# Future of Smart Classroom in the Era of Wearable Neurotechnology
††thanks: This research was partially supported by NSF awards CISE CRII
2105084 and CMMI 1739503. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not
necessarily reflect views of our funding agency.

Mojtaba Taherisadr, Berken Utku Demirel, Mohammad Abdullah Al Faruque, and
Salma Elmalaki  Department of Electrical Engineering and Computer Science
University of California, Irvine, California, USA  (taherisa, bdemirel,
alfaruqu<EMAIL_ADDRESS>
###### Abstract

Interdisciplinary research among engineering, computer science, and
neuroscience to understand and utilize the human brain signals resulted in
advances and widespread applicability of wearable neurotechnology in adaptive
human-in-the-loop smart systems. Considering these advances, we envision that
future education will exploit the advances in wearable neurotechnology and
move toward more personalized smart classrooms where instructions and
interactions are tailored to students’ individual strengths and needs. In this
paper, we discuss the future of smart classroom and how advances in
neuroscience, machine learning, and embedded systems as key enablers will
provide the infrastructure for envisioned smart classrooms and personalized
education along with open challenges that are required to be addressed.

###### Index Terms:

Smart Classroom, Personalized education, Human-in-the-loop, Wearable
neurotechnology, Embedded systems.

##  I Introduction

The disruption of face-to-face exchanges and the urgent digital transformation
of our everyday life activities, abide the COVID-19 hit, have spurred us to
shift many working paradigms to incorporate the new reality of the online and
remote collaborative environment. In particular, many sectors including the
education and the healthcare systems had to integrate new technology to assist
the continuation of their services even with remote interactions [1, 2].
Nevertheless, this forced digital transformation of many sectors is expected
to stay even in the post-COVID-19 era. In particular, in the education sector,
the e-learning market worldwide is forecast to surpass 243 billion U.S.
dollars by 2022 [3].

While the education sector has already taken technological leaps over the last
decades with the introduction of smart classroom concept [4, 5, 6], efficient
teacher-student interaction has to be envisioned especially with the
possibility of remote education. In particular, we envision that the future of
education using smart classrooms should move from a _one-size-fits-all_
approach to a _personalized_ process in which instructions and interactions
are tailored to students’ individual needs. In this paper, we argue that
advances in bio-sciences, machine learning, and embedded systems can play a
significant role in realizing the vision of personalized and smart education.

A cornerstone in the envisioned personalized learning is to replace the
traditional measures of education (e.g., quizzes, scores in exams, and teacher
evaluations) with real-time measurements of the student’s state of mind. Such
real-time signals can be then used to adapt the instructed materials to
maximize the student’s performance. The student’s stress level, drowsiness
level, readiness to learn new knowledge, cognitive load, and learning rate are
just examples of such real-time signals that can be used to adapt the
instructed materials to enhance the learning performance. Indeed, there is a
tremendous opportunity in attaining this vision. In particular, advances in
neuroscience and wearable sensors technology have widen the horizon to reveal
the fundamental processes of various human mental states [7, 8]. These
advancements resulted in tremendous research efforts among interdisciplinary
fields from engineering researchers, computer scientists, and healthcare
(neurologists, clinicians) to integrate the brain state (drowsiness, sleep
quality, readiness potential, and learning activity) as an additional sensor
modality to design adaptive human-in-the-loop smart systems [9]. Using such
advances, we envision the future smart classroom as shown in Figure 1. Future
smart classroom will be extended to include physical and remote students.
Students’ state will be used to adapt the education system through feedback
and personalized recommendation to enhance the student’s performance.

In what follows, we highlight two scientific and technological advances that
are key enablers for the envisioned personalized and smart classrooms along
with open challenges that need to be addressed.

Figure 1: Future smart classroom in the era of wearable neurotechnology.
Student’s mental state inferred from wearable sensors is used to provide
feedback and recommendation to the student and the teacher for both physical
and remote classes.

##  II Advancement in Neuroscience to understand cognitive processing

As discussed above, a fundamental challenge to achieve the aforementioned
vision is to measure the student’s mental state, in real-time. Indeed, recent
advances in neuroscience have opened the gate to unveil fundamental processes
in the human brain, such as the ability to generate emotions, memories, and
actions [10, 11]. These research efforts have become possible by the ability
to record and stimulate the human brain activity in the clinical setup by
neurologists with a very high accuracy [12]. Optimal cognitive processing is
central to all aspects of human activities, and recent advances in
neuroscience have provided critical insights on how the brain accomplishes
cognitive processing, spanning single neuron to neural population level
resolution [12].

In particular, recent studies using event-related functional magnetic
resonance imaging (fMRI) showed that there exists a high correlation between
the magnitudes of focal activation in the right prefrontal cortex and the
bilateral parahippocampal cortex during visual learning with the memory
processing [13]. In addition, recent findings in literature demonstrating that
brain electrophysiological activity observed before stimulus or new concepts
presentation profoundly influences subsequent behavior on fear conditioning
[14], motor responses [15], attention [16], and memory tasks [17].

While these studies gave us the fundamentals to measure the ability to
memorize visual content, and the specific parts in the brain that are
responsible for cognitive processing, the form factor of the utilized
machinery prevent the widespread use of this technology in classroom settings.
The next section reviews some technological advances that can provide a
solution to such a challenge.

##  III Advancement in wearable neurotechnology

With the development of efficient, intelligent sensing technology, detecting
small changes in electric signals becomes plausible. This technology paved the
way for developing and producing devices that are capable of monitoring the
brain’s electrical activity non-invasively with enough resolution and low cost
compared to the clinical setups. Although the traditional brain activity
monitoring systems suffer from motion artifacts, long preparation time due to
conductive gels or pastes, and large equipment (metal electrodes, long wires),
the advances in sensor technology allow a fully portable, wireless, long-term,
flexible scalp electronic system, incorporating a set of dry electrodes [18].
Moreover, the importance of wearable brain activity monitoring devices is that
they are feasible for envisioning a real-time human-in-the-loop education
system thanks to their portability, comfortability, and wireless data transfer
system.

A study by Dikker et al. [19] proves that using a portable
electroencephalogram (EEG) device for recording brain activity from a class of
12 high school students over a semester during regular activities, analyzing
the group-based neural coherence is possible where the brain activity is
synchronized across students in both student class engagement and social
dynamics.

Another study by Babini et al.[20] comparatively measured the learning of the
students in a virtual reality (VR) environment for using a wearable
electroencephalogram (EEG) [21]. In particular, they compared the brain
engagement level of a group of students while learning through a VR
environment compared to the regular two-dimensional environment. Their study
showed that the higher the brain engagement, the higher the attention level of
the students which led to better learning performance.

Moreover, the developments in wearable neurotechnology devices can enable
complex applications to be integrated to understand and predict cognitive
processing. For example, sleep has a pivotal role in cognitive functions, and
their relationship has been a topic of interest for over a century. Extensive
research on sleep studies has shown that better sleep is associated with a
myriad of superior cognitive functions in healthy adults [22, 23], including
better learning and memory [24]. Although the exact mechanisms behind the
relationship between sleep, memory, and learning are still research topics,
the general agreement is those specific synaptic connections that were active
during awake periods are strengthened during sleep, allowing for the
consolidation of memory, and inactive synaptic connections are weakened [25].
Consequently, sleep provides an essential function for memory consolidation;
in other words, it will enable us to remember the topics that have been
studied before[25]. It is now well established from a variety of studies that
sleep monitoring can be used to associate with the students’ performances.
Moreover, sleep monitoring should be long-term rather than a single night or
particular event recordings. Therefore, the recent developments on low-power
wearable devices have crucial importance in monitoring students’ sleep or
emotional situations in the long term to create the future smart classroom
environments [26].

