-CF

     333https://github.com/e-delaney/Instance-Based_CFE_TSC

Delaney et al. (2021) proposed using the K-nearest neighbors from the dataset
belonging to a different class as native guide to generate counterfactuals.
They propose three options for transforming the original time series with this
native guide: the plain native guide, the native guide with bary centering,
and transformation based on the native guide and class activation mapping.

CoMTE

     444https://github.com/peaclab/CoMTE

Ates et al. (2021) proposed CoMTE as a perturbation-based approach for
multivariate time series counterfactuals. The goal is to exchange the smallest
possible number of features with reference data by applying random-restart
hill climbing to obtain a different classification.

LEFTIST

     555https://www.dropbox.com/s/y1xq5bhpf0irg2h/code_LEFTIST.zip?dl=0

Agnostic Local Explanation for Time Series Classification by Guilleme et al.
(2019) adapted LIME for time series classification and proposed to use
prefixed shapelets as the interpretable components. Each shapelet is a non-
overlapping subsection of the original time series with a prefixed length. The
feature importance is provided on a shapelet basis.

TSR

     666https://github.com/ayaabdelsalam91/TS-Interpretability-Benchmark

Temporal Saliency Rescaling (Ismail et al. (2020)) calculates the importance
of each timestep, followed by the feature importance based on different
Saliency Methods, both back-propagation based and perturbation based. We refer
the reader to our code documentation for a complete list of implemented
methods. The implementation in TSInterpret is based on tf-explain (Meudec
(2021)), SHAP (Lundberg and Lee (2017)) and captum (Kokhlikyan et al. (2020)).

(a) NUN-CF.

(b) LEFTIST.

Figure 2: Interpretations obtained with TSInterpret the univariate ECG5000
dataset.

(a) CoMTE.

(b) TSR.

Figure 3: Interpretations obtained with TSInterpret the multivariate Epilepsy
dataset.

Returning to the use case described in section 2, we train a 1D-conv ResNet on
ECG5000 and Epilepsy. Figure 2 and Figure 3 show interpretations obtained with
TSInterpret for the 1D-conv ResNet on the first item of the testset for the
univariate ECG5000 dataset (fig. 2) and the multivariate Epileptical dataset
(fig. 3). With the help of the interpretations shown in Figure 2 a physician
can gather evidence on the functionality of the ResNet classification on ECG
5000. The feature attribution method LEFTIST (fig. 2(b)) identifies sections
with positive/ negative influence on the current time series prediction. In
this case, the first section of the time series has the most considerable
impact on the current prediction being a normal electrocardiogram. If a
physician wants to know why the electrocardiogram shows a normal instance and
not an abnormal beat, conclusions can be drawn from the counterfactual
approach NUN-CF. Compared to the original time series (in blue fig. 2(a))
classified as normal, the classification is changed if the drop at the
beginning is lower and the drop at the end is replaced with a rise (pink line
in fig. 2(a)). Figure 3 shows interpretations for the epilepsy dataset.
Suppose a physician wants to know how the original instance is predicted as
walking (blue) instead of running. In such a case, the practitioner applies
CoMTE to generate a counterfactual in the class direction of running,
resulting in a different acceleration of the x-axis in the first 200 time
steps (see fig. 3(a)). If the physician is only interested in seeing the most
important features TSR (fig. 3(b)) can be applied. With the help of
interpretability, a physician can gather evidence for the predicted
classification and gain trust in the system (Amann et al. (2020)). In general,
the visualization for the counterfactual approaches CoMTE and NUN-CF have a
similar style, although NUN-CF visualizes univariate and CoMTE multivariate
data. Note that CoMTE only visualizes the changed features. For the feature
attribution methods LEFTIST and TSR, the visualizations have similar styles
but a different color map. This is necessary as LEFTIST returns a positive and
negative influence (range $[-1,1]$), while TSR returns normalized time slices
and feature importances (range $[0,1]$).

The easy-to-use interface of TSInterpret allows the out-of-box usage of
interpretability models on time series data for different use cases, time
series types, and classification models. Further, the framework can be easily
extended by inheriting from either InterpretabilityBase or one of the more
customized classes (e.g., Feature Attribution or InstanceBased). All classes
below the Output layer also come with a default plot function to match the
visualizations obtained by the already implemented algorithms.

##  6 Outlook

TSInterpret provides a cross-backend unified API for the interpretability of
time series classifications enabling interpretation generation with just three
lines of code. In the first phase of the development, TSInterpret provides
four interpretability algorithms for both uni- and multivariate time series
classification and functions for visualizing the interpretation results.
However, the Framework is easily extensible by inheritance and provides a
simple API design. In the future, we will include support for additional
interpretability algorithms, time series prediction, as well as metrics to
progress into an easy-to-use benchmarking tool for time series
interpretability. In order to focus the research and application of time
series prediction models not only on performance but also on understanding the
model in a practical environment, a further development step is to integrate
TSInterpret into the landscape of existing frameworks and thus establish it as
an easily usable standard.

## Acknowledgments

This work was carried out with the support of the German Federal Ministry of
Education and Research (BMBF) withhin the project ”MetaLearn” (Grant
02P20A013).

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# Topological phase transitions via attosecond x-ray absorption spectroscopy

Juan F. P. Mosquera  Departamento de Química, Universidad Autónoma de Madrid,
28049 Madrid, Spain  Giovanni Cistaro  Departamento de Química, Universidad
Autónoma de Madrid, 28049 Madrid, Spain  Mikhail Malakhov  Departamento de
Química, Universidad Autónoma de Madrid, 28049 Madrid, Spain  M.N. Mikheev
Institute of Metal Physics of the Ural Branch of the Russian Academy of
Sciences, S. Kovalevskaya str. 18, 620108 Yekaterinburg, Russia  Emilio
Pisanty  ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of
Science and Technology, 08860 Castelldefels (Barcelona), Spain  Attosecond
Quantum Physics Laboratory, King’s College London, London WC2R 2LS, UK
Alexandre Dauphin  ICFO – Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
Luis Plaja  Grupo de Investigación en Aplicaciones del Láser y Fotónica,
Departamento de Física Aplicada, University of Salamanca, E-37008, Salamanca,
Spain  Alexis Chacón  Department of Physics and Center for Attosecond Science
and Technology, POSTECH, 7 Pohang 37673, South Korea; Max Planck POSTECH/KOREA
Research Initiative, Pohang, 37673, South Korea  Maciej Lewenstein  ICFO –
Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels (Barcelona), Spain  ICREA, Pg. Lluís Companys
23, 08010 Barcelona, Spain  Antonio Picón<EMAIL_ADDRESS>corresponding
author Departamento de Química, Universidad Autónoma de Madrid, 28049 Madrid,
Spain  Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de
Madrid, 28049, Madrid, Spain.

###### Abstract

We present a numerical experiment that demonstrates the possibility to capture
topological phase transitions via an x-ray absorption spectroscopy scheme. We
consider a Chern insulator whose topological phase is tuned via a second-order
hopping. We perform time-dynamics simulations of the out-of-equilibrium laser-
driven electron motion that enables us to model a realistic attosecond
spectroscopy scheme. In particular, we use an ultrafast scheme with a
circularly polarized IR pump pulse and an attosecond x-ray probe pulse. A
laser-induced dichroism-type spectrum shows a clear signature of the
topological phase transition. We are able to connect these signatures with the
Berry structure of the system. This work extend the applications of attosecond
absorption spectroscopy to systems presenting a non-trivial topological phase.

