# Recognising the importance of preference change: A call for a coordinated
multidisciplinary research effort in the age of AI

1Matija Franklin, 1Hal Ashton, 2Rebecca Gorman, 3Stuart Armstrong

###### Abstract

As artificial intelligence becomes more powerful and a ubiquitous presence in
daily life, it is imperative to understand and manage the impact of AI systems
on our lives and decisions. Modern ML systems often change user behavior (e.g.
personalized recommender systems learn user preferences to deliver
recommendations that change online behavior). An externality of behavior
change is preference change. This article argues for the establishment of a
multidisciplinary endeavor focused on understanding how AI systems change
preference: Preference Science. We operationalize preference to incorporate
concepts from various disciplines, outlining the importance of meta-
preferences and preference-change preferences, and proposing a preliminary
framework for how preferences change. We draw a distinction between preference
change, permissible preference change, and outright preference manipulation. A
diversity of disciplines contribute unique insights to this framework.

## Introduction

Modern Artificial Intelligence (AI) often uses Machine Learning methods that
can learn its users’ preferences (e.g., favorite artist) to personalize a
service to them (e.g., song recommendation; (Domshlak et al. 2011)). User
preference on online platforms is typically inferred from consumer behavior
(e.g., what they have listened to) or consumer ratings (e.g., did the user
click on the ”like” or ”dislike” button) (Ibrahima and Younisb 2018; Yakhchi
2021). The tailored recommendations change users’ online behavior (Jesse and
Jannach 2021). Behavior change practitioners use behavioral insights \- cause
and effect understanding of how factors change behavior at the level of a
population - to create changes in choice architecture \- the background of
people’s behaviors - to influence behavior (Ruggeri 2018). Research in
Behavioral science - the discipline concerned with the study of behavioral
insights - has led to a valid and reliable understanding of behavior and how
it changes, allowing for the development of accurate predictive models
(Michie, Van Stralen, and West 2011).

It is evident that user interactions with AI-powered systems whose design has
been informed by Behavioral Science lead to behavioral change. What is less
evident is that these practices also cause a preference change. Preference can
and does influence our behavior, but our behavior often predates and leads to
the emergence of new preference (Ariely and Norton 2008). This is an issue
given that behavior change practitioners ethically defend the practice by
arguing that they merely influence behavior, without limiting or forcing
options (Sunstein 2016). Tailoring to user preference with AI has been
defended with the intent to learn about a platform’s customers in order to
improve their user experience. Due to the present preference change, we argue
that the practices in AI and Behavioral Science can be much more manipulative
(Sunstein 2015). It is thus impossible to ensure the ethical and safe use of
AI without an understanding of the impact these technologies have on
preference. Furthermore, any attempt to align AI so that its ”objective is to
maximise the realization of human preferences” (Russell 2019) is futile
without the acknowledgment that AI can change human preference.

To this end, this article argues for the establishment of a multidisciplinary
endeavor focused on the operationalization of preference in the broader sense,
and an understanding of how it changes. In this article, we will refer to this
endeavor as Preference Science for brevity. It seeks to tackle issues central
to establishing this endeavor. First, it seeks to operationalize preference in
a way that incorporates features of the concept from various disciplines in
order to broaden our understanding of what it is. Second, it considers meta
preference \- people’s preferences over what their preferences are or shall be
in the future - and preference-change preference \- preferences about how
preferences are formed. Third, it proposes a framework for understanding how
preferences change. Finally, the article will propose a research direction for
science that seeks to understand how preferences change.

## Preference redux

Social sciences have operationalized and studied preference, and related
concepts, in ways that are specific to those disciplines111In research design,
operationalization is the process of defining a concept which cannot be
directly measured. Each of these disciplines illuminate on important
properties of preferences. A definition of preference must account for this,
to arrive at a more robust operationalization of preference.

We propose preference be defined as _any explicit, conscious, and reflective
or implicit, unconscious, and automatic mental process that brings about a
sense of liking or disliking for something_. This definition broadens the
scope of preference, allowing for the inclusion of other concepts that pertain
to conative psychological processes. Conative mental states are those which
aim to bring about something in the environment, rather than describe it.
Conative mental states inform us about what the world ought to be like, in
contrast to cognitive mental states, which inform us about what the world is
like (Schulz 2015).

## Preferences and conative states

In economics, preference is often operationalized as a choice between
alternative options. A choice of one option over another indicates a
preference towards it. This is an example of a ‘revealed preference’
(Samuelson 1938) – the preference is assumed to be revealed through the
agent’s actions. It is clear that this is but a partial model of preference,
as humans are not fully rational calculating machines with infinite time and
knowledge. Approaches have been developed to extract preferences that
acknowledge that humans are ’boundedly rational’ (Simon 1955) \- that is to
say mainly rational but with certain imperfections. Armstrong and Mindermann
(2018) formally demonstrate that, unless one makes normative assumptions, one
cannot learn an agent’s level of rationality _and_ their preferences from
observation.

‘Stated preferences’ (Kroes and Sheldon 1988) are often used in philosophy and
in some areas of social science: the preferences of an agent are elicited by
directly asking them. This has problems; people may not honestly report their
true preferences (Sunstein 2018b), and they may even be influenced in how the
questions are phrased (Thaler and Benartzi 2004; Špecián 2019).

Psychological research operationalizes preferences both in terms of a
decision, or in terms of a judgment towards an object; specifically, liking or
disliking it (Kahneman and Tversky 1982). The field acknowledges that
preferences can be influenced by a person’s physical and social environment,
as well as their previous behavior and life history (Michie, Van Stralen, and
West 2011; Cialdini 1987).

Psychologists make a conceptual distinction between preference and desire
(Pettit 1998; Schulz 2015). Desire is directed at a single object, while
preferences relate to a comparison between alternatives. A preference can thus
be operationalized as a comparison between desires. Each desire towards an
object will have a level of intensity. People display different judgments for
the same object when making relative judgments between alternatives versus
stating an absolute judgment towards a particular object (Bazerman,
Loewenstein, and White 1992; Azar 2011). Both conative states are thus
relevant for a preference science.

## Preferences in machine learning

In industry today, user preferences are heavily studied by machine learning
practitioners developing personalized recommender systems. Personalized
recommender engines seek to model a user’s preferences in order to provide
users with a tailored selection of options on a certain platform. To do this,
they use data on the user’s previous behavior – this is thus a form of
revealed preference222Inverse reinforcement learning makes this assumption
explicit, estimating user’s ‘reward function’ under the assumption that the
user is rational (Ng, Russell et al. 2000) or noisily rational (Ziebart et al.
2008).. Systems that include some of the user’s stated preferences, looking at
their rating of pieces of content on the platform, have been widely deprecated
due to a lower level of effectiveness at predicting user behavior.
Collaborative filtering, which has been found to be effective in recommender
systems, assumes users’ preferences are similar to those of other users with
similar behavior or background.

Commercial recommender engines are designed with the platform’s interest in
mind, and hence target the user’s behavior - aiming to get them to click
through, to buy, to spend more time on the platform, etc…. By nudging a user
to consume certain information on the website, recommender engines, through
the bidirectional causal relationship between preference and behavior, can
cause a runaway feedback-loop which over time changes the user’s preference
(Evans and Kasirzadeh 2021). Alfano et al. (2020) has found that such a system
will recommend extremist content to maintain user engagement, creating
internet echo chambers and polarization on social media platforms. Ashton and
Franklin (forthcoming) discuss the general problem of AI systems manipulating
user preferences in greater depth.

## Preferences in behavioral science

In behavioral science, notable frameworks of behavior change have identified
conative mental states that can be used to broaden our understanding of
preferences. The frameworks also lay out the requirements for establishing a
framework of preference change.