##  IV Personalized feedback for the future smart classroom

When these advancements in neuroscience and sensor technology combined with
the current evolution in machine learning and signal processing,
classification, and analysis of the massive amount of data streaming from
wearable devices have become practicable. The resourcefulness of machine
learning models provides the ability to analyze the complex recurring patterns
of neural activity and detect hidden patterns. However, there are several
challenges that need to be addressed to use these advancements to provide
personalized feedback to the students and adapt the teaching environment for
future smart classrooms.

  * •

Challenge 1: Decision making in the face of system variability: Even if we are
able to decode the brain activity in real-time to estimate the cognitive
processing, addressing the students’ individual needs to achieve personalized
feedback has a lot of challenges. In particular, there are intrinsic
variations that humans exhibit that can complicate the idea of a personalized
smart classroom. As discussed in one of our recent works, the same human’s
preferences and responses may change over time. Even for a small period of
time, the same human may produce different responses based on unmodeled
external effects (intra-human variability). Similarly, different humans may
have different responses under similar conditions (inter-human variability)
[27]. For example, adapting the pace of the teaching material or the time of a
quiz based on the cognitive load and the sleep quality of the student may not
be the same for a different student and even for the same student across time.
Moreover, when multiple humans interact in the same environment, their
preferences differ based on the number of people they interact with and the
type of this interaction. Moreover, when the education system adapts to one
student in a multi-student classroom (e.g., change the pace of the presented
material), it has a direct effect on the other students in the same classroom.
In addition to that, multiple students may require different feedback
adaptation actions based on their different preferences (multi-human
variability) which can raise a question of fairness of adaptation in future
smart classrooms [28, 29]. Hence, these human variations have to be considered
as an integral part of the design process of the adaptation algorithms for the
future of the smart classroom.

  * •

Challenge 2: Restructuring the pedagogical materials: Prior to applying the
vision of personalized feedback in the future smart classroom, the pedagogical
materials have to be reconstructed to be adaptable. For example, teaching
materials have to be redesigned into small modules that are self-contained to
enable quizzes and breaks at any time. Moreover, the same teaching materials
themselves have to be designed into different modalities, such as project-
based learning with a small group of peers, independent work to complete
complex tasks, using immersive technology (VR) to increase the engagement of
some students, and even recordings of the material to be watched at a
different pace. Quizzes have to redesigned be automated based on the time they
are taken and the student’s strengths. Projects and group assignments have to
take into account the group coherency and the average learning performance of
the group members. Even teachers have to be trained to keep an up-to-date
record from real-time data that provides a deep understanding of each
student’s individual strengths, needs, motivations, progress and goals [30].

  * •

Challenge 3: Internet-of-Things (IoT) design and robustness to errors:
Infusing technology into pedagogy to attain the vision of the future smart
classroom requires careful design of the enabling technology to ensure high
accuracy of real-time data and robustness to errors. Albeit the wide-ranging
adaptation of the Internet-of-Things (IoT) technology in different application
domains, low power and low energy design of wearable embedded systems are some
of the most critical challenges to realize this new paradigm of future smart
classrooms and to ensure its feasibility. Moreover, the need to process EEG
data at the edge level with high accuracy (popularly termed as machine
learning at the edge) due to privacy, security, and communication latency
require new hardware-software design techniques to achieve low cost for
accurate real-time local processing [26]. Current research in the area of
battery-aware task scheduling for wearables [31], post CMOS technology, and
Neuromorphic computing systems [32, 33] will play a significant role in
developing such low power, low energy, and high accuracy efficient IoT design
for future smart classrooms.

##  V Conclusion

This is an opportune time for smart classrooms. The increased demands of new
affordable yet accurate solutions for human-in-the-loop systems lead to rich
research areas, especially in human sensing. With the advancement in
neuroscience and wearable neurotechnology, new possibilities for improving the
quality of the education system by merging both fields have emerged. However,
new challenges are yet to be solved in all development phases of future smart
classrooms, from more neuroscience insights into the brain circuitry to design
phases of accurate sensing to efficient adaptation algorithms. This article
highlights the most prominent opportunities for future smart classrooms and
recent approaches to tackle them. Nevertheless, this is not an exhaustive list
leaving much to be discovered and done.

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# Chemical disorder induced electronic orders in correlated metals

Jinning Hou (gbsn侯晋宁)  Tsung-Dao Lee Institute & School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Yuting Tan
(gbsn谭宇婷)  Department of Physics and National High Magnetic Field Laboratory,
Florida State University, Tallahassee, FL 32306, USA  Wei Ku (bsmi顧威)
corresponding email<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Ministry of Education Key Laboratory of Artificial Structures and Quantum
Control, Shanghai 200240, China  Shanghai Branch, Hefei National Laboratory,
Shanghai 201315, China

###### Abstract

In strongly correlated metals, long-range magnetic order is sometimes found
only upon introduction of a minute amount of disordered non-magnetic
impurities to the unordered clean samples. To explain such anti-intuitive
behavior, we propose a scenario of inducing electronic (magnetic, orbital, or
charge) order via chemical disorder in systems with coexisting local moments
and itinerant carriers. By disrupting the damaging long-range quantum
fluctuation originating from the itinerant carriers, the electronic order
preferred by the local moment can be re-established. We demonstrate this
mechanism using a realistic spin-fermion model and show that the magnetic
order can indeed be recovered as a result of enhanced disorder once the length
scale of phase coherence of the itinerant carriers becomes shorter than a
critical value. The proposed simple idea has a general applicability to
strongly correlated metals, and it showcases the rich physics resulting from
interplay between mechanisms of multiple length scales.

Typically, random disorder is expected to suppress long-range orders in
materials, especially those with a characteristic length scale such as
antiferromagnetic order, antiferro-orbital order, or charge density order.
This is in part because of the damage to quantum phase coherence resulting
from the inhomogeneity in density, in addition to the direct disruption of the
preferred spatial periodicity of the long-range order. Indeed, in dirtier
samples with more impurities, one usually observes weaker magnetic Shen et al.
(2017); Cheong et al. (1991); Vajk et al. (2002); Delannoy et al. (2009),
superconducting Schneider et al. (2012); Mackenzie et al. (1998); Fujita et
al. (2005) and charge Sham and Patton (1976); Straquadine et al. (2019); Fang
et al. (2019, 2020) orders. Correspondingly, one often intuitively seeks
cleaner and more uniform samples for stronger long-range orders.

