# ZzzGPT: An Interactive GPT Approach to Enhance Sleep Quality

Yonchanok Khaokaew<EMAIL_ADDRESS>1234-5678-9012 ,  Thuc Hanh Nguyen
<EMAIL_ADDRESS>School of Computer Science and Engineering
University of New South Wales -NSWAustralia2052 ,  Kaixin Ji
<EMAIL_ADDRESS>,  Hiruni Kegalle
<EMAIL_ADDRESS>and  Marwah Alaofi
<EMAIL_ADDRESS>RMIT University-MelbourneAustralia

(2023)

###### Abstract.

In today’s world, sleep quality is pivotal for overall well-being. While
wearable sensors offer real-time monitoring, they often lack actionable
insights, leading to user abandonment. This paper delves into the role of
technology in understanding sleep patterns. We introduce a two-stage
framework, utilizing Large Language Models (LLMs), aiming to provide accurate
sleep predictions with actionable feedback. Leveraging the GLOBEM dataset and
synthetic data from LLMs, we highlight enhanced results with models like
XGBoost. Our approach merges advanced machine learning with user-centric
design, blending scientific accuracy with practicality.

Sleep Prediction; Large Language Model; Machine Learning; Interactive User
Interface;

††copyright: acmcopyright††journalyear: 2023††conference: ; ; ††ccs: Computing
methodologies Machine learning††ccs: Human-centered computing Human computer
interaction (HCI)††ccs: Applied computing Life and medical sciences

##  1\. Introduction

In the modern age, the adage ”health is wealth” becomes increasingly apparent,
not just in terms of physical but also mental. One critical but often
overlooked aspect of overall health is the quality of sleep. Although sleep is
seemingly physical, it could have far-reaching implications for mental health,
productivity, and overall quality of life. Technologies such as wearable
sensors have paved the way for real-time monitoring of sleep patterns,
promising insights into this complicated and crucial behaviour.

However, while wearable sensors have provided a technological leap in self-
quantification and health monitoring, they are not without their challenges.
Research has shown that people often abandon these devices due to
unsatisfactory results or a lack of actionable insights. However, these
sensors have the potential for lifestyle and health improvement, as poor sleep
is not just an isolated issue but often a symptom or cause of other health
problems, such as depression. Therefore, we studied essential factors that
impacted sleep quality and developed a graph-based interface and Large
Language Models that guide users to gain valuable insights into their
behaviour and detect ways of lifestyle improvement.

##  2\. MOTIVATION

The intricate relationship between sleep and health attributes, such as mental
health or physical activity, has garnered significant attention in scientific
literature. These studies have delved into correlation analysis of such
attributes and ushered in advanced medical research. Notably, they have
provided insights into illness detection, ranging from depression and diabetes
to Parkinson’s disease.

Another study (Chen et al., 2020) incorporated data from smartphones, Fitbit
devices, weather conditions, and personal calendars to predict sleep duration.
In this context, smartphone data gives information on proximity to other phone
users, screen time, communication history, and geo-location. Fitbit data
provided metrics like heart rate, physical activity, and sleep status. By
leveraging the General Linear Model (GLM) and the General Linear Mixed Model
(GLMM), they fine-tuned their input factors by eliminating multicollinearity
and considering feature importance. Their findings underscored the high
correlation between most studied factors and sleep duration, reporting a
correlation coefficient of 0.745 between actual and estimated sleep duration.
Although their primary focus was correlational analysis, the parallels between
their chosen input features and ours set the groundwork for our research.

Figure 1. Wordcloud of Quoras and Reddit Dataset

Building on the wearable devices narrative, Attig and Franke (2020) found that
many individuals abandoned these gadgets due to unsatisfactory results. To
better understand this phenomenon, we analyzed sleep-related questions on
popular online platforms like Reddit (Fionn Delahunty, 2018) and Quora
(DataCanary, hilfialkaff, Lili Jiang, Meg Risdal, Nikhil Dandekar, tomtung,
2017) (See Fig. 1). Common queries, such as ”How do I fix/change/maintain a
sleep pattern?” and ”I need help to get me to sleep; I start University soon
and can’t maintain a steady sleep pattern!” revealed a challenge. Such
abstract questions suggest that users might lack a clear understanding of
sleep, making it challenging to seek precise solutions. This knowledge gap
could demotivate users from using wearable devices for sleep tracking.

Drawing from these studies, an evident gap emerges. While the correlation and
prediction of sleep patterns have been substantially explored, there is a
pressing need for an integrated solution that not only predicts but also
offers actionable insights. Enter Large Language Models (LLMs). The advent of
LLMs, especially models like GPT-3, has reshaped the landscape of natural
language processing. These models enable intelligent interactions between
humans and technology, ushering in possibilities for improved user experience
and potential behavioral modifications. By addressing the limitations of
wearables with the capacity for personalized communication, LLMs have the
potential to re-engage users. Our research seeks to harness the capabilities
of LLMs, aiming to revolutionize sleep monitoring by combining accurate
predictions with actionable behavioral improvements.

Figure 2. Overview of the proposed framework

##  3\. Proposed framework

###  3.1. Overview

The primary focus of this project is to present a holistic, interactive model
to help users better understand their sleep. We propose a detailed two-stage
framework (Fig. 2). This framework consists of a Training Stage and a Demo
Stage, which functions collaboratively to build a predictive model and an
interactive user interface.

####  3.1.1. Training Stage

We used data and features from the GLOBEM dataset (Xu et al., 2022), which
contains diverse user behavioural information, including location, phone
usage, call logs, step count, Bluetooth, and Wi-Fi connection. We undertook
statistical analysis to identify variables that strongly correlate with sleep
quality and further filter features using feature importance ranking.