The COM-B model argues that for any behavior there are three components:
Capability, Opportunity, and Motivation (Michie, Van Stralen, and West 2011).
To perform a behavior, a person must feel they are able to, have the
opportunity for the behavior, and want or need to carry out that particular
behavior (more so than any other behavior). The three components interact, and
behavior change occurs as a result. An extension to this model is the Behavior
Change Wheel which describes how the components of COM-B get influenced by
Choice Architecture \- the omnipresent, and influential contexts in which
people behave (Michie et al. 2014)

Capability either relates to a person’s psychological or physical skills.
Opportunity relates to either the opportunity afforded by the social
environment, including social cues, cultural norms, social expectations, and
other people’s behavior, or by the physical environment, involving time,
resources, and location. Motivation describes the conative psychological
processes that are involved in behavior, and are thus applicable to our
broader definition of preference. It is a mental state which relates to
people’s initiation, continuation, and termination of certain behaviors at a
particular time. Motivation, in other words, brings about an explicit or
implicit liking or disliking for behaviors. COM-B uses a dual-process approach
to recognize that motivation can occur due to both automatic and reflective
processes. Reflective motivation includes conscious mental processes including
attitudes and beliefs, while automatic motivation is an umbrella term for less
conscious mental processes including desires, emotional reactions, and habits.

The Theoretical Domains Framework (TDF) – a list of factors that influence
behavior – identifies corresponding conative states that map onto the
components of COM-B’s reflexive and automatic motivation (Atkins et al. 2017).
Here reinforcement (i.e., an automatic response to a stimulus) and emotion map
onto automatic motivation. Social/professional role and identity, beliefs
about capabilities and consequences, intentions and goals map onto reflexive
motivation.

Three things become evident from the behavioral science literature. First,
concepts that have not been traditionally thought of as preferences can be as
they are conative states that bring about a sense of liking or disliking for
something. Second, these conative states are not only conscious but also often
automatic. Finally, preferences are shaped by multiple factors, thus, an
individual’s preferences will often change. Preference science must account
for these factors, as well as for people’s preferences towards their own
preferences and how they change.

## Meta-preference

People’s preferences can conflict with each other – self-regulation can be
seen as successfully privileging long-term goals over short-term pleasure
(Doerr and Baumeister 2010). People will have preferences over which of these
should win out, and at what cost; these preferences over preferences can be
termed meta or second-order preferences.

Humans can endorse or un-endorse some of their own preferences (Frankfurt
1988): for instance, people can harbor sexist or racist instincts which they
desire not to have. And they can want to be a specific sort of person - even
at a cost to themselves (Hewitt and Flett 1990).

Examples of preference conflict and resolution can happen when a subject is
confronted with a novel situation or thought experiment beyond their normal
experience (consider the trolley problem (Thomson 1976) or the repugnant
conclusion (Parfit 1984)). People often resolve these conflicts in a
contingent and non-consistent way, depending on the details of the
circumstances they’re in, their social environment, and how the issue is
presented (Payne et al. 1993; Schuman, Presser, and Ludwig 1981; Kleiman-
Weiner, Saxe, and Tenenbaum 2017). But they also seem to have a desire for
consistency in their ethics (Schuman and Presser 1996; Kleiman-Weiner, Saxe,
and Tenenbaum 2017). This desire for consistency can itself be seen as a meta-
meta or third-order preference.

Many fields define some form of idealized preferences (Arneson 1989; Sunstein
and Thaler 2003; Rosati 2009), and claim that truly serving a person’s
interest is to respect these idealized preferences. These idealized
preferences are typically those that the person would have with, for instance,
rational reflection, full information, no pressure, and enough time to ponder.

But those requirements are themselves meta-preferences, that determine the
idealized preferences. It is very possible, for instance, for humans to feel
that their true preferences emerge in the more emotional and dynamic parts of
their lives (Bazerman, Tenbrunsel, and Wade-Benzoni 1998; Caplan 2001).
Indeed, emotional decision-making can be more consistent than non-emotional
(Lee, Amir, and Ariely 2009). Thus the very definition of idealized
preferences – the definition of what it is to act in someone’s best interests
– depends on taking their meta-preferences into account333Though these must be
balanced against standard ‘first-order’ preferences as well; a weak meta-
preference should not automatically override a strongly held first-order
preference..

We believe meta-preferences are a necessary component for building any
framework concerning the ethics of preference change.

## Preference-Change Preferences

As well as meta-preferences, we must also recognise preferences concerning the
method of the formation process of any preference. Even if preferences meet
the meta-preference requirements, how they get there is important; the ends do
not always justify the means. This is recognised in the Article 5 of the draft
EU AI Act (CNECT 2021), which prohibits the use of any AI system that deploys
subliminal techniques…in order to materially distort a person’s behavior in a
manner that causes…that person or another person physical or psychological
harm. Colburn (2011) argues that subliminal or unconscious preference change
devices are wrong because they interfere with a user’s autonomy. They are
preferences which the user cannot correctly understand why they have, because
they were unaware of the process which caused them. With this account
preference manipulation and induced addiction are not acceptable practices
whilst self induced processes like learning and character planning are.

Guidance concerning what behavioral change mechanisms are and are not
acceptable can be considered preference-change preferences given the causal
relationship between behavior and preference. One guidance advocates for ”a
right not to be manipulated” (Sunstein 2021). Behavior change is said to be
manipulative if it does not engage with people’s capacity for reflective
choice (Sunstein 2015). Other guidance argues for ”Nudge, not sludge” – making
welfare-promoting behaviors easier to do, and removing sludge – behavior
change strategies that have the behavior change practitioner’s best interest
in mind rather than that of the target individual (Thaler 2018). Sludge is
said to take two forms: discouraging a person’s best interest or encouraging
self-defeating behaviors. A common mechanism used in sludge is friction –
making a behavior slower to do or unnecessarily complicated (Sunstein 2018a).
Anti sludge guidance proposes for sludge audits – identifying the mechanisms
that change behavior and preference, and eliminating those not in line with
our preference-change preferences (Sunstein 2020). A final guidance comes from
Libertarian Paternalism. Proponents of it argue that behavior change is
ethical when it avoids material incentives and coercion, and it is used to
give people a guidance for best practice given their own goals, rather than
changing their end goals (Thaler and Sunstein 2003). Altogether, there is a
strong reason to have a preference-change preference for preference change
mechanisms which are not manipulative, have one’s best interest in mind, and
aid towards one’s goals.

## Preference change: A framework

Preference science needs a framework that identifies the factors that
influence preference and defines the causal relationship between these
factors. The proposed framework is not a comprehensive list of factors, but
rather a statement about how these factors group into larger ones, and relate
to each other to bring about changes in preferences. Our proposed framework is
visually represented in Figure 1.

Figure 1: Preference Change Framework: Causal diagram showing the relationship
between preference, behavior, capability and choice architecture

The framework outlines how behavior, capability, and choice architecture shape
preference. Preference is operationalized with the aforementioned definition,
thus including all conative mental states, both reflective and automatic. In
the framework, preference is influenced by choice architecture, in that
conative states by definition get influenced by what is present in the
environment (Michie et al. 2014). Preferences are also influenced by
capability, in that people will on average prefer activities (as well as
objects associated with those activities) that are easier for them to do (Lee
and Benbasat 2011; Juvina et al. 2018). Finally, preference is in a
bidirectional causal relationship (i.e., feedback loop) with behavior
(Albarracín and McNatt 2005; Ariely and Norton 2008; Wyer Jr, Xu, and Shen
2012; Hill, Kusev, and Van Schaik 2019). Preference shapes behavior, but
behavior often predates and causes the emergence of new preferences.

Behavior is the activity of a human individual, interacting with their
environment (Popescu 2014). The way it is operationalized is context-specific.
Aside from behavior’s relationship with preference (see above), behavior is in
a bidirectional causal relationship with capability and choice architecture.
Evident in research showing how practice and learning result in skill
development, behavior influences capability (Billett 2010; Vaci et al. 2019).
Evidence for capability’s influence on behavior comes from research showing
that people tend to avoid activities that they find physically or
psychologically difficult (Kool et al. 2010; Feghhi and Rosenbaum 2021).
Behavior change practitioners use this insight - if you want to increase a
particular behavior in a population make it easier for people to do (Halpern
2015). Finally, behavioral science research has well documented the effects of
choice architecture on behavior (Michie, Van Stralen, and West 2011; Thaler
and Sunstein 2021), and individuals behaving in an environment will cause
changes to the environment (i.e., choice architecture).

