# Reclaiming Power over AI: Equipping Queer Teens as AI Designers for HIV
Prevention

William Liem<EMAIL_ADDRESS>0000-0002-5873-6470 Northwestern
University633 North St. Clair, Floor 20ChicagoIllinoisUSA60611 ,  Andrew Berry
Northwestern UniversityChicagoUSA<EMAIL_ADDRESS>and  Kathryn
Macapagal  Northwestern UniversityChicagoUSA<EMAIL_ADDRESS>
(2024)

###### Abstract.

In this position paper, we explore the potential of generative AI (GenAI)
tools in supporting HIV prevention initiatives among LGBTQ+ adolescents. GenAI
offers opportunities to bridge information gaps and enhance healthcare access,
yet it also risks exacerbating existing inequities through biased AI outputs
reflecting heteronormative and cisnormative values. We advocate for the
importance of queer adolescent-centered interventions, contend with the
promise of GenAI tools while addressing concerns of bias, and position
participatory frameworks for empowering queer youth in the design and
development of AI tools. Viewing LGBTQ+ adolescents as designers, we propose a
community-engaged approach to enable a group of queer teens with sexual health
education expertise to design their own GenAI health tools. Through this
collaborative effort, we put forward participatory ways to develop processes
minimizing the potential iatrogenic harms of biased AI models, while
harnessing AI benefits for LGBTQ+ teens. In this workshop, we offer
specialized community-engaged knowledge in designing equitable AI tools to
improve LGBTQ+ well-being.

generative AI, bias in AI, queer communities, sexual orientation, HIV
prevention

††copyright: acmcopyright††journalyear: 2024††doi: XXXXXXX.XXXXXXX††ccs:
Human-centered computing Human computer interaction (HCI)††ccs: Social and
professional topics Sexual orientation; Gender

##  1\. Introduction

The advent of GenAI introduces immense promise in personalized healthcare,
however, biased AI models risk further marginalizing historically vulnerable
populations such as the LGBTQ+ community. Particularly, LGBTQ+ adolescents
encounter distinct health disparities, highlighted by alarming rates of HIV
infections and challenges in discreetly accessing HIV preventive services,
often hindered by laws restricting minors from self-consenting (for Disease
Control, 2022a). GenAI tools like ChatGPT have the potential to enable LGBTQ+
adolescents to seek sexual health information anonymously. However, in
alignment with open problems for AI in designing for well-being, we grapple
with the relevance and effectiveness of AI outputs in addressing such
disparities. Given the complexity of personal health data and biases in AI
models, addressing these issues emerges as critical focal points. Moreover, we
argue that building LGBTQ+ adolescents’ AI literacy and involving them in co-
design processes are essential for sustaining diversity, inclusivity, and
adaptability in AI-driven healthcare designs.

Within this context, GenAI tools offer promising avenues for bridging
informational gaps and enhancing healthcare access. However, inherent biases
within AI models can cause undue harm to queer users. For example, AI models
are trained on cisnormative and heteronormative datasets (Costanza-Chock,
2018) and are capable of misgendering users (Keyes, 2018). This can exacerbate
emotional distress among transgender users who have historically faced
systemic identity neglect (Scheuerman et al., 2019). In this position paper,
we advocate for an approach aimed at co-developing safeguards against
potential AI harm to LGBTQ+ adolescents and exploring potential opportunities
for GenAI tools in addressing teen sexual health education. We highlight the
need for adolescent-centric HIV prevention strategies among LGBTQ+ adolescents
(section 2.1), survey the potential promise of ChatGPT for personalized LGBTQ+
inclusive healthcare (section 2.2), and propose a combinatory approach to co-
designing AI tools that mitigate potential harms of biased models (section
2.3). We conclude with a forthcoming research plan that responds to the
growing need for involving LGBTQ+ adolescents in the interrogation and
development of AI tools. We anticipate this work to demonstrate the potential
for AI scientists, policymakers, and practitioners to work with
disenfranchised queer populations in creating more equitable AI health tools.

##  2\. Background

###  2.1. Co-Designing HIV Prevention for LGBTQ+ Youth

In 2021, people aged 13 to 24 accounted for almost 20 percent of new HIV
infections, (for Disease Control, 2024) with the COVID-19 pandemic
significantly disrupting the availability of HIV health services, (Rick et
al., 2022) including pre-exposure prophylaxis (PrEP), a once-daily medication
with 99% effectiveness at preventing HIV transmission (for Disease Control,
2022b). This trend underscores the pressing need to address HIV prevention and
education strategies, particularly among this vulnerable population. Higher
HIV incidence rates may be attributed to the discontinuation of PrEP use by 1
in 7 young men who have sex with men aged 17-24 (Hong et al., 2022). Younger
people are particularly vulnerable, as they are more likely to discontinue
PrEP (Chow et al., 2021) and less likely to reinitiate PrEP (Rao et al.,
2022). This discontinuation trend is emblematic of broader challenges within
the LGBTQ+ youth community, which faces a disproportionately higher burden of
HIV compared to their heterosexual peers (Ocfemia et al., 2018). Multiple
studies reveal that only half of LGBTQ+ teens know about PrEP’s existence and
less than 3 percent have ever used it (Dunville et al., 2021; Macapagal et
al., 2021; Matson et al., 2021; Moskowitz et al., 2021, 2020; Thoma and
Huebner, 2018). In response to these challenges, innovative approaches such as
co-author Dr. Macapagal’s Teen Health Lab’s work in developing adolescent-
centered digital HIV prevention programs and social marketing campaigns have
emerged to empower youth in HIV prevention, advocacy, and education (Macapagal
et al., 2022). By involving youth in the creation of their own health
promotion efforts, participatory initiatives like the Teen Health Lab
represent a promising model and ethos for reducing HIV incidence among young
people by cultivating a sense of ownership and agency in HIV prevention
strategies.

However, systemic barriers still plague LGBTQ+ youth regarding PrEP access and
information. Parents are key gatekeepers for PrEP uptake as LGBTQ+ youth rely
on them for navigating health care and associated costs (McKay and Fontenot,
2020), however, there is a risk of disclosing their sexual identity in the
process of exploring PrEP (Fisher et al., 2018; Moskowitz et al., 2020).
Recent work examining experiences of PrEP uptake among 100 adolescent sexual
minority men in the US calls for interventions to enhance adolescents’ self-
efficacy skills and PrEP knowledge, such as sexual health care coaching to
discuss PrEP uptake with parents or providers, enabling them to pursue PrEP
independently if desired (Gordon et al., 2024). With 56% of young people aged
18-24 using GenAI tools for information searching (Research, [n. d.]), there
is a prime opportunity to explore the role of GenAI in addressing barriers to
PrEP information seeking among LGBTQ+ youth. In the next section, we review
innovative approaches to leverage GenAI tools for personalized healthcare, and
contextualize it within an LGBTQ+ adolescent context.

###  2.2. Generative AI for Personalized Healthcare

Recent studies have shown that AI chatbots potentially improve knowledge about
and uptake of PrEP (Braddock et al., 2023; Hassani and Young, 2022) and
ChatGPT specifically shows some promise in counseling people living with HIV
on treatment (Koh et al., 2024). Further, GenAI tools like ChatGPT are
promising for delivering simplified health information (Ayre et al., 2023;
Biswas, 2023) and promoting health literacy (Coşkun et al., 2024), however,
caution is needed to mitigate the potential harms of biased algorithms towards
the queer community (Bragazzi et al., 2023; Tomasev et al., 2021; McAra-
Hunter, 2024). Datasets used to train chatbots and internet-based assistance
are often riddled with gender biases (Gross, 2023; Mittermaier et al., 2023)
such as the conflation of sex and gender, discourse of toxic masculinity, and
erasure of non-binary identities (Seaborn et al., 2023), which can result in
harmful outputs such as misgendering. This is particularly concerning given
that many trans people already struggle with gender dysphoria, where gendered
outputs may exacerbate emotional distress related to their gender identity or
social experiences (Scheuerman et al., 2019). Further, these harms could cause
people to disengage with GenAI tools completely, preventing them from reaping
the potential benefits of accessible personalized health information.