Subsequently, we ventured beyond conventional feature analysis. A hypothesis
was proposed: Could synthetic data generated from state-of-the-art Large
Language Models (LLMs) enhance the model’s predictive power? These LLMs are
adept at replicating the natural variance of human behaviour and environmental
interactions. Therefore, by introducing this method, we aimed to test whether
such augmentation would bolster the model’s performance or inadvertently
introduce noise and complexities.

####  3.1.2. Demo Stage

The second stage is designed to offer an interactive experience for end-users
via a chat application111https://github.com/MarwahAlaofi/UbiComp23-student-
challenge/. This stage is divided into five distinct parts, as elaborated
below:

  * •

User Interaction via LLMs: The Demo Stage has been designed to simulate a
real-world application of the trained model. A chat application, fortified by
a Large Language Model (LLM) API, serves as the front-end interface. Users
will see a range of suggested questions when they interact with the chatbot,
including queries about their anticipated sleep quality for the night.

  * •

Sleep Quality Prediction: When a user opts to investigate their sleep quality,
the system swiftly deploys the predictive model honed during the Training
Stage. This model considers the top 20 important features ranked during
analysis to generate a prediction.

  * •

Testing Sample Acquisition: To enact this demonstration, we assume the passive
acquisition of relevant user data through the chatbot installed on their
devices. A representative testing sample will be culled from the GLOBEM
dataset, which comprises the features earmarked as important during the
Training Stage.

  * •

Predictive Responses and Interactive Graph: The system offers an interactive
element beyond text-only. The LLM API generates textual advice or comments
based on the predicted sleep quality. Furthermore, an interactive graphical
interface allows users to modify key features to see their real-time impact on
predicted sleep quality. (Fig. 3)

  * •

Personalized Recommendations: Finally, the application auto-generates
pragmatic suggestions tailored to the user’s lifestyle and environment. By
setting thresholds and manipulating feature values, the system explores how
minor adjustments can lead to significant improvements in sleep quality

Figure 3. Demo interface

###  3.2. Dataset

The GLOBEM dataset (Xu et al., 2022) offers a multi-year mobile and wearable
sensing data collection from 2018 to 2021. Over these four years, the dataset
captures information from various cohorts: INS-W_1 (2018), INS-W_2 (2019),
INS-W_3 (2020), and INS-W_4 (2021). This project exclusively uses the INS-W_1
set, which offers a glimpse into user behavior before the disruption of the
COVID-19 pandemic.

The core objective of the dataset is to serve as a foundational platform for
comparing behavior modeling algorithms, examining cross-dataset generalization
tasks, and fostering the development of more comprehensive longitudinal
behavior modelling algorithms by ML and UbiComp researchers.

The data was collected via a self-developed mobile app with the AWARE
framework, which continuously gathers data such as location, screen status,
Bluetooth scans, and call logs. The app is compatible with iOS and Android and
seamlessly works in the background once installed on participants’ phones.
Additionally, participants were equipped with Fitbits to record their physical
activity and sleep patterns. This combination ensured a passive 24/7 data
collection, with an average participation duration of 78 days per individual
each year. Surveys regarding physical and mental health, social issues, and
substance use were also administered during the study period.

###  3.3. Data Prepossessing

The dataset offers hundreds of features. To reduce feature dimensions, we
conduct a comprehensive, selective framework to develop our predictive machine
learning model. The task is to predict the ‘sleep efficiency’ on a coming day;
hence, we select the ‘sleep summary average efficiency’ calculated by the
Fitbit wearable as our target variable. The historical value for sleep
efficiency is derived to derive meaningful insights by shifting the target
variable by one row and eliminating the first index for every participant
(pid). This is essential since the initial index for each participant lacks
the preceding day’s data. Rows containing missing values for the target
variable were excluded.

Pertinent features from Bluetooth, call, location, screen, and steps were
incorporated to enrich our dataset. These features were meticulously chosen
based on their relevance and prominence in current literature related to sleep
patterns. Recognizing the potential of these features to provide a
comprehensive insight into sleep quality, we conducted an exhaustive review of
existing studies. The subsequent list details these features, supported by
pertinent literature to bolster their credibility and relevance:

  * •

surrounding Bluetooth devices: the proximity and number of surrounding
Bluetooth devices can offer insights into an individual’s social interactions
and ambient environment, both of which have been linked to sleep quality (Xu
et al., 2012).

  * •

phone call: Frequency, duration, and timing of phone calls can hint at an
individual’s social commitments and stress levels, which are known factors
affecting sleep (Shin et al., 2017; Tettamanti et al., 2020).

  * •

location: an individual’s location, particularly visits to green spaces, might
suggest relaxation activities beneficial for sleep. Conversely, nighttime
urban activity might disrupt sleep. Green spaces are linked with relaxation
and potentially improved sleep quality (Feng et al., 2020; Yang et al., 2020).

  * •

screen usage: excessive screen time, especially before bedtime, correlates
with sleep disturbances due to the blue light emitted from screens (Wu et al.,
2015; Christensen et al., 2016; Shin et al., 2017).

  * •

steps: physical activity levels, measured through steps, can provide an
understanding of fatigue levels, which directly impact sleep quality (Bisson
et al., 2019; Stepnowsky et al., 2011).

Understanding the rationale behind the selection of these features, grounded
in prior research, reinforces the robustness of our approach and provides a
credible foundation for our sleep prediction endeavors. It also avoids
overfitting issues by including a selective set of features.

Following this, we merged these datasets with the sleep dataset. Since the
dataset spans 6177 rows, ensuring that each feature presents substantial
information is imperative. As such, 49 columns inundated with ‘n/a’ values,
specifically those with more than 30% (about 1853) such entries, were removed.
We also removed the rows based on the outliers of some key features, such as
screen time, sleep duration, and step count. We used the Tukey method to
remove the outliers based on lower and upper fences. To deal with the missing
values of each feature, we filled it with the mean values of each participant.
Furthermore, columns with weak correlations ($r<.001$) with the target
variable were pruned.