As with the COM-B framework (Michie, Van Stralen, and West 2011), capability
refers to people’s psychological or physical skills. Capability is
operationalized in terms of changes in performance. Psychological skills
include, but are not limited to, knowledge, cognitive skills, memory,
interpersonal skills, attention, behavioral regulation, and willpower (Atkins
et al. 2017). Physical skills refer to various abilities for different
activities, as well as strength and stamina. Aside from capability’s
previously discussed relationship with preference and behavior, capability is
causally influenced by choice architecture. Certain environments promote more
(skill) learning than others (Land and Jonassen 2012; Gilavand 2016; Amundrud
et al. 2021). In behavior change, these insights are captured with boosts –
behavioral interventions which promote people’s capabilities (Franklin, Folke,
and Ruggeri 2019; Hertwig and Grüne-Yanoff 2017).

Choice architecture, for the present framework, is conceptualized with its
original definition, proposed by (Thaler and Sunstein 2008). Choice
architecture is the environment in which people behave (Thaler, Sunstein, and
Balz 2013). The use of choice architecture in the present framework differs
from how it is used in COM-B, in that it captures both opportunity and choice
architecture as described in the later framework (Michie, Van Stralen, and
West 2011). Choice architecture in the present framework thus captures the
resources needed for a behavior afforded by the physical (e.g., time,
location) and social (e.g., language) environment. It also captures aspects of
the physical and social environment that influence changes in capability,
behavior, and preference. Not all aspects of the environment are equally
influential, with different aspects having a larger or smaller effect size.
There have been several documented aspects of the physical environment being
influential with a full review being outside of the scope of the present paper
(Ruggeri 2018; Thaler and Sunstein 2021). Influence from the social
environment broadly relates to either influence from other people’s observable
behavior or stated preferences (Cialdini and Goldstein 2004; Cialdini and
Griskevicius 2010). In behavioral science, these sources of influence are
referred to as descriptive social norms –- how most people behave -– and
injunctive social norms –- what most people prefer and value (Reno, Cialdini,
and Kallgren 1993).

## Future research directions

A robust science of preference will benefit from contributions from the bodies
of knowledge of diverse fields, including but not limited to psychology,
cognitive science, computer science, marketing, economics, philosophy, and
law. Translational research using existing insights from these fields would be
beneficial. Such research could expand the scope of preference, as well as our
understanding of preference change, meta-preference, and preference-change
preference.

Preference science could benefit from new empirical and simulation studies.
This would involve developing paradigms focusing on preference as the main
dependent variable that can explore how controlled changes in capability,
behavior, and choice architecture predicatively change preference. Simulation
studies using valid assumptions for preference change mechanisms can also
allow researchers to test hypotheses.

Internal researchers in industries that own user data and can rapidly deploy
changes to user interfaces have the ability to conduct preference-science
research. However, any research which could be categorized as behavioral
experimentation must be accompanied by ethical and legal protocols developed
for such cases, even when conducted in a commercial context. Computer
Engineers must educate themselves about the ethics of user manipulation, and
it should be incorporated as a core part of their formal education to help the
effort in making any research conducted behind closed doors ethically and
legally sound. For this to happen, Preference Science should address when
preference change is ethically appropriate and legally permissible. That
altering preferences of users necessarily requires ethical consideration is a
major contribution of this paper.

## Conclusion

This article argues for a multidisciplinary effort to study preference change
processes motivated by the observation that AI and ML system owners are
altering user preferences for their own purposes. This is often brought about
through behavior change techniques. We have introduced a broad definition of
preferences and discussed the important issues of meta and preference-change
preferences. We present a framework that looks at the dynamic interplay
between choice architecture, capability, and behavior, and how this results in
preference change. We argue that society needs to develop legal and ethical
frameworks to identify acceptable from abusive preference change practices.

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# Correlation functions and characteristic lengthscales in flat band
superconductors

M. Thumin<EMAIL_ADDRESS>G. Bouzerar
<EMAIL_ADDRESS>Université Grenoble Alpes, CNRS, Institut NEEL,
F-38042 Grenoble, France

(August 28, 2024)

###### Abstract

The possibility of an unconventional form of high temperature
superconductivity in flat band (FB) material does not cease to challenge our
understanding of the physics in correlated systems. Recently, it has been
argued that the coherence length in FB compounds could be decomposed into a
conventional part of BCS type and a geometric contribution which characterises
the FB eigenstates, the quantum metric. Here, we propose to address this issue
in various FB systems and discuss whether the extracted characteristic
lengthscales such as the size of the Cooper pairs obey this conjecture. It is
found that the relevant lengthscales are less than one lattice spacing, weakly
sensitive to the strength of the electron-electron interaction, and more
importantly disconnected from the quantum metric.

##  I Introduction

Over the past ten years we are witnessing a rapidly growing interest for the
physics in dispersion-less bands Leykam_review ; Regnault2022 ; BERGHOLTZ ;
singular_FB ; photonic_Leykam ; photonic_Poblete ; Derzhko ; Efetov . In flat
band (FB) compounds, because the width of these bands is extremely narrow, the
Coulomb energy is left as the unique relevant energy scale. This places
naturally these systems in the class of highly correlated materials and opens
the access to exotic and unexpected physical phenomena and quantum phases.
Undeniably, one of the most striking feature is the possibility of high
critical temperature superconductivity (SC) in compounds where the Fermi
velocity vanishes Volovik_T_linear ;
Volovik_T_linear_Flat_bands_in_topological_media ; Peotta_Nature ;
Batrouni_Creutz ; Torma_revisiting ; Batrouni_Designer_Flat_Bands ;
Peotta_band_geometry ; Hofmann_PRB ; Iskin ; Thumin_EPL_2023 . This
unconventional form of SC is of inter-band nature and characterised by a
geometrical quantity known as the quantum metric (QM). The QM is connected to
the real part of the quantum geometric tensor Provost_metric ; Resta  and
provides a measure of the typical surface associated to the FB Bloch
eigenstates. So far, the unique experimental realisation of such an unusual
form of superconductivity is very likely the one that has been observed in
twisted bilayer of graphene (Moiré) in the vicinity of magic angles Cao2018_1
; Cao2018_2 ; Efetov ; Cao_t3g ; t3g ; McDo ; Torma_TBG .
It is well known that in conventional BCS systems where the SC is of intra-
band nature BCS ; Tinkham , the coherence length $\xi_{c}$ is given by
$\xi_{BCS}=\frac{\hbar v_{F}}{\Delta}$ where $v_{F}$ and $\Delta$ are
respectively the Fermi velocity and superconducting gap or pairing amplitude.
We recall that $\xi_{c}$ measures the size of the Cooper pair in real space.
Since, in the BCS regime (weak coupling) the SC gap is exponentially small,
$\xi_{c}$ is often extremely large, hence Cooper pairs are highly overlapping
with each other. On the other hand, in the strong coupling regime the Cooper
pairs can be assimilated to tightly bound non-overlapping composite bosons
which at low temperature leads to the well known Bose Einstein condensation
phenomenon (BEC)Leggett ; Pitaevskii .
A natural question arises: what about the case of FB superconductors?
Recently, it has been argued that the coherence length in these systems has
two contributions, the first is of conventional type and the other is purely
geometric in nature Law_GL ; Law_anomalous . More precisely, it is claimed
that the coherence length can be expressed as
$\xi_{c}=\sqrt{\xi^{2}_{BCS}+\langle g\rangle}$ where $\langle g\rangle$ is
the average of the QM. Hence, if the band is rigorously flat the first term
vanishes.

Figure 1: Schematic representation of (a) the stub lattice (StL), (b) the
sawtooth chain (SaL), (c) the Creutz ladder (CrL) and (d) the two-dimensionnal
$\chi$-lattice ($\chi$-L). The hoppings and the on-site Hubbard attractive
interaction term are depicted in the figure. In the case of the $\chi$-L (two
orbitals A and B per site) the hoppings are long range (see main text).

The purpose of the present study is to address this issue in several FB
lattices and discuss our findings in connection with these predictions and
with the existing literature. More precisely, we propose to consider four
different systems, three of them are one dimensional and the last one is two
dimensional: the stub lattice (StL), the sawtooth chain (SaL), the Creutz
ladder (CrL) and the $\chi-$lattice ($\chi$-L). These models are illustrated
in Fig.1. Notice that the $\chi$-L has been originally introduced in Ref.
Intro_chi . However, since no specific name has been attributed to this
peculiar model,$"\chi-$lattice" has been chosen. In this system, the range of
the extended hoppings is controlled by a single parameter ($\chi$) as it will
become more explicit in the next paragraph. The choice of these four different
systems is motivated by several intentions. It allows to estimate the impact
of (i) the bipartite character of the lattice, (ii) the tunability of the
quantum metric, (iii) the absence of dispersive bands in the spectrum, (iv)
and last the lattice dimension.