Nascent work on curbing gendered biases in large language models has called
for further work on creating “personalized guardrails” enabling end-users to
design the boundaries of AI tools via prompt engineering as a means to combat
gendered outputs of ChatGPT (Dwivedi et al., 2023). Scholars have sounded the
alarm for engaging marginalized communities in co-designing AI tools that meet
their needs (Bragazzi et al., 2023; Tomasev et al., 2021; McAra-Hunter, 2024;
Zytko et al., 2022; Buslón et al., 2023). In particular, recommendations
include promoting awareness of sex and gender biases in AI, advocating for
fair AI-personalized medicine, and including diverse LGBTQ+ perspectives in
either auditing or developing AI tools (Mahelona et al., 2023). We respond to
these recommendations by surveying approaches that have engaged marginalized
communities in the critique and interrogation of AI models in order to inform
our research agenda.

###  2.3. Community-Engaged AI Design

There is a pressing need to involve underrepresented and marginalized groups
in designing and developing AI solutions to improve one’s well-being. Such
involvement should build on the lived experiences of the user, incorporate the
diversity of queer communities, and accommodate the multitude of
considerations needed to design inclusive and equitable AI health tools.
However, inviting everyday individuals to interrogate and make AI systems more
equitable remains a perennial challenge, as it requires advanced technical
knowledge that often acts as a barrier to participation (Sloane et al., 2022)
GenAI tools like ChatGPT have opened a new format for an everyday audience to
influence AI outputs where each individual user can set parameters and
guidelines to shape the output. This presents a unique opportunity for
community participation in the critique of AI tools, however, we need to be
intentional with how participation is designed, as there are significant risks
with “participation washing” where shallow forms of engagement inhibit the
actual interrogation and redesign of these tools (Birhane et al., 2022; Agnew
et al., 2023). We draw from two promising frameworks for designing AI tools
with a non-technical audience: meta-design for resistance, and the
PARticipatory Queer AI Research in Mental Health (PARQAIR-MH) protocol.

####  2.3.1. Meta-design for resistance

Resistance as characterized by Agnew et al.,(Agnew et al., 2023) is a process
of adapting technologies to meet the community’s needs. This framework
addresses power imbalances inherent in technology-driven environments by
empowering individuals and communities to oversee data collection, algorithmic
decision-making, and AI model creation. This facilitates an environment where
individuals can actively oppose undesirable AI applications within their lives
and local settings. There are three levels of meta-design for resistance.

  1. (1)

Adversarial Attacks and Privacy Enhancing Technologies as AI Defenses: The
first level involves identifying tools to defend against AI harms, such as
exploiting how biased AI models can cause unintentional harm, and creating
guardrails to protect users from that.

  2. (2)

Collaboration for Resistance: The second level is collaboration for
resistance. Sociotechnical in nature, it explores how AI defenses and
collaborations can be designed together to empower resistance. An example is
when TikTok users replace words that would trigger algorithms to flag their
content as harmful, when they are trying to advocate for a social justice
issue (Karizat et al., 2021).

  3. (3)

Communities of Resistance: Supportive networks that facilitate collaboration,
knowledge sharing, and coordinated actions to resist AI harms, fostering
empowerment and grassroots efforts to challenge and mitigate the negative
impacts of AI technologies within marginalized communities. This bridges the
gap between research and practical application, while also shaping broader
cultural expectations of contestability and ownership in the face of data and
AI-related challenges.

Agnew et al., argue that central to this framework is the concept of designing
AI tools that align with the values of the community. This is necessary
because one goal of this framework is to enable end users to provide feedback
and iterate so that if there is value drift, communities have the ability to
realign the technology or stop its deployment. In order to achieve a consensus
on these values to center in the design process, we draw from the
PARticipatory Queer AI Research in Mental Health (PARQAIR-MH) protocol.

####  2.3.2. PARQAIR-MH

This community-engaged Delphi process incorporates queer perspectives on AI
design in mental health care (Joyce et al., 2023). This protocol aims to
deliver a toolkit that will help ensure that the specific needs of the LGBTQ+
communities are accounted for in mental health applications of data-driven
technologies. The protocol starts with a comprehensive literature review of
the problem space, with the goal of sharing findings back to queer
stakeholders to ground foundational knowledge of the AI problem space.
Subsequently, through a series of Delphi rounds, the group refines a moral
framework for how AI tools should be used and developed that meet the needs of
their community. It assists researchers, healthcare organizations, and
policymakers in deciding how to collect and use data on sensitive
characteristics appropriately.

We adapt this process to be inclusive, but not limited to, mental health care
in AI. We believe this protocol embodies scientific rigor through a Delphi
process for achieving a consensus across diverse perspectives, while also
adhering to the central tenets of community-based participatory research. We
conclude this position paper with a proposed research plan, one where we use
the aforementioned frameworks to engage an adolescent queer community to be AI
designers and have them explore concerns and opportunities as it relates to
LGBTQ+ health.

##  3\. Proposed Research Direction

Building across our multidisciplinary team in digital sexual health,
community-engagement, and values-aligned technology, we plan on engaging a
Youth Advisory Council (YAC) managed by the Teen Health Lab (led by co-author
Dr. Macapgal) housed within Northwestern’s Institute of Sexual and Gender
Minority Health and Wellbeing. In the near term, we aim to grow a GenAI
working group between our team and YAC. The YAC uses a private, teen-only
Discord server facilitated by young LGBTQ research staff where LGBTQ high
school students across the U.S. have offered meaningful input into four teen-
centered digital and multimedia sexual health projects over the last year.
This YAC has co-designed a text-messaging intervention aimed at promoting
engagement in sexual healthcare among LGBTQ+ adolescents and offered feedback
on an adolescent sexual consent intervention. We have worked with this group
on co-designing PrEP4teens.com (PrEP4Teens, [n. d.]), a website for promoting
and informing PrEP.

The YAC is representative of a wide variety of sexual, gender, and racial
identities–members who are historically excluded from the participation of AI
tools, which has been repeatedly cited to be a significant contributor to
biased AI algorithms (Ayre et al., 2023; Biswas, 2023; Coşkun et al., 2024).
Our research aims to leverage the vested interests of the YAC and create an
opportunity for them to shape AI tools in the way they deem necessary for
equitable HIV information seeking and dissemination. We plan on following an
adapted version of the PARQAIR-MH protocol to build on our existing
infrastructure and equip the YAC with a foundational knowledge of GenAI,
informing them about documented use cases and harms towards the queer
community. This will prepare them to explore what opportunities, concerns, and
considerations must be made to leverage these tools for HIV prevention and
information dissemination equitably, resulting in a moral framework for co-
design.

Following suit, we follow the meta-design for resistance framework where we
will lead a series of co-design workshops and usability testing to identify
requirements for a queer-centric AI health assistant on ChatGPT and conduct
one-on-one usability tests to exploit unintended harms. Through this, we will
present back findings to the YAC to elicit reactions and create a robust list
of guidelines to safeguard queer adolescents who may use ChatGPT for HIV
prevention information. We plan on finding opportunities with the YAC for
dissemination channels that would promote AI literacy.

We believe this is a promising approach to participatory methods in designing
personalized AI health tools. Through engaging a diverse cast of queer teens
who are experts in sexual health, we can build interventions to minimize the
potential iatrogenic harms of biased AI models. In this workshop, we are
poised to contribute our expertise in designing with marginalized communities
to cultivate the development of equitable AI tools. We anticipate that this
project will significantly benefit from the insightful discussions facilitated
by the one-day workshop on designing with AI for well-being.

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Emergent flat band and topological Kondo semimetal
driven by orbital-selective correlations

Lei Chen1, Fang Xie1, Shouvik Sur1, Haoyu Hu2, Silke Paschen3,1, Jennifer
Cano4,5, and Qimiao Si1,∗

1Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice
University, Houston, Texas 77005, USA

2Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018
Donostia-San Sebastian, Spain

3Institute of Solid State Physics, Vienna University of Technology, Wiedner
Hauptstr. 8-10, 1040 Vienna, Austria

4Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY
11794, USA

5Center for Computational Quantum Physics, Flatiron Institute, New York, NY
10010, USA

Flat electronic bands are expected to show proportionally enhanced electron
correlations, which may generate a plethora of novel quantum phases and
unusual low-energy excitations. They are increasingly being pursued in
$d$-electron-based systems with crystalline lattices that feature destructive
electronic interference, where they are often topological. Such flat bands,
though, are generically located far away from the Fermi energy, which limits
their capacity to partake in the low-energy physics. Here we show that
electron correlations produce emergent flat bands that are pinned to the Fermi
energy. We demonstrate this effect within a Hubbard model, in the regime
described by Wannier orbitals where an effective Kondo description arises
through orbital-selective Mott correlations. Moreover, the correlation effect
cooperates with symmetry constraints to produce a topological Kondo semimetal.
Our results motivate a novel design principle for Weyl Kondo semimetals in a
new setting, viz. $d$-electron-based materials on suitable crystal lattices,
and uncover interconnections among seemingly disparate systems that may
inspire fresh understandings and realizations of correlated topological
effects in quantum materials and beyond.