###  3.4. Data Generation

In our next stage, we employed the predictive capabilities of Large Language
Models (LLMs). Our primary goal was to emulate the intricate patterns seen in
the actual participant data. Starting with sample selection, a subset
comprising 20 samples, equivalent to roughly three weeks of data, was
meticulously chosen for each participant. This duration was perceived as
adequate for encapsulating the significant behavioral tendencies of an
individual.

These chosen samples were methodically transformed into a table-format prompt
to guide the LLM’s data generation task. This structured presentation offers
clarity and ensures the LLM can seamlessly pick up on the dataset’s inherent
temporal patterns and subtle nuances. With these prompts at the ready, our
next course of action involved the OpenAI API, harnessing the power of LLMs to
extrapolate from the given data. For each structured prompt based on one
participant’s data, the model was asked to generate new data for five
additional days, taking cues from the preceding week’s patterns. However, not
all generated data met our rigorous standards. In situations where the
generated data did not encompass all requisite features, we manually pruned
such data to maintain the dataset’s integrity and coherence.

Once generated, this new set of data was integrated with the original training
set. This augmented compilation provides a larger volume of data and
potentially encompasses a more diverse set of behavioral variations. After the
augmentation process, we conducted a rigorous evaluation to gauge the impact
of the LLM-generated data on model performance, keenly observing if it played
a role in enhancing the training outcomes. This endeavor aimed to shed light
on the potential merits of using LLMs for data augmentation in behavioral
predictive modeling.

##  4\. Experiment and Result

###  4.1. Experiment Setting

For our experimental design, we partitioned the data into distinct sets:
$64\%$ was allocated for training, $21\%$ for validation, and $17\%$ (last 14
days) for testing. Given the nature of our research objective, we approached
this task as a regression problem. We evaluated various regression models,
including Random Forest (RF) (Biau, 2012), CatBoost (CatB) (Prokhorenkova et
al., 2018), and Gradient Boosting (XGB) (Chen and Guestrin, 2016). The target
variable is not normally distributed ($std=.058$). Thus, we use Mean as the
baseline; it is expected that the mean baseline achieves a comparable
accuracy.

Two feature selection approaches were compared: manually curated features and
the top-$K$ features. As our project aims to provide sleep insights, we
carefully select features by considering whether the users can understand and
adjust to improve their sleep. Then, the top-$K$ is chosen by comparing the
influence on the performance with different $K$ and the best-perform model.
Lastly, the top 20 features (from CatB) perform the best. We have also tried
the permutation feature importance and attention-based feature importance from
a Long short-term memory (LSTM) model (Sathyanarayana et al., 2016); however,
the results are not better than the ML model-based approach. The inclusion of
LLM-generated data in the training set was also investigated for its potential
performance boost.

The evaluation metrics reported are Root Mean Square Error (RMSE), Mean
Absolute Error (MAE), and $R^{2}$ score. These metrics collectively offer a
comprehensive view of model accuracy, error magnitude, and the proportion of
the variance for the dependent variable explained by independent variables.

###  4.2. Result & Discussion

Features (N) | Hand-picked(66) | Top-k(20) | Top-k(20)+G
---|---|---|---
Model/Metric | RMSE | MAE | R2 | RMSE | MAE | R2 | RMSE | MAE | R2
Mean Baseline | $.059$ | $.032$ | $-.007$ | $.059$ | $.032$ | $-.008$ | $.059$ | $.032$ | $-.008$
Random Forest | $.059$ | $.033$ | $.018$ | $.054$ | $.032$ | $.013$ | $.055$ | $.032$ | $.032$
XGBoost | $.057$ | $.037$ | $-.003$ | $.050$ | $.035$ | $-.150$ | $.050$ | $.034$ | $-.153$
Catboost | $.054$ | $.031$ | $.116$ | $.049$ | $.030$ | $.124$ | $.050$ | $.030$ | $.102$

Table 1. Test Performance of models. G: LLM-generated training data.

####  4.2.1. Performance of Different Features

Overall, the CatB model achieves the best performance across all rounds. With
the 66 hand-picked features, all the models achieved similar RMSE and higher
$R^{2}$ than the baseline. We then select the top 20 features computed by the
CatB model. With the top-20 features, all models have much lower RMSE, MAE,
and higher $R^{2}$. CatB is still the best model.

####  4.2.2. Performance of Specific Models

After integrating the generative data, particularly in the experiment with
top-$K$ features, an intriguing pattern emerged in the performance metrics.
XGB markedly improved, while CatB and RF exhibited slight performance
degradation. This divergence necessitates a deeper examination into the
inherent qualities of these models.

XGBoost, known for its paralleled gradient boosting mechanism, inherently
benefits from larger datasets. The added generative data, which emulates
diverse patterns from the original set, likely complemented XGB’s capability
to train numerous models on diverse subsets and then amalgamate their
performances. Furthermore, the regularization aspects intrinsic to XGB aid in
generalizing the model, making it adept at handling the nuances the generative
data introduces.

On the contrary, CatB is renowned for its symmetric tree building and ordered
boosting. Although efficient and a guard against overfitting, the symmetric
trees might not be as flexible as the more dynamic trees that XGB creates.
While CatB’s ordered boosting often prevents target leakage and overfitting,
it’s possible that the subtle intricacies of the generated data, which are
designed to capture complex behavioral patterns, may not align perfectly with
CatB’s strict ordering and thus lead to a slight performance dip.

Random Forest, an ensemble of decision trees, prioritizes diversity in its
trees. While the generative data introduces diversity, it may also introduce
noise or intricacies that Random Forest doesn’t capture as adeptly as
gradient-boosted models like XGB.

In summary, while CatB displayed the best overall performance, the results
hint at the potential of our framework to be particularly synergistic with
specific models, like XGB, significantly when leveraging generative data. This
underscores the significance of understanding model intricacies when
augmenting datasets, highlighting the potential for future research in bespoke
data augmentation tailored to specific model architectures.