††preprint: APS/123-QED

##  I Introduction

Topological states of matter permit the possibility to create materials that
are insulators within the bulk, but they hold conducting states on the
surface, known as edge states. Edge states are symmetry protected by the
intrinsic properties of the bulk, providing a new landscape for developing
unique optoelectronics applications with no precedence. In the last years, it
has been enormous progress in the development of these modern materials, such
as topological insulators Hasan2010 ; Bansil2016 . Interestingly, with the
advent of ultrashort intense lasers, some materials that are topologically
trivial insulators in equilibrium can be driven out of equilibrium into
topologically non-trivial insulators Oka2009 ; Inoue2010 ; Sie2015 ;
McIver2020 ; JimenezGalan2020 ; JimenezGalan2024 ; Tyulnev2024 . Those light-
induced topological insulators, also known as Floquet topological insulators,
only live during the laser pulse length, that is only around several
femtoseconds (10-15 s). Generating topological states of matter is as
important as characterizing them Ma2021 . In equilibrium, detecting if an
insulator is topologically trivial or not trivial is possible by measuring the
energy bands of the edge states via angle-resolved photoemission spectroscopy
(ARPES) Bansil2016 ; Lv2019 ; Sobota2021 . Characterizing Floquet topological
insulators is much more challenging, as we need an ultrafast probe in the
femtosecond timescale in order to capture the ultrafast topological phases.
One possibility is performing time-resolved ARPES and follow the edge-state
energy bands in time Lv2019 ; Freericks2009 ; Sentef2015  or spin-resolved
ARPES Sobota2021 ; Schuler2020 ; Schuler2020B . However, the use of
photoelectron spectroscopy techniques in the ultrafast regime have the
drawback that reducing the probe pulse length decreases the photoelectron
energy resolution. Additionally, if the system under investigation is
interacting with a moderate intense IR pulse that drives the system out of
equilibrium, the photoelectron spectrum becomes hard to interpret due to the
laser effects on the ejected electron Cavalieri2007 ; Siek2017 ; Ossiander2018
. Another possibility is the use of high-order harmonic generation (HHG)
Ghimire2019 . HHG is an extreme non-linear process in which several IR photons
of the driving laser are absorbed and emitted by the material in the form of a
high-frequency photon. Recent theoretical developments have shown that HHG is
sensitive to the Berry curvature of the system Liu2017 ; Luu2018 , and the
measurement of topological phases has been demonstrated Bauer2018 ;
JimenezGalan2020 ; Silva2019 ; Chacon2020 ; Bai2020 ; Schmid2021 ;
Baykusheva2021 ; Heide2022 . Despite the enormous progress in the last years,
resolving ultrafast topological phases in periodic systems are far from being
resolved.

Here, we present a complementary scheme to ARPES that enables us to directly
measure the topological phase by using x-ray attosecond absorption
spectroscopy. This scheme relies on a dispersive absorption measurement, which
it is suitable for ultrashort probe pulses as the energy resolution is not
diminished by the probe pulse length Leone2016 . Attosecond absorption
spectroscopy has been successfully applied in different bulk and thin
materials, from insulators to semimetals, to investigate carrier dynamics,
phononic effects, and excitonic interactions Schultze2014 ; Lucchini2016 ;
Zuerch2017 ; Moulet2017 ; Schlaepfer2018 ; Volkov2019 ; Lucchini2021 ;
Buades2021 . In this work, we extend the application of attosecond absorption
spectroscopy to probing topological phase transitions.

Figure 1: Ultrafast x-ray scheme and system under investigation. (a) Two
ultrashort laser pulses, separated by a time delay, interact with a boron
nitride monolayer. The pump pulse is in the range of mid IR and is circularly
polarized, and it is intense in order to drive a strong intra-band current.
The probe pulse is in the range of soft x-rays and is linearly polarized, and
it excites transitions from the K-edge of boron. (b) Graphical representation
of the first ($\gamma$) and second order hopping ($t_{2}e^{\pm i\phi_{0}}$)
terms between different lattice sites. (c), (d) and (e) represent the energy
dispersion of the system. The red and blue arrows represent the IR and x-ray
transitions, respectively. The IR pulse couples the conduction and valence
bands, while the x-ray pulse couples the core band (1s orbital of boron) with
the bands around the Fermi level. The second-order hopping $t_{2}$ controls
the topological phase, which depends on the parameter
$t_{s}=\Delta/6\sqrt{3}$, being $\Delta$ the gap of the system without second-
order hopping. (b) and (d) are insulators with a trivial and non-trivial
topology, respectively, and their direct bandgap is the same. In (d) the
conduction and valence bands join at the K point.

The absorption of an attosecond pulse occurs in a very short timescale, in
which the dynamics are dominated by the electron response. The induced
electron motion is therefore much faster than the nuclei motion, and we may
consider neglecting the coupling with phonons at such timescale. We can then
conceive experiments that take advantage of the coherent electron dynamics
Mandal2021 ; Xue2022 ; Moitra2023 . In a previous work Cistaro2021 , we
demonstrated that the attosecond absorption is very sensitive to Van Hove
singularities, and it is possible to extract information around these points,
not only of the energy-dispersion structure, but also of the Berry structure.
By performing numerical simulations with a recently developed approach to
describe the electron dynamics out of equilibrium, the EDUS code Cistaro2023 ,
we show here that attosecond absorption spectroscopy can indeed capture
topological phase transitions. This work is based on a Chern insulator
described by a Haldane Hamiltonian Haldane1988 , in which we control the
topological phase by changing a second-order hopping. We correlate the
features of the absorption to the laser-driven coherent electron dynamics by
using a semiclassical approach Cistaro2021 ; Picon2019 ; Dong2022 . This
enables us to get a further insight of the effects of the Berry structure on
the absorption spectrum around van Hove singularities. Our numerical and
theoretical study opens the door to further investigations on relevant systems
for optoelectronics applications, such as topological or Floquet insulators.

##  II  Ultrafast laser-induced x-ray dichroism

We aim at studying the signatures of electron dynamics on the absorption of
attosecond x-ray pulses for different topological phases. We start with a
Haldane Hamiltonian Haldane1988  for a tight-binding model of boron-nitride
monolayer (hBN), and by adding a second-order hopping that breaks time-
reversal symmetry (TRS) we manipulate the topological phase. The system
interacts with a linearly-polarized x-ray pulse and a circularly-polarized
intense IR pulse. The Hamiltonian is expressed as $H(t)=H_{0}+V(t)$, where
$H_{0}$ is the Haldane Hamiltonian and $V(t)$ is the light-matter interaction
coupling. More details are given in appendix A. From previous studies
Picon2019 ; Cistaro2021 ; Dong2022 , we know that the lineshape of an
attosecond absorption spectrum is modified by the laser-driven electron
dynamics, particularly at the energies corresponding with van Hove
singularities. Importantly, the energy and Berry structure of those points are
sensitive to the topological phase. This inspires us to use a laser-induced
x-ray dichroism scheme as the one illustrated in Fig. 1. The IR pulses induces
strong intra-band currents, making electrons to follow circular trajectories.
By changing the polarization handedness, and observing the difference in the
absorption lineshape, i.e. a dichroism-type absorption spectrum, we infer
information of the Berry structure (connected with the topological phase).
This is because left-handed and right-handed laser-driven trajectories are not
so different in energy, overall, but they are with respect to the Berry
structure due to the second-order hopping, see the illustration in Fig. 1(b),
as we show in the following.

In the Haldane Hamiltonian, when the second-order hopping is
$t_{2}=t_{s}=\Delta/6\sqrt{3}$, being $\Delta$ the gap of the system without
second-order hopping, the conduction and valence bands join at the K points,
as illustrated in Fig. 1(d). For larger (smaller) hopping, the gap opens, and
the system presents a non-trivial (trivial) topology. The x-ray pulse mainly
promotes core electrons from the boron site to the conduction band, as the
valence band is fully occupied before the arrival of the IR pulse. The
attosecond x-ray pulse of 188 eV photon energy is linearly polarized, with a
small intensity to ensure a first-order perturbation excitation. The
polarization is out-of-plane in order to couple the 1s-2p transitions. The
pulse length is 80-as and an intensity of 1010 W/cm2, short enough in order
that the bandwidth, with a full-width half-maximum (FWHM) of 188 as, covers
the energy bands around the Fermi level. Core electrons are further driven
along the conduction band in the presence of the IR pulse before the core-hole
relaxation that occurs approximately in 6.91 fs, translated to a decay width
$\Gamma_{ch}=0.095$ eV. In this short timescale, the electron dynamics is
coherent and the accumulated dynamical phase is imprinted in the attosecond
absorption spectrum Cistaro2021 . The IR pulse of 0.41 eV photon energy
(3000-nm wavelength and period of T$\sim 10$ fs) is circularly polarized, with
an intensity of 1011 W/cm2, and a pulse duration of 7 cycles, with an envelope
modeled by a $\sin^{2}$ profile. Note that the time delay between both pulses
are taken with respect to the maxima of each pulse envelope. Therefore, a
negative time delay means that the probe maximum arrives to the system before
the pump maximum.