Introduction.  Certain crystalline lattices feature flat bands, via
frustration caused by destructive interference in electron motion 1, which are
increasingly being explored in $d$-electron-based systems 2, 3. The reduced
bandwidth correspondingly enhances the effect of electron correlations. In
addition, such flat bands are often topologically nontrivial. As such, these
systems represent a new platform to uncover novel physics for both correlation
and topology as well as their interplay 4. For example, kagome metals may host
flat bands and have been the subject of considerable recent interest for
realizing unusual forms of charge-density-wave order 5, 6, 7, 8, 9, 10. They
have also been implicated to exhibit a type of strange metal behavior 11, 12,
13 that resembles what has been extensively studied in quantum critical heavy
fermion metals 4, 14, 15.

In order to strongly influence the low-energy physics, the flat bands need to
be placed near the Fermi energy. However, this typically is not the case.
There have been considerable recent experimental efforts to tune the flat
bands to the vicinity of the Fermi energy. With the rare exception coming from
materials search 11, the tuning study has met with only a limited success 16,
17. Because the flat bands are associated with $d$-electrons, the energy
scales that determine the flat-band placement (with respect to the Fermi
energy) are relatively large and, as a result, it is challenging to achieve
the required tuning. We are thus motivated to ask the following important
questions: Can electron correlations generate flat bands at the Fermi energy
in $d$-electron-based systems? And, if so, to what extent do the resulting
phases display nontrivial correlation and/or topological physics?

We address both issues in a Hubbard model in which the noninteracting limit
features a topologically nontrivial flat band that is far away from the Fermi
energy. Due to a well-separated hierarchy in the widths of the flat band and
wide bands that it is coupled to, orbital-selective Mott correlations develop
13. We show that such orbital-selective correlations lead to emergent flat
bands that are pinned to the Fermi energy. Moreover, based on considerations
of symmetry constraints 18, 19, 20, 21, 22 in interacting settings 23, we
demonstrate that the emergent flat bands lead to a topological Kondo
semimetal. The latter is in the same family as Weyl Kondo semimetals that
appear in models 24, 25 and materials 26, 27 of both existing and designed 28,
23 heavy fermion systems. The qualitative physics is illustrated in Figs.
1(a-c). Our results motivate a design principle for the Weyl Kondo semimetals
in the new setting of $d$-electron-based systems, and point to the realization
of fractional Chern insulators 29 in transition-metal compounds.

One-orbital Hubbard model on the clover lattice.  For a proof-of-principle
demonstration, we consider a variant of the kagome lattice, the two-
dimensional (2D) clover lattice. As shown in Fig. 1(d), it contains five
sublattices per unit cell. This lattice features a flat band [see the
Supplementary Information (SI), Sec. II]. As a case study, the model is
simplified while preserving the topological nature of the flat band; we do so
by removing the $C_{3}$ symmetry of the clover lattice (see the Methods),
leaving only a mirror symmetry $M_{x}$. There is one orbital per site. The
Hubbard model takes the form $\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{1}$,
where $\mathcal{H}_{0}$ is the kinetic term and $\mathcal{H}_{1}$ represents
the onsite Hubbard interaction. We consider the generic setting that has not
been analyzed before, namely with the flat band of the noninteracting
Hamiltonian being far away from the Fermi energy, as illustrated in Fig. 1(a)
and shown in Fig. 2(a). Further details of the model are given in the Methods
and in the SI (Sec. I).

The lattice can be divided into two groups of sublattices [denoted by blue and
yellow dots in Fig. 1(d)], which contain different numbers of sites per unit
cell. The flat-band formation can be seen by considering only the nearest-
neighbor hopping between the blue and yellow sites, reflecting a destructive
interference of the electronic wavefunction on the lattice 30 (see the SI,
Sec. II). The flat band overlaps with the wide bands.

Molecular orbitals, effective extended Hubbard model and the solution method.
A flat band that is topologically nontrivial cannot by itself be represented
by exponentially localized symmetry-preserving (Kramers-doublet) Wannier
orbitals 22. Such a wannierization only becomes possible when other bands are
considered along with the flat band. In addition, a flat band coming from
destructive interference comprises states from multiple (inequivalent) atomic
sites. If one Wannier orbital is to primarily capture this flat band, this
Wannier orbital (and, by extension, the others accompanying it) must involve
multiple atomic orbitals. In other words, in this case, the Wannier orbitals
are necessarily molecular orbitals.

In our case, we can restrict to three bands (see the SI, Sec. III). We find
the centers of the three localized Kramers-doublet Wannier orbitals to be
located near the geometric center of the unit cell, which forms a triangular
lattice [see Fig. 1(e)]. Importantly, one Wannier orbital primarily captures
the flat band; it is the most localized and is denoted as the $d$ orbital. The
other two Wannier orbitals are dominated by the wide bands; they decay much
more slowly and are marked as $c$ orbitals. The large difference in the width
of the flat band ($D_{\rm flat}$) and the wide bands ($D_{\rm wide}$) opens up
a range of interactions that are in between. In this range, the electron
correlations are strongly orbital-selective and the system affords a
Kondo/Anderson-lattice description 13.

We project the Hubbard model of the original lattice to the Wannier basis.
This leads to the effective model expressed in terms of the $d$ and $c$
Wannier orbitals with $H_{\mathrm{eff}}=H_{0}+H_{int}$. The kinetic term is
specified in the Methods. For the interaction terms, we keep the most dominant
interactions on the Wannier basis. They include the onsite Hubbard interaction
among the $d$ electrons and the density-density interactions between the $d$
and $c$ electrons:

$\displaystyle H_{\rm int}$ $\displaystyle=H_{d}+H_{F}$ (1)
$\displaystyle=\sum_{i}\frac{u}{2}\left(n_{i\uparrow}^{d}+n_{i\downarrow}^{d}-1\right)^{2}+\sum_{i,\alpha}F_{\alpha}\,n_{i}^{d}\,n_{i}^{c,\alpha}$

where $n_{i,\sigma}^{a}=a^{\dagger}_{i\sigma}a_{i\sigma}$, with
$a=d,c_{\alpha}$, $\alpha=1,2$, and $n_{i}^{a}=\sum_{\sigma}n_{\sigma}^{a}$.
The onsite Hubbard interaction on the $d$-orbital, $u$, is the most dominant
one, given the much more localized nature of this Wanner orbital. The density-
density interactions between the $d$ and $c$ electrons, $F_{\alpha}$, are
weaker but also sizable: $F_{1}/u\approx 0.3$ and $F_{2}/u\approx 0.25$. These
effective interaction parameters are determined by those of the original
Hubbard model (see the SI, Sec. III). The interactions among the $c$ electrons
are relatively small compared to their bandwidths and, accordingly, will be
unimportant.

To take into account the effect of the interactions, we use the U(1) slave
spin (SS) method 31. Given that only the onsite interaction of the $d$ orbital
is important, we need to introduce a SS representation for the $d$ orbital
only: $d_{i\sigma}^{\dagger}=o^{\dagger}_{i\sigma}f_{i\sigma}^{\dagger}$,
where the auxiliary bosonic and fermionic operators, $o^{\dagger}$ and
$f_{\sigma}^{\dagger}$, carry the charge and spin degrees of freedom,
respectively. We treat the SS formulation at the saddle-point level and self-
consistently solve the corresponding Hamiltonians for the SS and the auxiliary
fermion parts. The SS method is also used to obtain the contributions to the
single-electron excitations from the (interaction-driven) incoherent part of
the spectrum. The details are found in the Methods and SI (Sec. V).

Emergent flat band at the Fermi energy.  We are now in position to discuss the
effect of interactions on the single-electron excitations. Consider first the
density of states (DOS). In the noninteracting case, as shown in Fig. 2(a),
the $d$ electron DOS (the red curve) has a sharp peak compared with the
background (purple color) $c$ electron component. This reflects the $d$
electrons as primarily describing the flat band. As can be seen, the peak is
located far away from the Fermi level, which also reflects its origin from the
noninteracting flat band (see the SI, Sec. I).