Figure 4. RMSE test results. N: number of features. G: LLM-generated training
data.

##  5\. Conclusion

To promote the usage of wearable devices by leveraging explainability and
interactability, this project proposes a novel two-stage framework. The
framework merges Machine Learning and Large Language Models with an engaging
user interface. It includes rigorous data analysis and instant user feedback,
producing promising results and practical accessibility. This framework
pioneers a fresh approach to sleep research and personalized healthcare
domains. We believe our exploration shed light on the impact of integrating
synthetic data and offered valuable insights into its implications for
predictive modelling in sleep quality.

## ACKNOWLEDGMENTS

This research is partially supported by the Sponsor Australian Research
Council Centre of Excellence for Automated Decision Making and Society
https://www.arc.gov.au/ (Grant #CE200100005).

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## Ethics Statement

We’re exploring the realms of sleep health by integrating Machine Learning and
Large Language Models, utilizing data from the GLOBEM dataset. This technology
unfolds a panorama where users and researchers can peek into the intricate
world of sleep patterns, enabling them to draw profound insights and correlate
sleep with various lifestyle aspects. The ultimate goal is to foster better-
informed decisions about sleep health and overall wellness. Nonetheless, our
journey through technology isn’t without hurdles. Ensuring the advice and
insights offered by our system remain accurate and dependable is pivotal to
upholding our initiative’s integrity and genuinely supporting health-related
decision-making.

Addressing the aforementioned challenges is critical, and to do so, we stand
firmly on two pillars: Ensuring Data Security and Maintaining Model Accuracy.
Implementing and routinely auditing strong data security measures guard
against potential data manipulation and upholding the reliability of our
insights. Rather than implementing a live checking system, our approach
involves employing more accurate predictive models when they become available,
ensuring our technology remains state-of-the-art and offers the most reliable
and accurate insights possible. Thus, we propel our initiative forward,
prioritizing a solid, ethical foundation in all our technological and research
endeavours. 0


# Correlation Driven Magnetic Frustration and Insulating Behavior of TiF3

Gayanath W. Fernando<EMAIL_ADDRESS>Department of Physics,
University of Connecticut, Storrs, Connecticut, 06269 USA  Donal Sheets
Department of Physics, University of Connecticut, Storrs, Connecticut, 06269
USA  Jason Hancock  Department of Physics, University of Connecticut, Storrs,
Connecticut, 06269 USA  Arthur Ernst  Max Planck Institute of Microstructure
Physics, Weinberg 2, D-06120 Halle, Germany  Institute for Theoretical
Physics, Johannes Kepler University, Altenberger Strasse 69, 4040 Linz,
Austria  R. Matthias Geilhufe<EMAIL_ADDRESS>Department of
Physics, Chalmers University of Technology, 412 96 Göteborg, Sweden

###### Abstract

We investigate the halide perovskite TiF3, renowned for its intricate
interplay between structure, electronic correlations, magnetism, and thermal
expansion. Despite its simple structure, understanding its low-temperature
magnetic behavior has been a challenge. Previous theories proposed
antiferromagnetic ordering. In contrast, experimental signatures for an
ordered magnetic state are absent down to 10 K. Our current study has
successfully reevaluated the theoretical modeling of TiF3, unveiling the
significance of strong electronic correlations as the key driver for its
insulating behavior and magnetic frustration. In addition, our frequency-
dependent optical reflectivity measurements exhibit clear signs of an
insulating state. Analysis of the calculated magnetic data gives an
antiferromagnetic exchange coupling with a net Weiss temperature of order 25 K
as well as a magnetic response consistent with a $S$=1/2 local moment per
Ti3+. Yet, the system shows no susceptibility peak at this temperature scale
and appears free of long-range antiferromagnetic order down to 1 K. Extending
ab initio modeling of the material to larger unit cells shows a tendency for
relaxing into a non-collinear magnetic ordering, with a shallow energy
landscape between several magnetic ground states, promoting the status of this
simple, nearly cubic perovskite structured material as a candidate spin
liquid.

##  I Introduction

Perovskite materials are known to exhibit fascinating physical properties such
as colossal magnetoresistance, high-temperature superconductivity and negative
thermal expansion. Fluoride perovskites, with the typical formula unit MF3,
only require a trivalent M site in contrast to a typical AMO3 for oxide
perovskites. Transition metal trifluoride perovskites are less well studied
compared to their oxide counterparts, and known to display marked differences
such as the strong negative thermal expansion (NTE) in ScF3. Doping of such
MF3 perovksites has been the focus of the experimental work of Morelock et al.
[1] which provides a fairly comprehensive structural study of the material
class Sc1-xTixF3. Such materials have both fundamental and practical value.
When designing composite materials, controlling and tuning their thermal
properties is highly desirable to avoid thermal fracture/fatigue and related
issues.

Most of the materials mentioned above also show a cubic to rhombohedral
structural transition as temperature is lowered. Above 350-370 K, the
trifluoride TiF3 has the cubic perovskite structure AMX3 with no cation A
present. Each Ti is at the center of corner sharing octahedrons of fluorine
atoms. Perturbations of the strong NTE state using metallic substitution A on
the trivalent Sc+3 site in Sc1-xAlxF3 has revealed the NTE behavior to be
present at elevated temperatures for A = Al, Y, Ti, Fe over a significant
range of compositions. The opportunity to compositionally introduce electronic
degrees of freedom in a system with a structurally critical ground state is an
intriguing prospect, particularly in light of the high interest in low-carrier
superconductivity in perovskites SrTiO3 and KTaO3. There are reasons to
believe that the structural phase transition from cubic to rhombohedral is not
directly tied to magnetic properties of the metal atom A since even AlF3
exhibits it. In fact, induced dipole-dipole interactions of the F ions are
believed to drive this transition [2].