Figure 2: Laser-induced x-ray absorption dichroism for different topological
phases (a) $t_{2}=\frac{1}{2}t_{s}$, (b) $t_{2}=\frac{3}{4}t_{s}$, (c)
$t_{2}=\frac{5}{4}t_{s}$, and (d) $t_{2}=\frac{3}{2}t_{s}$. The changes are
localized around van Hove singularities (K, K’, M’, and $\Gamma$ points).
$\Delta A$ expressed in percentage indicates the difference of absorption
intensity normalized with respect to the maximum absorption calculated without
the presence of the IR pulse.

The out-of-equilibrium dynamics induced by the lasers is calculated using our
EDUS code Cistaro2023 . From the calculated electron dynamics, we obtain the
polarization of the system in time, and from this one we obtain the absorption
of the system. The laser-induced x-ray absorption dichroism for different
topological phases is shown in Fig. 2. This has been calculated, for a fixed
time delay ($\tau_{d}=$ 0.0 fs) between the IR and the x-ray pulse, by
computing the absorption spectrum when the IR pulse is left-handed and right-
handed circularly polarized and then taking the difference between them. If
the system has no second-order hopping, i.e. $t_{2}=0$, then there is no
dichroism signal. When $t_{2}\neq 0$, the absorption dichroism is localized
around the van Hove singularities. We clearly observe that the energies
corresponding to the K, K’, and M’ points are those with stronger signal. The
M’ point is located in different points of the reciprocal space depending on
the topological phase, see Fig. 3. The M’ point coincides with the M point
when there is no second-order hopping and represents a saddle point. Note that
the M’ points are between the K’ and K points in the trivial phase, and
between K and $\Gamma$ points in the topological phase.

When the system is a trivial or non-trivial insulator, the lineshape of the
absorption dichroism around the K point is quite different, having a stronger
multipeak structure when the system is a trivial insulator. The signal around
M’ and K’ overlap, but in general do not show such strong variation when the
topology of the system changes. The energy positions of the van Hove
singularities changes due to topology, this is expected because the energy
dispersion depends on $t_{2}$, see Fig. 3. Interestingly, in Fig. 2 we show
two different examples, top and bottom panel, in which the energy gaps are the
same, but still the calculated lineshapes are very different when the system
has a trivial or a topological phase. Note that the signals are around 1-4 %
units of the maximum calculated absorption without the presence of the IR
pulse, so these changes are notable. We have also explored different IR
intensities. As higher the intensity, the signal is higher and the multipeak
structure extends in energy (showing more modulations), but in general the
same behavior is observed around the K points during a topological phase
transition.

Figure 3: Energy dispersion of the conduction band for (a)
$t_{2}=\frac{1}{2}t_{s}$, (b) $t_{2}=\frac{3}{4}t_{s}$, (c)
$t_{2}=\frac{5}{4}t_{s}$, and (d) $t_{2}=\frac{3}{2}t_{s}$. M’ is a saddle
point and changes with $t_{2}$. Note that the energy color scale takes as a
0-eV reference the K point, in which the direct gap is located for each case.

The laser-induced dichroism shown in Fig. 2 is for a fixed time delay between
the two laser pulses. We also explore different time delays $\tau$, on which a
negative value means that the maximum of the probe pulse occurs before the
maximum of IR pump pulse. Thus, a zero time delay $\tau=0$ is understood as
the maxima of both pulses overlap. The ultrafast laser-induced dichroism at
different time delays is shown in Fig. 4 for trivial and topological phases.
The time-dependent changes around the M’ and K’ points are located at
different energies when the system is trivial ($t_{2}=t_{s}/2$) or topological
($t_{2}=3t_{s}/2$). This is because of the difference in the energy
dispersion. However, the time-dependent changes are quite similar. The time-
dependent changes around the K point are quite different for the two different
phases though, the topological phase presents two (positive, red, and
negative, blue) clear peaks, while the trivial phase presents a higher
modulation (multipeak structure). These changes are also influenced by the
Stark shift induced by the laser pulse and depends on the IR intensity.

In conclusion, the used ultrafast scheme clearly shows a signature when the
topological phase changes. In the next section, we analyze the absorption
dichroism using a semiclassical model that enables us to understand the laser-
induced lineshapes and correlate them to the Berry structure.

Figure 4: Laser-induced x-ray absorption dichroism for different pump-probe
time delays. (a) Scheme of the IR pumb pulse for right- and left-handed
polarized fields. Ultrafast laser-induced x-ray dichroism for (b) a trivial,
$t_{2}=\frac{1}{2}t_{s}$, and (c) a topological, $t_{2}=\frac{3}{2}t_{s}$,
phase. The density of states (DOS) is represented on the side.

##  III Correlating the Berry structure with the laser-induced dichroism

In our previous work Cistaro2021 , we developed a semiclassical approach to
calculate the attosecond absorption spectrum. In this approach, the
polarization of the system is calculated by the sum of semiclassical electron
trajectories driven by the IR vector potential. The approach assumes that an
electron starts in a particular ${\bf k}$ point of the conduction band, i.e.
the excitation by the attosecond pulse is sudden, and the quasi-momentum
evolves as ${\bf K}={\bf k}-{\bf A}(t)$, where ${\bf A}(t)$ is the vector
potential of the IR pulse. Hence, the intra-band current is described by the
evolution of the quasi momentum, and the contribution of the valence band is
neglected. Along the semiclassical trajectory, the electron acquires a phase
given by

$\displaystyle\Delta\varphi(t,t_{0},{\bf
K})=i\int_{t_{0}}^{t}dt^{\prime}[(\epsilon_{c}({\bf K}+{\bf
A}(t^{\prime}))-\epsilon_{c}({\bf K}))]\hskip 14.22636pt$
$\displaystyle+i\int_{t_{0}}^{t}dt^{\prime}[\Phi_{S}({\bf K}+{\bf
A}(t^{\prime}),t^{\prime})]\hskip 14.22636pt$ (1)

where the dynamical, action Berry phase $\Phi_{S}$ is $\Phi_{S}({\bf
k},t)=\frac{1}{2}\boldsymbol{\varepsilon}_{IR}(t)\cdot\partial_{\bf
k}\phi({\mathbf{k}})[\cos\theta({\mathbf{k}})+1]$, being $\epsilon_{c}({\bf
k})$ the energy of the conduction band, and the angles $\theta({\bf k})$ and
$\phi({\bf k})$ are related to the Berry structure of the system, see appendix
A. Note that the coherent phase mainly depends on the conduction band and not
at all on the core-hole band $\epsilon_{ch}({\bf k})$. This is due to the fact
that the core orbitals are well localized and, therefore, the electronic
structure is not ${\bf k}$ dependent.

Figure 5: Semiclassical approach for the laser-induced x-ray dichroism.
(a),(b) Comparison of the laser-induced x-ray dichroism between first-
principle and semiclassical calculations for trivial $t_{2}=\frac{1}{2}t_{s}$
and topological $t_{2}=\frac{3}{2}t_{s}$ phases. (c),(d) The laser-induced
x-ray dichroism computed by the semiclassical approach with and without
considering the Berry structure for trivial and topological phases. The red
(black) line is when energy and Berry phase are included (when energy is only
included) in the accumulated dynamical phase, see Eq. (1).