Importantly, under the influence of electron correlations, a new flat band
emerges. This is demonstrated in Fig. 2(b) with $u=1.6$. The emergent flat
band is pinned to the Fermi energy, as captured by the coherent peak (the red
solid lines) in the DOS. The background DOS associated with the conduction $c$
electrons is largely unchanged from its noninteracting counterpart. Varying
the interaction strongly influences the spectral weight of the emergent flat
band (see the SI, Sec. VI): This part of the spectral weight is reduced as the
interaction increases [comparing Fig. 2(b) and Fig. S5(a)]; the reduced
spectral weight is transferred to the incoherent part (the red dashed lines).
This form persists until the weight of the coherent peak is completely lost
and the system goes through an orbital-selective Mott transition. When that
happens, the incoherent parts of the single-electron excitations develop into
the full-fledged lower and upper Hubbard bands [see the SI, Sec. VI and Fig.
S5(b)].

Orbital-selective Mott correlations.  To expound the origin of the emergent
flat band, we further analyze the orbital-selective Mott correlations in the
regime of interactions of our interest, viz. $D_{\rm flat}<u<D_{\rm wide}$. As
shown in Fig. 3, the metal-to-insulator transition of the $d$ electrons occurs
at $u_{c}=2.4$. We reiterate that the range of the interactions being
considered here is weaker than the width of the wide bands associated with the
$c$ electrons (see the SI, Fig. S3). Thus, the $c$ electrons are fully
itinerant. This justifies the neglect of the interactions among the $c$
electrons, as we have done, so that the quasiparticle weight of the
$c$-electrons remains to be $1$ as seen in Fig. 3. The existence of an
orbital-selective Mott transition is further illustrated in the nature of the
Fermi surface. As shown in Fig. 3 (insets), the Fermi surface undergoes a
dramatic change across the transition. This change parallels the electron
localization-delocalization (Kondo destruction) physics of heavy fermion
systems 32, 33, 34, 35, 36. The phase with the $d$-electrons being itinerant
corresponds to the Kondo-screened phase, in which the local moment is
converted into (fragile, or heavy) electronic excitations that hybridize with
the conduction electrons to form the quasiparticles. By contrast, the orbital-
selective Mott phase (OSMP) is analogous to the Kondo-destroyed phase of the
heavy fermion systems, in which the Fermi surface is formed entirely from the
conduction electrons. The fact that our noninteracting flat band is, to begin
with, far away from the Fermi energy makes the parallel with the heavy fermion
systems especially clear. Our analysis of this flat-band system provides a
realization of the Kondo physics in a one-band Hubbard model that physically
describes a $d$-electron-based system.

More specifically, the value of the interaction illustrated in Fig. 2(b) is
marked by an arrow in Fig. 3. The differentiation between the quasiparticle
weights of the $d$ and $c$ electrons at this interaction characterizes the
orbital-selective nature of the electron correlations. This is reflected in
the $d$-electron spectral weight: as seen in Fig. 2(b), the incoherent peaks
(the red dashed lines) are well formed, which corresponds to the precursor of
the lower and upper Hubbard bands of the OSMP [see the SI, Fig. S5(b)]. The
coherent spectral weight, i.e. the central peak [the red solid line of Fig.
2(b)] is thus described in terms of the Kondo resonance of a Kondo lattice
model, in which the local moments correspond to the effective spin degrees of
freedom associated with the lower and upper Hubbard bands. This description
makes precise the notion that the flat band at the Fermi energy is emergent,
driven by the orbital-selective Mott correlations.

Topological Kondo semimetal.  The energy dispersion of the electronic states
is shown in Fig. 4. From the dispersion of the interacting ($u=1.6$) case, we
again see that a Fermi-energy-bound flat band emerges in the interacting case.

We are then in position to analyze the symmetry constraints18, 19, 20, 21, 22.
In the noninteracting limit, the three Wannier orbitals have different $M_{x}$
eigenvalues. The flat $d$ orbital has the $M_{x}$ eigenvalue $+1$, while the
two $c$ orbitals have the $M_{x}$ eigenvalues of $-1$ and $+1$ respectively
13. In the presence of time-reversal symmetry, along the $\Gamma-K$ line, the
flat band from the $d$ orbital has a symmetry-protected Dirac crossing with
the $c$ orbital of the opposite mirror eigenvalue. This same symmetry
constraint also applies to the Kondo-driven flat band. The Dirac node for the
emergent flat band is shown in Fig. 4(b). A node from the flat band close to
the Fermi energy allows a high tunability. As shown in Fig. 4(d), a relatively
small Zeeman coupling (illustrated here with $m=0.03$, which is small compared
to the width of the emergent flat band of $\sim 0.1$) causes a substantial
separation of the nodes. These nodes now have two-fold degeneracy.

The orbital-selective Mott correlations are caused by local correlations.
While we have provided a case study of how such correlations give rise to
emergent flat bands in a particular 2D model, a similar conclusion is expected
in general cases, including for models in three dimensions (3D). There is an
important distinction though. In 3D, topological nodes develop in the presence
of SOC under symmetry constraints 37, 22. For noncentrosymmetric systems, or
for centrosymmetric systems with the breaking of time-reversal symmetry, we
can then expect the emergent flat bands to feature Weyl nodes leading to a
Weyl Kondo semimetal.

Design principle for Weyl Kondo semimetals in physical $d$-electron systems.
Weyl Kondo semimetals have so far been explored in $f$-electron-based
materials, Ce3Bi4Pd3 26, 27 and several newly proposed Ce-, Pr- and U-based
compounds 28, 23. The present work leads us to propose a design principle for
realizing Weyl Kondo semimetals in a new setting. Importantly, our theoretical
results are expected to be robust against the effect of the residual
interactions among the quasiparticles. This is so because the origin of the
topological nodes lies in the symmetry constraints, which have recently been
shown to operate on the eigenvectors of the matrix associated with the exact
single-electron Green’s function of an interacting system 23.

Accordingly, our proof-of-principle demonstration enables us to advance a new
materials design procedure for Weyl Kondo semimetals in the setting of
$d$-electron-based systems. The procedure would start from 3D lattices that
can host flat bands from quantum interference. Examples include the pyrochlore
lattice 38, the perovskite lattices 39, and other 3D versions of the bipartite
crystalline lattices 40. We seek materials with $d$-elements, and utilize
orbital-selective correlations to drive interacting flat bands that are Fermi-
energy-bound. Symmetry constraints can then lead to either Dirac or (with the
breaking of inversion or time-reversal symmetry) Weyl nodes in these emergent
flat bands. The latter case corresponds to a Weyl Kondo semimetal.

The procedure for this materials identification approach goes beyond that for
Weyl Kondo semimetals in $f$-electron-based systems 28. In addition to the
requirement for both correlations and crystalline symmetry constraints, it
also involves the crystal lattice conditions for the formation of flat bands
in the bare dispersion. We reiterate that the noninteracting flat bands are
not required to be near the Fermi energy. This is an important feature in the
proposed materials design principle, given that the noninteracting flat bands
in relevant materials are generically away from their Fermi energy.

Implications for fractional Chern insulators.  Fractional Chern insulators,
with a fractional quantum Hall effect and the associated fractional charge in
a lattice setting, have been proposed in correlated models with an appropriate
filling ratio of a flat band when the latter crosses the Fermi energy 41, 42,
43. Experimental evidence has recently been identified in twisted bilayer
graphene 29, in which the moiré bands are located near the Fermi energy, in a
small external magnetic field. Our results on the Fermi-energy-bound emergent
flat bands raise the possibility of another potential platform to realize the
fractional Chern insulators, namely in $d$-electron-based 2D systems. Indeed,
when a spin-orbit coupling is included in the 2D model, the Dirac node is
gapped leading to flat $Z_{2}$ topological bands (see the SI, Sec. IV). The
combination of the flatness of the associated bands (see the SI, Fig. S4) and
the residual interactions among the heavy quasiparticles represents a
condition that is similar to what happens in the moiré systems 29; however,
the $Z_{2}$ nature of the flat bands makes them distinct and rare 44.
Accordingly, with appropriate fillings, our results suggest that the
corresponding $d$-electron-based 2D materials provide a new setting for
realizing a fractional Chern insulator. The naturalness of the emergent flat
band crossing the Fermi energy makes our proposal robust. Thus, this
represents a promising new direction for a systematic examination.