However, some exceptional cases exist in which long-range order, for example
magnetic order, can emerge from the introduction of disorders, such as non-
magnetic impurities. A well-known example is the emergence of
antiferromagnetic (AFM) order in Sr2RuO4 Braden et al. (2002); Minakata and
Maeno (2001); Ishida et al. (2003); Pucher et al. (2002) when a minute amount
($\sim 3\%$) of Ru4+ are substituted by non-magnetic Ti4+ ions. Similarly,
iron-based superconductor LaFePO also develops antiferromagnetism upon As
substitution of P Kitagawa et al. (2014); Lai et al. (2014); Mukuda et al.
(2014). The indications that AFM order could emerge from unordered systems
also have been found in hole-doped cuprates via Zn substitution of Cu, as
measured by muon spin resonance ($\mu$SR) Watanabe et al. (2000) and neutron
scattering experiments Hirota et al. (1997); Kimura et al. (2003); Suchaneck
et al. (2010). Such an anti-intuitive behavior appears to contradict the above
fundamental consideration of quantum phase coherence, and thus poses a great
challenge to our generic basic understanding.

Theoretically, in a strongly correlated and highly polarizable environment, it
is natural to expect the development of local effective moments around even
non-magnetic impurities Wessel et al. (2001); Yasuda et al. (2001); Bobroff et
al. (2009). Such an effective moment surely would have a large impact on the
local correlation, such as modifying its temporal fluctuation or inducing a
spatial standing-wave pattern through reflection against impurities Martiny et
al. (2019); Gastiasoro and Andersen (2015); Zinkl and Sigrist (2021); Song et
al. (2020). Nonetheless, since these effects are primarily local in nature and
centered around random location of the impurities, it is unlikely that they
can provide positive contributions to the formation of long-range order,
especially those with a characteristic spatial period, such as an
antiferromagnetic order. Therefore, a generally applicable mechanism for the
observed seemingly anti-intuitive behavior is desperately needed for such a
long-standing puzzle.

Here, we propose a generic scenario of inducing electronic order via a small
amount of chemical disorder in strongly correlated metals. Accepting that most
of unordered correlated metals only fail to order due to the influence of
itinerant carriers Tan  _et al._ (2022); Tam  _et al._ (2015), we suggest that
chemical impurities can suppress the damaging carrier-induced long-range
quantum fluctuation and in turn allow the local moments to order. We
demonstrate this generic mechanism using a realistic spin-fermion model
derived from FeSe as a prototypical case with a failed antiferromagnetic (AFM)
order Tan  _et al._ (2022). Using the linear response as a measure of the
stability of the AFM ordered state, we find that with a stronger disorder the
long-range magnetic order indeed establishes. Further analysis indicates that
the main physical effect of impurity scattering is equivalent to shortening
the length scale of carrier-induced quantum fluctuation, such that the
correlation of local moments is no longer overwhelmed at long range Tan  _et
al._ (2022). Our study demonstrates a typical example of the rich interplay
between mechanisms of multiple length scales present in most strongly
correlated metals, to which our proposed simple idea can be applied in
general.

Figure 1 illustrates our proposed scenario to resolve the long-standing puzzle
of electronic ordering upon the introduction of chemical disorder in
correlated metals. The key theoretical question here is how disorder, a
generic source of incoherence, can induce a coherent long-range order. Our
proposal is based on a “failed order” scenario Tan  _et al._ (2022) in which
the long-range order preferred by the correlation between local moments is
disrupted by the long-range quantum fluctuation induced by itinerant carriers
Tam  _et al._ (2015). Such quantum fluctuation can be quite effective in
general since in contrast to the exponential decay of the order-related
correlation in three dimensions, the carrier-induced fluctuation has generic
power-law decay, due to the discontinuity at the Fermi surface of the
fermionic carriers Jagannathan  _et al._ (1988); Bulaevskii and Panyukov
(1986); Sobota  _et al._ (2007). We therefore propose that by restricting the
fluctuation to a short enough finite length scale, the presence of disorder
can play a positive role in promoting the long-range order of the local
moments.

Below we proceed to demonstrate this generic mechanism using a realistic spin-
fermion model. We first integrate out the influence of the itinerant carriers
to second order, which associates their long-range fluctuation with the
effective interaction between the local moments. We then demonstrate the
system’s “failed order” nature using the linear response of the ordered state
as a measure of its instability. After that, we simulate the disorder effect
numerically and confirm the establishment of long-range order. Finally, we
analyze the various emergent length scales in our result and provide an
intuitive microscopic picture for the leading physics.

Figure 1: Key question of the study: How can chemical disorder induce
electronic orders in materials? Our proposal: Charge disorder can suppress the
damaging long-range fluctuation of a failed order and in turn allow the
intrinsic electronic order to emerge.

As a generic example, consider a realistic spin-fermion model consisting of
coupled local moments affected by itinerant carriers Yin et al. (2010); Tam
_et al._ (2015); Tan  _et al._ (2022); Lv  _et al._ (2010); Kou et al. (2009);
Liang et al. (2012):

$\begin{split}H&=\sum_{i\neq
i^{\prime}}J_{ii^{\prime}}\mathbf{S}_{i}\cdot\mathbf{S}_{i^{\prime}}\\\
&-J_{\mathrm{H}}\sum_{im\nu\nu^{\prime}}\mathbf{S}_{i}\cdot
c^{\dagger}_{im\nu}\mathbf{\sigma}_{\nu\nu^{\prime}}c_{im\nu^{\prime}}\\\
&+\sum_{jj^{\prime}mm^{\prime}\nu}t_{jmj^{\prime}m^{\prime}}c^{\dagger}_{jm\nu}c_{j^{\prime}m^{\prime}\nu},\end{split}$
(1)

where the local moments $\mathbf{S}_{i}$ at site $i$ and $i^{\prime}$ couple
via $J_{ii^{\prime}}$ such that a magnetic stripe ($\pi,0$) order is preferred
by the local moments Tan  _et al._ (2022). The non-trivial physics emerges
when these local moments couple ferromagnetically to the itinerant carriers
$c_{im\nu}^{\dagger}$ of orbital $m$ and spin $\nu$ at the same site $i$ via
coupling constant $J_{\mathrm{H}}$, where $\mathbf{\sigma}_{\nu\nu^{\prime}}$
are the Pauli matrices. This is because the itinerant carriers can propagate
between sites with kinetic parameter $t_{jnj^{\prime}n^{\prime}}$ and are thus
able to mediate an effective long-range interaction Jagannathan  _et al._
(1988); Bulaevskii and Panyukov (1986); Sobota  _et al._ (2007) between the
local moments at longer time scale (or lower energy) relevant to the slower
spin dynamics. Note that we consider a general case in which the fermion
orbitals at sites $j$ can reside at the same site $i$ as the local moments or
those without (such as ligand sites).