##  II Theory and methods

Electrons are described by the attractive Hubbard model which reads,

$\begin{split}\hat{H}=\sum_{i\lambda,j\eta,\sigma}t^{\lambda\eta}_{ij}\;\hat{c}_{i\lambda,\sigma}^{\dagger}\hat{c}_{j\eta,\sigma}-\mu\hat{N}-|U|\sum_{i\lambda}\hat{n}_{i\lambda,\uparrow}\hat{n}_{i\lambda,\downarrow},\end{split}$
(1)

where $\hat{c}_{i\lambda,\sigma}^{\dagger}$ creates an electron of spin
$\sigma$ at site $\textbf{r}_{i\lambda}$, $i$ being the cell index and
$\lambda$ the orbital index ranging from 1 to $n_{orb}$.
$\hat{N}=\sum_{i\lambda,\sigma}\hat{n}_{i\lambda,\sigma}$, $\mu$ is the
chemical potential and $|U|$ is the strength of the on-site attractive
electron-electron interaction. The hoppings are very short ranged in StL, SaL
and CrL as depicted in Fig.1. On the other hand, in the $\chi$-L the situation
differs, the hoppings are long-ranged, restricted to $(A,B)$-pairs, and given
by
$t^{AB}_{ij}=-\frac{t}{N_{c}}\sum_{\textbf{k}}e^{i\textbf{k}.\textbf{r}}e^{i\alpha_{\textbf{k}}}$
where $\alpha_{\textbf{k}}=\chi(\cos(k_{x}a)+\cos(k_{y}a))$,
$\textbf{r}=\textbf{r}_{j}-\textbf{r}_{i}$, and $N_{c}$ being the number of
unit cells. The parameter $\chi$ controls both the range of the hoppings and
the QM which is given by $\langle g\rangle=\chi^{2}a^{2}/8$ Chi_QMC .

In this work, we treat the interaction term within the Bogoliubov de Gennes
(BdG) approach which consists in the following decoupling scheme,

$\begin{split}\hat{n}_{i\lambda,\uparrow}\hat{n}_{i\lambda,\downarrow}\overset{BdG}{\simeq}&\langle\hat{n}_{i\lambda,\downarrow}\rangle\hat{n}_{i\lambda,\uparrow}+\langle\hat{n}_{i\lambda,\uparrow}\rangle\hat{n}_{i\lambda,\downarrow}\\\
+\;&\,\frac{\Delta_{i\lambda}}{|U|}\hat{c}^{\dagger}_{i\lambda,\uparrow}\hat{c}^{\dagger}_{i\lambda,\downarrow}+\frac{\Delta_{i\lambda}^{*}}{|U|}\hat{c}_{i\lambda,\downarrow}\hat{c}_{i\lambda,\uparrow},\\\
\end{split}$ (2)

where the self-consistent parameters $\langle\hat{n}_{i\lambda,\sigma}\rangle$
and
$\Delta_{\lambda}=-|U|\langle\hat{c}_{i\lambda\downarrow}\hat{c}_{i\lambda\uparrow}\rangle$
are respectively the orbital dependent occupations and pairings.
$\langle\ldots\rangle$ corresponds to the grand canonical average. Notice,
that the total carrier density is defined as $n=N_{e}/N_{c}$, where $N_{e}$ is
the total number of electrons, hence $n$ varies from 0 to $2\,n_{orb}$.

Before we discuss our calculations, we propose to provide some arguments that
justify that our approach is meaningful. We first start with the shortcomings.
It is well established that the BdG Hamiltonian being quadratic, it is
inappropriate to calculate reliably two particles correlation functions (CFs)
such as the pairing-pairing correlation function
$f_{P}(\textbf{r}_{i}-\textbf{r}_{j})=\langle\hat{\Pi}^{\dagger}_{i}\hat{\Pi}_{j}\rangle$
where the on-site pairing operator (s-wave)
$\hat{\Pi}^{\dagger}_{i}=\hat{c}^{\dagger}_{i\uparrow}\hat{c}^{\dagger}_{i_{\downarrow}}$.
In the case of the attractive Hubbard model in two dimensional systems, one
expects the correlation function $f_{P}(\textbf{r})$ to decay algebraically
with a $T$-dependent power for $T<T_{BKT}$, and exponentially when
$T>T_{BKT}$, where $T_{BKT}$ is the Berezinskii-Kosterlitz-Thouless transition
temperature Berezinsky_1972 ; Kosterlitz_1972 ; Kosterlitz_1973 . On the other
hand, the one-particle CF of the form
$f^{\sigma}_{sp}(\textbf{r}_{i}-\textbf{r}_{j})=\langle\hat{c}^{\dagger}_{i\sigma}\hat{c}_{j\sigma}\rangle$
always decays exponentially both in the superconducting phase and in the
normal phase. Mean Field theory such as the BdG approach can not describe the
change of behaviour of $f_{P}(\textbf{r})$ across the BKT transition, since
through Wick’s theorem two-particles CFs reduce to products of one-particle
CFs only. However, in FB systems, one expects the single particle CFs to be
well captured within the BdG theory. For instance, it has been shown, that the
local occupations, the pairings and the superfluid weight calculated by the
numerically unbiased DMRG are in excellent agreement with the mean field
values in the CrL and in the SaL Batrouni_Creutz ; Batrouni_sawtooth . It
should be emphasised that the agreement found concerns both the weak and the
strong coupling regime. In what follows it will be shown that it is as well
the case for correlations functions.

To study the characteristic lengthscales in the SC phase at $T=0$, we define
the normal and anomalous CFs,

$\displaystyle G_{\alpha\beta}(\textbf{r})$
$\displaystyle=\langle\hat{c}^{\dagger}_{i\alpha,\sigma}\hat{c}_{j\beta,\sigma}\rangle,$
(3) $\displaystyle\,K_{\alpha\beta}(\textbf{r})$
$\displaystyle=\langle\hat{c}_{i\alpha,\uparrow}\hat{c}_{j\beta,\downarrow}\rangle,$
(4)

where the index $i$ (respectively $j$) refers to the unit cell position
$\textbf{r}_{i}$ (respectively $\textbf{r}_{j}$), $\alpha$ (resp. $\beta$)
labels the orbitals and $\textbf{r}=\textbf{r}_{j}-\textbf{r}_{i}$. Here, the
spin index $\sigma=\uparrow,\downarrow$ is irrelevant, the SC phase being non
magnetic. The CF $K_{\alpha\beta}$ is particularly of interest since it allows
the extraction of the Cooper pair size. Indeed, in the case of a single one
dimensional dispersive band problem (conventional SC) it can be shown
analytically that
$K_{\alpha\alpha}(\textbf{r})\simeq\frac{1}{\sqrt{|\textbf{r}|}}e^{-|\textbf{r}|/\xi_{BCS}}$
for $|\textbf{r}|\rightarrow\infty$ as addressed in the next paragraph.

Figure 2: $\xi^{(K)}_{AA}$ as a function of the averaged pairing
$\Delta_{avg}$ in the quarter filled sawtooth chain (SaL). The red thick line
is the BCS formula $\frac{\hbar v_{F}}{\Delta_{avg}}$ where $\hbar v_{F}=2at$.
The first inset (top right) shows the correspondence between $|U|$ and
$\Delta_{avg}$ and the other one illustrates the density of states for
$|U|=0$, with $E_{F}=-2t$ for the quarter filling. The BCS regime corresponds
to $\xi^{(K)}_{AA}\gg a$ and BEC to $\xi^{(K)}_{AA}\leq a$.

##  III Results and discussions

###  III.1 Coherence length in dispersive bands

Before we discuss in details the case where the Fermi energy coincides with
that of the FB, it is interesting to analyse the situation where it is located
inside the dispersive bands. To illustrate this scenario, we consider the
quarter filled SaL. This density corresponds to the half-filling of the lower
dispersive band.

