# NU HLT at CMCL 2022 Shared Task:
Multilingual and Crosslingual Prediction of Human Reading Behavior in
Universal Language Space

Joseph Marvin Imperial
Human Language Technology Lab (NU HLT)
National University
Manila, Philippines
<EMAIL_ADDRESS>

###### Abstract

In this paper, we present a unified model that works for both multilingual and
crosslingual prediction of reading times of words in various languages. The
secret behind the success of this model is in the preprocessing step where all
words are transformed to their universal language representation via the
International Phonetic Alphabet (IPA). To the best of our knowledge, this is
the first study to favorably exploit this phonological property of language
for the two tasks. Various feature types were extracted covering basic
frequencies, n-grams, information theoretic, and psycholinguistically-
motivated predictors for model training. A finetuned Random Forest model
obtained best performance for both tasks with 3.8031 and 3.9065 MAE scores for
mean first fixation duration (FFDAvg) and mean total reading time (TRTAvg)
respectively111https://github.com/imperialite/cmcl2022-unified-eye-tracking-
ipa.

##  1 Introduction

Eye movement data has been one of the most used and most important resource
that has pushed various interdisciplinary fields such as development studies,
literacy, computer vision, and natural language processing research into
greater heights. In a technical point of view, correctly determining
theoretically grounded and cognitively plausible predictors of eye movement
will allow opportunities to make computational systems leveraging on these
properties to be more human-like Sood et al. (2020).

Common human reading prediction works make use of the standard Latin alphabet
as it is internationally used. However, investigating eye movement and reading
patterns in other non-Anglocentric writing systems such as Chinese and Bengali
is as equally as important Share (2008); Liversedge et al. (2016).
Fortunately, there is a growing number of previous works exploring
multilinguality in eye tracking prediction both in data collection and novel
prediction approaches. The study of Liversedge et al. (2016) was the first to
explore potential crosslinguality of Chinese, English and Finnish which differ
in aspects of visual density, spacing, and orthography to name a few. The
results of the study favorably support possible universality of representation
in reading. In the same vein, Hollenstein et al. (2021) was the first to try
use of large finetuned multilingual language models like BERT Devlin et al.
(2019) and XLM Conneau and Lample (2019) in a crosslingual setting to predict
eye tracking features across English, Dutch, German, and Russian. Data-wise,
the published works of Siegelman et al. (2022) for MECO, Pynte and Kennedy
(2006) for the Dundee corpus, and Cop et al. (2017) for GECO have made
significant impact in the field where they covered curation and collection of
eye-tracking corpus for other languages in addition to English.

##  2 Task Definition and Data

The CMCL 2022 Shared Task Hollenstein et al.
(2022)222https://cmclorg.github.io/shared_task describes two challenges:
predicting eye-tracking features in a multilingual and crosslingual setup. The
eye movement dataset for this Shared Task contains sentences written in six
languages: Mandarin Chinese Pan et al. (2021), Hindi Husain et al. (2015),
Russian Laurinavichyute et al. (2019), English Luke and Christianson (2018);
Hollenstein et al. (2018, 2020), Dutch Cop et al. (2017), and German Jäger et
al. (2021). The mean first fixation duration (FFDAvg) and mean total reading
time (TRTAvg) as well as their corresponding standard deviations (FFDStd and
TRTStd) are the four main eye-tracking features that need to be predicted by
the participants through proposed computational means. For the multilingual
task, the training, validation, and testing datasets conform to the identified
six languages. While for the crosslingual task, a surprise language (Danish)
is provided as the test dataset.

Figure 1: The proposed unified approach to multilingual and crosslingual human
reading pattern prediction in universal language space via IPA.

##  3 Eye-Tracking Prediction in Universal Language Space

The proposed solution in this work is inspired by both classical and recent
previous works in speech recognition systems Schultz and Waibel (1998, 2001);
Dalmia et al. (2019) with multilingual and crosslingual capabilities through
the transformation of words or similar sounding units in one global shared
space using the International Phonetic Alphabet (IPA). This functionality
allows models to generalize and adapt parameters to new languages while
maintaining a stable vocabulary size for character representation. By
definition, the IPA contains 107 characters for consonants and vowels, 31 for
diacritics for modifying said consonants and vowels, and 17 signs to emphasize
suprasegmental properties of phonemes such as stress and intonation
Association et al. (1999).

Figure 1 describes the unified methodology used for tackling both the
multilinguality and crosslinguality challenge of the Shared Task. The backbone
of this proposed solution lies with the phonetic transcription preprocessing
step to convert the raw terms from the data written in Mandarin Chinese,
Hindi, Russian, English, Dutch, and German to their IPA form. We used Epitran
by Mortensen et al. (2018) for this process. The surprise language for the
crosslingual task, Danish, is not currently supported by Epitran. We instead
resorted to use Automatic Phonetic
Transcriber333http://tom.brondsted.dk/text2phoneme/, a paid transcription
service that caters the Danish language. The transcription cost of the Danish
test data is €15.

###  3.1 Feature Extraction

After obtaining the phonetic transcriptions, a total of fourteen features
based on various types were extracted spanning general frequencies, n-grams,
based on information theory, and based on motivations from psycholinguistics.

Frequency and Length Features. The simplest features are frequency and length-
based predictors. Studies have shown that the length of words correlate with
fixation duration as long words would obviously take time to read Rayner
(1977); Hollenstein and Beinborn (2021). For this study, we extracted the (a)
word length (word_len), (b) IPA length (ipa_len), (c) IPA vowels count per
term (ipa_count), and (d) normalized IPA vowel count per term over length
(ipa_norm).