This emergent interaction can be obtained by integrating out the faster
itinerant electron degrees of freedom. For simplicity, we stick to the weak
coupling regime where $J_{\mathrm{H}}$ can be considered a perturbation that
renormalizes Tan  _et al._ (2022); Tam  _et al._ (2015); Lv  _et al._ (2010)
the linear spin-wave theory Anderson (1952); Sup  corresponding to the
preferred long-range order. Represented in the second quantized magnon
creation operator $a_{i}^{\dagger}$ associated with the Holstein-Primakoff
transformation Holstein and Primakoff (1940), the resulting spin-wave
Hamiltonian reads:

$\begin{split}H^{\mathrm{SW}}&=\sum_{i}\tilde{K}_{i}a_{i}^{\dagger}a_{i}\\\
&+\sum^{\mathrm{FM}}_{i\neq
i^{\prime}}\tilde{J}_{ii^{\prime}}(a_{i}^{\dagger}a_{i^{\prime}}+a_{i}a^{\dagger}_{i^{\prime}})+\sum^{\mathrm{AF}}_{i\neq
i^{\prime\prime}}\tilde{J}_{ii^{\prime\prime}}(a_{i}^{\dagger}a_{i^{\prime\prime}}^{\dagger}+a_{i}a_{i^{\prime\prime}}),\end{split}$
(2)

where
$\tilde{K}_{i}=2\sum^{\mathrm{AF}}_{i^{\prime\prime}}\tilde{J}_{ii^{\prime\prime}}-2\sum^{\mathrm{FM}}_{i^{\prime}}\tilde{J}_{ii^{\prime}}$
ensures the preservation of the Goldstone mode of the ordered system. Here the
summation is split into those between the parallel (FM) and anti-parallel (AF)
pairs of spins, with their coupling being renormalized by
$\tilde{J}_{ii^{\prime}}=J_{ii^{\prime}}+A_{ii^{\prime}}$ and
$\tilde{J}_{ii^{\prime\prime}}=J_{ii^{\prime\prime}}+B_{ii^{\prime\prime}}$,
respectively. Represented in momentum $q$ space,

$\begin{split}A(q)=&\frac{J_{\mathrm{H}}^{2}}{2S}\sum_{kll^{\prime}}\frac{(f_{l}(k)-f_{l^{\prime}}(k+q))(E_{l}(k)-E_{l^{\prime}}(k+q))}{(E_{l}(k)-E_{l^{\prime}}(k+q))^{2}+\delta^{2}}\\\
&\times\left|\sum_{m}U^{l^{\prime}\star}_{m\downarrow}(k+q)U^{l}_{m\uparrow}(k)\right|^{2}\mathrm{,\
and}\\\
B(q)=&\frac{J_{\mathrm{H}}^{2}}{2S}\sum_{kll^{\prime}}\frac{(f_{l}(k)-f_{l^{\prime}}(k+q))(E_{l}(k)-E_{l^{\prime}}(k+q))}{(E_{l}(k)-E_{l^{\prime}}(k+q))^{2}+\delta^{2}}\\\
&\times\sum_{mm^{\prime}}U^{l^{\prime}\star}_{m\downarrow}(k+q)U^{l}_{m\uparrow}(k)U^{l\star}_{m^{\prime}\downarrow}(k)U^{l^{\prime}}_{m^{\prime}\uparrow}(k+q),\end{split}$
(3)

where $E_{l}(k)$ denotes the eigenvalues with momentum $k$ and band index $l$
(that absorbs the spin index as well) and $U_{m\nu}^{l}(k)$ denotes the
eigenvectors in the basis of orbital $m$ with spin $\nu=\uparrow$ or
$\downarrow$. $f_{l}(k)=\frac{1}{1+e^{\beta(E_{l}(k)-\mu)}}$ is the standard
Fermi-Dirac distribution function for a given chemical potential $\mu$, and
$S$ the effective magnitude of the local moments. The typical numerical
broadening of $\delta=0^{+}$ is not necessary here since we are only
interested in the zero-frequency limit of the renormalization.

We now seek a “failed order” state as the unordered state prior to the
introduction of chemical disorder. It was recently suggested Tan  _et al._
(2022) that the semi-metallic FeSe is such a failed ordered system whose AFM
order only appears under external pressure greater than 1GP when the carrier
density decreases. In essence, the reduction of carrier density weakens the
carrier-induced long-range fluctuation and in turn allows the long-range AFM
order of the local moments to emerge. The fact that the failed order state can
be overcome by mere 1GPa of pressure implies that the long-range fluctuation
is close to being overcome by the ordering, making it an ideal model system
for our demonstration. We thus take the parameters of Eq.(1) from the previous
density-function based study Tan  _et al._ (2022), which incorporates
$t_{jmj^{\prime}m^{\prime}}$ of five $d$-orbitals and three $p$-orbitals,
$J_{\mathrm{H}}=0.8$ eV, $S=1.7$, and $J$=19meV and 12meV for the nearest and
next nearest neighbors, respectively. A discrete $500\times 500$ momentum mesh
and a 10meV thermal broadening are used to ensure a good convergence.

Figure 2: Emergence of long-range magnetic order via the introduction of
disorder to a failed ordered metal with large magnetic moments. Magnetic
susceptibility is shown here as a measure of the stability of the preferred
ordered state. (a) In the clean system ($W_{\mathrm{max}}=0$) imaginary (shown
as negative) frequency appears near ($\pi$,0) as magnified in the inset,
indicating the ($\pi$,0) AFM order is unstable due to carrier-induced
fluctuation at long range beyond $2\pi/\Delta q$. (b) Introduction of disorder
with $W_{\mathrm{max}}=20$meV weakens the fluctuation (corresponding to a
smaller $\Delta q$), allowing the correlation to extend to a longer
$2\pi/\Delta q$ range. (c) By $W_{\mathrm{max}}=30$meV, all excitations
energies become positive ($\Delta q=0$), indicating the correlation is now
long-range and the preferred ($\pi$,0) magnetic order is a stable phase. That
is, the electronic long-range order is induced by the introduction of chemical
disorder.

Let’s first verify the “failed order” state prior to the introduction of
chemical disorder, by examining the stability of the ordered state via its
linear response. Figure 2(a) and its inset show that in the absence of
disorder, the obtained spin wave energy-momentum dispersion displays no
positive-energy excitation in the vicinity of ($\pi$,0). Such lack of
positive-energy excitation in the linear response is a direct indication that
the assumed AFM ordered state is unstable, in this case due to the carrier-
induced long-range fluctuation that overwhelms the correlation at length scale
longer than $2\pi/\Delta q$. In other words, we have verified that our
starting point is indeed a failed order state, in which the local moments are
unable to establish long-range order even at the zero temperature limit.

Figure 3: An example of disorder configurations containing 1664 sites with
$W(\mathbf{x})$ randomly sampled from a uniform distribution within
[0,$W_{\mathrm{max}}$]. The orientation of the periodicity of the disorder
configuration (denoted by the stilted square) is randomly chosen for each
configuration so that the associated artifacts can be suppressed in the
ensemble average.

We now proceed to include the effect of chemical disorder-induced scattering
on the itinerant carriers and investigate its effect on the long-range order.
Specifically, we aim to calculate the linear response of the long-range
ordered state by ensemble-averaging over a large number of chemically
disordered configurations. It is well-established Jagannathan  _et al._
(1988); Bulaevskii and Panyukov (1986); Sobota  _et al._ (2007) that the main
effect of disorder on the magnetic quantum fluctuation of itinerant carriers
is to introduce incoherent phase shifts along its propagation without
affecting its power-law spatial decaying profile. We therefore approximate the
incoherent phase shifts in the fluctuation within each configuration according
to Bulaevskii and Panyukov (1986)

$\tilde{J}_{ii^{\prime}}\longrightarrow\tilde{J}_{ii^{\prime}}\cos{\phi_{ii^{\prime}}},$
(4)

where

$\phi_{ii^{\prime}}=\frac{2}{\hbar
v_{F}}\int_{\mathbf{x}_{i^{\prime}}}^{\mathbf{x}_{i}}dsW(\textbf{x}),$ (5)

accumulates phase shift from scattering against spatially random potential
$W(\textbf{x})$ along a straight path from position $\textbf{x}_{i^{\prime}}$
of site $i^{\prime}$ to position $\textbf{x}_{i}$ of site $i$. (see
Supplementary Sup  for detail on discretization of the disorder strength and
its path integration.) The strength of the disorder potential $W(\textbf{x})$
is randomly sampled from a uniform distribution between 0 and
$W_{\mathrm{max}}$. We apply Eq.(4) to disorder configurations with large
systems (typically containing around 1600 sites) of various shapes and
orientations in the simulation Berlijn et al. (2014); Wang et al. (2013);
Berlijn et al. (2012, 2011). (See Fig. 3 for an example.) For each
configuration containing different phase shifted $\tilde{J}_{ii^{\prime}}$ for
each pair of $i$ and $i^{\prime}$ (Eq. 5), the magnon spectral function is
then calculated via numerical bosonic Bogoliubov diagonalization Tsallis
(1978); Sup  of $H^{\mathrm{SW}}$ followed by the unfolding procedure Ku et
al. (2010) before being averaged over the ensemble.