Once we calculate the phase for different trajectories at different points of
the reciprocal space, then we calculate the polarization of the system as

$\displaystyle{\bf P}(t)\\!\propto\\!\sum_{{\bf K}}\,$
$\displaystyle\left[\frac{\left|\boldsymbol{\xi}_{20}(\textbf{K}+\textbf{A}(t))\right|}{\left|\boldsymbol{\xi}_{20}(\textbf{K}+\textbf{A}(t_{0}))\right|}\right.$
$\displaystyle\left.e^{-i(\epsilon_{ch}({\bf K})-\epsilon_{c}({\bf
K})-i\Gamma_{ch}/2)(t-t_{0})+\Delta\varphi(t,t_{0},{\bf K})}+c.c.\right],$

where we sum over all the ${\bf K}$ points, and $\boldsymbol{\xi}_{20}$ is the
inter-band Berry connection matrix that couples the core and conduction band,
see more details in appendix B and C. The previous equation can be easily
understood as the sum of all the oscillating dipole terms in the reciprocal
space, which deviates by a phase $\Delta\varphi$ that arises from the light-
induced electron motion during the core-hole decay, represented by the width
$\Gamma_{ch}$.

Once we have the semiclassical polarization, Eq. (III), we obtain from it the
attosecond absorption spectrum Cistaro2023 , both when the IR laser pulse is
left-handed and right-handed circularly polarized, and take the difference in
order to obtain the laser-induced absorption dichroism, see Fig. 5 for the
case of a trivial and topological phase. The agreement with the first-
principle calculations obtained with EDUS, see Figs. 5a,b, is excellent. This
simple semiclassical model is then enough to describe the observed features of
the dichroism spectrum. The advantage of the semiclassical approach is that it
enables us to distinguish the effects of the energy and the Berry structure of
the system. We compute the laser-induced absorption dichroism, but now by not
including the dynamical Berry phase in the accumulated phase of Eq. (1), see
the black line in Figs. 5b,c.

The latter calculations show that the dichroism features around the K point
are mainly because of the Berry structure of the material. The accumulated
energy phase is very similar for left-handed and right-handed circular
polarization. This phase induces an energy shift towards higher energies. The
accumulated action Berry phase changes sign due to the handedness of the
polarization. The part of the dynamical Berry phase that depends on the angle
$\theta$ changes the sign for the topological case compared with the trivial
case. This change of sign has consequences when is added to the other phase
terms. In the topological phase, the dynamical Berry phase is comparable to
the energy phase and results in the two well-separated (positive and negative)
peaks, Fig. 5b, while for the trivial phase, the Berry phase is weak and
results in the multipeak structure, Fig. 5a.

We can get a different point of view by relating the dynamical Berry phase
with the Berry curvature. If we assume a closed electron trajectory, i.e. the
electron is in the same position in the reciprocal space at the initial time
$t_{i}$ and at the final time $t_{f}$, then the accumulated dynamical phase is
$\int_{t_{i}}^{t_{f}}dt^{\prime}[\Phi_{S}({\bf K}+{\bf
A}(t^{\prime}),t^{\prime})]=\int_{\Delta S}d{\bf S}\cdot\bm{\Omega}({\bf K})$,
where $\Delta S$ is the area enclosed by the trajectory, ${\bf S}$ is a normal
vector to the area $\Delta S$, and $\bm{\Omega}({\bf K})$ is the Berry
curvature of the conduction band. If the enclosed trajectory is small enough
that the Berry curvature is constant within the area, then one expects that
the accumulated phase is proportional to the Berry curvature. The Berry
curvature changes sign around the K point when there is a topological phase
transition.

Around the K’ and M’ points, we observe that both the energy and the dynamical
Berry phase play an important role. The laser-driven electron trajectories
perceive then a different energy-dispersion landscape depending on the
handedness of the polarization, as it takes place around the M’ point, see the
energy dispersion represented in Fig. 3. Hence, the energy dispersion has an
important effect on the signal around the K’ and M’ energy points.

In conclusion, the semiclassical approach enables us to correlate the changes
in the absorption spectrum with the laser-driven dynamics. This correlation
allows us to understand the observed features around the different van Hove
singularities as a function of the topological phase.

##  IV Saddle Point Approximation

To get a clear physical insight of the absorption spectrum around the K point
(van Hove singularity with stronger differences under a phase transition), we
have studied the nature of different trajectories driven by the IR pulse in
the framework of the semiclassical approach, knowing that the polarization of
the system would be given by $P(\omega)=\int_{-\infty}^{\infty}dte^{-i\omega
t}P(t)$. Due to the nature of our specific pump-pulse scheme, it is possible
to assume that the exponential action $e^{-iS}\equiv
e^{-iS(\textbf{k},t,t_{0})}$ (defined in detail and its connection to the
polarization in the appendix C) oscillates rapidly as a function of the
crystal momentum k. Therefore, one can perform a saddle-point approximation
(SPA) Becker2006 ; Nayak2019 ; Lewenstein2020 ; Milosevic2020 , to find the
trajectories in the reciprocal space that satisfy with the conditions

$\displaystyle\Delta\textbf{R}={\bf r}_{d}({\bf
K},t,t_{0})-\boldsymbol{\alpha}({\bf
K})+\boldsymbol{\alpha}^{\parallel}(\textbf{K}-{\bf A}(t)+{\bf A}(t_{0}))=0,$
(3) $\displaystyle\epsilon_{c}({\bf K})-\epsilon_{ch}({\bf
K})+\boldsymbol{\varepsilon}_{IR}(t)\cdot\left[{\bf r}_{d}({\bf
K},t,t_{0})+\boldsymbol{\alpha}^{\parallel}(\textbf{K}-{\bf A}(t)+{\bf
A}(t_{0}))\right]=\omega,$ (4)

which are similar to the ones obtained in the study of the recollision model
for HHG in solids Gaarde2020 ; Yue2022 . In this approximation, we are
neglecting the effects of the core-hole decay $\Gamma_{ch}$. Also, it is more
convenient to rewrite the dynamical action Berry phase Eq. (1) explicitly as a
function of the inter- and intra-band Berry connection matrix elements to
study the contribution of each of these terms, see more details in Appendix D.
These equations are easy to intepret. On one hand, the first equation is
interpreted as the distance between the electron-hole pair,
$\Delta\textbf{R}$. When this distance is zero, then the electron and hole may
recombine. The trajectories satisfying this condition are the most relevant in
the action then. Interestingly, the term ${\bf r}_{d}(\textbf{K},t,t_{0})$
contains the ballistic velocity, which just depends on the energy dispersion
of the conduction band, but also the anomalous velocity, which depends on the
Berry curvature of the conduction band. The Berry distances
$\boldsymbol{\alpha}({\bf k})$ depend purely of the intra- and inter-band
Berry connections. These terms modify when the recombination may occur. This
is an important term to describe the so-called imperfect recollisions in HHG
Gaarde2020 . On the other hand, the second equation is interpreted as the
energy during recombination. It will provide the energy $\omega$ of the
absorption when the recombination occurs.

To compute the time evolution of different trajectories at different initial
points in the k-space, we use a discretized k-grid in the neighborhood of the
K point, as the guess for the initial positions of the electron in the
conduction band, and then evolve them using the quasimomentum K. Only those
trajectories which recombine ($\Delta\textbf{R}=0$) at a certain time
contribute to the polarization. Thus, the recollision times $t$ and the
initial crystal momenta k can be found using equation (3). Among all possible
trajectories, just the ones that recollide at a certain energy $\omega$, see
Eq. (4), are shown in Fig. 6. The different dots represent the found
trajectories at their corresponding recollision energy $\omega$, both for
right-handed circular polarization (red dots) and left-handed circular
polarization (blue dots). In order to better visualize the number of
trajectories per energy, we add a broadening to each one that sum up
incoherently. In the top of that, we have added, in green, the value of the
corresponding action, $S(\textbf{k},t,t_{0})$ (accumulated imaginary phase) at
the given recollision time of each trajectory.

Figure 6: In the top panels, the dots represent the trajectories that satisfy
the SPA conditions for (a) a trivial $t_{2}=\frac{1}{2}t_{s}$ and (b) a
topological $t_{2}=\frac{3}{2}t_{s}$ phase. The trajectories for right-handed
(left-handed) circular polarization are highlighted by red (blue) dots. In
order to better visualize the number of trajectories per energy, we add a
broadening to each one that sum up incoherently. We also represent the
accumulated phase of each trajectory at their corresponding recolision time in
green. In the bottom panels, the laser-induced x-ray dichroism spectrum around
the neighborhood of K is shown for comparison.