Discussion and summary.  Our work opens a new bridge between topological flat
bands and correlation physics. The interaction effect tends to localize the
molecular orbital that has the most overlap with the flat band. As a result,
these molecular orbitals play the role of local moments, by analogy with the
local spins of Kondo systems. Our work provides a rare non-perturbative way to
address the interplay between correlations and topology effects in such flat
band systems and, as such, promises to elucidate the physics of correlated
kagome transition-metal compounds 2, 3, 11, 12 and related materials. We also
expect that our analysis will inspire new understandings of moiré structures,
which are increasingly being viewed from a Kondo perspective 45, 46, 47, 48,
49. Our work has also allowed us to advance a new materials design principle
to identify Weyl Kondo semimetals in the new setting of $d$-electron-based
systems. We expect the interconnections that our work reveals among seemingly
disparate systems to inspire new realizations and understandings of correlated
topological effects in a wide variety of quantum materials and structures.

Methods
Hubbard model on the clover lattice.  The clover lattice has been discovered
in real materials such as the van der Waals system $\mathrm{Fe_{5}GeTe_{2}}$
30, 50. As shown in Fig. 1(d), it contains five sites in each unit cell, which
are reclassified into two groups as marked by the yellow and blue colors. For
an illustrative purpose, we restrict our model to have only a $d_{z^{2}}$
orbital on each site. The case with other $d$-orbitals, such as
$d_{xz}/d_{yz}$, have a similar realization of the geometry-induced flat bands
30. We consider the Hubbard model written as
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{1}$, where $\mathcal{H}_{0}$ is the
kinetic term that connects the two different groups of sublattices and
$\mathcal{H}_{1}$ represents the onsite one-orbital Hubbard interaction. We
label the orbitals based on the group of sublattices to be $A/B$ and $C/D/E$
respectively. For each site, we consider the onsite Hubbard interaction,

$\mathcal{H}_{1}=U\sum_{i,\alpha}n_{i,\uparrow}^{\eta_{\alpha}}n_{i,\downarrow}^{\eta_{\alpha}}\,,$
(2)

where $\eta_{\alpha}$ ($\alpha=1\sim 5$) goes through all the five orbitals in
each unit cell. The kinetic Hamiltonian is written as

$\mathcal{H}_{0}=\sum_{ij,\alpha\beta,\sigma}t\eta_{i\alpha\sigma}^{\dagger}\eta_{j\beta\sigma}-\mu_{0}\sum_{i\alpha\sigma}\eta^{\dagger}_{i\alpha\sigma}\eta_{i\alpha\sigma}+\sum_{i,\sigma,\alpha\in\\{C,D,E\\}}m\eta_{i\alpha\sigma}^{\dagger}\eta_{i\alpha\sigma}+\sum_{i\sigma\alpha\in\\{D,E\\}}\gamma\eta^{\dagger}_{i\alpha\sigma}\eta_{i\alpha\sigma}\,.$
(3)

Here $t$ denotes the nearest-neighbor hopping between the two sites that are
connected by the solid lines shown in Fig. 1(d), and $\mu_{0}$ is the chemical
potential. In addition, $m$ denotes the energy splitting between the two
groups of sublattices, which have a different local environment and thus
generically have different energy levels. Finally, $\gamma$ represents an
additional energy splitting between $C$ and $D/E$, which breaks the $C_{3}$
rotational symmetry. As mentioned earlier, we work with the case that breaks
$C_{3}$ symmetry to simplify the symmetry characterization and, thus, the
Wannier construction. It is possible that this $C_{3}$ symmetry breaking
spontaneously appears as a result of interactions that drive a nematic order,
although, for our illustrative purpose, we do not pursue this route
specifically. A detailed analysis of the dispersion is shown in the SI (Sec.
I).

Effective extended Hubbard model.  We project the Hubbard model of the
original lattice to the Wannier basis. This leads to the effective model
expressed in terms of the $d$ and $c$ Wannier orbitals with
$H_{\mathrm{eff}}=H_{0}+H_{\rm int}$. The kinetic term takes the following
form:

$\displaystyle H_{0}$ $\displaystyle=H_{d}+H_{c}+H_{V}$ (4)
$\displaystyle=\sum_{ij,\sigma}t_{ij}\left(d_{i\sigma}^{\dagger}d_{j\sigma}+h.c.\right)-\sum_{i}\mu
d_{i\sigma}^{\dagger}d_{i\sigma}$
$\displaystyle+\sum_{ij,\alpha\beta,\sigma}t_{ij}^{\alpha\beta}\left(c_{i\alpha\sigma}^{\dagger}c_{j\beta\sigma}+h.c.\right)+\sum_{i\alpha\sigma}(\Delta_{\alpha}-\mu)c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$
$\displaystyle+\sum_{ij,\alpha\sigma}V_{ij}^{\alpha}\left(d_{i\sigma}^{\dagger}c_{j\alpha\sigma}+h.c.\right)\,.$

Here, $d^{\dagger}_{i\sigma}$ ($c^{\dagger}_{i\alpha\sigma}$ ) creates a heavy
(light) electron at the position $i$ with spin $\sigma$ (and orbital
$\alpha$). In addition, $t_{ij}$ ($t_{ij}^{\alpha\beta}$) denotes the hopping
parameter between the $d$ ($c$) electrons at the positions $i$ and $j$
(orbitals $\alpha$ and $\beta$ ). Moreover, $V^{\alpha}$ represents the
hybridization between the light orbital $\alpha$ and the heavy orbital.
Finally, $\Delta_{\alpha}$ describes the difference in the energy levels
between the $d$ orbital and $c$ orbitals, and $\mu$ specifies the chemical
potential. We will focus on the case with the $d$-electron level being deep
below the Fermi energy and will show that the interaction effect creates a
heavy band near the Fermi energy. The noninteracting dispersion is shown in
Fig. 4(a), where there is a Dirac crossing between the flat band and lower
wide band located deep below the Fermi energy; the crossing is protected by
the $M_{x}$ lattice symmetry 22.

Slave spin method and self-consistent equations.  We describe the U(1) slave
spin approach 31. Because the bandwidth of the heavy orbital is much smaller
than those of the light orbitals, the interaction effect is most pronounced on
the $d$ orbital. We therefore only introduce the SS representation on the $d$
orbital: $d_{i\sigma}^{\dagger}=o^{\dagger}_{i\sigma}f_{i\sigma}^{\dagger}$.
The auxiliary bosonic field $o^{\dagger}_{\sigma}=P^{+}S^{+}_{\sigma}P^{-}$ is
represented by the spin operator accompanied with the projection operators
$P^{\pm}_{i\sigma}=\frac{1}{\sqrt{1/2\pm S^{z}_{i\sigma}}}$ suitable for a
system that is away from half filling. We treat the SS formulation at the
saddle-point level by fully decoupling the SS and auxiliary fermion operators.
This leads to the decoupled Hamiltonian:

$\displaystyle H^{f}$ $\displaystyle=\sum_{\bm{k},\sigma}\langle
O_{\sigma}\rangle^{2}\epsilon_{\bm{k}}^{f}f_{\bm{k}\sigma}^{\dagger}f_{\bm{k}\sigma}+\sum_{\bm{k}\sigma}\left(-\mu-\lambda_{\sigma}+\lambda_{\sigma}^{0}\right)f_{\bm{k}\sigma}^{\dagger}f_{\bm{k}\sigma}$
(5) $\displaystyle+\sum_{\bm{k},\alpha\sigma}\left(V_{\bm{k}}^{\alpha}\langle
O_{\sigma}^{\dagger}\rangle
f_{\bm{k}\sigma}^{\dagger}c_{\bm{k}\alpha\sigma}+h.c.\right)+H_{c}$
$\displaystyle H^{S}$
$\displaystyle=\sum_{\sigma}\left[\tilde{\epsilon}_{f}\left(\langle
O_{\sigma}\rangle
O_{\sigma}^{\dagger}+h.c.\right)+\sum_{\alpha}\left(\tilde{V}_{\alpha}O_{\sigma}^{\dagger}+h.c.\right)\right]$
$\displaystyle+\sum_{\sigma}\lambda_{\sigma}\left(S^{z}_{\sigma}+\frac{1}{2}\right)+H_{\rm
int}^{S}\,,$

where

$H_{\rm
int}^{S}=\frac{u}{2}(\sum_{\sigma}S^{z}_{\sigma})^{2}+\sum_{\alpha}F_{\alpha}\langle
n^{\alpha}_{c}\rangle\sum_{\sigma}S^{z}_{\sigma}\,,$ (6)