N-Gram Features. Language model-based features is a classic in eye-tracking
prediction research as they capture word probabilities through frequency. We
extracted raw count of unique n-grams per word (bigram_count, trigram_count),
raw count of total n-grams per term (bigram_sum, trigram_sum), and normalized
counts over word length (bigram_norm, trigram_norm) for character bigrams and
trigrams in IPA form guided by the general formula for n-gram modelling below:

$P(w_{n}\mid
w_{n-N+1}^{n-1})=\frac{C(w_{n-N+1}^{n-1}w_{n})}{C(w_{n-N+1}^{n-1})}$ (1)

Psycholinguistially-Motivated Features. Features with theoretical grounding
are more practical to use when invetigating phenomena in human reading. In
line with this, we extracted two psycholinguistically-motivated features:
imageability and concreteness. When reading, humans tend to visualize words
and scenarios as they are formed in context. This measure of ease of how words
or phrases can easily be visualized in the min from a verbal material is
quantified as imageability Lynch (1964); Richardson (1976). On the other hand,
concreteness is a measure of lexical organization where words are easily
perceived by the senses. In the example of Schwanenflugel et al. (1988), words
such as chair or computer are better understood than abstract words like
freedom. Words with high concreteness scores are better recalled from the
mental lexicon than abstract words as they have better representation in the
imaginal system Altarriba et al. (1999). We use these two features as we posit
that the visualization and retrieval process of imageability and concreteness
respectively can contribute to the reading time in milliseconds.

For this task, we used the crosslingual word embedding-based approximation for
all the seven languages present in the dataset from the the work of Ljubešić
et al. (2018).

Information Theoretic Features. Features inspired by information theory such
as the concept of surprisal have thoroughly used in human reading pattern
prediction Hale (2001); Levy (2008); Demberg and Keller (2008, 2009); Goodkind
and Bicknell (2018). Surprisal describes that processing time of a word to be
read is proportional to its negative log based on a probability given by
context as shown below:

$\textrm{surprisal}(w_{i})=-\textrm{log}_{2}\>P(w_{i}\mid w_{1}...w_{i-1})$
(2)

Thus, if a word is more likely to occur in its context, it is read more
quickly Shannon (1948). For this task, since words are converted to a
universal language space, the correct terminology in this case is bits per
phoneme or phonotactic complexity as coined by Pimentel et al. (2020).

While surprisal quantifies the word’s predictability or processing cost during
reading, we also obtain the entropy $H$ of each word $x$ from the corpus. The
entropy quantifies the expected value of information from an event as shown in
the formula below:

$H(X)=-\sum_{i=1}^{n}\>(\frac{count_{i}}{N})\>\textrm{log}_{2}\>(\frac{count_{i}}{N})$
(3)

where $count_{i}$ is the count of character $n_{i}$ and each word $N$ consists
of $n$ characters. With this measure, a higher entropy score entails higher
uncertainty for a word, thus, leading to increased reading time at the
millisecond level.

###  3.2 Model Training Setup

We used four machine learning algorithms via WEKA Witten and Frank (2002) for
modelling the features with FFDAvg and TRTAvg: linear regression (LinReg),
multilayer perceptron (MLP), random forest (RF), and k-Nearest Neighbors
(kNN). We only used the finetuned RF model for the prediction of FFDAvg and
TRTAvg. Meanwhile, FFDStd and TRTStd are obtained by using the top models of
all the four algorithms, re-running them to get FFDAvg and TRTAvg, and
calculating the standard deviation. For TRTAvg, we added the predicted FFDAvg
from the best model as an additional feature as we posit that the first
fixation duration is a contributor to the overall reading time.

Model | FFDAvg | TRTAvg
---|---|---
MAE | RMSE | MAE | RMSE
LinReg (k=10, M5)*${\dagger}$ | 5.2361 | 6.7267 | 4.3419 | 7.0546
LinReg (k=10, greedy) | 5.2361 | 6.7267 | 4.3420 | 7.0545
LinReg (k=10, none) | 5.2363 | 6.7274 | 4.3429 | 7.0594
MLP (k=10, lr=0.005, m=0.2)*${\dagger}$ | 4.9898 | 6.4169 | 4.1744 | 6.2140
MLP (k=10, lr=0.5, m=0.2) | 6.7916 | 8.3791 | 4.8475 | 7.0840
MLP (k=10, lr=0.005, m=0.002) | 5.0018 | 6.4299 | 4.1862 | 6.2177
MLP (k=10, lr=0.5, m=0.002) | 6.4447 | 8.0110 | 4.9528 | 6.9668
MLP (k=10, lr=0.0005, m=0.0002) | 5.5024 | 7.0474 | 4.2956 | 6.3823
RF (k=10, iters = 100)* | 3.8031 | 5.2750 | 3.9600 | 5.8446
RF (k=10, iters = 100, 50% feats) | 3.8045 | 5.2766 | 3.9094 | 5.8015
RF (k=10, iters = 100, 75% feats${\dagger}$) | 3.8056 | 5.2762 | 3.9065 | 5.8006
kNN (k=10, nn=5, dist=euc)* | 4.3335 | 5.9651 | 4.2953 | 6.3741
kNN (k=10, nn=10, dist=euc) | 4.4263 | 6.0133 | 4.2053 | 6.2436
kNN (k=10, nn=20, dist=euc)${\dagger}$ | 4.5646 | 6.1284 | 4.1793 | 6.2432
Table 1: Results of predicting mean first fixation duration (FFDAvg) and mean
total reading time (TRTAvg) using hyperparameter-tuned traditional supervised
models. The tuned Random Forest (RF) model achieved the best performance which
was used for both tasks of multilingual and crosslingual prediction. Top
performing models from the four algorithm class were used for predicting the
held-out test data to get the standard deviation of FFDAvg (*) and TRTAvg
(${\dagger}$).

##  4 Results

Table 1 describes the main results of the experiments for predicting FFDAvg
and TRTAvg using multiple finetuned supervised techniques evaluated through
mean absolute error (MAE) and root mean squared error (RMSE). As mentioned
previously, since the methodology used in this study cuts across multilingual
and crosslingual tasks, the results reported in this applied are applicable to
both. From the Table, the RF models outperformed the other three models in
predicting FFDAVg and TRTAvg using 100% and 75% random selected features
respectively and across 100 iterations. The RF model’s effectivity can be
attributed to its structure of multiple decision trees which normalize
overfitting Ho (1995). Following RF in performance is kNN using Euclidean
distance observing the same pattern as RF with different hyperparameter values
such as 5 and 20 for the nearest neighbor for predicting FFDAvg and TRTAvg. On
the other hand, both LinReg and MLP have no improvements regardless of
hyperparameter values. For LinReg, using an M5 feature selection only provides
extremely minor improvement in performances for FFDAvg and TRTAvg prediction.
For MLP, using default values in WEKA for momentum and learning rate obtained
the best performance similarly for for FFDAvg and TRTAvg prediction.