For both phases, trivial and topological, the contribution of the Berry
curvature in the integral of the displacement ${\bf r}_{d}({\bf K},t,t_{0})$
is small. Also, the contribution of the intra-band Berry connections in
$\boldsymbol{\alpha}(\textbf{K})$ (or
$\boldsymbol{\alpha}^{\parallel}(\textbf{K})$) is larger than the inter-band
Berry connections for computing $\Delta\textbf{R}$ in the trajectories that
recollide. Therefore, the trajectories that satisfy the first condition of the
SPA are mostly described by the intra-band Berry connection
$\boldsymbol{\xi}_{22}(\textbf{K})$ and the dispersion velocity
$\textbf{v}(\textbf{K})=\nabla_{\bf K}\Delta E(\textbf{K})=\nabla_{\bf
K}\epsilon_{c}(\textbf{K})$. The latter is responsible for most of the
effects, i.e. the highest contribution is given by how electrons move along
the dispersion energy rather than by how the bands are coupled by the infrared
pulse. Thus, the energies $\omega$ in Eq. (4) are described by the energy
$\epsilon_{c}({\bf k})$ and $\boldsymbol{\alpha}(\textbf{K})$
($\boldsymbol{\alpha}(\textbf{K})\approx\boldsymbol{\xi}_{22}(\textbf{K})$),
the latter modulated by the electric field at the recollision time.

For the topological phase in the neighborhood of K, the trajectories whose
energies are lower than $1.7$ eV (their initial k are not further $0.1$ a.u.
from K), their recollision times are larger than $11$ fs., then the stronger
contribution to the energy $\omega$ is just given by $\epsilon_{c}({\bf k})$.
On the other hand, for those trajectories whose energy is larger than $1.7$
eV, both energy and intra-band Berry connection contribute, the latter
modulated by the amplitude of the electric field, which groups the
trajectories together towards smaller energies for the left-handed polarized
IR pulse and towards larger energies for the right-handed case. This may
qualitatively explain the two-single peak structure of the light-induced
dichroism spectrum just in terms of the number of trajectories, see Fig. 6a.

For the trivial phase in the region of energies around K, the contribution
arising from the intra-band Berry connections is more notorious (both for
those closer and further from the K point), and its effect, again modulated by
the amplitude of the electric field at the different recollision times,
results in trajectories recolliding at different energies, but being
overlapped, see the trajectories in Fig. 6a. This may qualitatively related to
the multi-peak structure in the light-induced dichroism spectrum, see Fig. 6b.

We note that the accumulated phase of a single SPA trajectory may play an
important role in the absorption spectrum. The analysis above is mainly
restricted to the number of trajectories satisfying the SPA conditions, but
each trajectory indeed accumulate a phase given by the function of the action.
Because the absorption is sensitive to the imaginary part of the polarization
Cistaro2021 , those trajectories approaching to zero accumulated phase may
have a small contribution in the absorption spectrum. The accumulated phase is
indicated in Fig. 6 for each trajectory.

In conclusion, the most important effect in the dynamical phase is coming from
the contribution of the energy dispersion, but this is mainly modified by the
intra-band Berry connection, which depends on the amplitude and polarization
of the electric field at different recollision times. This latter effect is
different for the case of trivial and topological phase.

##  V Conclusions

In this work we perform a numerical experiment and develop a semiclassical
theory to investigate the effects of the topological phase on the absorption
of an x-ray attosecond pulse. In particular, we propose an ultrafast laser-
induced x-ray dichroism scheme to study a Chern insulator. The ultrafast
scheme consists of an x-ray attosecond pulse, which excites core electrons
into the conduction band, and a secondary IR pulse, which produces a strong
intra-band motion. The IR pulse is left-handed and right-handed circularly
polarized, and by calculating the x-ray absorption spectrum for the two IR
polarizations, and then taking the difference between them, we obtain the
absorption dichroism. We show that this scheme is very sensitive to the change
of topological phase. The absorption dichroism features are localized around
the energies corresponding to van Hove singularities. We develop a
semiclassical theory to understand these features, correlate them with the
electron dynamics around the van Hove singularities, and associate them with
an accumulated dynamical phase that depends on the topology of the system. We
show that the main changes due to the topological phase are because the Berry
structure of the system. To further simplify the semiclassical model, we use
the saddle-point approximation. This enables us to qualitatively study the
main changes in the action and identify the intra-band Berry connection of the
conduction band as the relevant contribution to the topological phase.

This work opens the door to further investigations in capturing coherent
electron dynamics via attosecond absorption spectroscopy and infer relevant
properties such as the Berry structure or the topological phase. Future
investigations may involve the study of relevant systems for optoelectronic
applications, such as Floquet and topological insulators Lein2022 .

##  Appendix A Haldane hamiltonian

We consider the protypical Haldane Hamiltonian Haldane1988  in order to
describe different topological phases. This Hamiltonian breaks time-reversal
symmetry via a second-order hopping, typically used on a 2D hexagonal system.
Here we consider a tight-binding hBN model Rivas2013 , with lattice vectors
${\bf a}_{1}=\frac{a}{2}(\sqrt{3},-1)$ and ${\bf
a}_{2}=\frac{a}{2}(\sqrt{3},1)$, being $a=2.5$ Å the lattice constant. For
periodic boundary conditions, the Haldane Hamiltonian in the reciprocal space
is given by

$\hat{H}_{s}(\textbf{k})=B_{0}(\textbf{k})\hat{\sigma}_{0}+\sum_{i=1}^{3}B_{i}(\textbf{k})\hat{\sigma}_{i},$
(5)

where we have used the fact that any Hermitian complex $2\times 2$ matrix can
be expressed in terms of the identity $\hat{\sigma}_{0}$ and the Pauli
matrices $\hat{\sigma}_{i}$. The set of $B_{0}(\textbf{k})$ and
$B_{i}(\textbf{k})$ is read as

$\displaystyle B_{0}(\textbf{k})$
$\displaystyle=2t_{2}\cos{\left(\phi_{0}\right)}\sum_{i}{\cos{\left(\textbf{k}\cdot\textbf{b}_{i}\right)}},$
(6) $\displaystyle B_{1}(\textbf{k})$
$\displaystyle=\gamma\sum_{i}{\cos{\left(\textbf{k}\cdot\boldsymbol{\delta}_{i}\right)}},$
(7) $\displaystyle B_{2}(\textbf{k})$
$\displaystyle=\gamma\sum_{i}{\sin{\left(\textbf{k}\cdot\boldsymbol{\delta}_{i}\right)}},$
(8) $\displaystyle B_{3}(\textbf{k})$
$\displaystyle=\frac{\Delta}{2}-2t_{2}\sin{\left(\phi_{0}\right)}\sum_{i}{\sin{\left(\textbf{k}\cdot\textbf{b}_{i}\right)}}.$
(9)

being $\textbf{b}_{i}$ the second neighbors atoms and
$\boldsymbol{\delta}_{i}$ are the nearest neighbors, which can be writen as

$\begin{matrix}[l]\boldsymbol{\delta}_{3}=\frac{a}{\sqrt{3}}(1,0),&\boldsymbol{\delta}_{2}=\boldsymbol{\delta}_{3}-\textbf{a}_{1},&\boldsymbol{\delta}_{1}=\boldsymbol{\delta}_{3}-\textbf{a}_{2},\end{matrix}$
(10)

and

$\begin{matrix}[l]\textbf{b}_{1}=-\textbf{a}_{1},&\textbf{b}_{1}=\textbf{a}_{2},&\textbf{b}_{1}=\textbf{a}_{1}-\textbf{a}_{2}.\end{matrix}$
(11)

The energy gap is $\Delta=5.9$ eV, and $\gamma$ and $t_{2}e^{i\phi_{0}}$ are
the first and second order hopping, respectively, given by $\gamma=3.16$ eV
and $\phi_{0}=\pi/2$.