and $\epsilon_{f}(\bm{k})$ and $V^{\alpha}_{\bm{k}}$ are the Fourier
transforms of $t_{ij}$ and $V^{\alpha}_{ij}$, respectively,
$\tilde{\epsilon}_{f}=\sum_{\bm{k}}\epsilon_{f}(\bm{k})\langle
f_{\bm{k}}^{\dagger}f_{\bm{k}}\rangle$, and
$\tilde{V}_{\alpha}=\sum_{\bm{k}}V_{\bm{k}}^{\alpha}\langle
f_{\bm{k}}^{\dagger}c_{\bm{k}}\rangle$. In addition, $O_{\sigma}=\langle
P^{+}\rangle S_{\sigma}^{+}\langle P^{-}\rangle$,
$\lambda_{\sigma}^{0}=2\left(\bar{\epsilon}_{f}+\sum_{\alpha}\bar{V}_{\alpha}\right)\eta_{\sigma}$
with $\bar{\epsilon}_{f}=\tilde{\epsilon}_{f}\langle O_{\sigma}\rangle\langle
O_{\sigma}^{\dagger}\rangle+c.c.$, $\bar{V}=\tilde{V}\langle
O_{\sigma}^{\dagger}\rangle+c.c.$ and
$\eta_{\sigma}=\frac{1}{2}\frac{n^{f}_{\sigma}-1/2}{(1-n^{f}_{\sigma})n^{f}_{\sigma}}$.
Finally, we introduce the Lagrangian multiplier $\lambda_{\sigma}$ to remove
the unphysical Hilbert space [See the SI (Sec. V)]. The pseudo-spin carries
the U(1) charge degree of freedom; the quasiparticle weight associated with
the coherent part near the Fermi level is described by $Z=\langle
O_{\sigma}\rangle\langle O^{\dagger}_{\sigma}\rangle$. An (orbital-selective)
Mott localization transition happens when some quasiparticle weight $Z$ goes
to zero.

In addition to the coherent quasiparticle peak, the SS method also calculates
the contributions from the incoherent excitations. The Green’s function of the
$d$ electron $G_{d}$ is obtained by the convolution of the SS and the
auxiliary fermions, with
$G(\bm{k},i\omega_{n})=\sum_{i\Omega_{m}}G_{S}(i\Omega_{m})G_{f}(\bm{k},i\omega_{n}-i\Omega_{m})$,
where $G_{S}(\tau)=\langle o_{\sigma}(\tau)o^{\dagger}_{\sigma}(0)\rangle$ and
$G_{f}(\tau)=-\langle f_{\sigma}(\tau)f_{\sigma}^{\dagger}(0)\rangle$. The
spectral function is then obtained from $A(\bm{k},\omega)=\frac{-1}{\pi}\Im
G_{d}^{R}(\bm{k},\omega)$.

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Acknowledgments
We thank Gabriel Aeppli, Joseph Checkelsky, Han Wu, Yonglong Xie and Ming Yi
for useful discussions. Work at Rice has primarily been supported by the U.S.
DOE, BES, under Award No. DE-SC0018197 (L.C. and F.X.), by the Air Force
Office of Scientific Research under Grant No. FA9550-21-1-0356 (Q.S.), and by
the Robert A. Welch Foundation Grant No. C-1411 (S.S.). The majority of the
computational calculations have been performed on the Shared University Grid
at Rice funded by NSF under Grant EIA-0216467, a partnership between Rice
University, Sun Microsystems, and Sigma Solutions, Inc., the Big-Data Private-
Cloud Research Cyberinfrastructure MRI-award funded by NSF under Grant No.
CNS-1338099, and the Extreme Science and Engineering Discovery Environment
(XSEDE) by NSF under Grant No. DMR170109. H.H. acknowledges the support of the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (Grant Agreement No. 101020833). Work in
Vienna was supported by the Austrian Science Fund (project I 5868-N - FOR 5249
- QUAST) and the ERC (Advanced Grant CorMeTop, No. 101055088). J.C.
acknowledges the support of the National Science Foundation under Grant No.
DMR-1942447, support from the Alfred P. Sloan Foundation through a Sloan
Research Fellowship and the support of the Flatiron Institute, a division of
the Simons Foundation. Q.S. acknowledges the hospitality of the Aspen Center
for Physics, which is supported by NSF grant No. PHY-1607611.

Author contributions
Q.S. conceived the research. L.C., F.X., S.S., H.H., J.C. and Q.S. carried out
model studies. L.C., S.P., J.C. and Q.S. contributed to the development of the
design principle for correlated topological phases in $d$-electron systems.
L.C. and Q.S. wrote the manuscript, with inputs from all authors.

Competing interests
The authors declare no competing interests.

Additional information
Correspondence and requests for materials should be addressed to Q.S.
<EMAIL_ADDRESS>
Figure 1: Illustration of the bare and emergent flat bands and lattice
geometry. a, In the noninteracting case, a flat band (red solid line) appears
far away from the Fermi energy. b, In the presence of orbital-selective
correlations, an interaction-driven flat band emerges at the Fermi energy (red
solid line), while leaving incoherent excitations far away from the Fermi
energy (red dashed lines). c, The emergent flat band crosses a dispersive
band, leading to a topological Kondo semimetal with symmetry-protected
Dirac/Weyl nodes that are pinned close to the Fermi energy, within an
effective Kondo energy scale. d, Geometry of the clover lattice with 5
sublattices per unit cell. The lattice does not have inversion symmetry. This
can be seen from the mismatch between the (dark) blue sublattices and their
inversion counterparts (dots in light blue). e, The Wannier orbitals are near
the geometric centers of the unit cells, which form a triangular lattice.

Figure 2:  Bare and emergent flat bands. a, The DOS of the $d$ and $c$
electrons for the noninteracting case. b, The corresponding result at $u=1.6$;
here the red solid (dashed) lines denote the coherent (incoherent) part of the
$d$-electron excitations. The purple backgrounds mark the DOS of the
conduction $c$ electrons.

Figure 3:  Orbital-selective correlations. The quasiparticle weights of the
$d$ and $c$ electrons, $Z_{d}$ and $Z_{c}$, as a function of the effective
interaction $u$. The Left (right) inset plots the Fermi surface for the values
of interactions right below (above) $u_{c}=2.4$.

Figure 4: Topological Kondo semimetal. a, The noninteracting band structure.
b, The dispersion of the single-electron excitations at $u=1.6$. The red solid
curve denotes the emergent flat band close to the Fermi energy. The grey lines
mark the incoherent single-electron excitations. c, The zoomed-in view of the
emergent flat band. d, The band structure at $u=1.6$ with a Zeeman splitting
$m_{z}=0.03$.

Supplemental Information

##  I Band structure of in the original Hamiltonian

In this section, we show the dispersion in the original lattice. We fix the
reference energy scale $t=2$. The noninteracting band structure is shown in
Fig S1(a), in which the bands are clearly separated into the lower two bands
and upper three bands. The spectral function is provided in Fig. S1(b), where
the sharp peak denotes the position of the flat band. We focus on the upper
three bands for our Wannier construction.

Figure S1:  The noninteracting electronic structure of the original Hubbard
model. a, The band structure in the original clover lattice, with $\mu=2.2$,
$m=2$ and $\gamma=-0.2$. b, The corresponding density of states.

##  II  Destructive interference and the origin of the flat band

In this section, we discuss the origin of the flat band in the clover lattice.
Consider the nearest-neighbor hoppings between the blue and yellow
sublattices. The kinetic part of the Hamiltonian takes the form as

$\mathcal{H}_{0}(\bm{k})=\begin{pmatrix}0_{2\times 2}&\mathcal{H}_{\bm{k}}\\\
\mathcal{H}_{\bm{k}}&0_{3\times 3}\end{pmatrix}\,,$ (S1)

where $\mathcal{H}_{\bm{k}}$ is the hopping matrix between the two groups of
sublattices. The localized wavefunction takes the following form 30:

$\psi(k_{x},k_{y})=\begin{pmatrix}0&0&e^{\frac{k_{y}}{3}}-e^{-\frac{2}{3}k_{y}}&e^{-\frac{k_{x}}{2\sqrt{3}}-\frac{k_{y}}{6}}-e^{\frac{k_{x}}{\sqrt{3}}+\frac{k_{y}}{3}}&e^{\frac{k_{x}}{2\sqrt{3}}-\frac{k_{y}}{6}}-e^{-\frac{k_{x}}{\sqrt{3}}+\frac{k_{y}}{3}}\end{pmatrix}\,.$
(S2)

Here, the amplitude is identically zero on the yellow sites because of the
destructive interference effect. The real space wavefunction is shown in Fig.
S2.