FFDAvg | TRTAvg
---|---
bigram_norm | -0.1751 | FFDAvg | 0.8068
trigram_norm | -0.1393 | bigram_count | 0.2219
word_len | -0.1334 | trigram_count | 0.2156
bigram_sum | -0.1304 | phonetic_comp | -0.2107
trigram_sum | -0.1101 | ipa_ent | 0.1925
imageability | 0.1101 | ipa_len | 0.1921
concreteness | 0.1044 | trigram_norm | -0.1886
Table 2: Top 7 predictors for FFDAvg and TRTAvg with the highest correlation
coefficients.

###  4.1 Feature Importance

Viewing the results in a correlation analysis perspective, Table 2 shows the
top 50% of the predictors, total 7, which are significantly correlated with
FFDAvg and TRTAvg respectively. Only one predictor is common for both values,
the normalized trigrams in IPA space which is fairly high in FFDAvg along with
normalized bigrams than in TRTAvg. This may hint that normalized n-gram
features may be plausible features of eye movement only for first passes over
the word and not with the total accumulated time of fixations. Likewise, the
psycholinguistically-motivated features, imageability and concreteness, were
only seen in the FFDAvg section as well proving their potential plausibility
for the same observation. All the length-based features such as word, IPA,
bigram, and trigram-based counts were considered as top predictors for FFDAvg
and TRTAvg. This unsurprisingly supports the results from the classical work
of Rayner (1977) on correlation of lengths with fixations. Lastly, the strong
correlation of first fixation duration with the total reading time with a
score of $r$ = 0.8068 proves the theoretical grounding of the proposed
methodology as stated in Figure 1 albeit in post-hoc.

##  5 Conclusion

Precise eye movement datasets in multiple languages are considered one of the
most important contributions that benefit various interdisciplinary fields
such as psycholinguistics, developmental studies, behavioral studies, computer
vision, and natural language processing. In this paper, we present a novel
method of transforming multilingual eye-tracking data (English, Mandarin,
Hindi, Russian, German, Dutch, and Danish) to their IPA equivalent, enforcing
a single vocabulary space which allows competitive results for both
multilingual and crosslingual tasks in a regression analysis setup. Future
directions of this paper can explore more cognitively and theoretically
plausible features that can be extracted as well as deeper interpretation
studies of the predictive models trained.

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# Ultrafast metal-to-insulator switching in a strongly correlated system

Francesco Grandi  Department of Physics, University of Erlangen-Nürnberg,
91058 Erlangen, Germany  Martin Eckstein  Department of Physics, University of
Erlangen-Nürnberg, 91058 Erlangen, Germany

(August 27, 2024)

###### Abstract

Light-manipulation of correlated electronic phases in solids offers the
tantalizing prospect of realizing electronic devices operating at the
ultrafast time-scale. In this context, the experimental realization of non-
equilibrium transitions from a metal to a band or Mott insulator has shown to
be particularly elusive. Using dynamical mean-field theory, we study a simple
model representing the main physical properties of the oxygen-enriched
compound LaTiO3+x. By properly optimizing the photo-doping of electrons from a
low-energy band into the valence states of the system, we show it is possible
to induce a valence transition from a correlated metallic state to a Mott
insulator at ultrashort time scales and to contain the heating during this
process, with the final non-thermal valence insulator having almost the same
effective temperature of the starting metal.

###### pacs:

##  I Introduction

Transition metal compounds (TMC) are an intriguing class of materials in
condensed matter physics due to the conspiracy of charge, spin, orbital and
lattice degrees of freedom Khomskii (2014). Many representatives of this group
show a metal-to-insulator transition upon changing intensive variables such as
the temperature $T$, the external pressure or the chemical potential Morin
(1959); Imada  _et al._ (1998). The equilibrium insulator-to-metal phase
transition is typically realized upon increasing $T$ if it involves some form
of symmetry breaking (lattice dimerization, orbital order, antiferromagnetism,
etc.) Torrance  _et al._ (1992); McWhan  _et al._ (1973); Park  _et al._
(2013); Frandsen  _et al._ (2016). On the other hand, if both metal and
insulator have the same symmetry, the transition to the insulator might occur
with increasing instead of decreasing $T$, simply because the entropy of the
Mott state can be higher than that of the metallic state. For example, after
locking the charge, the remaining spin degree of freedom in a spin-disordered
Mott insulator implies a high entropy $S_{\text{Mott}}\sim\ln\left(2\right)$
per lattice site Georges  _et al._ (1996). In contrast, the entropy in a Fermi
liquid vanishes for low temperatures like $S_{FL}\left(T\right)\propto T$ so
that the transition to a Mott state can be driven by a temperature increase if
the spin order remains frustrated.

While some of the equilibrium properties of these materials are still not
fully understood, a large interest in their non-equilibrium behavior has grown
recently Giannetti  _et al._ (2016); Basov  _et al._ (2017). In particular,
the tantalizing prospect of realizing a Mottronic switch between insulating
and metallic states has attracted much attention Guiot  _et al._ (2013); Mazza
_et al._ (2016); Li  _et al._ (2017); Ronchi  _et al._ (2020); del Valle  _et
al._ (2019); Janod  _et al._ (2015). Even if ultrafast insulator-to-metal
phase transitions have been reported in several materials Cavalleri  _et al._
(2004); Stojchevska  _et al._ (2014); Wegkamp  _et al._ (2014); Lantz  _et
al._ (2017); Mogunov  _et al._ (2019), the opposite phase change is harder to
achieve. Since a laser excitation typically increases the effective
temperature and the total entropy of a system, it is generally easier to move
from a symmetry broken insulating state to a metallic one. Nevertheless, one
can imagine several pathways to laser-induce the opposite transition from
metallic to insulating behavior if the final insulating state is of the Mott
kind. Firstly, with the entropic argument given above, the metal-to-insulator
phase transition may be induced in some specific cases simply by a laser-
induced heating of the electrons. An alternative scenario is that a photo-
induced transfer of charges from other bands into a previously doped valence
band can bring the latter to commensurate filling. While this will not make
the system insulating as a whole (because holes are generated in other
previously filled bands), Mott physics can arise in the valence band. In the
following, we will term this transient state a valence Mott insulator. Again,
the main difficulty is that a rapid charge transfer typically leads to a
strong increase of the entropy, working against correlation effects. The
question is, therefore, whether suitably optimized protocols can be designed
to generate such a valence Mott state within short times.