The eigenvalues of $\hat{H}_{s}$ are given by
$E_{\pm}(\textbf{k})=B_{0}(\textbf{k})\pm\sqrt{B_{1}(\textbf{k})^{2}+B_{2}(\textbf{k})^{2}+B_{3}(\textbf{k})^{2}}$.
Hence, the conduction energy is given by
$\epsilon_{c}(\textbf{k})=E_{+}(\textbf{k})$ and the valence energy by
$\epsilon_{v}(\textbf{k})=E_{-}(\textbf{k})$. The eigenvectors can be written,
in a suitable gauge for the rest of the formulas, as

$\displaystyle\left|u_{-}(\textbf{k})\right>$
$\displaystyle=\begin{pmatrix}e^{-i\phi({\textbf{k})}/2}\sin{\frac{\theta(\textbf{k})}{2}}\\\\[6.0pt]
-e^{+i\phi({\textbf{k})}/2}\cos{\frac{\theta(\textbf{k})}{2}}\end{pmatrix},$
$\displaystyle\left|u_{+}(\textbf{k})\right>$
$\displaystyle=\begin{pmatrix}e^{-i\phi({\textbf{k})}/2}\cos{\frac{\theta(\textbf{k})}{2}}\\\\[6.0pt]
e^{+i\phi({\textbf{k})}/2}\sin{\frac{\theta(\textbf{k})}{2}}\end{pmatrix},$
(12)

where the angles $\theta(\textbf{k})$ and $\phi(\textbf{k})$ are the spherical
coordinates angles between the vector formed by $(B_{1},B_{2},B_{3})$ and are
given by

$\displaystyle\tan\phi(\textbf{k})$
$\displaystyle=\frac{B_{2}(\textbf{k})}{B_{1}(\textbf{k})},$ (13)
$\displaystyle\cos\theta(\textbf{k})$
$\displaystyle=\frac{B_{3}(\textbf{k})}{\sqrt{B_{1}^{2}(\textbf{k})+B_{2}^{2}(\textbf{k})+B_{3}^{2}(\textbf{k})}}.$
(14)

To include core orbitals to describe the x-ray interactions in our ultrafast
laser-driven dichroism scheme, we have added the 1$s$ orbitals (as a flat
band) of boron to the Haldane model, where we have assumed that the 1$s$
orbitals are well localized and there is no overlapping with the others. Thus,
the x-ray photon has enough energy to promote an electron in the 1$s$ orbital
(or core band) to the conduction/valence band. Finally, the extended
Hamiltonian can be written as

$\hat{H}=\begin{pmatrix}E_{1s}-E_{f}&|&0&0\\\\[3.0pt] \hline\cr
0&|&H_{11}(\textbf{k})&H_{12}(\textbf{k})\\\\[6.0pt]
0&|&H_{12}^{*}(\textbf{k})&H_{22}(\textbf{k})\end{pmatrix},$ (15)

where the different matrix elements $H_{ij}(\textbf{k})$ are the ones given by
the Haldane Hamiltonian in Eq. (5).

A summary of the used parameters are in Table (1).

Parameter | Value
---|---
Lattice constant, $a$ | 2.5 Å
Band gap, $\Delta$ | 5.9 eV
First-order hopping, $\gamma$ | 3.16 eV
Second-order hopping |
$\phi_{0}$ | $\pi/2$
$t_{s}$ |  $0.567728$ eV
Core-Hole decay, $\Gamma_{ch}$ | 0.00396892 au
Core band energy |  $-188.03074636$ eV
Table 1: Used parameters in the Haldane model.

##  Appendix B Berry Connection

It is useful to represent the Berry connection in a Wannier basis to avoid
discontinuities presented in the eigenstate basis, i.e. the one that
diagonalizes the unperturbed Hamiltonian in the reciprocal space. In this
representation, the Berry connection acquires the very simple form

$\boldsymbol{\xi}^{(w)}=\begin{pmatrix}0&0&\textbf{r}_{1s,2p}\\\ 0&0&0\\\
\textbf{r}_{1s,2p}&0&0\end{pmatrix},$ (16)

on which have used the symmetry and parity of the orbitals (1$s$ for the core
band and 2$p$ for the valence and conduction bands) and the atomic gauge to
simplify the expression presented above. We have defined $\textbf{r}_{1s,2p}$
as the Berry connection between the core orbital (1$s$) and the outer orbital
that forms the chemical bond (2$p$). Note that the coupling between the 1$s$
and 2$p$ orbitals

$\textbf{r}_{1s,2p}=\int
d^{3}\textbf{r}\phi_{1s}^{*}(\textbf{r})\textbf{r}\phi_{2p}(\textbf{r}),$ (17)

is related to the x-ray pulse. Thus, only the out-of-plane component (named
from now $\textbf{r}_{a}$) will be non-zero and is equal to $0.041$ Å.

For representing the Berry connection in the eigenstate basis, it is necessary
to compute a transformation similar to

$\boldsymbol{\xi}(\textbf{k})=\hat{U}(\textbf{k})\boldsymbol{\xi}^{(w)}(\textbf{k})\hat{U}^{\dagger}(\textbf{k})-i\left(\partial_{\textbf{k}}\hat{U}(\textbf{k})\right)\hat{U}^{\dagger}(\textbf{k}),$
(18)

being $\hat{U}(\textbf{k})$ the unitary matrix formed by the eigenfunctions of
the Haldane Hamiltonian spanned over the columns, see (Eq. 15). In this case,
this kind of transformation must be done to include a phase factor that
preserves the gauge covariance Silva2019PRB . Thus, the Berry connection in
the eigenstate basis is then given by

$\displaystyle\boldsymbol{\xi}={\scriptscriptstyle\left(\begin{array}[]{ccc}0&-\textbf{r}_{a}e^{i\phi(\textbf{k})/2}\cos\frac{\theta(\textbf{k})}{2}&\textbf{r}_{a}e^{i\phi(\textbf{k})/2}\sin\frac{\theta(\textbf{k})}{2}\\\\[3.0pt]
-\textbf{r}_{a}e^{-i\phi(\textbf{k})/2}\cos\frac{\theta(\textbf{k})}{2}&-\frac{1}{2}\cos\theta(\textbf{k})\partial_{\textbf{k}}\phi(\textbf{k})&\frac{1}{2}\left(\sin\theta(\textbf{k})\partial_{\textbf{k}}\phi(\textbf{k})-i\partial_{\textbf{k}}\theta(k)\right)\\\\[3.0pt]
\textbf{r}_{a}e^{-i\phi(\textbf{k})/2}\sin\frac{\theta(\textbf{k})}{2}&\frac{1}{2}\left(\sin\theta(\textbf{k})\partial_{\textbf{k}}\phi(\textbf{k})+i\partial_{\textbf{k}}\theta(\textbf{k})\right)&\frac{1}{2}\cos\theta(\textbf{k})\partial_{\textbf{k}}\phi(\textbf{k})\\\
\end{array}\right)}.$

This matrix is a vector-valued matrix. It means that the inter-band matrix
elements, which relate the core with the valence and conduction bands, have
out-of-plane components. In the other hand, the intra-band and inter-band
matrix elements of the valence and conduction bands have in-plane components
coming from the k-derivative.