In our opinion, the magnetic and insulating properties of TiF3 have not been
well understood amidst several controversial claims. The extra valence
$d$-electron that Ti carries compared to Sc ([1, 3]) gives rise to unusual
electronic and magnetic properties.

Several theoretical studies have been reported in the past, discussing the
magnetic ground state of TiF3. Perebeinos and Vogt employing the FLAPW method
[4] showed that without correlation effects, TiF3 would be a ferromagnetic
half-metal. If the Hubbard interaction strength $U$ exceeds a value of
$4J_{H}$, with $J_{H}$ being the Hund-coupling, the material undergoes a
transition into a Mott insulator. Similar behavior was later confirmed, e.g.,
by Mattsson and Paulus [5], using a slightly revised computational approach.
In contrast, Qin et al. [6] claimed that in metal trifluorides, itinerant
electrons play a central role in controlling thermal expansion while ignoring
strong electron correlation effects mentioned in the FLAPW work above [4].
Even though theory work hints towards an antiferromagnetic state, it has been
mentioned that low temperature neutron scattering experiments have not
revealed any such transition [4]. More recent experiments by Sheets et al. [7]
show clear indications for some kind of magnetic transition below 10 K,
hinting towards a more complex magnetic order or magnetic frustration.

In the following, we discuss the option of magnetic frustration by extending
the theoretical modeling of TiF3 towards non-collinear magnetic
configurations. We show that such a solution gives rise to an insulating
ground state in agreement with experiment. In addition, it explains the
absence of a clear antiferromagnetic transition.

##  II Phase Transitions - phenomenological theory

At high temperatures, TiF3 is cubic (space group Pm$\overline{3}$m). Each Ti
ion contains an unpaired electron occupying a $t_{2g}$ orbital. Due to the
partial occupation of the $t_{2g}$ band, the system can lower its free energy
by a rhombohedral distortion and a resulting symmetry breaking into the space
group $R\overline{3}c$ taking place at temperatures below 350 K. This lowering
of symmetry lifts the degeneracy of the $t_{2g}$ band, imposing a split into
two bands with $e_{g}$ and $a_{g}$ character which corresponds to the point
group $D_{3d}$ (Figures 1 (a) and (b)). However, this splitting is very weak
$\approx 6$ meV as opposed to the total band width of $\approx 2$ eV and an
accompanying band narrowing effect of $\approx 18$ meV [4]. The corresponding
basis functions transform as [4, 8, 9]

$\left|a_{g}\right>=\frac{\left|xy\right>+\left|yz\right>+\left|zx\right>}{\sqrt{3}},$
(1)

and

$\left|e_{g};1\right>=\frac{\left|yz\right>-\left|zx\right>}{\sqrt{2}},\quad\left|e_{g};2\right>=\frac{2\left|xy\right>-\left|yz\right>-\left|zx\right>}{\sqrt{6}}.$
(2)

Figure 1: Illustration of symmetry breaking order in TIF3. (a) in the normal
state and in the absence of interactions, TIF3 would behave as a ferromagnetic
metal. (b) due to the partial occupation of the $t_{2g}$ band the total energy
of the system can be lowered by a Jahn-Teller-like symmetry breaking, lifting
the band degeneracy. (c) strong Coulomb repulsion due to flat bands impose a
significant band splitting between the $e_{g}$ and $a_{g}$ sates, favoring an
insulating state with antiferromagnetic or frustrated magnetic order.

To first order, the transition is described as a tilting of corner sharing
octahedrons surrounding each metal atom. TiF3 has been observed to undergo the
phase transition from rhombohedral to cubic with noticeable positive thermal
expansion below the transition and negligible thermal expansion above it.
Fluorine is bonded in a linear geometry to two equivalent Ti atoms with a bond
length of 1.97 Å in the cubic phase. The tilting angle has been observed to
vary between the cubic value (i.e., zero degrees) to a maximum of about
sixteen degrees (continuously) in the rhombohedral phase as temperature is
lowered, and could play a role in controlling a possible magnetic super-
exchange interaction between Ti atoms. Other Ti doped Sc-trifluorides
(Sc1-xTixF3) also display similar phase transitions occurring at lower
temperatures[1]. However, note that there are trifluorides such as AlF3 with
no $d$-electrons where similar tilting of octahedrons has been observed and
discussed in terms of electrostatic dipole-dipole interactions due to
displaced Fluorine ions [2]. While it is likely that such dipole-dipole
interactions are present in TiF3, we believe that the resulting gap and
magnetism is directly tied to electronic correlations.

The magnetism in TiF3 caused by the unpaired $d$ electrons at Ti ions, is
strongly influenced by the strong electronic correlations present in the
material. First and foremost, this is a result of the narrow width of the
$t_{2g}$ band, where the suppressed kinetic energy fosters the dominance of
interaction effects. In the framework of perturbation theory applied to the
Hubbard model, Perebeinos and Vogt argued that a transition between
ferromagnetic and antiferromagnetic ordering takes place at sufficiently high
Hubbard interaction [4]. However, here we argue that the favored magnetic
ordering might instead be non-collinear. To support this idea, we use the
double-exchange model [10] described by the following Hamiltonian, in the
absence of interactions

$\hat{\mathrm{H}}_{0}=\sum_{ij}\sum_{ab}\sum_{\gamma}t_{ij}^{ab}\hat{\mathrm{c}}^{\dagger}_{ia\gamma}\hat{\mathrm{c}}_{jb\gamma}-J\sum_{i}\boldsymbol{S}_{i}\hat{\mathrm{\boldsymbol{s}}}_{i}.$
(3)