Even if the entropy of the system as a whole is expected to grow after the
pulse, the entropy might transiently decrease in a given subsystem, leaving
the remainder at high entropy. This concept has been explored theoretically
during the past years Denny  _et al._ (2015); Nava  _et al._ (2018); Fabrizio
(2018); Werner and Murakami (2020) as a pathway to induce or enhance
electronic orders. Recently, it has been observed that also a charge transfer
between different bands can realize this entropy redistribution Werner  _et
al._ (2019). By transferring (“evaporating”) holes from the valence band to
localized core levels, entropy is reshuffled between the valence and the core
states, with the latter having practically zero entropy before the arrival of
the pulse. In Werner  _et al._ (2019), a single-band strongly correlated metal
is considered as the initial state. By photo-doping from a narrow core level,
the occupation of the valence band becomes almost commensurate with the
lattice, and the effective temperature of the valence subsystem can decrease
and eventually reach values below the equilibrium antiferromagnetic Neél
temperature. This mechanism has therefore been termed cooling by photo-doping.
With a similar mechanism, an enhancement of the $\eta$-paired superconducting
susceptibility in the half-filled single-band Hubbard model has been predicted
Li  _et al._ (2020).

In this paper, we aim to investigate the cooling by photo-doping procedure in
a multi-band setup. While the simulations will be based on a simplified two-
band model in order to make them feasible, this model is designed to capture
the important properties of the $3$d1 perovskite series. The latter includes
the metallic compounds BaVO3, SrVO3, and CaVO3, and the Mott insulators LaTiO3
and YTiO3 Nishimura  _et al._ (2014); Bannikov (2016); Pavarini  _et al._
(2005, 2004). In all these materials, the transition metal atoms sit at the
centre of a corner-sharing O6 octahedron, while the cations occupy the
interstitial space between the octahedra. The cubic crystal field splits the
$3$d manifold in two higher energy $\text{e}_{\text{g}}$ and three lower
energy $\text{t}_{2\text{g}}$ levels, filled with a single electron. We
particularly focus on LaTiO3, the first insulating member of the series. In
this compound, a distortion of the GdFeO3 kind lowers the lattice symmetry to
orthorhombic and further splits the $\text{t}_{2\text{g}}$ states into a lower
energy $\text{a}_{\text{g}}$ level and a higher energy
$\text{e}_{\text{g}}^{\prime}$ doublet Ishida and Liebsch (2008). The material
has a Mott-gap of $\sim 0.2\ \mathtt{eV}$, which is small compared to the
bandwidth of approximately $W\sim 2.1\ \mathtt{eV}$ Arima  _et al._ (1993);
Okimoto  _et al._ (1995). A broad band, mainly derived from oxygen p-orbitals,
lies well below the $\text{t}_{2\text{g}}$ manifold El-Mellouhi  _et al._
(2013). Below temperature $T_{N}\sim 146\ \mathtt{K}$, the system undergoes a
transition from a paramagnet to G-type antiferromagnetism Komarek  _et al._
(2007).

The small value of the energy gap observed in this material suggests its
proximity to a metal-insulator phase boundary, making it interesting for
Mottronics Kubota  _et al._ (2004). Moreover, this compound shows a doping-
driven phase transition by growing it under oxygen atmosphere so that the
high-temperature non-stoichiometric compound LaTiO${}_{3+\text{x}}$, with
$x\sim 0.04$, is a correlated metal with well defined quasi-particle
properties Scheiderer  _et al._ (2018); Okimoto  _et al._ (1995); Taguchi  _et
al._ (1999); Lichtenberg  _et al._ (1991). In the following, we explore the
possibility to drive the d-valence bands in the oxygen-enriched compound
LaTiO${}_{3+\text{x}}$ from the metallic to the Mott insulating state by
photo-doping electrons into the energy states around the Fermi level from the
lower-energy p-derived band. In other words, the oxygen p-bands are supposed
to take the role of the core bands in the cooling by photo-doping scenario, in
spite of these bands being certainly not localized, but in contrast even
broader than the valence bands. Depending on the form of the excitation pulse,
we indeed find that the valence band state can be turned into a Mott insulator
with an effective temperature almost equal to the one of the starting metal on
ultrafast timescales due to an entropy increase in the oxygen bands, similar
to the cooling by photo-doping protocol.

The article is organized as follows. In Sec. II, we define a minimal model for
the description of the photo-doping process in LaTiO${}_{3+\text{x}}$ (Sec.
II.1), discuss the excitation protocol (Sec. II.2) and the solution of the
model using non-equilibrium dynamical mean-field theory (DMFT) (Sec. II.3). In
Sec. III, we present our results, showing the properties of the charge
transfer process (Sec. III.1) and the characteristics of the final state for
several pulse lengths and optimized pulse parameters (Sec. III.2). Finally,
Sec. IV is devoted to concluding remarks.

##  II Model and Method

###  II.1 Two-band model

This section aims to define a minimal model that can describe the photo-
induced metal-to-valence insulator transition in the oxygen-enriched strongly
correlated compound LaTiO3+x. As we already commented, the parent compound
LaTiO3 has the three $\text{t}_{2\text{g}}$-derived states (the lower energy
$\text{a}_{\text{g}}$ level and the $\text{e}_{\text{g}}^{\prime}$ doublet) at
the valence filled with a single electron. Instead of three bands we consider
two, where the degeneracy between the states is lifted by a crystal field
interaction that mimics the effect of the GdFeO3-like distortion. This way, we
neglect the double degeneracy of the $\text{e}_{\text{g}}^{\prime}$ states,
which should however not alter the qualitative picture we aim to present
because these states are mostly empty. Indeed, having two bands is already
enough to explore the role of orbital imbalance in this system and to describe
the nearly inter-band gap predicted in several theoretical analysis Ishida and
Liebsch (2008); Pavarini  _et al._ (2005). In our two-band model, the undoped
Mott insulator corresponds to quarter filling (one electron per site), and the
total filling per site will be fixed to $0.92$ for the doped compound. In
LaTiO${}_{3+\text{x}}$, a valence occupation $0.92$ would correspond to
oxygen-doping $x=0.04$, enough for the material to show well-defined
correlated metal properties.