The Berry curvature of the conduction state, the one took into account for the
Saddle point approach and in the main text, is given by

$\Omega(\textbf{K})=\nabla_{\textbf{K}}\times\boldsymbol{\xi}_{22}(\textbf{K})=\frac{1}{2}\left(\nabla_{\textbf{K}}\cos\theta(\textbf{k})\right)\times\left(\nabla_{\textbf{K}}\phi(\textbf{k})\right)$

##  Appendix C Semiclassical Model for the Absorption

We present here the derivation of Eq. III for the polarization of the system.
This expression is used to obtain a physical interpretation of first-principle
calculations. We start deriving the light-induced dipole response of the
system by using the following ansatz:

$\left|\Psi(t)\right>=\sum_{\textbf{K}}\left[b_{0}(\textbf{K},t)\left|g,\textbf{K}\right>+b_{1}(\textbf{K},t)\left|1,\textbf{K}\right>\right],$
(20)

where
$\left|g,\textbf{K}\right>=\hat{a}_{ch}^{\dagger}(\textbf{K})\left|0\right>$
and
$\left|1,\textbf{K}\right>=\hat{a}_{c}^{\dagger}(\textbf{K})\hat{a}_{ch}(\textbf{K})\left|g,\textbf{K}\right>$,
being $\hat{a}_{\alpha}^{\dagger}(\textbf{k})$ the operator that creates a
particle with quasi-momentum k in the $\alpha$ band. We are not considering
excitations from valence to conduction because the difference in energy is too
small for the electric field.
The time-dependent evolution of the amplitudes can be found by solving the
time-dependent Schrödinger equation. The Hamiltonian is written as
$H(t)=H_{0}+V_{I}(t)$, where $H_{0}$ is the unperturbed Hamiltonian, showed in
the Appendix A, and $V_{I}(t)$ is the coupling between the electric field and
the system in the length gauge. Assuming a sudden excitation at $t=t_{0}$ from
the x-ray pulse (it couples the core with the conduction band),
$\boldsymbol{\varepsilon}_{x}(t)=\boldsymbol{\varepsilon}_{x0}\delta(t-t_{0})$,
then the evolution of the amplitudes is governed by

$\displaystyle b_{0}(\textbf{K},t)$
$\displaystyle=e^{-i\int_{t_{0}}^{t}dt^{\prime}\left[E_{0}(\textbf{K}+\textbf{A}(t^{\prime}))+\boldsymbol{\varepsilon}_{IR}(t^{\prime})\cdot\tilde{\boldsymbol{\xi}}_{00}(\textbf{K}+\textbf{A}(t^{\prime}))\right]}b_{0}(\textbf{K},t_{0}),$
$\displaystyle b_{1}(\textbf{K},t)$
$\displaystyle=e^{-i\int_{t_{0}}^{t}dt^{\prime}\left[E_{1}(\textbf{K}+\textbf{A}(t^{\prime}))-i\Gamma_{ch}/2+\boldsymbol{\varepsilon}_{IR}(t^{\prime})\cdot\tilde{\boldsymbol{\xi}}_{11}(\textbf{K}+\textbf{A}(t^{\prime}))\right]}b_{1}(\textbf{K},t_{0}).$

Using the Haldane Hamiltonian and the Berry connection elements, we define
$\tilde{\boldsymbol{\xi}}_{00}=\boldsymbol{\xi}_{00}$,
$\tilde{\boldsymbol{\xi}}_{11}=\boldsymbol{\xi}_{22}+\boldsymbol{\xi}_{00}$,
$\tilde{\boldsymbol{\xi}}_{01}=\boldsymbol{\xi}_{02}$ and
$E_{0}(\textbf{k})=\epsilon_{ch}(\textbf{k})$,
$E_{1}(\textbf{k})=\epsilon_{c}(\textbf{k})$. $\epsilon_{c}$ and
$\epsilon_{ch}$ are the energies for the conduction and core bands,
respectively. $\Gamma_{ch}$ is the lifetime of the core-excited states and
$\boldsymbol{\varepsilon}_{IR}(t)$ is the external IR laser field. Using the
Ansatz described in Eq, (20), the polarization can be written as

${\bf P}(t)\propto\sum_{{\bf
K}}\tilde{\boldsymbol{\xi}}_{10}(\textbf{K}+\textbf{A}(t))b_{1}^{*}(\textbf{K},t)b_{0}(\textbf{K},t)+c.c.,$
(21)

and this last expression can be rewritten as

${\bf P}(t)\propto\sum_{{\bf
K}}\left|\tilde{\boldsymbol{\xi}}_{10}(\textbf{K}+\textbf{A}(t))\right|e^{-iS(\textbf{K},t,t_{0})}e^{-\Gamma_{ch}(t-t_{0})/2}\;b_{1}^{*}(\textbf{K},t_{0})b_{0}(\textbf{K},t_{0})+c.c.,$
(22)

where we define

$\displaystyle\tilde{\boldsymbol{\xi}}_{10}(\textbf{K})$
$\displaystyle=\left|\tilde{\boldsymbol{\xi}}_{10}(\textbf{K})\right|e^{i\phi_{10}(\textbf{K})},$
$\displaystyle S(\textbf{K},t,t_{0})$
$\displaystyle=\int_{t_{0}}^{t}dt^{\prime}\left[E_{0}(\textbf{K}^{\prime})-E_{1}(\textbf{K}^{\prime})+\boldsymbol{\varepsilon}_{IR}(t^{\prime})\cdot\tilde{\boldsymbol{\xi}}_{00}(\textbf{K}^{\prime})-\boldsymbol{\varepsilon}_{IR}(t^{\prime})\cdot\tilde{\boldsymbol{\xi}}_{11}(\textbf{K}^{\prime})+\frac{d}{dt^{\prime}}\phi_{10}(\textbf{K}^{\prime})\right],$
$\displaystyle=-\int_{t_{0}}^{t}dt^{\prime}\left[\Delta
E(\textbf{K}^{\prime})+\boldsymbol{\varepsilon}_{IR}(t^{\prime})\cdot\Delta\tilde{\boldsymbol{\xi}}(\textbf{K}^{\prime})-\frac{d}{dt^{\prime}}\phi_{10}(\textbf{K}^{\prime})\right],$

and $\textbf{K}^{\prime}=\textbf{K}+\textbf{A}(t^{\prime})$. The last term is
the gauge invariant semiclassical action $S(\textbf{K},t,t_{0})$, in which the
term
$\Delta\tilde{\boldsymbol{\xi}}(\textbf{K})=\tilde{\boldsymbol{\xi}}_{11}(\textbf{K})-\tilde{\boldsymbol{\xi}}_{00}(\textbf{K})$
involves the intra-band matrix elements of the Berry connection. Using the
matrix elements of the Haldane model, the action takes the form

$S(\textbf{K},t,t_{0})=-\int_{t_{0}}^{t}dt^{\prime}\left[\epsilon_{c}(\textbf{K}^{\prime})-\epsilon_{ch}(\textbf{K}^{\prime})+\frac{1}{2}\boldsymbol{\varepsilon}_{IR}(t^{\prime})\cdot\partial_{\textbf{k}}\phi(\textbf{K}^{\prime})\left[\cos\theta(\textbf{K}^{\prime})+1\right]\right].$
(23)

##  Appendix D Derivation of the Saddle Point Equations for the Polarization

We present here the derivation of Eqs. (3) and (4) of the main text, which are
used to simplify the integrals over ${\bf k}$ obtained for the absorption
within the semiclassical model. The SPA enables to solve easily an integral of
the type

$\displaystyle
I_{n}(\sigma)=\int_{C}f(\mathbf{z})e^{\sigma\varphi(\mathbf{z})}\mathrm{d}\mathbf{z}\approx\sum_{s}\left(\frac{2\pi}{\sigma}\right)^{\frac{n}{2}}\frac{f\left(\mathbf{z}_{s}\right)}{\sqrt{\operatorname{det}\left(-\varphi_{n}^{\prime\prime}\left(\mathbf{z}_{s}\right)\right)}}e^{\sigma\varphi\left(\mathbf{z}_{s}\right)}=\sum_{s}\sqrt{\frac{(2\pi
i)^{n}}{\operatorname{det}\left[\phi_{n}^{\prime\prime}\left(\mathbf{z}_{s}\right)\right]}}f\left(\mathbf{z}_{s}\right)e^{i\phi\left(\mathbf{z}_{s}\right)}$

where the points ${\bf z}_{s}$ are those that satisfy $\nabla_{\bf
z}\varphi({\bf z})|_{\bf z_{s}}=0$.