Figure S2:  Wavefunction for the electronic states of the flat band.
Illustrated here is the real space representation of the electronic
wavefunction for the flat band in the clover lattice.

##  III Effective Multi-orbital Hubbard Model

For the kinematic hoppings of the effective molecular orbitals, we refer to
Table II in the SM of Ref 13. However, we focus on the case such that, in the
noninteracting case, the flat band lies considerably below the Fermi energy.
In Fig. S3, we present the DOS for the $d$ and $c$ electrons in the absence of
any hybridization between the two orbitals. For the interaction part, the
ratio between the Hubbard interaction of the effective $d$ electron and the
one on the original lattice is $u/U=0.15$ 13. For the same reason that the
onsite interaction among the $c$-orbitals is unimportant, the Hund’s coupling
does not play an important role here 13.

Figure S3: The noninteracting density of states projected to the Wannier
orbitals. a,b, The DOS of the $d$ and $c$ electrons, respectively, in the
absence of any hybridization between the two orbitals.

##  IV Analysis of the Effective Multi-orbital Hubbard Model with a Spin-Orbit
Coupling

In this section, we discuss the influence of the spin-orbit coupling (SOC).
For our purpose, it is adequate to consider a SOC that is small compared to
the other energy scales. We can then analyze the effect of the SOC
perturbatively, by using the same Wannier orbitals described above. We thus
project the SOC of the original lattice onto the Wannier orbital basis. We
consider the SOC along the $z$ direction of the original clover lattice, which
takes the form as

$\mathcal{H}_{\rm soc}=\sum_{i,\sigma=\pm}it_{soc}\sigma\left(\eta_{i\sigma
C}^{\dagger}\eta_{i\sigma D}+\eta_{i\sigma D}^{\dagger}\eta_{i\sigma
E}+\eta_{i\sigma E}^{\dagger}\eta_{i\sigma C}\right)+h.c.\,.$ (S3)

Here, $t_{soc}$ is the strength of the SOC, and $\sigma=\pm$ represents spin
up and spin down, respectively. The hopping parameters on the basis of the
three Wannier orbitals are shown in Table. 1 and Table. 2.

$(i,j)$ | $(0,1)$ | $(-1,0)$ | $(1,0)$ | $(0,-1)$
---|---|---|---|---
$t_{ia_{1}+ja_{2}}$ | $-0.04566+i0.01088$ | $-0.04566-i0.01088$ | $-0.04566+i0.01088$ | $-0.04566-i0.01088$
| $\pm(2,-2)$ | (0, -2) | (0, 2) | (2,0)
| 0.04991 | $0.03701-i0.0031$ | $0.03701+i0.0031$ | $0.03701+i0.0031$
| $(-2,0)$ | (-1, 2) | (1,2) | (-2,1)
| $0.03701-i0.0031$ | $-0.03314-i0.00061$ | $-0.03314+i0.00061$ | $-0.03314+i0.00061$
| $(2,1)$ |  $\pm$(1, -1) |  |
| $-0.03314-i0.00061$ | $0.02032$ |  |
$(i,j)$ | $(1,0)$ | $(0,-1)$ | $(-1,2)$ | $(-2,1)$
$V_{ia_{1}+ja_{2}}^{1}$ | $0.13301+i0.01893$ | $-0.13301+i0.01893$ | $0.09264-i0.00299$ | $-0.09264-i0.00299$
| $(1,-2)$ | $(2,-1)$ | $(0,-2)$ | $(2,0)$
| $0.07903-i0.00749$ | $-0.07903-i0.00749$ | $-0.06361+i0.00909$ | $0.06361+i0.00909$
| $(-2,3)$ | $(-3,2)$ | $(1,1)$ | $(-1,-1)$
| $0.05494-i0.00431$ | $-0.05494-i0.00431$ | $-0.05082-i0.007$ | $0.05082-i0.007$
$(i,j)$ | $(1,-1)$ | $(-1,1)$ | $(0,1)$ | $(-1,0)$
$V_{ia_{1}+ja_{2}}^{2}$ | $-0.15053$ | $-0.14577$ | $0.096-i0.00896$ | $0.096+i0.00896$
| $(0,-1)$ | $(1,0)$ | $(0,0)$ | $(2,-2)$
| $-0.09143+i0.01766$ | $-0.09143-i0.01766$ | $-0.07918$ | $0.07563$
| $(1,1)$ | $(-1,-1)$ | $(-1,2)$ | $(-2,1)$
| $-0.07542+i0.00449$ | $-0.07542-i0.00449$ | $0.07311+i0.00509$ | $0.07311-i0.00509$
Table 1: Parameters for the noninteracting Hamiltonian in the Wannier basis – part I. Shown here are the hopping parameters for the $d$ orbitals and the hybridization between the $d$ and $c$ orbitals for spin up when $t_{\rm soc}=0.06$. The parameters in the spin-down case are the Hermitian conjugate of their spin-up counterparts.  $(i,j)$ | $(0,-1)$ | $(0,1)$ | $(1,0)$ | $(-1,0)$
---|---|---|---|---
$t^{11}_{ia_{1}+ja_{2}}$ | $-0.35388+i0.01913$ | $-0.35388-i0.01913$ | $-0.35388-i0.01913$ | $-0.35388+i0.01913$
| $\pm(1,-1)$ | $\pm(2,-2)$ | (-1,-1) | (1,1)
| $-0.15797$ | $-0.08055$ | $-0.07438+i0.00105$ | $-0.07438-i0.00105$
$(i,j)$ | $(0,-1)$ | $(0,1)$ | $(1,0)$ | $(-1,0)$
$t^{22}_{ia_{1}+ja_{2}}$ | $0.27789-i0.00826$ | $0.27789+i0.00826$ | $0.27789+i0.00826$ | $0.27789-i0.00826$
| $(0,-2)$ | $(0,2)$ | $(-2,0)$ | $(2,0)$
| $-0.07658+i0.0008$ | $-0.07658-i0.0008$ | $-0.07658+i0.0008$ | $-0.07658-i0.0008$
| $(-1,-1)$ | $(1,1)$ |  |
| $0.05848+i0.0014$ | $0.05848-i0.0014$ |  |
$(i,j)$ | $(0,-1)$ | $(1,0)$ | $(1,-2)$ | $(2,-1)$
$t^{12}_{ia_{1}+ja_{2}}$ | $0.43550-i0.00395$ | $-0.43550-i0.00395$ | $0.1959-i0.00079$ | $-0.1959-i0.00079$
| $(0,1)$ | $(-1,0)$ | $(2,-3)$ | $(3,-2)$
| $0.15083-i0.00138$ | $-0.15083-i0.00138$ | $0.11537-i0.00017$ | $-0.11537-i0.000179$
| $(0,2)$ | $(-2,0)$ | $(-1,2)$ | $(-2,1)$
| $0.09257+i0.00133$ | $-0.09257+i0.00133$ | $0.09107+i0.00516$ | $-0.09107+i0.00516$
| $\mu$ | $\Delta_{1}$ | $\Delta_{2}$ |
| $0.15$ | $1.81$ | $2.708$ |
Table 2: Parameters for the noninteracting Hamiltonian in the Wannier basis –
part II. Shown here are the hopping parameters between the $c$ electrons for
spin up when $t_{\rm soc}=0.06$, the chemical potential $\mu$ and the
effective crystal field differences $\Delta_{\alpha}$. The parameters for the
spin-down case are the Hermitian conjugate of their spin-up counterparts.

Because the value of the SOC is small, we use the same saddle-point parameters
as we obtained in the case without the SOC. The renormalization factors $Z$
for the different orbitals will then renormalize the SOC accordingly.

As shown in Fig. S4, the SOC generically gaps out the Dirac nodes (located
along $\Gamma-K$) for our 2D model, both for the cases with and without
interactions. This leads to flat $Z_{2}$ topological bands. We stress that,
the flatness of the emergent bands close to the Fermi energy is robust against
the SOC.

Figure S4: The electronic dispersion in the presence of the SOC. Shown here
are the electronic dispersions for $u=0$ (a) and for $u=1.6$ (b), with the SOC
in the original lattice to have a strength of $t_{soc}=0.06$.