In Fig.2, $\xi^{(K)}_{AA}$ is plotted as a function of the averaged pairing
$\Delta_{avg}$ in the quarter filled SaL where
$\Delta_{avg}=\frac{1}{2}(\Delta_{A}+\Delta_{B})$ (A and B sites are
inequivalent). This characteristic lengthscale is obtained from a fit of the
form $\frac{1}{\sqrt{|\textbf{r}|}}e^{-|\textbf{r}|/\xi^{(K)}_{AA}}$ of the
long distance behaviour of the anomalous CF $K_{AA}(\textbf{r})$. The BCS-like
expression (red thick line in the figure) is defined as $\frac{\hbar
v_{F}}{\Delta_{avg}}$. Here the Fermi velocity
$v_{F}=\frac{2a\,t}{\hbar}\sin(k_{F}a)$ where $k_{F}a=\frac{\pi}{2}$ for the
quarter filled SaL. It is striking to see that the excellent agreement found
between the numerical data and the BCS expression is not restricted to the
weak coupling regime ($\Delta_{avg}\ll t$). Indeed, remarkably the agreement
is obtained for values of the average pairing that varies over four decades
(see inset of Fig.2), which corresponds to $|U|/t$ that varies from $1$ to
$8$.

###  III.2 The case of half-filled bipartite lattices

We consider the specific case of half-filled bipartite lattices where the
number of orbitals in one sublattice is larger than that of the other,
implying that at least one FB is located at $E=0$. We propose to demonstrate
the following remarkable property, valid for any |U|,

$\displaystyle G_{\lambda\lambda}(\textbf{r})=\frac{1}{2}\delta(\textbf{r}).$
(5)

In a recent study Bouzerar_SciPost_2024  it has been shown that the Bogoliubov
quasi-particle (QP) eigenstates present an interesting symmetry in half-filled
systems. If $\mathcal{A}$ (resp. $\mathcal{B}$) denotes the first (resp.
second) sublattice which contain $\Lambda_{\mathcal{A}}$ (resp.
$\Lambda_{\mathcal{B}}$) orbitals per unit cell, the QP eigenstates can be
subdivided in two families $\mathcal{S^{+}}$ and $\mathcal{S^{-}}$ defined in
what follows. First, a generic QP eigenstate (in momentum space) has the form
$|\Psi\rangle=(|\Psi^{\uparrow}\rangle,|\Psi^{\downarrow}\rangle)^{t}$ where
the first $\Lambda_{\mathcal{A}}$ (resp. next $\Lambda_{\mathcal{B}}$) rows of
$|\Psi^{\sigma}\rangle$ are the components on sublattice $\mathcal{A}$ (resp.
$\mathcal{B}$). This eigenstate belongs to the subspace $\mathcal{S^{+}}$
(resp. $\mathcal{S^{-}}$) if
$|\Psi^{\downarrow}\rangle=\hat{M}|\Psi^{\uparrow}\rangle$ (resp.
$|\Psi^{\downarrow}\rangle=-\hat{M}|\Psi^{\uparrow}\rangle$) where the matrix
$\hat{M}=\text{diag}(\hat{1}_{\Lambda_{\mathcal{A}}},-\hat{1}_{\Lambda_{\mathcal{B}}})$.
Additionally, for any finite $|U|$, it has been shown in Ref.
Bouzerar_SciPost_2024  that the subset $\mathcal{S^{-}}$ (respectively
$\mathcal{S^{+}}$) consists exactly in $\Lambda_{\mathcal{B}}$ (respectively
$\Lambda_{\mathcal{A}}$) eigenstates of positive or zero energy and
$\Lambda_{\mathcal{A}}$ (respectively $\Lambda_{\mathcal{B}}$) eigenstates of
strictly negative energy.

Now, start with the definition
$G_{\lambda\lambda}(\textbf{r})=\frac{1}{N_{c}}\sum_{\textbf{k}}e^{i\textbf{k}.\textbf{r}}\langle\hat{O}_{\lambda\textbf{k},\uparrow}\rangle$
where ,
$\hat{O}_{\lambda\textbf{k},\uparrow}=c^{\dagger}_{\textbf{k}\lambda,\uparrow}c_{\textbf{k}\lambda,\uparrow}$.
At $T=0$, its grand canonical average is given by,

$\displaystyle\langle\hat{O}_{\lambda\textbf{k},\uparrow}\rangle=\sum_{m}\langle\Psi^{<}_{m\textbf{k}}|\hat{O}_{\lambda\textbf{k},\uparrow}|\Psi^{<}_{m\textbf{k}}\rangle,$
(6)

where $|\Psi^{<}_{m\textbf{k}}\rangle$ are the QP eigenstates of the BdG
Hamiltonian of negative energy, $m$ being band index. Using the closure
relation,
$\sum_{m,s=<,>}|\Psi^{s}_{m\textbf{k}}\rangle\langle\Psi^{s}_{m\textbf{k}}|=1$,
where the sum runs over QP eigenstates with positive ($s=>$) and negative
energy ($s=<$) and the symmetry mentioned above one can show that,

$\displaystyle\sum_{m}\langle\Psi^{<}_{m\textbf{k}}|\hat{O}_{\lambda\textbf{k},\uparrow}|\Psi^{<}_{m\textbf{k}}\rangle=\sum_{m}\langle\Psi^{>}_{m\textbf{k}}|\hat{O}_{\lambda\textbf{k},\uparrow}|\Psi^{>}_{m\textbf{k}}\rangle,$
(7)

which combined with Eq.(6) leads to
$\langle\hat{O}_{\lambda\textbf{k},\uparrow}\rangle=\frac{1}{2}$ and
demonstrates Eq.(5).
It is interesting to remark that our proof can be straightforwardly extended
to the case of disordered systems that preserve the bipartite character of the
lattice, such as the presence of vacancies or bond disorder.

Figure 3: (a) $K_{CC}$ as a function of $r$ in the stub lattice (StL) for
several values of $|U|/t$ (1, 2.5 and 5). (b) $\xi^{(K)}_{CC}$ as a function
of $|U|$. The (dark-green) horizontal line depicts the square root of the
quantum metric $\langle g\rangle$. The inset shows $\xi^{(K)}_{CC}$ for
$|U|\gg t$. Here, $\alpha$ is set to $0.5$ (see Fig.1) and the carrier density
is fixed to $n=3$ which corresponds to half-filling.

###  III.3 The Stub lattice

The StL is bipartite and offers the possibility to tune the QM without
changing the of the nature of the compact localized eigenstates. The QM is
controlled by the A-C hopping ($\alpha t$)(see Fig.1) and given by $\langle
g\rangle=\frac{1}{2|\alpha|\sqrt{2+\alpha^{2}}}$ Giant_boost . The StL has
been studied in great details in Refs. Thumin_PRB_2023 ; Thumin_EPL_2023 .
Here, we restrict our study to the case $\alpha=0.5$ and $n=3$ which
corresponds to a half-filled FB with $\langle g\rangle\simeq 0.7$.
First, one can already conclude from the previous section that the
conventional CFs ($G_{\alpha\alpha}$) are given by Eq.(5), which is indeed
what we find numerically for any $|U|$ and any $\alpha$. Figure 3(a) depicts
the anomalous CF $K_{CC}$ as a function of $|\textbf{r}|$ for several values
of $|U|$ which correspond to weak, intermediate and strong coupling regime. As
it can be clearly seen, in all cases this CF decays exponentially with a
lengthscale $\xi^{(K)}_{CC}$ (Cooper pair size) that reduces rapidly as $|U|$
increases. The variation of the extracted lengthscale $\xi^{(K)}_{CC}$ is
plotted as a function of $|U|/t$ in Fig.3(b). In the limit of vanishing
$|U|/t$ it is approximately (for this value of $\alpha$) $2a$, then it
increases and reaches a maximum for $|U|/t=1.5$ and beyond it decreases
continuously. There is no simple explanation for the origin of this maximum,
since for larger values of $\alpha$ it disappears. The inset represents, its
behaviour in the large $|U|/t$ limit. It is found that
$\xi^{(K)}_{CC}\rightarrow 0.125\,a$. As it can be seen, $\xi^{(K)}_{CC}$
crosses $\sqrt{\langle g\rangle}=0.7\,a$ at $|U|/t\approx 4$ and converges to
a much smaller value. The large $|U|/t$ behaviour, is consistent with the fact
that in the BEC regime, the Cooper pair size is expected to be very small.
Remark that $K_{BB}$ and $K_{AA}$ vary similarly with the same lengthscale.