For our formalism, the integrals over ${\bf K}$ and time $t$ in the
calculation of the dipole response in frequency domain can be reduced by SPA.
For this, we calculate the critical points of the action, given by
$\nabla_{\bf K}S(\textbf{K},t,t_{0})=0$ and
$\partial_{t}S(\textbf{K},t,t_{0})=0$. For convenience, we re-write the action
defined above as

$\displaystyle S(\textbf{K},t,t_{0})=-\left[\int_{t_{0}}^{t}\Big{(}\Delta
E(\textbf{K}-{\bf
A}(t)+\textbf{A}(t^{\prime}))+\boldsymbol{\varepsilon}_{IR}(t^{\prime})\cdot\Delta\tilde{\boldsymbol{\xi}}(\textbf{K}-{\bf
A}(t)+\textbf{A}(t^{\prime}))\Big{)}dt^{\prime}\right.$
$\displaystyle\left.+\arg\Big{(}\tilde{\boldsymbol{\xi}}_{10}(\textbf{K})\Big{)}-\arg\Big{(}\tilde{{\xi}}_{10}^{\parallel}\big{(}\textbf{K}-{\bf
A}(t)+{\bf A}(t_{0})\big{)}\Big{)}\right],$

by including the coupling between the core and the conduction band, on which
we explicitly write the inter- and intra-band elements of the Berry connection
and the dependence of each term with respect to the quasi-momentum K and the
IR vector potential at different times. The parallel component
$\tilde{{\xi}}_{10}^{\parallel}$ is taken along the axis of polarization of
the x-ray pulse. In this expression,
$\arg\left(\tilde{\boldsymbol{\xi}}_{10}\right)$ is the phase of the inter-
band Berry connection matrix elements, i.e. $\phi_{10}(\textbf{k})$. We define
the semiclassical action as a vector
($\arg\left(\tilde{\boldsymbol{\xi}}_{10}\right)$) meaning that to calculate
the $\alpha$-th component of the polarization we need the $\alpha$-th
component of the phase.

The critical points of the action must satisfy the equations

$\displaystyle{\bf r}_{d}({\bf K},t,t_{0})-\boldsymbol{\alpha}({\bf
K})+\boldsymbol{\alpha}^{\parallel}(\textbf{K}-{\bf A}(t)+{\bf A}(t_{0}))=0,$
$\displaystyle\epsilon_{c}({\bf k})-\epsilon_{ch}({\bf
k})+\boldsymbol{\varepsilon}_{IR}(t)\cdot{\bf r}_{d}({\bf K},t,t_{0})+$
$\displaystyle\qquad\,+\boldsymbol{\varepsilon}_{IR}(t)\cdot\boldsymbol{\alpha}^{\parallel}(\textbf{K}-{\bf
A}(t)+{\bf A}(t_{0}))=\omega,$

which are found by considering the conditions $\nabla_{\bf K}\left(\omega
t+S(\textbf{K},t,t_{0})\right)=0$ and $\partial_{t}\left(\omega
t+S(\textbf{K},t,t_{0})\right)=0$. In the previous equations we define ${\bf
r}_{d}({\bf K},t,t_{0})$ as the relative distance between the created
electron-hole pair

$\displaystyle{\bf r}_{d}({\bf K},t,t_{0})=\int_{t_{0}}^{t}\Big{(}\nabla_{\bf
K}\epsilon_{c}(\textbf{K}-{\bf A}(t)+\textbf{A}(t^{\prime}))$
$\displaystyle+\bm{\varepsilon}_{IR}(t^{\prime})\times\Omega(\textbf{K}-{\bf
A}(t)+\textbf{A}(t^{\prime}))\Big{)}dt^{\prime}.$

The terms inside the time integral can be interpreted as the instant velocity.
The first term is the ballistic velocity that contains the derivative with
respect to ${\bf k}$ of the energy dispersion, while the second term is the
anomalous velocity that contains the Berry curvature of the conduction band
$\Omega(\textbf{K})=\nabla_{\textbf{K}}\times\boldsymbol{\xi}_{22}(\textbf{K})$.
Also, we define other variables, we call them Berry distances, that purely
depend on the intra- and inter-band Berry connections

$\displaystyle\boldsymbol{\alpha}({\bf K})$
$\displaystyle=\bm{\xi}_{22}(\textbf{K})-\nabla_{\bf
K}[\arg\Big{(}\bm{\xi}_{20}(\textbf{K})\Big{)}],$
$\displaystyle\boldsymbol{\alpha}^{\parallel}({\bf K})$
$\displaystyle={\bm{\xi}}_{22}(\textbf{K})-\nabla_{\bf
K}[\arg\Big{(}{\boldsymbol{\xi}}^{\parallel}_{20}(\textbf{K})\Big{)}].$

These expressions have been obtained by using that
$\nabla\left(\textbf{F}\cdot\textbf{G}\right)=\textbf{F}\times\left(\nabla\times\textbf{G}\right)+\textbf{G}\times\left(\nabla\times\textbf{F}\right)+\left(\textbf{F}\cdot\nabla\right)\textbf{G}+\left(\textbf{G}\cdot\nabla\right)\textbf{F}$.
Note that that the Berry distances modify the condition for the recombination
of the electron-hole pair. If those are zero, then we have the condition ${\bf
r}_{d}({\bf K},t,t_{0})=0$, and it can be interpreted as the creation of the
electron-hole pair at $t=t_{0}$, when the attosecond pulse excites the system,
and the propagation of both until their distance is zero. But the Berry
structure of the material makes this process more complex, and the Berry
distances modify this condition.

###### Acknowledgements.

J.F.P.M, G.C., M.M., and A.P. acknowledge Comunidad de Madrid through TALENTO
grant refs. 2017-T1/IND-5432 and 2021-5A/IND-20959, and the Spanish Ministry
of Science, Innovation and Universities & the State Research Agency through
grants refs. PID2021-126560NB-I00 and CNS2022-135803 (MCIU/AEI/FEDER, UE), and
the ”María de Maeztu” Programme for Units of Excellence in R&D
(CEX2023-001316-M), and FASLIGHT network (RED2022-134391-T), and computer
resources and assistance provided by Centro de Computación Científica de la
Universidad Autónoma de Madrid (FI-2021-1-0032), Instituto de Biocomputación y
Física de Sistemas Complejos de la Universidad de Zaragoza (FI-2020-3-0008),
and Barcelona Supercomputing Center (FI-2020-1-0005, FI-2021-2-0023,
FI-2021-3-0019). This publication is based upon work from COST Action NEXT,
CA22148 supported by COST (European Cooperation in Science and Technology). M.
Malakhov’s work also carried out within the state assignment of Ministry of
Science and Higher Education of the Russian Federation (theme “Quantum” No.
122021000038-7). E.P. acknowledges Royal Society funding under URF\R1\211390,
RF\ERE\210255 and RF\ERE\231081. ICFO and ex-ICFO co-authors acknowledge
Europea Research Council AdG NOQIA; MCIN/AEI (PGC2018-0910.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, Plan National STAMEENA PID2022-139099NB, I00, project
funded by MCIN/AEI/10.13039/501100011033 and by the “European Union
NextGenerationEU/PRTR” (PRTR-C17.I1), FPI); QUANTERA MAQS PCI2019-111828-2);
QUANTERA DYNAMITE PCI2022-132919, QuantERA II Programme co-funded by European
Union’s Horizon 2020 program under Grant Agreement No 101017733); Ministry for
Digital Transformation and of Civil Service of the Spanish Government through
the QUANTUM ENIA project call - Quantum Spain project, and by the European
Union through the Recovery, Transformation and Resilience Plan -
NextGenerationEU within the framework of the Digital Spain 2026 Agenda;
Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social
Fund FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2023-3-0024); Funded by the
European Union. Views and opinions expressed are, however, those of the
author(s) only and do not necessarily reflect those of the European Union,
European Commission, European Climate, Infrastructure and Environment
Executive Agency (CINEA), or any other granting authority. Neither the
European Union nor any granting authority can be held responsible for them
(HORIZON-CL4-2022-QUANTUM-02-SGA PASQuanS2.1, 101113690, EU Horizon 2020 FET-
OPEN OPTOlogic, Grant No 899794), EU Horizon Europe Program (This project has
received funding from the European Union’s Horizon Europe research and
innovation program under grant agreement No 101080086 NeQSTGrant Agreement
101080086 — NeQST); ICFO Internal “QuantumGaudi” project; European Union’s
Horizon 2020 program under the Marie Sklodowska-Curie grant agreement No
847648; “La Caixa” Junior Leaders fellowships, La Caixa” Foundation (ID
100010434): CF/BQ/PR23/11980043.

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