##  V Slave spin method

In this section, we derive the slave spin equations we used in the main text.
In the slave spin method, the creation operator for the correlated orbital is
re-expressed as the product of a SS operator and an auxiliary fermionic
operator: $d_{\sigma}^{\dagger}=f^{\dagger}_{i\sigma}o^{\dagger}_{i\sigma}$,
where $o_{i\sigma}^{\dagger}=P^{+}_{i\sigma}S^{+}_{i\sigma}P^{-}_{i\sigma}$
and $P^{\pm}_{i\sigma}=\frac{1}{\sqrt{1/2\pm S^{z}_{i\sigma}}}$. To remove the
unphysical Hilbert space, the SS and the auxiliary fermion need to follow the
constraint:

$S^{z}_{i\sigma}+\frac{1}{2}=n^{f}_{i\sigma}.$ (S4)

The interaction term can be expressed by the SS operators as described by Eq.
6 in the main text. The noninteracting part of the Hamiltonian now takes the
following form:

$\displaystyle H_{0}$
$\displaystyle=\sum_{ij\sigma}t_{ij}\left(f_{i\sigma}^{\dagger}o_{i\sigma}^{\dagger}o_{j\sigma}f_{j\sigma}+h.c.\right)-\sum_{i}\mu
f_{i\sigma}^{\dagger}f_{i\sigma}$ (S5)
$\displaystyle+\sum_{ij,\alpha\sigma}V^{\alpha}_{ij}\left(f_{i\sigma}^{\dagger}o_{i\sigma}^{\dagger}c_{j\alpha\sigma}+h.c.\right)$
$\displaystyle+\sum_{ij,\alpha\beta,\sigma}t_{ij}^{\alpha\beta}\left(c_{i\alpha\sigma}^{\dagger}c_{j\beta}+h.c.\right)+\sum_{i\alpha\sigma}(\Delta_{\alpha}-\mu)c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}\,.$

We decouple the $f$ and $o$ at the saddle-point level with
$f_{i\sigma}^{\dagger}o_{i\sigma}^{\dagger}o_{j\sigma}f_{j\sigma}=f_{i\sigma}^{\dagger}f_{j\sigma}\langle
o_{i\sigma}^{\dagger}o_{j\sigma}\rangle+\langle
f_{i\sigma}^{\dagger}f_{j\sigma}\rangle
o_{i\sigma}^{\dagger}o_{j\sigma}+const.$ and
$f_{i\sigma}^{\dagger}o_{i\sigma}^{\dagger}c_{j\alpha\sigma}=\langle
f_{i\sigma}^{\dagger}c_{j\alpha\sigma}\rangle o^{\dagger}_{i\sigma}+\langle
o^{\dagger}_{i\sigma}\rangle f_{i\sigma}^{\dagger}c_{j\alpha\sigma}+const$. We
take the single site decoupling $\langle
o_{i}o_{j}^{\dagger}\rangle\approx\langle o_{i}\rangle\langle
o_{j}^{\dagger}\rangle$, and Taylor expand $o^{\dagger}_{i\sigma}$ at the
saddle point as follows:

$\displaystyle o^{\dagger}_{\sigma}$ $\displaystyle\approx\langle
P^{+}_{\sigma}\rangle S^{+}\langle P^{-}_{\sigma}\rangle+\langle
P^{+}_{\sigma}\rangle\langle S^{+}_{\sigma}\rangle\langle
P^{-}_{\sigma}\rangle\frac{1}{2}\left(S^{z}_{\sigma}-\langle
S^{z}_{\sigma}\rangle\right)\left(\frac{-1}{n_{\sigma}}+\frac{1}{1-n_{\sigma}}\right)$
(S6) $\displaystyle=O^{\dagger}_{\sigma}+\langle
O^{\dagger}_{\sigma}\rangle\frac{1}{2}\frac{n_{\sigma}-1/2}{(1-n_{\sigma})n_{\sigma}}\left(2S^{z}_{\sigma}-(2n_{\sigma}-1)\right)$
$\displaystyle=O^{\dagger}_{\sigma}+\langle
O^{\dagger}_{\sigma}\rangle\eta_{\sigma}\left[2S^{z}_{\sigma}-(2n_{\sigma}-1)\right]\,,$

where

$\displaystyle O_{\sigma}^{\dagger}$ $\displaystyle=\langle
P^{+}_{\sigma}\rangle S^{+}_{\sigma}\langle P^{-}_{\sigma}\rangle$ (S7)
$\displaystyle\eta_{\sigma}$
$\displaystyle=\frac{1}{2}\frac{n_{\sigma}-1/2}{(1-n_{\sigma})n_{\sigma}}\,.$

We notice that $\langle o_{\sigma}\rangle=\langle O_{\sigma}\rangle$.
Combining the above equations and introducing the Lagrangian multipler
$\lambda_{\sigma}$ to enforce the constraint in Eq. S4, we obtain the self-
consistent equations as described in Eq. 5 of the main text.

The Green’s function of the $d$ electron is obtained by the convolution of the
SS and the auxiliary fermion propagators, with

$G(\bm{k},i\omega_{n})=\sum_{m}G_{S}(i\Omega_{m})G_{f}(\bm{k},i\omega_{n}-i\Omega_{m})\,.$
(S8)

After an analytical continuation, we have the spectral function

$\displaystyle A_{d}(\bm{k},\omega)$
$\displaystyle=\frac{2\pi}{Z}\sum_{\sigma}\sum_{n,m}\langle
n|o_{\sigma}|m\rangle\langle m|o_{\sigma}^{\dagger}|n\rangle\delta(\omega-
E_{nm}-\epsilon_{\lambda\bm{k}})|\Lambda_{\lambda\bm{k}}^{f}|^{2}$ (S9)
$\displaystyle\times\left[e^{-\beta E_{m}}(1-f_{\lambda\bm{k}})+e^{-\beta
E_{n}}f_{\lambda\bm{k}}\right]\,,$

where $\lambda$ labels the bands solved from $H^{f}$, and
$\Lambda_{\lambda\bm{k}}^{f}$ represents the corresponding $f$-component in
the eigenvectors. In addition, $n,m$ denote the eigenvectors of $H^{S}$, and
$E_{nm}=E_{n}-E_{m}$ is the energy difference between the two slave spin
states. Finally,
$f_{\lambda\bm{k}}=\frac{1}{e^{\beta\epsilon_{\lambda\bm{k}}}+1}$ is the
Fermi-Dirac distribution function.

##  VI Evolution of the spectral functions across the orbital-selective Mott
transition

In this section, we provide further details on the evolution of the single-
electron spectral weight when the interactions further increase from the case
of $u=1.6$ shown in Fig. 2(b), approaching and into the OSMP. The evolution is
primarily seen in the $d$-electron DOS (red lines). The $c$-electron DOS
(marked in purple) is not markedly changed.

Figure S5(a) shows the DOS for $u=2.2$, which is smaller than but close to the
critical value ($u_{c}=2.4$) for the orbital-selective Mott transition.
Compared to the case of $u=1.6$ shown in Fig. 2(b), there is a significant
transfer of the $d$-electron spectral weight from the coherent part (the
central red peak near the Fermi energy) to the incoherent part (the side peaks
marked by the red dashed lines).

Fig. S5(b) displays the DOS for $u=2.5$, where the system is in the OSMP (at
$u>u_{c}=2.4$). Here, for the $d$-electron DOS, the coherent peak has
vanished. What is left is the incoherent part that comprises the lower and
upper Hubbard bands marked by the red dashed lines.

The slave spin method incorporates the nonlocal spin correlations in the
system 51, which is important in capturing the orbital-selective Mott
transition. The result here has some similarities with its counterpart arising
in the context of the Fe-based superconductors 52, 53. There is however one
important distinction: the existence of the flat band in the noninteracting
dispersion means that the difference between the bandwidths ($D_{\rm flat}$
and $D_{\rm wide}$) is especially large here.

Figure S5:  Evolution of the DOS across the orbital-selective Mott transition.
a, The DOS of the $d$ and $c$ electrons for $u=2.2$, which is just below the
critical value $u_{c}=2.4$. The coherent part of the $d$-electron spectrum
corresponds to the central peak (the red solid line), and its incoherent part
appears as side peaks (the red dashed lines). The $c$-electron spectral
function is shown in purple. b, The case of $u=2.5$, which is larger than
$u_{c}$ and, thus, the system lies inside the OSMP. The $d$-electrons are in a
Mott insulating state, and their spectrum contains only the lower and upper
Hubbard bands.