The Hamiltonian of the two-band model is given by the sum of a local
($H_{\text{loc}}$) and a non-local ($H_{\text{hop}}$) contribution,

$H=H_{\text{hop}}+H_{\text{loc}}\;.$ (1)

The local part is given by

$\begin{split}H_{\text{loc}}=&-\frac{\Delta}{2}\sum_{\bm{R}}\left(n_{\bm{R},1}-n_{\bm{R},2}\right)-\mu\sum_{\bm{R}}n_{\bm{R}}\\\
&+U\sum_{\bm{R},a}n_{\bm{R},a,\uparrow}n_{\bm{R},a,\downarrow}+U^{\prime}\sum_{\bm{R}}n_{\bm{R},1}n_{\bm{R},2}\;,\end{split}$
(2)

where the index $a=1,2$ labels orbitals; although this is a simplified model,
we will refer to orbital $1$ and $2$ as $\text{a}_{\text{g}}$ and
$\text{e}_{\text{g}}^{\prime}$, respectively. The spin takes values
$\sigma=\uparrow,\downarrow$; $n_{\bm{R}}=\sum_{a,\sigma}n_{\bm{R},a,\sigma}$
is the occupation on site $\bm{R}$, $\Delta$ is the crystal field splitting
coupled to the orbital imbalance
$\tau^{z}_{\bm{R}}=\frac{1}{2}\left(n_{\bm{R},1}-n_{\bm{R},2}\right)$, $U$ is
the intra-band Hubbard interaction, and $U^{\prime}$ is the inter-band Hubbard
repulsion. The Hund’s coupling is relevant only for doubly occupied states
and, moreover, is expected to be small compared to the bandwidth Grandi  _et
al._ (2020), so its contribution is neglected. The chemical potential $\mu$ is
used to fix the average occupation $\langle n_{\bm{R}}\rangle=0.92$ on each
site.

The hopping part in Eq. (1) is written as

$H_{\text{hop}}=-\sum_{\langle\bm{R},\bm{R^{\prime}}\rangle,\sigma}\sum_{a,a^{\prime}}v_{\bm{R},a;\bm{R^{\prime}},a^{\prime}}\left(c^{\dagger}_{\bm{R},a,\sigma}c_{\bm{R^{\prime}},a^{\prime},\sigma}+\text{H.c.}\right)\;.$
(3)

The operator $c_{\bm{R},a,\sigma}$ ($c^{\dagger}_{\bm{R},a,\sigma}$)
annihilates (creates) an electron on site $\bm{R}$ in orbital $a$ with spin
$\sigma$, and the summation extends just over nearest-neighbor sites. The
hopping matrix $v_{\bm{R},a;\bm{R^{\prime}},a^{\prime}}$ allows both intra-
and inter-band charge transfers, so that the occupation of each band is not
individually conserved. We take the intra-band hopping
$v_{\bm{R},a;\bm{R^{\prime}},a}=v/\sqrt{z}$ equal for all nearest neighbors
($z$ is the coordination number). The inter-band hopping
$v_{\bm{R},1;\bm{R^{\prime}},2}=v_{\bm{R},2;\bm{R^{\prime}},1}$ has absolute
value $v^{\prime}/\sqrt{z}$ and phases such that

$\sum_{\bm{R}^{\prime}}v_{\bm{R},1;\bm{R^{\prime}},2}=0\;,$ (4)

so that the two bands do not mix at the $\Gamma$-point. Below, we will solve
the model on an infinitely-coordinated Bethe lattice, mainly because this
implies a closed form of the DMFT self-consistency (as described in Sec. II.3)
and therefore allows to perform simulations up to longer times. It can be
expected that the simulations on the Bethe lattice give qualitatively the same
physics as on a cubic lattice if the bandwidth and energy gaps are properly
matched and if the local environment [Eq. (4)] of every site is correctly
represented.

The parameters in Eq. (1) are chosen so that the bandwidth and energy gaps are
comparable to the LaTiO3-derived compound: the energy scale is set to the
half-bandwidth, which is $2v$, and time is measured in units of
$\hbar/\left(2v\right)$. Because the bandwidth in LaTiO3 is $2.1\ \mathtt{eV}$
Ishida and Liebsch (2008); Pavarini  _et al._ (2005), we can almost read the
energy units in our plots in electronvolts. Further, we fix the crystal field
to $\Delta=0.15$ Haverkort  _et al._ (2005); Pavarini  _et al._ (2004);
Sclauzero  _et al._ (2016) and the inter-band hopping $v^{\prime}=0.15$
Pavarini  _et al._ (2005). Finally, we set the values of the intra- and inter-
orbital Hubbard repulsion to $U=3.10$ and $U^{\prime}=2.294$, respectively, to
reproduce experimentally obtained energy gap of the quarter-filled (undoped)
system, as well as the inter-band nature of the gap. This way, the ratio
$U^{\prime}/U$ has the same value as obtained in more sophisticated
calculations Haverkort  _et al._ (2005). The initial inverse temperature of
the problem is $\beta=10$, well above the antiferromagnetic critical
temperature.

Figure 1:  Spectral functions for the two orbitals $A_{1}\left(\omega\right)$
(red) and $A_{2}\left(\omega\right)$ (green), and the non-interacting $p$-band
$A_{p}\left(\omega\right)$ (black), together with their occupations (shaded
areas) for the undoped (panel (a)) and the doped (panel (b)) systems, where we
consider an average filling per site $n=1$ and $n=0.92$, respectively. The
effect of a finite hopping frequency $\Omega$ is to shift the fermionic bath
up in energy, as schematically shown by the arrow and the dashed line in panel
(b). The frequencies $\omega$ are shown with respect to the Fermi level of the
system. The values of the double occupancies and of the orbital imbalance of
the systems are given by $d\sim 0.0300$ and $\tau_{z}\sim 0.2197$ for $n=1$,
and by $d\sim 0.0234$ and $\tau_{z}\sim 0.2077$ for $n=0.92$.

Figure 1(a) shows the orbitally-resolved spectral functions of the model
obtained using DMFT (Sec. II.3) at $\beta=10$. The system features a lower
band of mostly $a_{g}$-character, a Hubbard gap of about $0.2$, and an upper
Hubbard band (UHB) which is of predominantly $e_{g}^{\prime}$ character at the
lower edge. Upon hole-doping (Fig. 1b), a quasiparticle band is formed at the
lower Hubbard band edge. In addition to the $d$-bands, we add an additional
low-lying noninteracting band with semi-elliptic density of states,
representing the $p$-band manifold (depicted by $A_{p}(\omega)$ in Fig. 1).