###  III.4 The sawtooth chain

In contrast to the StL, the SaL as illustrated in Fig.1(b), is a non bipartite
lattice and does not allow the tuning of the QM. The FB exists only when the
AB-hoppings (1st and 2nd neighbours) are $-\sqrt{2}t$. The superconductivity
in the StL has been addressed in details in Ref. Batrouni_sawtooth  using a
numerically exact method: the DMRG. It has been shown that the BdG approach
reproduces accurately the exact results, for both the pairings and the
superfluid weight. In Fig.4(a), both $G_{AA}$ and $K_{AA}$ are plotted as a
function of $|\textbf{r}|$, for different values of $|U|$. Here, the electron
density is set to $n=3$ which corresponds to the half-filled FB. As it can be
seen, the lengthscales associated to the decay of $G_{AA}$ and $K_{AA}$ are
almost identical both in the weak and strong coupling regime. Additionally,
the slope appears to vary weakly. Notice that $G_{BB}$ and $K_{BB}$ behave
similarly. Fig.4(b) depicts the variation of $\xi^{(K)}_{AA}$ as a function of
$t/|U|$. The inset describes the weak coupling regime. In this regime,
$\xi^{(K)}_{AA}\approx 0.735\,a$ and almost insensitive to $|U|$. As |U|
increases further, $\xi^{(K)}_{AA}$ decays monotonously. As seen in the case
of the StL, $\xi^{(K)}_{AA}$ crosses $\sqrt{\langle g\rangle}$ when
$t/|U|\approx 0.05$ and converges towards $0.2\,a$. In the SaL, it can be
shown that the minimal QM is $\langle g\rangle=\frac{1}{4\sqrt{3}}$. We should
mention as well that our values of $G_{AA}(r)$ are consistent with the DMRG
calculations of Ref. Batrouni_sawtooth .

Figure 4:  (a) $|G_{AA}|$ and $|K_{AA}|$ as a function of $r$ in the sawtooth
chain (SaL) for several values of $|U|$. For the sake of clarity, $|G_{AA}|$
and $|K_{AA}|$ have been multiplied by 10-2,10-4 and 10-6 for $|U|=0.5$, 1 and
5 respectively. The carrier density is $n=3$ (half-filled FB). DMRG data for
$|U|=1$ from Ref. Batrouni_sawtooth  are shown as well. (b) $\xi^{(K)}_{AA}$
as a function of $t/|U|$. The horizontal lines depicts the square root of the
minimal quantum metric $\langle g_{min}\rangle$. The inset represents
$\xi^{(K)}_{AA}$ as a function of $|U|$ for small values of $|U|$. The dashed
red line is a linear fit for $\frac{t}{|U|}\leq\,0.03$.

###  III.5 The Creutz ladder

The CrL depicted in Fig1(c) is particularly interesting since its dispersion
consists only in FBs, located at $E=\pm 2t$ in the non-interacting case. As a
consequence of the uniform pairings, these bands remain flat when $|U|$ is
non-zero. The superconductivity in the CrL have been addressed exactly, within
the DMRG approach in Refs Batrouni_Creutz ; Batrouni_sawtooth . As in the case
of the sawtooth chain, it has been revealed that pairings and superfluid
weight are accurately captured by the BdG theory. The A and B sites being
equivalent, we focus our attention on $|K_{AA}|$ and $|G_{AA}|$. In addition,
we consider the case of the quarter filled ladder (half-filled lower FB) which
corresponds to $n=1$. Both CFs are plotted in Fig.5 as a function of
$|\textbf{r}|$ for several values of $|U|$ ranging from weak to strong
coupling regime. As it can be seen these two CFS behave similarly.
Surprisingly, it is found that there are only two non-vanishing values
corresponding respectively to $|\textbf{r}|=0$ and a. For larger distances,
$|K_{AA}|$ and $|G_{AA}|$ are zero within the numerical accuracy. This is
illustrated in the inset of Fig.5(b) where for $|\textbf{r}|=2a$ the CF
$|G_{AA}|$ drops by 16 orders of magnitude. It is found as well that
$|K_{AA}|(|\textbf{r}|=a)$ decays very rapidly as $|U|\geq 1$ and eventually
vanishes when $|U|\rightarrow\infty$. Thus, the Cooper pair size varies
between $1$ and $0$ where $0$ corresponds to $|U|=\infty$.
In the Appendix A, we demonstrate analytically in the case of weak coupling
that the CFs are given by,

$\displaystyle\begin{split}&G_{AA}(r)=K_{AA}(r)=\frac{1}{4}\delta_{r,0}-\frac{i}{8}\delta_{r,a}+\frac{i}{8}\delta_{r,-a},\\\
&G_{AB}(r)=K_{AB}(r)=\frac{1}{8}(\delta_{r,a}+\delta_{r,-a}).\end{split}$ (8)

We point out the fact that the analytic expression found for $G_{AA}(r)$ is
consistent with the exact results obtained from DMRG calculations
Batrouni_Creutz . Indeed, it has been found (see Fig.10 in this manuscript)
that for $r\geq 2a$, $G_{AA}\leq 10^{-12}$.

Figure 5:  (a) $|K_{AA}|$ and (b) $|G_{AA}|$ as a function of $r$ in the
Creutz ladder (CrL) for several values of $|U|$. The charge density is fixed
$n=1$. For $r\geq 2\,a$, both $|K_{AA}|$ and $|G_{AA}|$ are zero within our
numerical precision. The inset in (a) represents $|K_{AA}|$ in log scale. The
inset in (b) shows $|G_{AA}|$ as a function of $|U|$ for $r=0$ and $r=a$.
Diamonds are our calculations and circles are the DMRG data of Ref.
Batrouni_Creutz

###  III.6 The $\chi$-Lattice

The $\chi$-L is a two dimensional system in which both electronic bands are
dispersion-less and located at $E=\pm t$. As mentioned earlier this system has
been introduced originally in Ref. Intro_chi . The superconductivity has been
addressed within the Quantum Monte Carlo (QMC) method in Ref. Chi_QMC  and
within a mean field approach in Ref. Law_anomalous . We recall that the
dimensionless parameter $\chi$ controls both the range of the hoppings and the
value of the QM. Here, we focus on the quarter filled system which corresponds
to a charge density $n=1$. As it is the case in the CrL, the orbitals A and B
are equivalent, pairings are identical on both sites. In addition, because the
long range hoppings connects A to B sites only, this lattice is bipartite as
well.

Figure 6:  (a) and (b) $|G_{AA}|$ and $|K_{AA}|$ as a function of $r$ (along
the $x-$direction) in the $\chi$-L for several values of $\chi$. (c) same as
in (a) and (b) for the off-diagonal correlation functions $|G_{AB}|$ and
$|K_{AB}|$. The carrier density is $n=1$ and the Hubbard parameter $|U|=1$.

Let us now discuss our results. First, for any value of both $|U|$ and $\chi$,
with high numerical accuracy we find,

$\displaystyle\frac{4}{n}G_{\alpha\alpha}(\textbf{r})=\frac{|U|}{\Delta}K_{\alpha\alpha}(\textbf{r})=\delta(\textbf{r}),$
(9)

where $\alpha=A,B$. These features are illustrated in Fig.6 (a) and (b). It
should be emphasised that the property given in Eq.(5) concerns only the case
of half-filled bipartite lattices. Here, our system is quarter filled, which
means that our findings are specific to the $\chi$-L. As a consequence, for
any $|U|$ the Cooper pair size is zero. In the Appendix B, we have
demonstrated analytically Eq.(9) in the weak coupling regime.