The first term describes a conventional electron hopping between sites $i$ and
$j$, orbitals $a$ and $b$ and spin states $\gamma$. The second term is the
double-exchange term, describing the local interaction of the mean-field spin
of the Ti atom $\boldsymbol{S}_{i}$ with the spin
$\hat{\mathrm{\boldsymbol{s}}}_{i}$ of a quasi-free electron. Hence,
electronic spins tend to align with the local spin, leading to the absence of
a good global quantization axis. In a two-site model, we can construct
$\boldsymbol{S}_{1}$ to point along the $z$-axis, while $\boldsymbol{S}_{2}$
is rotated by an angle $\theta$. Instead of up and down spins, we introduce
$\left|\alpha_{i}\right>$ and $\left|\beta_{i}\right>$ given by

$\displaystyle\left|\alpha_{1}\right>$
$\displaystyle=\left(\begin{array}[]{c}1\\\
0\end{array}\right),\qquad\left|\beta_{1}\right>=\left(\begin{array}[]{c}0\\\
1\end{array}\right),$ (8) $\displaystyle\left|\alpha_{2}\right>$
$\displaystyle=\left(\begin{array}[]{c}\cos\left(\frac{\theta}{2}\right)\\\
\sin\left(\frac{\theta}{2}\right)\end{array}\right),\qquad\left|\beta_{2}\right>=\left(\begin{array}[]{c}\sin\left(\frac{\theta}{2}\right)\\\
-\cos\left(\frac{\theta}{2}\right)\end{array}\right).$ (13)

The hopping between the two sites is shown in Figure 2(a). In contrast to the
collinear case, there is always a finite hopping amplitude between both sites,
regardless of the spin-state.

Figure 2: Comparison of a noncollinear magnetic ground state in the double-
exchange Hubbard model and density functional theory. (a) Electron hopping in
the two-site double-exchange Hubbard model with mean-field background spins
shown in green. (b) Three different virtual hopping processes and their energy
increase starting from an occupation with a single electron in the $a_{g}$
orbital with spin state $\alpha_{i}$ on each of the two sites. (c) and (d)
total energy versus the angle $\theta$ between the magnetization directions of
the two Ti atoms in the primitive unit cell. (e) The angle versus energy plot
for the DFT calculations shows jumps at various critical angles which can be
resolved from Lifshitz transitions in the density of states, shown in (f).

The electron-electron interactions are described by the Hubbard model,

$\hat{\mathrm{H}}=\hat{\mathrm{H}}_{0}+\frac{U}{2}\sum_{i}\sum_{a}\hat{\mathrm{n}}_{ia\alpha_{i}}\hat{\mathrm{n}}_{ia\beta_{i}}+\frac{U^{\prime}}{2}\sum_{i}\sum_{b}\sum_{a\neq
b}\hat{\mathrm{n}}_{ia\alpha_{i}}\hat{\mathrm{n}}_{ib\beta_{i}}\\\
-\frac{J_{H}}{2}\sum_{i}\sum_{b}\sum_{a\neq
b}\sum_{\gamma}\hat{\mathrm{n}}_{ia\gamma}\hat{\mathrm{n}}_{ib\gamma_{i}}.$
(14)

Besides the non-interacting double exchange Hamiltonian
$\hat{\mathrm{H}}_{0}$, the Hubbard Hamiltonian contains three terms. The
first term describes the repulsion of electrons occupying the same site and
same orbital with opposite spins by an energy $U$. The second term describes
the repulsion of electrons occupying different orbitals at the same site. The
third term, the Hund’s coupling, is an attractive term supporting the
occupation of different orbitals with the same spin states on the same site.
Equation (14) cannot be solved directly, but is discussed using perturbation
theory. We assume a ground state with a single electron on each Ti atom,
occupying the $a_{g}$ orbital and spin state $\left|\alpha_{i}\right>$.
Similar to Ref.[4], we only consider $(dd\pi)$ hopping with a small hopping
strength $t\approx 0.225$ eV. From the orbital overlap (1) and spin overlap
(13), the hopping amplitude for a process
$\left|a_{g}\alpha\right>\rightarrow\left|a_{g}\alpha\right>$ between two
neighboring sites is $\frac{2}{3}t\sin\left(\frac{\theta}{2}\right)$. The
energy increase due to Hubbard interaction is $\delta
E=\sin\left(\frac{\theta}{2}\right)U$. In second order perturbation theory,
this leads to an energy lowering of $\Delta
E=-\sum_{i=1}^{6}\frac{\left|\left<a_{g}\alpha\right|\hat{\mathrm{H}}_{0}\left|a_{g}\alpha_{i}\right>\right|^{2}}{\delta
E}=-\frac{8\sin\left(\frac{\theta}{2}\right)t^{2}}{3U}$. Adding the
contributions for the processes
$\left|a_{g}\alpha\right>\rightarrow\left|e_{g};i\,\alpha\right>$ and
$\left|a_{g}\alpha\right>\rightarrow\left|e_{g};\,i\beta\right>$, the total
energy lowering is given by

$\Delta
E=-\frac{8}{3}\frac{\sin\left(\frac{\theta}{2}\right)t^{2}}{U}-\frac{4}{3}\frac{\cos\left(\frac{\theta}{2}\right)t^{2}}{U^{\prime}-J}-\frac{4}{3}\frac{\sin\left(\frac{\theta}{2}\right)t^{2}}{U^{\prime}}.$
(15)

In the limit of ferromagnetism ($\theta=0$) and antiferromagnetism
($\theta=\pi$), equation (15) gives

$\displaystyle\Delta E^{\text{FM}}$
$\displaystyle=-\frac{4}{3}\frac{t^{2}}{U^{\prime}-J_{H}},$ (16)
$\displaystyle\Delta E^{\text{AFM}}$
$\displaystyle=-\frac{8}{3}\frac{t^{2}}{U}-\frac{4}{3}\frac{t^{2}}{U^{\prime}},$
(17)

which is in agreement with the previously derived expressions in Ref. [4].
Note that the following relationship holds in the atomic limit,
$U=U^{\prime}+2J_{H}$.