###  II.2 Excitation pulse

The system is excited with a time-dependent hybridization $v_{pd}(t)=AE(t)$
between the $d$ and $p$ bands, representing a dipolar transition driven by the
electric field of a laser $E(t)$, with the dipolar matrix element $A$;
choosing $A=1$ in the following sets the unit of the electric field. As in
Ref. Werner  _et al._ (2019), we will investigate the use of chirped laser
pulses Strickland and Mourou (1985) for controlling the system. The functional
form for the electric field is taken as

$E(t)=E_{0}f_{\text{env}}(t)\text{sin}\left(\Omega(t)t\right),$ (5)

characterized by the amplitude $E_{0}$, a linearly chirped frequency

$\Omega\left(t\right)=\Omega_{\text{st}}+\frac{\Omega_{\text{end}}-\Omega_{\text{st}}}{t_{\text{pulse}}}t$
(6)

with pulse duration $t_{\text{pulse}}$ and start (end) frequency
$\Omega_{\text{st}}$ ($\Omega_{\text{end}}$), and a smooth envelope of
duration $t_{\text{pulse}}$,
$f_{\text{env}}(t)=\left[(1+e^{\alpha(t-t_{\text{pulse}})+6})(1+e^{-\alpha
t+6})\right]^{-1}$. In the following, we consider pulses with different
durations $t_{\text{pulse}}$, and optimize all parameters $\alpha$, $E_{0}$,
$\Omega_{\text{st}}$ and $\Omega_{\text{end}}$ to achieve certain final states
(Sec. III).

###  II.3 DMFT solution

We solve the model using nonequilibrium DMFT Georges  _et al._ (1996); Aoki
_et al._ (2014); Schüler  _et al._ (2020). In DMFT, the original Hamiltonian
Eq. (1) is mapped onto a single-site impurity with local properties that
correspond to Eq. (2). The impurity is hybridized with a bath characterized by
a self-consistent hybridization matrix
$\Delta_{a,b}\left(t,t^{\prime}\right)$, with orbital indexes $a$, $b$.
Because we consider relatively high temperatures and interaction strengths, we
can to first approximation apply the non-crossing approximation (NCA) as
impurity solver Eckstein and Werner (2010). Our choice of the hopping matrix
in Eq. (3), and in particular the condition in Eq. (4), allows to solve the
problem entirely in terms of the diagonal components
$\Delta_{a}\left(t,t^{\prime}\right)=\Delta_{a,a}\left(t,t^{\prime}\right)$ of
the hybridization matrix. On the Bethe lattice, the self-consistency relation
takes a closed form,

$\Delta_{a}\left(t,t^{\prime}\right)=v^{2}G_{a}\left(t,t^{\prime}\right)+v^{\prime
2}G_{\bar{a}}\left(t,t^{\prime}\right)+\Delta_{p}\left(t,t^{\prime}\right)\;,$
(7)

where $G_{a}\left(t,t^{\prime}\right)$ is the local contour-ordered electronic
Green’s function for band $a$, $\bar{a}=2$ for $a=1$ and vice versa, and
$\Delta_{\text{p}}\left(t,t^{\prime}\right)$ is the hybridization function due
to the coupling of the valence bands to the non-interacting $p$-band. The
latter is given by

$\Delta_{\text{p}}\left(t,t^{\prime}\right)=v_{pd}(t)\left(t\right)G^{0}_{\text{p}}\left(t,t^{\prime}\right)v_{pd}(t^{\prime})\;,$
(8)

where $G^{0}_{\text{p}}\left(t,t^{\prime}\right)$ is the local Green’s
function of the $p$-band (with a semi-elliptic density of states of bandwidth
$4$, as shown in Fig. 1), and $v_{pd}=E(t)$ is the field-controlled inter-band
hybridization. The time-dependent spectral and occupation functions are
computed from the retarded and lesser components of the Green’s functions via
backward Fourier transformation:

$\begin{split}&A_{a}\left(\omega,t\right)=-\frac{1}{\pi}\text{Im}\int_{0}^{s_{\text{max}}}ds\
G^{R}_{a}\left(t,t-s\right)e^{-i\omega s}\;,\\\
&A^{<}_{a}\left(\omega,t\right)=\frac{1}{\pi}\text{Im}\int_{0}^{s_{\text{max}}}ds\
G^{<}_{a}\left(t,t-s\right)e^{-i\omega s}\;.\end{split}$ (9)

The total spectral function is defined as the sum,
$A\left(\omega,t\right)=A_{1}\left(\omega,t\right)+A_{2}\left(\omega,t\right)$,
and similarly for $A^{<}\left(\omega,t\right)$.

##  III Results

###  III.1 Time-dependent charge transfer

Figure 2:  Time dependence of the occupation per site $n=n_{1}+n_{2}$ (panel
(a)), of the orbital imbalance $\tau_{z}$ (panel (b)) and of the double
occupation $d$ (panel (c)) for several pulse durations $t_{\text{pulse}}$.
Each pulse is tailored to minimize the occupation of the UHBs in the final
state. Figure 3:  Optimized pulse parameters $\alpha$ (panel (a)), $E_{0}$
(panel (b)) and $\Omega_{\text{min}}$, $\Omega_{\text{max}}$ (panel (c)) as a
function of the pulse duration $t_{\text{pulse}}$, for the pulses used in the
simulations shown in Fig. 2. Figure 4:  Panel (a): spectral functions (lines)
of band $1$ at different times and respective occupations (shaded areas)
during and after the pulse with $t_{\text{pulse}}=40$ analyzed in Fig. 2.
Panel (b): same as for panel (a), but for band $2$. The starting time for the
simulations in this figures is $t=-30$, and the pulse arrives at $t=0$.