More strikingly, we have found that the off-diagonal correlation functions
$|G_{AB}|$ and $|K_{AB}|$ exhibit an unexpected behaviour as it can be clearly
seen in Fig.6(c). First, one finds that $|G_{AB}|$ and $|K_{AB}|$ are very
similar for any value of $\chi$. Furthermore, for a given $\chi$, one can
distinguish two distinct regimes. First, for $|\textbf{r}|\leq\chi a$ the CFs
oscillates as $|\textbf{r}|$ increases. Secondly, when $|\textbf{r}|\geq\chi
a$ it decays monotonously as the distance increases. However, any attempt to
fit the tail by a function of the form $r^{-\alpha}e^{-|\textbf{r}|/\beta}$ is
unsuccessful. Hence, one cannot extract any characteristic lengthscale from
these off-diagonal correlation functions. In the Appendix B, we have
calculated analytically the $G_{AB}$ and $K_{AB}$ as a function of r in the
limit of small values $|U|$. It is shown that
$G_{AB}(\textbf{r})=K_{AB}(\textbf{r})=\frac{1}{4N_{c}}\sum_{\textbf{k}}e^{i\textbf{k}.\textbf{r}}e^{-i\alpha_{\textbf{k}}}$.
This means that for this specific lattice the off-diagonal CFs coincides up to
a coefficient with the (A,B) hoppings in real-space. In addition, in the limit
of large $|\textbf{r}|$ along the $x$-direction, it is shown that,

$\displaystyle
G_{AB}(\textbf{r})\propto(-i)^{n_{x}}J_{0}(\chi)\frac{1}{\sqrt{2\pi
n_{x}}}e^{n_{x}.ln(\frac{e\chi}{2n_{x}})},$ (10)

where $\textbf{r}=(n_{x}a,0)$ and $J_{0}$ is the Bessel function of the first
kind and order 0. This clarifies why we could not extract a typical
lengthscale from the numerical data plotted in Fig.6(c).

###  III.7 Connection with recent studies

In this paragraph, we would like to discuss the connection between our
findings and recent studies Law_GL ; Law_anomalous . It is claimed in these
articles, that the coherence length in quasi FBs can be expressed as,
$\xi_{c}=\sqrt{\xi^{2}_{BCS}+\langle g\rangle}$ where $\langle g\rangle$ is
the average of the quantum metric (minimal). The BCS contribution vanishes
when the band is rigorously flat. In Ref. Law_anomalous , the authors have
illustrated their point by considering the $\chi-$L. Based on the fact that
the coherence length is extracted from the long distance decay of
$K_{\alpha\alpha}$ our findings clearly contradicts their prediction. Indeed,
in the specific case of the $\chi-$L we have found that for any $|U|$ and any
$\chi$, thus any QM, the size of the Cooper pair is always zero. What is the
origin of this contradiction? In Ref. Law_anomalous , the authors have treated
the electron-electron correlation at the mean field level, similar to ours.
However, their decoupling of the Hubbard term is performed in momentum space
instead of real space as done within the BdG approach. Additionally, the
authors have performed a projection of the fermionic operators onto the lowest
FB which is, in their work, the key step so that the QM can emerge. In
contrast, in our BdG approach we perform no projection and treat essentially
the electron-electron correlation at the same level, thus it is striking that
our results differ from those of Ref. Law_anomalous . This may suggest that
the origin of the disagreement could be the projection procedure. Notice that,
in their work, the authors have considered the pair-pair correlation function
of the form,
$C_{\alpha\beta}(\textbf{r})=\langle\hat{c}^{\dagger}_{i\alpha\uparrow}\hat{c}^{\dagger}_{i\beta\downarrow}\hat{c}_{0\beta\downarrow}\hat{c}_{0\alpha\uparrow}\rangle$
which within mean field reduces to,
$C_{\alpha\beta}(\textbf{r})\xrightarrow{MF}G_{\alpha\alpha}(\textbf{r})G_{\beta\beta}(\textbf{r})+|K_{\alpha\beta}(\textbf{0})|^{2}$.
Thus, our conclusion does not change, since the diagonal CFs
$G_{\alpha\alpha}(\textbf{r})$ ($\alpha=A$,$B$) reduce to $\delta$-functions
as given in Eq.(9).

##  IV Conclusion

To conclude, we have investigated the normal and anomalous correlations
functions in various flat band systems and extracted the associated
characteristic lengthscales. It is found that the size of the Cooper pairs is
less than one lattice spacing, both in the weak and strong coupling regime. We
have shown as well that the normal correlation function reduces to a Dirac
function in the case of half-filled bipartite lattices. Finally, in contrast
with a recent claim it appears that the coherence length is disconnected from
the quantum metric.

##  V Aknowledgement

We thank George Batrouni, Si Min Chan and Benoit Grémaud for kindly sending us
their DMRG data.

## Appendix A: The correlation functions in the Creutz ladder

In this appendix we propose to derive analytically the correlations functions
G and K as defined in the main text in the quarter filled Creutz ladder. We
focus our attention on small values of $|U|$. We restrict our calculation to
$T=0$. The BdG Hamiltonian reads,

$\displaystyle\hat{H}_{BdG}=\sum_{k}\hat{\Psi}_{k}^{\dagger}\begin{pmatrix}\hat{h}^{\uparrow}_{\textbf{k}}&\Delta\hat{1}_{2\times
2}\\\ \Delta^{*}\hat{1}_{2\times
2}&-\hat{h}^{\downarrow*}_{\textbf{-k}}\end{pmatrix}\hat{\Psi}_{k},$ (A.1)

where we have introduced the Nambu spinor
$\hat{\Psi}_{k}^{\dagger}=(\hat{c}_{Ak\uparrow},\hat{c}_{Bk\uparrow},\hat{c}_{A-k\downarrow},\hat{c}_{B-k\downarrow})^{t}$
and the block matrix,

$\displaystyle\hat{h}^{\uparrow}_{\textbf{k}}=\begin{pmatrix}-2t\sin(ka)-\tilde{\mu}&-2t\cos(ka)\\\
-2t\cos(ka)&2t\sin(ka)-\tilde{\mu}\end{pmatrix},$ (A.2)

where we have introduced $\tilde{\mu}=\mu+\frac{|U|}{4}n$. Because of time
reversal symmetry
$\hat{h}^{\downarrow*}_{\textbf{-k}}=\hat{h}^{\uparrow}_{\textbf{k}}$. Notice
as well that the pairing $\Delta$, uniform because A and B sites are
equivalent, can be taken real. Here, the total carrier density
$n=2\,n_{A}=2\,n_{B}$ is set to 1.

First, we consider the case $|U|=0$ for which the chemical potential
$\mu=\mu_{0}=-2\,t$. The quasi-particle (QP) eigenvalues are $E_{1,4}=\pm
4\,t$, and $E_{2,3}=0$ which is doubly degenerate. The corresponding QP
eigenstates are of the form,
$|\Psi_{i}\rangle=(|\psi^{\uparrow}_{i}\rangle,|\psi^{\downarrow}_{i}\rangle)^{t}$,
where $i=1,..,4$.
More precisely they are given by,
$|\Psi^{0}_{1}\rangle=(0,|\phi^{+}_{0}\rangle)^{t}$,
$|\Psi^{0}_{2}\rangle=(|\phi^{-}_{0}\rangle,0)^{t}$,
$|\Psi^{0}_{3}\rangle=(0,|\phi^{-}_{0}\rangle)^{t}$, and
$|\Psi^{0}_{4}\rangle=(|\phi^{+}_{0}\rangle,0)^{t}$, where,

$\displaystyle|\phi^{\pm}_{0}\rangle=\frac{1}{\sqrt{2}}\frac{1}{\sqrt{1\pm\sin(ka)}}\begin{pmatrix}-\cos(ka)\\\
\sin(ka)\pm 1\end{pmatrix}.$ (A.3)

When the Hubbard term is switched on, we apply a pertubation theory for
degenerate pair eigenstates $(|\Psi^{0}_{2}\rangle,|\Psi^{0}_{3}\rangle)$ that
leads to, $E_{\pm}=\pm\sqrt{(\delta\tilde{\mu})^{2}+\Delta^{2}}$ where
$\delta\tilde{\mu}=\tilde{\mu}-\mu_{0}$. The corresponding QP eigenstates are,