The total energy (15) evaluated for various values of the Hund’s coupling
$J_{H}$ and Hubbard repulsion $U$ is shown in Figure 2. The value $J_{H}=0.9$
eV corresponds to the value obtained in Ref. [4]. From (16) and (17) it can be
determined that the transition between ferromagnetic and antiferromagnetic
ground state takes place at $U/J\approx 4$. Indeed, we can see that this
behavior is reflected in our results for the limiting cases of $\theta=0,\pi$.
However, it becomes apparent that the lowest energy configuration for each Ti-
Ti pair is a noncollinear order, with an optimal magnetization angle depending
on the microscopic parameters $U$ and $J_{H}$. The asymptotic value of
$\theta$ in the limit $U\rightarrow\infty$ is
$\theta_{U\rightarrow\infty}=2\arctan 3\approx\frac{4}{5}\pi$. The non-
collinear order results from the multi-orbital nature of TiF3. In the absence
of the $e_{g}$ orbital, only the first term in the total energy (15) remains.
This term is minimized for the antiferromagnetic order ($\theta=\pi$), which
corresponds to a Mott insulator at half-filling.

We verify this behavior by performing non-collinear density functional theory
calculations for the primitive lattice with 2 Ti atoms, using the Vienna Ab
Initio Simulation Package (VASP) [11]. The exchange-correlation functional was
approximated by the generalized gradient approximation (GGA) [12]. We used a
$10\times 10\times 8$ $\boldsymbol{k}$-point mesh. We computed the total
energy for a constrained magnetization angle varying between $100^{\circ}$ and
$180^{\circ}$ degrees, introducing a Lagrange multiplyer of
$\lambda=0.2~{}\text{eV}/\mu_{B}$ [13]. Furthermore, we used a Hubbard-U of 3
eV and a Hund coupling of 0.2 eV. The cut-off energy was 290 eV. In agreement
with our model, the total energy is minimized for a magnetization angle of
$\frac{3}{4}\pi$ between the two Ti atoms in the primitive cell, as shown in
Figure 2(b) and Figure 2(c). In contrast to the simplified model, the ab
initio calculations show sharp steps in the total energy, as can be seen in
Figure 2(c). To explain these steps, we calculated the density of states for
three angles, $\theta=\frac{\pi}{2},\theta=\frac{2}{3}\pi$, and
$\theta=\frac{3}{4}\pi$. While the system remains insulating for angles larger
than $\theta=\frac{3}{4}\pi$ ($\approx$ antiferromagnetic) it cuts a band for
smaller angles. Interestingly, several such transitions seem to take place,
leading to a step-wise increase in the total energy.

##  III Experimental evidence for insulating behavior

To examine possible insulating behavior of TiF3, we present frequency-
dependent far-infrared reflectivity conducted on a hydraulically-pressed
pellet using a combination of beamsplitters and helium-cooled bolometer
detectors to cover the spectral range where strong signatures of single-phonon
lattice excitations and significant free carrier contributions are expected to
be obvious, if present. The clear presence of strong reststrahlen bands shows
that vibrations dominate the polarizability of this material. These strong
lattice vibrations can be fit to a set of five Lorentz oscillators in this
spectral range and are similar to those observed in single crystals of the
related compound ScF3[14]. Kramers-Kronig consistency in guaranteed by the
form of the Lorentz oscillator. Importantly, no free carrier contribution is
used in the fitting procedure, consistent with a vanishingly small Drude
weight expected for an insulating material. We conclude that the infrared
response of TiF3 is inconsistent with a scenario of metallic conduction and is
consistent with our attempts to measure directly the resistivity of the
sample. In addition, an analysis of magnetic susceptibility data appears to
show no long-range antiferromagnetic order down to 10 K [4, 7].

Figure 3: Optical Reflectivity of hydraulically-pressed pellets of TiF3. Two
broad reststrahlen bands show that the system is insulating.

##  IV First Principles Electronic and Magnetic Structure

Figure 4: Density of states (DOS) from non-collinear magnetic GGA+U VASP
calculations using a unit cell of Ti6F18 for selected values of $U$ and
$J_{H}~{}0.0-0.4$. The Fermi level is chosen to be at zero energy for $U\leq
1$ and is in the gap above zero energy for higher U. Note the metallic
behavior at $U=0,1$ and the insulating behavior for $U\geq 3$ eV.

In order to reveal the metal-insulator transition depending on the interaction
strength and non-collinear magnetism, we have performed additional VASP
calculations both on a full rhombohedral unit cell (space group
R$\overline{3c}$, $\\#$167) and on a primitive cell. In particular, we have
used a larger unit cell containing 6 Ti and 18 F atoms, in order to probe
possible non-collinear magnetism. For the large cell, a $6\times 6\times 6$
$\boldsymbol{k}$-space $\Gamma$ centered mesh according to Monkhorst and Pack
[15] was used during the self-consistent cycle. Structural optimization was
performed until the Hellman-Feynman forces acting on the atoms were
negligible.

The density of states plots in Fig.4 illustrate a metal-insulator transition
taking place as a function of the Hubbard $U$ parameter in the extended cell.
At higher $U$ values, the insulating gap is seen to increase due to increasing
$d$-orbital repulsion. As we see here, a large $U$ value such as $U=8.1$ eV
used in Ref. [4], is not necessary to yield an insulating state. This value is
also significantly higher than the Hubbard repulsion $U=3.3$ eV used by
Mattsson and Paulus [5]. The latter value is more consistent with typical
values of Hubbard repulsion $U\approx 3-4$ eV obtained for various transition
metal trihalides [16, 17, 18]. We note that recent frequency dependent far-
infrared reflectivity measurements support an insulating state in TiF3 [7].