In this section, we present the results obtained by solving the model Eq. (1)
using nonequilibrium DMFT, starting from the equilibrium configuration
depicted in Fig. 1(b). The dipolar coupling of the valence bands with the
p-band has the net effect of increasing the filling of the valence bands, see
Fig. 2(a). Since our main aim is to induce a metal-to-Mott insulator phase
transition in the valence bands, we target the nearest commensurate filling of
the valence bands, $n=1$. By applying laser pulses of different durations, we
realize that the dynamics of $n$ can be divided essentially into two steps:
(i) A growth from the initial value $n\sim 0.92$ to an occupation $n>1$ larger
than the target one, and then (ii), a convergence of $n$ to the desired
occupation $n\sim 1$, see Fig. 2(a). The maximum value of the occupation
reached during the dynamics is higher for shorter pulses. In passing, we note
that each pulse is tailored to minimize the occupation of the UHBs in the
final state. The resulting pulse parameters for the pulses used in Fig. 2 are
shown in Fig. 3, while the optimization procedure is described in Sec. III.2.

Once the p-band is decoupled from the valence bands, the number of electrons
in the valence bands stays fixed because direct charge transfer processes are
activated only in the presence of the electric field. Other recombination
channels, which are not included in the model, should be weak in the real
material and therefore not relevant on the timescale of the present study. In
particular, the main recombination channels would be either phonon-assisted,
which is slow because of the large energy mismatch between phonons and the p-d
energy separation, and p-d electron-electron scattering, which is weak if the
p-hole and the photodoped electron are mainly located on different atoms
(oxygen and titanium).

The double occupation $d$ shows qualitatively similar dynamics as the density,
see Fig. 2(c). The final value of $d$ decreases for longer pulses, suggesting
a final state with a smaller number of excitations. Finally, the charge
distribution of the model is also characterized by the orbital imbalance
$\tau^{z}$, shown in Fig. 2(b). At earlier times, $\tau^{z}$ changes roughly
opposite to the total occupation: When $n$ increases, $\tau^{z}$ decreases and
vice versa. After the pulse, the orbital imbalance continues to decrease, in
particular after the shorter pulses $t_{\text{pulse}}=10,15,20$, indicating
inter valence-band thermalization processes.

To get more insight into what happens during the dynamics, we show in Fig. 4
some snapshots of the spectral functions of the valence bands during one of
the longer pulses considered in Fig. 2. The quasiparticle peaks of the two
bands observed in equilibrium are gradually suppressed, and entirely vanish
once the pulse is over. Thus, while at the initial stage of the dynamics the
system is in a correlated metallic state, after the photo-doping the system
reaches a Mott state. As seen in Fig. 1, the initial metallic state already
shows a preformed Mott gap within the unoccupied region of the spectrum. This
gap is reduced at intermediate stages of the dynamics due to the hybridization
of the valence with the lower energy band (see the intermediate spectral
functions shown in Fig. 4). In the final state, a well-defined Mott gap is
restored.

###  III.2 Final state and pulse optimization

Following the original aim of transferring the valence bands into a Mott
state, we now analyze the final state after the pulse as a function of the
pulse parameters. The insulating character of the state after the pulse
strongly depends on the pulse duration $t_{\text{pulse}}$. Figures 5(a) and
(b) show the spectral functions of the Mott insulators reached after pulses of
duration $t_{\text{pulse}}=10$ and $t_{\text{pulse}}=40$, respectively. It
becomes apparent that for the shorter pulse, the final state looks like a Mott
insulator with both occupation in the UHB and hole occupation in the lower
Hubbard band (LHB), and still some signatures of metallic character such as
quasiparticle peaks in particular on the upper edge of the lower Hubbard band.
This state can be called a photo-doped Mott-insulator Dasari  _et al._ (2020).
In contrast, for the longer pulse, the final insulating state resembles a low-
temperature Mott insulator.

One could attempt to extract an effective temperature of these states (and
also for the transient states) by analyzing the distribution function
$F\left(\omega,t\right)=\frac{A^{<}\left(\omega,t\right)}{A\left(\omega,t\right)}$
in terms of a Fermi-Dirac fit with a given inverse temperature
$\beta_{\text{eff}}(t)$. This would also tell if an effective temperature is
well-defined during the dynamics, i.e., whether the laser induced charge
transfer can be viewed as an adiabatic process. However, even for the longer
pulses, we find a substantial mismatch between the time-dependent distribution
function and the Fermi-Dirac distribution during the pulse. Also after the
pulse, a Fermi-Dirac fit is difficult: for shorter pulses, the simultaneous
hole occupation in the LHB and electron occupation in the UHB (as in Fig. 5a)
does not fit a Fermi distribution, while for longer pulses, where the system
becomes closer to a low-temperature Mott insulating state, a Fermi-Dirac fit
is not possible simply because a distribution cannot be properly defined in
the region of the gap and is either zero or unity outside.

Figure 5: Spectral functions for the two orbitals $A_{1}\left(\omega\right)$
(red) and $A_{2}\left(\omega\right)$ (green), and respective occupations
(shaded areas) computed by a backward Fourier transformation at time $t=130$
after the pulses $t_{\text{pulse}}=10$ (panel (a)) and $t_{\text{pulse}}=40$
(panel (b)) analyzed in Fig. 2.

We therefore proceed by characterizing the final state in terms of its energy.
We compare the state of the system with a thermal state sharing the same
internal energy and a total valence occupation $n=1$. The inverse temperature
$\beta_{\rm eff}$ of this equilibrium reference state is shown in Fig. 6(a).
We can distinguish two main regimes here: When the pulse is short, the
effective temperature of the system is increased with respect to the
temperature of the initial equilibrium system $\beta=10$, while with longer
pulses there is the possibility to at least keep the effective temperature
close to the initial value. Essentially, we have a transition from a regime
where the effect of the laser pulse is to produce a photo-doped Mott insulator
to a range where the entropy increase in the valence bands remains controlled,
as in the cooling by photo-doping protocol. The red squares shown in Fig. 6(c)
and (d) represent the values of the orbital imbalance and the double
occupation, respectively, as a function of the pulse duration obtained from
the equilibrium reference state with the inverse temperatures shown in Fig.
6(a). The black circles in Fig. 6(c) and (d) are instead the values of
$\tau_{z}$ and $d$ at the end of the total simulation time for the curves
shown in Fig. 2(b) and (c). The crossing of the curves around
$t_{\text{pulse}}\sim 30$ separates two regimes: Below $t_{\text{pulse}}\sim
30$, the system looks warmer than the corresponding equilibrium state if we
look at the double occupation, while it appears colder if we look at the
orbital imbalance; above $t_{\text{pulse}}\sim 30$, the opposite behavior
occurs. While the double occupation is essentially constant in the state
reached once the pulse is over, see Fig. 2(c), the orbital imbalance evolves
towards the corresponding equilibrium value. Thus it keeps decreasing for
short pulses (characterized by $t_{\text{pulse}}\leq 20$), and slowly
increasing for longer pulses, see Fig. 2(b).