$\displaystyle|\Psi_{\pm}\rangle=\frac{1}{\sqrt{N^{\pm}}}\Big{(}\Delta|\Psi^{0}_{2}\rangle+(\delta\tilde{\mu}\pm\sqrt{(\delta\tilde{\mu})^{2}+\Delta^{2}})|\Psi^{0}_{3}\rangle\Big{)},$
(A.4)

where
$N^{\pm}=2\Big{(}\delta\tilde{\mu}^{2}+\Delta^{2}\pm\delta\tilde{\mu}\sqrt{(\delta\tilde{\mu})^{2}+\Delta^{2}}\Big{)}$.
Using the self-consistent equations for the carrier density which for each
spin sector is $1/4$ and the gap equation one finds in the limit of small
$|U|$,

$\displaystyle\delta\tilde{\mu}=0+o(|U|^{2}),$ (A.5)
$\displaystyle\Delta=\frac{|U|}{4}+o(|U|^{2}).$ (A.6)

Thus, the QP eigenstates take the simple form
$|\Psi_{\pm}\rangle=\frac{1}{\sqrt{2}}(|\Psi^{0}_{2}\rangle\pm|\Psi^{0}_{3}\rangle)$,
their respective energy being $E_{\pm}=\mp\frac{1}{4}|U|$.
Using the expressions of $|\phi^{\pm}_{0}\rangle$ as given in Eq.(A.3), one
finds, $\langle c^{\dagger}_{Ak,\uparrow}c_{Ak,\uparrow}\rangle=\langle
c^{\dagger}_{Ak,\uparrow}c^{\dagger}_{Ak,\downarrow}\rangle=\frac{1}{4}(1+\sin(k))$.
After a trivial Fourier transform, we finally end up with,

$\displaystyle
G_{AA}(r)=K_{AA}(r)=\frac{1}{4}\delta_{r,0}-\frac{i}{8}\delta_{r,a}+\frac{i}{8}\delta_{r,-a}.$
(A.7)

In addition for the off-diagonal CFs it is found that,

$\displaystyle G_{AB}(r)=K_{AB}(r)=\frac{1}{8}(\delta_{r,a}+\delta_{r,-a}).$
(A.8)

These results explain the data plotted in Fig. 5 of the present manuscript. We
recall that our proof is restricted to $|U|\leq t$.

## Appendix B: The correlation functions in the $\chi$-lattice

In this appendix, our purpose is to derive analytically the correlation
functions G and K in the quarter filled $\chi-$Lattice. The BdG calculations
are performed for small values of the Hubbard parameter $|U|$ at
$T=0\leavevmode\nobreak\ K$. The BdG Hamiltonian has the same form as that
given in Eq.(A.1) of the Appendix A, with $\hat{h}_{\textbf{k}}^{\uparrow}$
now given by,

$\displaystyle\hat{h}_{\textbf{k}}^{\uparrow}=\begin{pmatrix}-\mu-\frac{|U|}{4}n&-te^{-i\alpha_{\textbf{k}}}\\\
-te^{i\alpha_{\textbf{k}}}&-\mu-\frac{|U|}{4}n\end{pmatrix},$ (B.1)

where $\alpha_{\textbf{k}}=\chi(\cos(k_{x}a)+\cos(k_{y}a))$.
Notice that the $\chi-$Lattice is both bipartite and time reversal symmetric
as well which implies
$\hat{h}_{\textbf{-k}}^{\downarrow*}=\hat{h}_{\textbf{k}}^{\uparrow}$.
To calculate the QP eigenstates, we use the same notation as those of Appendix
A. At $|U|=0$, the quasi-particle (QP) eigenstates are located at $E_{1,4}=\pm
2\,t$, and $E_{2,3}=0$ which is doubly degenerate, the chemical potential
being $\mu=\mu_{0}=-\,t$. The one particle eigenstates read,

$\displaystyle|\phi^{\pm}_{0}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\mp
e^{-i{\frac{\alpha_{\textbf{k}}}{2}}}\\\
e^{i{\frac{\alpha_{\textbf{k}}}{2}}}\end{pmatrix}.$ (B.2)

The equations (A.4), (A.5) and (A.6) of Appendix A are valid as well in the
case of the $\chi-L$ at quarter filling. Thus one straightforwardly gets,
$\langle\hat{c}^{\dagger}_{A\textbf{k},\uparrow}\hat{c}_{A\textbf{k},\uparrow}\rangle=\frac{1}{4}$,
$\langle\hat{c}^{\dagger}_{A\textbf{k},\uparrow}\hat{c}_{B\textbf{k},\uparrow}\rangle=\frac{1}{4}e^{-i{\alpha_{\textbf{k}}}}$,
$\langle\hat{c}^{\dagger}_{A\textbf{k},\uparrow}\hat{c}^{\dagger}_{A-\textbf{k},\downarrow}\rangle=\frac{1}{4}$
and
$\langle\hat{c}^{\dagger}_{A\textbf{k},\uparrow}\hat{c}^{\dagger}_{B-\textbf{k},\downarrow}\rangle=\frac{1}{4}e^{-i{\alpha_{\textbf{k}}}}$.
It follows that,

$\displaystyle
G_{AA}(\textbf{r})=K_{AA}(\textbf{r})=\frac{1}{4}\delta_{\textbf{r},\textbf{0}},$
(B.3)

and the off-diagonal CFs are,

$\displaystyle
G_{AB}(\textbf{r})=K_{AB}(\textbf{r})=\frac{1}{4}f_{AB}(\textbf{r}),$ (B.4)

where, we have introduced
$f_{AB}(\textbf{r})=\frac{1}{N_{c}}\sum_{\textbf{k}}e^{i\textbf{k}.\textbf{r}}e^{-i\alpha_{\textbf{k}}}$.
Thus, $G_{AB}(\textbf{r})$ and $K_{AB}(\textbf{r})$ coincide, up to a
constant, with the (A,B) hoppings. We now propose to calculate the analytic
expression of $f_{AB}(\textbf{r})$ for both $|\textbf{r}|/a\leq\chi$ and
$|\textbf{r}|/a\gg\chi$.
Let us write $\textbf{r}=(n_{x},n_{y})$, $f_{AB}(\textbf{r})$ can be rewritten
as the following product,

$\displaystyle f_{AB}(\textbf{r})=I_{n_{x}}(-i\chi)\cdot I_{n_{y}}(-i\chi),$
(B.5)

where
$I_{n}(i\chi)=\frac{1}{2\pi}\int^{+\pi}_{-\pi}e^{in\theta}e^{i\chi\cos(\theta)}$
is the modified Bessel function of the first kind and order $n$. We can now
rely on the properties of the Bessel functions such as
$I_{n}(-i\chi)=(-i)^{n}J_{n}(\chi)$ which leads to,

$\displaystyle f_{AB}(\textbf{r})=(-i)^{n_{x}+n_{y}}J_{n_{x}}(\chi)\cdot
J_{n_{y}}(\chi).$ (B.6)

In the regime where $|\textbf{r}|\leq\chi a$ one can expand the Bessel
function Bessel ,

$\displaystyle
J_{n}(\chi)\simeq\sqrt{\frac{2}{\pi\chi}}\cos(\chi-n\frac{\pi}{2}-\frac{\pi}{4}),$
(B.7)

and similarly for $J_{m}(\chi)$. This clearly explains the presence of the
oscillations observed in Fig. 6 of the manuscript.
In the opposite limit, more precisely for $\chi\ll\sqrt{|n_{x}|+|n_{y}|}$, one
has,

$\displaystyle
J_{n}(\chi)\simeq\frac{1}{\Gamma(n+1)}\Big{(}\frac{\chi}{2}\Big{)}^{n}.$ (B.8)

According to the well known Stirling formula, for $n\gg 1$ one can write
$\Gamma(n+1)\simeq\frac{1}{\sqrt{2\pi n}}(\frac{n}{e})^{n}$. Thus, along the
$x-$direction for instance, it implies the following result,

$\displaystyle f_{AB}(\textbf{r})=(-i)^{n_{x}}\frac{J_{0}(\chi)}{\sqrt{2\pi
n_{x}}}e^{n_{x}\ln{(\frac{e\chi}{2n_{x}}})}.$ (B.9)

This equation explains (i) the rapid decay observed in Fig. 6 of our
manuscript and (ii) the impossibility to extract a characteristic lengthscale
from the decay at large distance of the off-diagonal correlation functions.

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