Figure 2(b) and (c) gives the resulting magnon energy-momentum dispersion
under increasing disorder strength. Since the main effect of disorder is
through the phase shift of the carrier-induced long-range fluctuation, the
physical broadening Berlijn et al. (2014); Wang et al. (2013); Berlijn et al.
(2012, 2011) in energy and momentum due to the lack of translational symmetry
is not apparent. Interestingly, at $W_{\mathrm{max}}=20$meV [panel (b)] the
momentum region without positive frequency reduces to a smaller one,
indicating an increase of the length scale in which the ordering persists.
Most importantly, at $W_{\mathrm{max}}=30$meV [panel (c)] the magnon spectrum
shows well-defined positive frequency in the entire momentum space, indicating
that the proposed stripe ($\pi$,0) AFM order is a stable state of the system!
This confirms our proposal (cf. Fig. 1) that by disturbing enough the carrier-
induced long-range fluctuation via (chemical) disorder scattering, a strong
electronic order can emerge from the previous failed order state.

Figure.2 also shows a clear trend about the emergence of long-range order. As
the disorder increases, $\Delta q$ systematically decreases, reflecting the
fact that the correlation is able to extend to a longer length scale $\sim
2\pi/\Delta q$. Associated with it is the systematical reduction of the
strength of the “negative” frequency (representing imaginary frequency)
associated with the unstable magnon mode, indicating that the damaging long-
range fluctuation systematically becomes weaker. At the point when the
strength is no longer able to negate the magnon frequency, $\Delta q$ becomes
zero and the correlation can extend to the system size and establish the long-
range order.

Figure 4: Magnon energy-momentum dispersion upon restricting the length scale
of carrier propagation via different scattering rate $\eta$. With larger
scattering rate (shorter length scale of coherent propagation), the region
with imaginary frequency and the associated $\Delta q$ reduces.
Correspondingly, the momentum, $q^{\chi}$, at which the energy turns imaginary
increases, indicating a longer $2\pi/\Delta q$ correlation resulting from a
shorter length scale $2\pi/q^{\chi}$ of the carriers’ damaging fluctuation. By
$\eta>$5meV, the excitation energy becomes fully positive, indicating the
stability of the ($\pi$,0) AFM ordered state.

To gain further microscopic insight on how disorder scattering produces this
unusual effect, notice that according to Eq.(4), the main effect of the
scattering is to induce a phase shift proportional to the path integral.
Therefore, one would expect the coherence of the renormalization of
$\tilde{J}_{ii^{\prime}}$ to suffer systematically at longer range.
Particularly, beyond a characteristic length scale that emerges when the
random fluctuation of the phase reaches the order of 2$\pi$, the power-law
tail of the carrier-induced fluctuation should no longer be effective.

To verify this simple intuition and to make a better connection with the
underlying carrier dynamics, we investigate the effects of finite length scale
of the carrier propagation on their quantum fluctuation (without the above
disorder-induced phase shift) and in turn the influence on the magnon
dispersion. This is easily implemented by imposing a finite one-body
scattering rate $\eta$ in Eq.(3) via $\delta=2\eta$. Figure 4 summarizes the
resulting magnon dispersion for $\eta=1$meV to 7meV. Indeed, the strength of
the imaginary frequency becomes weaker systematically as the scattering rate
increases, and eventually all magnon frequency becomes positive at
$\eta>5$meV, when the long-range order becomes a stable phase. Notice that the
momentum region without positive frequency and its associated $\Delta q$ scale
reduces systematically, just like in the above cases with disorder. As
expected, in the aspect of allowing the correlation to grow in range and
finally reach a long-range order, a reduction in the length scale of carrier
propagation leads to a suppression of the long-range fluctuation similar to
that caused by the disorder.

Figure 5: Correspondence of length scales in carrier-induced fluctuation and
carrier propagation in unit of the lattice constant $a_{0}$. The length scale
of carrier-induced damaging fluctuation, $\lambda^{\chi}=2\pi/q^{\chi}$,
follows perfectly the scale of the mean-free path,
$\tilde{l}^{\mathrm{MFP}}\propto v_{F}/\eta$. It also roughly corresponds to
the length scale of the variation of the emerged long-range coupling,
$\lambda^{\Delta\tilde{J}}$, defined as
$\frac{\partial\widetilde{J}(r,\eta)}{\partial\eta}\bigg{|}_{r=\lambda^{\Delta\tilde{J}}}=0$.
The light green region denotes the emergence of a stable ($\pi$,0) AFM ordered
phase.

Figure.5 provides a more quantitative comparison between several relevant
length scales in our results. First, notice that in this length-scaled
controlled picture, our results display a well-defined $q^{\chi}$ at which the
magnon dispersion starts to become “negative”. It turns out that its
corresponding length scale, $\lambda^{\chi}=2\pi/q^{\chi}$, follows perfectly
the length scale of the mean-free path $\propto
v_{F}/\eta\equiv\tilde{l}^{\mathrm{MFP}}$. A similar consistency is also found
in the length scale of the variation of the emerged long-range coupling Sup ,
$\lambda^{\Delta\tilde{J}}$, defined through
$\frac{\partial\widetilde{J}(r,\eta)}{\partial\eta}\bigg{|}_{r=\lambda^{\Delta\tilde{J}}}=0$.
In essence, the limitation of the coherent length scale of the carrier-induced
fluctuation, either through its phase-disordering or via scattering of carrier
propagation itself, leads to a similar suppression of its effectiveness at
long range, thereby allowing the correlation to extend to a longer range and
eventually establish a long-range order (in the green region in Fig.5).

While the above example concerns only the magnetic order, the underlying
principles are generic to almost all symmetry breaking ordering since they
mostly make use of only the general behavior of various mechanisms at long
length scale. For example, typical long-range order are driven by short-range
many-body couplings that produce non-local correlation with a exponential
decay at long range. On the other hand, due to the discontinuity associated
with the Fermi surface, the fermionic carrier-induced fluctuations usually
have a long power-law tail that trumps the exponential decay of the above
correlation. This makes our proposed failed order more common than one might
realize. Indeed, in strongly correlated materials one often finds a rapid
demise of finite-momentum long-range order upon enhancing metallicity, even
though the correlations remain very strong at short range. As long as the
damaging carrier-induced fluctuation only marginally overwhelms the ordering,
our proposed mechanism would apply. By suppressing via chemical disorder the
carriers’ ability to coherently interfere with the ordering at long range, the
system has a chance to reveal its preferred long-range electronic order, in
magnetic, orbital, charge, or other channels.