Figure 6:  (a) Effective inverse temperature $\beta_{\text{eff}}$ of the
system after the action of a time dependent pulse of duration
$t_{\text{pulse}}$. The inverse temperature is computed by looking at the
internal energy of the system after the pulse. (b) Occupation of the upper
Hubbard band as a function of the duration of the pulse $t_{\text{pulse}}$.
Panel (c) and (d): orbital imbalance and double occupation of the system,
respectively, at the latest time of our simulations (circles) and in an
equilibrium state with the same temperature as in panel (a) (squares).

As expected, different pulse shapes Eq. (5) lead to different final states. To
choose the specific form of the laser pulse in the simulations so far, we
optimized the parameters $\Omega_{\text{end}}$, $\Omega_{\text{st}}$,
$\alpha$, and $E_{0}$ for each pulse duration $t_{\text{pulse}}$, keeping the
final occupation $n=1$ fixed. For the optimization, we start using Eq. (5) for
a non-chirped pulse close to
$\Omega_{\text{end}}=\Omega_{\text{st}}=\Omega\sim 6.65$ (set by the p-d
splitting) and $\alpha\sim 1$, and determine $E_{0}$ to impose the desired
total occupation $n\sim 1$. Considering several values of $\Omega$, we find
the configuration that minimizes the occupation of the UHB at given $\alpha$.
Repeating this procedure for several values $\alpha$, we optimize the number
of excitations in the system for this parameter too. Once we are in the best
possible state according to our requirements, we allow $\Omega_{\text{end}}$
to be different from $\Omega_{\text{st}}$, permitting us to optimize the
excitations number respect to the unchirped configuration slightly. (Certainly
an automatic multi-parameter pulse optimization based on machine learning
techniques would be intriguing, but computationally costly due to the
potentially large number of simulations required in the optimization.) The
optimized parameters are shown in Fig. 3. The optimal pulse shape is indeed
found to be slightly chirped, with $\Omega_{\text{end}}$ smaller than
$\Omega_{\text{st}}$, and a chirping amplitude
$\Delta\Omega_{\text{ch.}}=\Omega_{\text{st}}-\Omega_{\text{end}}$ comparable
to the gap size, see Fig. 3(c). Figure. 7 shows the occupation of the UHB
$n_{\text{UHB}}$ against $\Delta\Omega_{\text{ch.}}$ for pulses characterized
by $t_{\text{pulse}}=20$, $\alpha=1.15$ and $\Omega_{\text{end}}=6.700$. Each
point in Fig. 7 is computed at a different amplitude $E_{0}$ to impose the
total occupation of the final state $n\sim 1$. The minimum of $n_{\text{UHB}}$
occurs at $\Delta\Omega_{\text{ch.}}=0.126$, corresponding to
$\Omega_{\text{st}}=6.826$, as reported in Fig. 3(c).

Figure 7:  Occupation of the UHB as a function of the chirping amplitude
$\Delta\Omega_{\text{ch.}}$ for pulses with $t_{\text{pulse}}=20$,
$\alpha=1.15$ and $\Omega_{\text{end}}=6.700$.

A final question we would like to address is at which speed the transition
from the metallic to the Mott insulating regime can be achieved. Since we
target to keep a small number of excitations, we plot the number of electrons
in the UHB once the pulse is over as a function of the pulse duration, as
shown in Fig. 6(b). Not surprisingly, the shorter the pulse, the higher the
number of excitations in the system. We expect the curve to follow the
exponential decay $Ce^{-E_{g}^{\text{fit}}t_{\text{pulse}}}$, where
$E_{g}^{\text{fit}}$ should be of the order of the bandgap $E_{g}$ for the
best possible pulse process. By fitting the curve Fig. 6(b) with the
exponential function, we obtain $C\sim 0.102$ and $E_{g}^{\text{fit}}\sim
0.132$, with the second parameter of the same order of magnitude of the gap
size $E_{g}\sim 0.2$. We notice that, while the three points at
$t_{\text{pulse}}\leq 20$ in Fig. 6(b) are well fitted, the last three depart
from the fitting line. For this, one also must keep in mind that the
exponentially small excitation densities become increasingly difficult to
measure numerically. The decrease in $n_{\text{UHB}}$ observed in Fig. 6(b)
over several order of magnitudes indicates what should be the approximate
pulse length to get the desired number of excitations in the system.

##  IV Conclusions

In this study, we analyzed the possibility to induce an ultrafast metal-to-
valence Mott insulator transition in a model system relevant to the physics of
the oxygen-enriched transition metal compound LaTiO${}_{3+\text{x}}$. Through
non-equilibrium dynamical mean-field theory (with the non-crossing
approximation as an impurity solver), we considered the effect of the dipolar
photo-doping of electrons from a core state representative of the broad
p-derived oxygen band to the valence states that mimic the
$\text{t}_{2\text{g}}$ manifold at the Fermi level. An external laser pulse
opens a charge transfer channel from the oxygen related bands to the valence
bands. At each given pulse duration, we optimized the shape of the pulse to
reach the desired final occupation and simultaneously minimize the number of
excitations in the upper Hubbard bands. Depending on the pulse length, we get
a crossover from a regime where the final state is a Mott insulator
characterized by an effective temperature almost equal to the one of the
initial correlated metallic system to a regime where this effective
temperature is substantially larger. The crossover is at pulse durations as
short as $30$ hopping times.

Although the model under investigation is highly simplified, this suggests
that a photo-induced charge transfer to a valence Mott-insulator can be
realized in LaTiO${}_{3+\text{x}}$ on the $100\ \mathtt{fs}$ timescale without
substantially heating the system. As in the cooling by photo-doping mechanism,
the limited heating can be related to the fact that a large amount of entropy
is stored in the oxygen related bands after the photo-induced charge transfer.

## Acknowledgements

We thank Philipp Werner for useful discussions. This work was supported by the
ERC starting grant No. 716648. The authors gratefully acknowledge the
computational resources and support provided by the Erlangen Regional
Computing Center (RRZE).

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