First principles-based non-collinear magnetic calculations of TiF3 extend the
capability of revealing the interplay between electronic and magnetic
properties. Typically, non-collinear structures are found in materials where
the topology/geometry of the arrangement excludes collinear order [19, 20, 21,
22]. This could be possibly due to frustrations arising from a competition of
ferromagnetic and antiferromagnetic interactions, for example, as in
triangular lattices. Previous computational work discussing the electronic and
magnetic structure of TiF3 confined the magnetism to collinear magnetic order
[4, 5]. In agreement with Ref. [4], we find that for a collinear spin
arrangement with structural optimization, Ti2F6 VASP calculations predict
metallic, ferromagnetic behavior at $U=1$ eV, $J_{H}=0.2$ eV while at $U=3$ eV
or higher, with similar $J_{H}$ values, an insulating state is seen from the
density of states. The magnetic moment in the ferromagnetic phase is about 1
$\mu_{B}$ per Ti atom.

When a larger unit cell (Ti6F18) is used to carry out first principles-based
VASP GGA+U calculations, non-collinear magnetism was found. In Ti, the spin-
orbit interaction is not large and of the order of 20-30 meV or less. However,
this is most likely sufficient to introduce changes in magnetic order at low
temperature. With small $U$ values (of the order of $2$ eV), a metallic state
is seen with some magnetic order which is neither completely ferromagnetic nor
antiferromagnetic. These local moments appear to lie on a certain
crystallographic plane (as seen in Fig. 5(a) - $(1{\bar{1}}$1) plane) with
pairs of Ti atoms antiferromagnetically aligned at different angles with
respect to each pair but on a planar configuration. However, there could be
some sensitivity to the chosen initial magnetic configuration here. At larger
$U$ values ($U\geq 3$ eV), the system is insulating with some of these moments
pointing out of the above-mentioned plane, indicating possible magnetic
canting and frustration.

Figure 5:  (a) Magnetic structure at $U=2$ eV for Ti6F18 from non-collinear
magnetic calculations using VASP. These moments (shown in red) lie in a plane
for $U=2$ eV while at larger $U$ values some of the moments point out of this
plane indicating possible magnetic frustration. Magnetic order seen here
depends on the initial choice of the magnetic orientations used. (b) Fourier
transform of the Heisenberg exchange parameters ($U=3.3$ eV). The maximum
between the $L$ and $W$ points in the Brillouin zone points towards an
incommensurate magnetic order or frustration. (c) Magnon spectrum computed for
an antiferromagnetic ground state.

To estimate the magnetic ground structure, we have calculated Heisenberg
exchange coupling constants $J_{ij}$ using the magnetic force theorem
implemented within a first-principles Green function method [23, 24]. We note
that this method is complementary to our VASP calculations and has been
consitently shown to give similar results in electronic structure calculations
[24]. The full Fourier transform of the exchange constants,
$J(\boldsymbol{q})$, is presented in Fig. 5(b). $J(\boldsymbol{q})$ is
proportional to the transversal magnetic susceptibility and its maximum
corresponds to the total energy minimum. A maximum of $J(\boldsymbol{q})$ at a
high symmetry point would reflect a commensurate magnetic structure of ferro-
(if the maximum is at $\Gamma$ point) or an antiferromagnetic order. In the
case of TiF3 the maximum of $J(\boldsymbol{q})$ was found between
$L=(1/2,1/2,1/2)$ and $W=(1,1/2,0)$ high symmetry points, which corresponds to
a noncollinear magnetic order with the ordering vector
$\boldsymbol{Q}=(1,0.32,0.18)$. The origin of the noncollinear structure is
the short-range nature of the magnetic interaction. For the case of
$U=3.3~{}\text{eV}$, the exchange interaction between the nearest magnetic
moments is about -0.78 meV, while other coupling constants were found to be
negligibly smaller (magnitude $<0.02~{}\text{meV}$). Thus, a dominant
antiferromagnetic interaction between the nearest moments leads to a
frustration of magnetic moments and forming a noncollinear magnetic order. Our
calculations showed that the obtained $J_{ij}$ depends weakly on the choice of
the Hubbard parameter $U$ ($2~{}\text{eV}\leq U\leq 6~{}\text{eV}$). Thereby,
the critical transition temperature was estimated in the range of 18-22 K.

##  V Conclusions

In agreement with conclusive experimental reflectivity measurements, we argue
that TiF3 is an insulator, with a gap due to strong electron-electron
correlations. Due to the absence of a clear antiferromagnetic signal in the
material [7], we explored a potential magnetic frustration or the tendency
towards non-collinear magnetism. Here we extended the model of Perebeinos and
Vogt [4], showing that a non-collinear arrangement of magnetic moments between
neighboring Ti ions might be favorable. This claim is furthermore corroborated
by density functional theory calculations that include explicit correlations.
First, we observed a non-collinear magnetic ordering by performing total
energy calculations using a pseudopotential method. Furthermore, we computed
Heisenberg exchange parameters and found a strong antiferromagnetic nearest
neighbor coupling with all other exchange parameters being vanishingly small.
The Fourier transform of the Heisenberg exchange clearly points towards an
incommensurate magnetic order. Yet, the exact magnetic configuration seems to
be highly sensitive to input parameters, leading to open questions such as the
possible existence of a spin liquid phase.

R.M.G. acknowledges support from the Swedish Research Council (VR starting
Grant No. 2022-03350) and Chalmers University of Technology. J.N.H.
acknowledges support from the U.S. National Science Foundation, award No. NSF-
DMR-1905862. A.E. acknowledges funding by Fonds zur Förderung der
Wissenschaftlichen Forschung (FWF) Grant No. I 5384. We also acknowledge the
computing resources provided by the Center for Functional Nanomaterials, which
is a U.S. DOE Office of Science Facility, at Brookhaven National Laboratory
under Contract No. DE-SC0012704 and the Swedish National Infrastructure for
Computing (SNIC) via the National Supercomputer Centre (NSC).

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