In short, to resolve the long-standing puzzle of the emergence of electronic
order via the introduction of chemical disorder widely observed in strongly
correlated metals, we propose a “failed order” scenario and verify it through
a realistic spin-fermion model and a stability analysis based on linear
response of the ordered state. In essence, we propose that many of these
strongly correlated metals are in a failed order state, in which the preferred
order of the local moments is overwhelmed by carrier-induced long-range
fluctuation. The main effect of the chemical disorder is to efficiently reduce
the coherent length scale of the damaging fluctuation and thereby allow the
intrinsic long-range electronic order to emerge. Our study demonstrates a
typical example of the rich interplay between mechanisms of multiple length
scales present in most strongly correlated metals, to which our proposed
simple idea can be applied in general.

We thank Zhiqiang Mao, Vladimir Dobrosavljević, Zi-Jian Lang, Athony Hegg,
Fangyuan Gu, and Ruoshi Jiang for helpful discussions. This work is supported
by the National Natural Science Foundation of China (NSFC) No. 11674220 and
12042507.

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Supplementary materials: Chemical disorder induced electronic orders in
correlated metals

Jinning Hou, Yuting Tan, and Wei Ku

This supplementary provides additional detailed information about our
calculation that is based on well-established methods in the literature.

##  I Linear spin wave theory

The spin Hamiltonian only contains collinear local moments

$H^{\mathrm{loc}}=\sum_{i\neq j}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j},$ (S1)

where the local moments $\mathbf{S}_{i}$ at site $i$ and $i^{\prime}$ couple
via $J_{ii^{\prime}}$ ferromagneticlly ($J_{ii^{\prime}}<0$) or
antiferromagneticlly ($J_{ii^{\prime}}>0$). It is convenient to transform the
collinear spin operators from local frame to lab frame via a spin rotation

$S_{i}^{x}=\tilde{S}_{i}^{x},\ S_{i}^{y}=\kappa_{i}\tilde{S}_{i}^{y},\
S_{i}^{z}=\kappa_{i}\tilde{S}_{i}^{z},$ (S2)

where $\kappa_{i}=e^{i\mathbf{Q}\cdot\mathbf{r}_{i}}=\pm 1$. The spin
operators also can be expresses using raising operator $\tilde{S}^{+}_{i}$ and
lowering operator $\tilde{S}_{i}^{-}$

$\tilde{S}_{i}^{+}=\tilde{S}_{i}^{x}+i\tilde{S}_{i}^{y},\
\tilde{S}_{i}^{-}=\tilde{S}_{i}^{x}-i\tilde{S}_{i}^{y}.$ (S3)

In such rotated frame, all local moments point towards the “up” direction.
Using the lowest-order Holstein-Primakoff (HP) transformation ($S$ is the
magnitude of spin in the following discussion)

$\tilde{S}_{i}^{z}=S-a_{i}^{\dagger}a_{i},\ \tilde{S}_{i}^{+}=\sqrt{2S}a_{i},\
\tilde{S}_{i}^{-}=\sqrt{2S}a_{i}^{\dagger},$ (S4)

we can obtain the quadratic linear spin-wave Hamiltonian with bosonic creation
operator $a_{i}^{\dagger}$and annihilation operator $a_{i}$ from local moments

$H^{\mathrm{\mathrm{loc}}}=\sum_{i}K_{i}a_{i}^{\dagger}a_{i}+\sum^{\mathrm{FM}}_{i\neq
i^{\prime}}J_{ii^{\prime}}(a_{i}^{\dagger}a_{i^{\prime}}+a_{i}a^{\dagger}_{i^{\prime}})+\sum^{\mathrm{AF}}_{i\neq
i^{\prime\prime}}J_{ii^{\prime\prime}}(a_{i}^{\dagger}a_{i^{\prime\prime}}^{\dagger}+a_{i}a_{i^{\prime\prime}}),$
(S5)

where
$K_{i}=2\sum^{\mathrm{AF}}_{i^{\prime\prime}}J_{ii^{\prime\prime}}-2\sum^{\mathrm{FM}}_{i^{\prime}}J_{ii^{\prime}}$
ensures preservation of the Goldstone mode of the ordered system. Here the
summation is split into those between the parallel (FM) and anti-parallel (AF)
pairs of spins. The bosonic operators satisfy the commutation relations

$[a_{i},a^{\dagger}_{i^{\prime}}]=\delta_{ii^{\prime}},\quad[a_{i}^{\dagger},a_{i}^{\dagger}]=[a_{i},a_{i^{\prime}}]=0.$
(S6)

A simple one-band spin wave Hamiltonian represented in momentum $q$ space is

$H^{\mathrm{loc}}=\sum_{q}J^{A}(q)(a_{q}^{\dagger}a_{q}+a_{-q}a^{\dagger}_{-q})+J^{B}(q)(a_{q}^{\dagger}a_{-q}^{\dagger}+a_{q}a_{-q}),$
(S7)

where $J^{A}(q)$ is the coefficient after Fourier transformation of
$a_{i}^{\dagger}a_{i^{\prime}}$ ($i=i^{\prime}$ or $i\neq i^{\prime}$) and
$J^{B}(q)$ is the coefficient after Fourier transformation of
$a_{i}^{\dagger}a_{i^{\prime\prime}}^{\dagger}$. The spin-wave dispersion is

$\omega(q)=\sqrt{(J^{A}(q))^{2}-(J^{B}(q))^{2}}.$ (S8)

##  II Bogoliubov diagonalization of general quadratic bosonic Hamiltonian

If the spin system does not have a simple translational symmetry and is
difficult to diagonalize by hand, we can use a general method Tsallis (1978)
to diagonalize the quadratic bosonic Hamiltonian. Considering a general
quadratic bosonic Hamiltonian

$H=\sum_{ij}t_{ij}a_{i}^{\dagger}a_{j}+\tau_{ij}a^{\dagger}_{i}a_{j}^{\dagger}+\tau^{\star}_{ij}a_{i}a_{j},$
(S9)

where $t_{ij}$ is the hopping parameter for a boson from index $j$ to $i$ and
$\tau_{ij}$ is the parameter for creating two bosons with index $i$ and $j$.
If Hamiltonian $H$ is Hermitian, $t_{kj}=t_{ij}^{\star}$.
$\tau_{ij}=\tau_{ji}$ here. We aim to get the diagonal Hamiltonian

$H=\sum_{i}\epsilon_{i}b^{\dagger}_{i}b_{i},$ (S10)

where $\epsilon_{i}$ is the eigenvalues of index $i$ and bosonic operators
$b_{i},b^{\dagger}_{i}$ satisfy the commutation relations Eq.(S6). The new
operators are the linear combination of the previous operators

$b_{i}^{\dagger}=\sum_{j}a^{\dagger}_{j}T^{N}_{ji}+\sum_{j}a_{j}T^{A}_{ji},\quad
b_{i}=\sum_{j}a_{j}T^{N\star}_{ji}+\sum_{j}a_{j}^{\dagger}T^{A\star}_{ji},$
(S11)

where $T^{N}_{ji}$ and $T^{A}_{ji}$ are the eigenvectors. We can define a
redundant and over complete basis:

$A_{I}^{\dagger}=\left\\{\begin{array}[]{cc}a_{i}^{\dagger}&;I\in U\\\
a_{i}(i=I-\\#\mathrm{\ of\ }i)&;I\in D\end{array}\right.,\quad
B_{I}^{\dagger}=\sum_{J}A_{J}^{\dagger}T_{JI},$ (S12)

where $U$ and $D$ means the upper and down channel respectively. $T_{JI}$ is
the matrix of eigenvectors

$T_{JI}\longrightarrow\left(\begin{array}[]{cc}T_{ji}^{N}&T^{A\star}_{ji}\\\
T_{ji}^{A}&T^{N\star}_{ji}\end{array}\right).$ (S13)

Since the bosonic satisfy the commutation relations Eq.(S6), there is

$[B_{I},B^{\dagger}_{J}]=c_{I}\delta_{IJ};\mathrm{\ where\
}c_{I}=\left\\{\begin{array}[]{cc}1&;I\in U\\\ -1&;I\in D\end{array}\right..$
(S14)

Substituting Eq.(S12) into Eq.(S14), we can obtain the rule of
orthonormalizing eigenvectors

$\sum_{I^{\prime}J^{\prime}}[A_{I^{\prime}},A^{\dagger}_{J^{\prime}}]T^{\star}_{I^{\prime}I}T_{J^{\prime}J}=\sum_{K}c_{K}T^{\star}_{KI}T_{KJ}=c_{I}\delta_{IJ}.$
(S15)

Using the commutation relations Eq.(S6), the result that the Hamiltonian
Eq.(S9) commute with $B_{I}^{\dagger}$ is

$[H,B_{I}^{\dagger}]=c_{I}\epsilon_{I}B^{\dagger}_{I}=\sum_{J}A_{J}^{\dagger}T_{JI}c_{I}\epsilon_{I}.$
(S16)

Eq.(S16) can be expressed using Eq.(S12) as

$\sum_{K}[H,A_{K}^{\dagger}]T_{KI}=\sum_{k\in
U}\left(\sum_{j}t_{jk}a_{j}^{\dagger}+\sum_{j}(\tau^{\star}_{kj}+\tau^{\star}_{jk})a_{j}\right)T_{KI}-\sum_{k\in
D}\left(\sum_{j}t_{kj}a_{j}+\sum_{j}(\tau_{kj}+\tau_{jk})a_{j}^{\dagger}\right)T_{KI}.$
(S17)

Combinding Eq.(S16) and Eq.(S17), we find

$T_{JI}c_{I}\epsilon_{I}=\sum_{K}M_{JK}T_{KI}\mathrm{\ ;\ where\
}M_{JK}=\left(\begin{array}[]{cc}t_{jk}&-2\tau_{jk}\\\
2\tau_{jk}^{\star}&-t_{kj}\end{array}\right).$ (S18)

The matrix $M$ is the non-Hermition matrix that we diagonalize and we can get
the eigenvalues $\epsilon_{I}$ and corresponding eigenvectors.

##  III Integrating out the carriers

Here, we derive the effective linear spin-wave Hamiltonian via integrating out
the influence of itinerant carriers Lv  _et al._ (2010); Tam  _et al._ (2015);
Tan  _et al._ (2022). In general, the spin-fermion Hamiltonian contains local
moments and itinerant carriers

$H=H^{\mathrm{loc}}+H^{\mathrm{it}}+H^{\mathrm{H}},$ (S19)

where $H^{\mathrm{loc}}$ is the spin Hamiltonian of local moments,
$H^{\mathrm{it}}$ is the Hamiltonian of itinerant carriers and
$H^{\mathrm{H}}$ describes the coupling between local moments and itinerant
carriers. For simplicity, we use one-band linear spin-wave Hamiltonian Eq.(S7)
and treat the Hund’s coupling $J_{\mathrm{H}}$ between the itinerant and local
degrees of freedom $\frac{J_{\mathrm{H}}}{2S}$ as perturbation (in unit of
$S^{2}$)

$H^{\mathrm{H}}=-\frac{J_{\mathrm{H}}}{2S}\sum_{im\nu\nu^{\prime}}\mathbf{S}_{i}c^{\dagger}_{im\nu}\boldsymbol{\sigma}_{\nu\nu^{\prime}}c_{im\nu},$
(S20)

where $\boldsymbol{\sigma}_{\nu\nu^{\prime}}$ are the Pauli matrices and
$c^{\dagger}_{im\nu}$ represents creating an itinerant carrier at site $i$ of
orbital $m$ with spin $\nu$. Applying a canonical transformation to the
Hamiltonian Eq.(S19)

$e^{\Delta}He^{-\Delta}=H+[\Delta,H]+\frac{1}{2}[\Delta,[\Delta,H]]+\dots$
(S21)

results in the renormalized linear spin-wave Hamiltonian from its quadratic
components

$H^{\mathrm{SW}}=H^{\mathrm{loc}}+<(H^{\mathrm{H}})^{2}+\frac{1}{2}[\Delta,(H^{\mathrm{H}})^{(1)}]>_{e}=\sum_{q}[\tilde{J}^{A}(q)(a_{q}^{\dagger}a_{q}+a_{-q}a_{-q}^{\dagger})+\tilde{J}^{B}(q)(a_{q}^{\dagger}a_{-q}^{\dagger}+a_{-q}a_{q})]$
(S22)

up to the second order in $1/S$, where

$\tilde{J}^{A}(q)=J^{A}(q)+A(q),\quad\tilde{J}^{B}(q)=J^{B}(q)+B(q),$ (S23)

and

$\begin{split}A(q)=&\frac{J_{H}^{2}}{2S}\sum_{kll^{\prime}}\frac{f_{l}(k)-f_{l^{\prime}}(k+q)}{E_{l}(k)-E_{l^{\prime}}(k+q)}\times\left|\sum_{m}U^{l^{\prime}\star}_{m\downarrow}(k+q)U^{l}_{m\uparrow}(k)\right|^{2}\mathrm{,\
and}\\\
B(q)=&\frac{J_{H}^{2}}{2S}\sum_{kll^{\prime}}\frac{f_{l}(k)-f_{l^{\prime}}(k+q)}{E_{l}(k)-E_{l^{\prime}}(k+q)}\times\sum_{mm^{\prime}}U^{l^{\prime}\star}_{m\downarrow}(k+q)U^{l}_{m\uparrow}(k)U^{l\star}_{m^{\prime}\downarrow}(k)U^{l^{\prime}}_{m^{\prime}\uparrow}(k+q),\end{split}$
(S24)

where $E_{l}(k)$ denotes the eigenvalues with momentum $k$ and band index $l$
(that absorbs the spin index as well) and $U_{m\nu}^{l}(k)$ denotes the
eigenvectors in the basis of orbital $m$ with spin $\nu=\uparrow$ or
$\downarrow$. $f_{l}(k)=\frac{1}{1+e^{\beta(E_{l}(k)-\mu)}}$ is the standard
Fermi-Dirac distribution function for a given chemical potential $\mu$, and
$S$ the effective magnitude of the local moments. Note that $J^{A}(q)$
contains the a constant correction term
$\frac{J_{H}}{2S}\sum_{kl}f_{l}(k)\sum_{m\nu}\nu|U^{l}_{m\nu}(k)|^{2}$ that is
from the Hund’s coupling.

We also can get similar result using a dynamic method. Using the perturbation
theory with Green’s function to integrate out the carrier channel, we can
obtain the susceptibility in real part (see Eq.(3) in manuscript)

$\begin{split}A(q)=&\frac{J_{H}^{2}}{2S}\sum_{kll^{\prime}}\frac{(f_{l}(k)-f_{l^{\prime}}(k+q))(E_{l}(k)-E_{l^{\prime}}(k+q))}{(E_{l}(k)-E_{l^{\prime}}(k+q))^{2}+(2\eta)^{2}}\times\left|\sum_{m}U^{l^{\prime}\star}_{m\downarrow}(k+q)U^{l}_{m\uparrow}(k)\right|^{2}\mathrm{,\
and}\\\
B(q)=&\frac{J_{H}^{2}}{2S}\sum_{kll^{\prime}}\frac{(f_{l}(k)-f_{l^{\prime}}(k+q))(E_{l}(k)-E_{l^{\prime}}(k+q))}{(E_{l}(k)-E_{l^{\prime}}(k+q))^{2}+(2\eta)^{2}}\times\sum_{mm^{\prime}}U^{l^{\prime}\star}_{m\downarrow}(k+q)U^{l}_{m\uparrow}(k)U^{l\star}_{m^{\prime}\downarrow}(k)U^{l^{\prime}}_{m^{\prime}\uparrow}(k+q),\end{split}$
(S25)

where the scattering rate $\eta$ of the carrier-induced fluctuation is
dominated by the disorder effect, and is therefore set $0^{+}$ in the absence
of disorder for a clearer comparison. Diagonalization of the spin-wave
Hamiltonian gives the spin-wave dispersion

$\omega(q)=\sqrt{(\tilde{J}^{A}(q))^{2}-(\tilde{J}^{B}(q))^{2}}.$ (S26)

Using the renormalized linear spin wave Hamiltonian, we can obtain the
renormalized Hamiltonian in real space in terms of Eq.(S5) via Fourier
transformation.

##  IV Weak disorder on the emerged long-range couplings

Our goal is to investigate the effect of weak charge disorder on the spin
channel. It is well-known Jagannathan  _et al._ (1988); Bulaevskii and
Panyukov (1986); Sobota  _et al._ (2007) that the main effect of disorder-
induced scattering on the magnetic quantum fluctuation of itinerant carriers
is to introduce incoherent phase shifts along its propagation without
affecting its power-law spatial decaying profile. When the Fermi wavevector
$k_{F}$ is well-defined, the oscillations with weak non-magnetic disorders can
be expressed as Bulaevskii and Panyukov (1986) $J(r)\cos{(2k_{F}r+\phi_{r})}$,
where $J(r)$ is the magnitude with power-law decaying, $r$ is the distance
from site $i$ to site $i^{\prime}$ in real space. In discrete lattice,
Eq.(S22) can be transformed into Eq.(S5) via Fourier transformation,
$J_{ii^{\prime}}\cos{(2\mathbf{k_{F}}(\mathbf{x}_{i}-\mathbf{x}_{i^{\prime}}))}$.
In realistic systems, however, the Fermi surface is typically not perfectly
nested and thus the oscillation in $J_{ii^{\prime}}$ is not with a fixed
$2k_{F}$ period, but instead it displays a rather complicated pattern. We
therefore approximate the disorder-induced phase shift via

$\tilde{J}_{ii^{\prime}}\longrightarrow\tilde{J}_{ii^{\prime}}\cos{\phi_{ii^{\prime}}},$
(S27)

where $\tilde{J}_{ii^{\prime}}$ contains the power-law decaying term and
oscillating term. The disorder-dependent phase is

$\phi_{ii^{\prime}}=\frac{2}{\hbar
v_{F}}\int_{\mathbf{x}_{i}}^{\mathbf{x}_{i}^{\prime}}dsW(\textbf{x}),$ (S28)

where $W(\textbf{x})$ denotes the strength of the spatial disorder randomly
sampled from a uniform distribution between 0 and $W_{\mathrm{max}}$ and the
integration is along a straight path from position $\textbf{x}_{i}$ of site
$i$ to position $\textbf{x}_{i}^{\prime}$ of site $i^{\prime}$. Here $v_{F}$
is the Fermi velocity and $\hbar$ is 1 in the atomic unit. Then we discretize
the disorder-dependent phase factor from a continuum space to a discrete
lattice.

Figure S1: An example path integral of the disorder-dependent phase factor in
a discrete 8$\times$8 lattice from light yellow site (1,2) to dark yellow site
(4,4). Along the straight path, it would pass a set of squares with different
depths of blue color which represent the strength of potential energy
$W(\bf{x}_{i})$.

An example of discretizing the phase factor is shown as Fig.S1. Every site at
the lattice have different random potential energy from zero to a maximum
potential energy $W_{\mathrm{max}}$. We treat the potential energy
$W_{\textbf{x}_{i}}$ dominating a square range around the site
$\textbf{x}_{i}$. The total phase factor is the summation of
$W_{\textbf{x}_{i}}\times ds$ from $\textbf{x}_{i}$ to
$\textbf{x}_{i^{\prime}}$. $ds$ is the length in the square range around the
disorder site. We generate random potential $W_{\textbf{x}_{i}}$ in lattice
with different sizes and orientations in the range of $(0,W_{\mathrm{max}})$.
Three kinds of disorder configurations as examples as shown in Fig.S2. The
Fermi velocity is estimated via the derivation of the Hamiltonian along $y$
direction around Fermi energy.

Figure S2: Examples of disorder configurations with the maximum value
$W_{\mathrm{max}}=20$meV in a 80$\times$80 lattice. The black dotted squares
show disordered patterns which have different sizes and orientations.

##  V Length scale of variation of the emerged long-range coupling

We get the renormalized spin wave Hamiltonian by integrating out the itinerant
carriers with different scattering rate $\eta$:

$H^{\mathrm{SW}}=\sum_{q}\tilde{J}^{A}(q)(a^{\dagger}_{q}a_{q}+a_{-q}a^{\dagger}_{-q})+\tilde{J}^{B}(q)(a^{\dagger}_{q}a^{\dagger}_{-q}+a_{-q}a_{q}).$
(S29)

The Hamiltonian in momentum space can be transformed into real space Eq.(S5)
with long-range couplings. Since our system is mainly unstable along
$q_{y}-$direction, we summate the contributions of $\tilde{J}_{ij}$ along
$x-$direction and obtain the fluctuating decaying couplings along
$y-$direction at long distance.

Figure. S3 shows the renormalized spin-spin interaction $\tilde{J}$ at long
distance with different $\eta=1$meV and $\eta=7$meV as example. The coupling
is suppressed at long range and enhanced at short range with increasing
damping energy. Since lines with different $\eta$ would cross with each other,
we define a distance $\lambda^{\Delta\tilde{J}}$ defined as
$\frac{\partial\widetilde{J}(r,\eta)}{\partial\eta}\bigg{|}_{r=\lambda^{\Delta\tilde{J}}}=0$
and estimate the range.

Figure S3: Renormalized spin-spin interaction $\tilde{J}$ along $y$ direction
at the range from 10$a_{0}$ to 70$a_{0}$ with scattering rate 1meV and 7meV.
$\tilde{J}$ is oscillating decaying. The two different lines across at this
range that shows the long-range coupling is suppressed when disorder is
stronger.

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