# A Paradigm Shift in Neuroscience driven by Big Data
State-of-the-Art, Challenges, and Proof-of-Concept

Zi-Xuan Zhou
State Key Laboratory of Cognitive Neuroscience and Learning,
Beijing Normal University, Beijing 100875, China
Developmental Population Neuroscience Center,
IDG/McGovern Institute for Brain Research,
Beijing Normal University, Beijing 100875, China  Xi-Nian Zuo
State Key Laboratory of Cognitive Neuroscience and Learning,
Beijing Normal University, Beijing 100875, China
Developmental Population Neuroscience Center,
IDG/McGovern Institute for Brain Research,
Beijing Normal University, Beijing 100875, China
National Basic Science Data Center, Beijing 100190, China
Correspondence to Xi-Nian Zuo<EMAIL_ADDRESS><EMAIL_ADDRESS>
(8 Dec 2022)

###### Abstract

A recent editorial in Nature noted that cognitive neuroscience is at a
crossroads where it is a thorny issue to reliably reveal brain-behavior
associations. This commentary sketches a big data science way out for
cognitive neuroscience, namely population neuroscience. In terms of design,
analysis, and interpretations, population neuroscience research takes the
design controls to an unprecedented level, greatly expands the dimensions of
the data analysis space, and paves a paradigm shift for exploring mechanisms
of brain-behavior associations.

##  1 Introduction

Greene and colleagues [1] demonstrated that the relationships between patterns
of brain networks and behavioral traits vary markedly across subgroups, and
the complexity of the relationships leads to the systematic failure of a
widely used predictive modeling protocol [2] in cognitive neuroscience. The
accompanying editorial [3] perceptively pointed out that this study [1] and
the study of Marek et al. [4] published five months beforehand marked a
crossroads in cognitive neuroscience. In this comment, we call for a big data
science way out for the current dilemma facing cognitive neuroscience,
representing a paradigm shift from cognitive neuroscience to population
neuroscience.

Approximately a decade ago, Dr. Tomáš Paus sensitively saw the benefits of
combining genetics and epidemiology with cognitive neuroscience to emphasize
the demand for tackling the genetic and environmental factors that shape the
processes leading to a particular brain state, and thus proposed population
neuroscience [5]. Falk et al. [6] further pointed out the benefits of
population neuroscience, such as recruiting representative samples in
neuroscience research with the help of population science and opening the
neural ‘black box’ in population science research with the help of
neuroscience. They stressed the profound influence of context and culture on
human behavior, and the risks of ignoring these factors and assuming uniform
brain-behavior relationships. Greene and colleagues [1] validated the
complexity of the brain-behavior relationships and the urgency of shifting to
population neuroscience.

##  2 State-of-the-Art: The Time is Ripe

Openly shared neuroimaging data have grown by orders of magnitude over the
past decade, and this big-data momentum continues. With the huge amount of in
vivo brain imaging data, cognitive neuroscientists are now able to conduct
previously unattainable studies. Notably, several studies based on available
large-scale datasets have also revealed the limitations of some common
practices in the field. For instance, using a total of approximately 50,000
samples from three datasets, Marek et al. [4] found that the effect sizes of
cross-sectional brain-behavior links are smaller than the effect sizes
commonly reported in previous literature. This result provoked disputes and
defenses [7, 8, 9, 10], leading to reflections on the importance of improving
research design [7, 8, 10], optimizing analysis methods [8, 10, 11], and
incorporating multimodal data [12]. Later, as mentioned above, Greene et al.
[1] demonstrated the flaws in previously established predictive models of
brain-behavior associations. The status is reminiscent of Minsky and Papert’s
proof in 1969 that it is impossible for single-layer perceptrons to learn an
exclusive or (XOR) function [13], which then triggered the first cold winter
in artificial neural network research.

Is the current understanding and utilization of neuroimaging data good enough?
Insights from population neuroscience are that rich population information
remains enclosed in the big data looking forward to being decoded from the
perspective of population science. Now, with the increasing number of
neuroimaging data acquisition sites, the popularization of smart terminals and
wearable devices for epidemiological information collection, as well as the
maturity of gene-wide association analysis technology [14], a door tailored
for population neuroscience has been opened. A prominent example is the study
by Bethlehem et al. [15], which aggregated the big data of magnetic resonance
imaging (MRI) from more than 100,000 participants to model brain charts over
the human lifespan. Due to the rich population information encoded, the charts
with such big data can serve as ‘microscopes’ for population neuroscience to
increase study power and accuracy (see Proof-of-Concept: A Paradigm with Brain
Charts).

##  3 More Opportunities than Challenges: Potential Perspectives

Potential perspectives from big data-driven cognitive neuroscience can be
categorized into three aspects: design, analysis, and interpretations.

###  3.1 Design controls

Traditionally, design controls in cognitive neuroscience primarily include
controlled tasks or stimuli. In recent years, naturalistic designs, which
combine the advantages of controlled tasks and the resting state, have
emerged. This paradigm is easy to apply to large-scale samples across the
human lifespan and thus can be mutually reinforced with population
neuroscience, although the methodological research is still in its infancy.
Research with naturalistic designs shows that fMRI data can capture
information at the semantic level [16], demonstrating the potential of highly
controlled neuroimaging data in reflecting attributes concerned by population
neuroscience. Given this opportunity, cognitive neuroscientists should attach
importance to demographic and epidemiological information, encompassing
factors that affect human cognition and behavior at the family, community,
macropolicy, and cultural levels. With these multiple levels of external
information, the analysis and interpretations of the acquired neuroimaging
data can be supported to the greatest extent.

In addition to paradigm innovations in the control of multilevel
sociodemographic variables, population neuroscience also emphasizes the
importance of multimodal data (including genetics), extensive sampling, and
longitudinal measurements. When population neuroscience was first introduced,
Dr. Tomáš Paus emphasized the importance of longitudinal studies, as they
provide knowledge of how various factors shape or regulate the developmental
processes of the human brain. The perspective of development across the human
lifespan has led to the emergence of developmental population neuroscience
[17, 18].

Although the ultimate goal of population neuroscience must rely on highly
controlled large-scale datasets (i.e., deep and big data) that are not
currently available, perspectives from population science can already promote
cognitive neuroscience in multiple ways. First, existing large-scale datasets
can be examined to chart the associations between the brain and the known
basic demographic variables [15]. Since the basic demographic variables are
simple and unambiguous, we can expect that these charts are robust and
therefore can lead to reliable insights. Second, these charts provide
references for accurately assessing confounds caused by demographic variables
in brain-wide association studies (BWAS). In other words, the charts can
inform small-scale BWAS by interpreting variations rooted in demographic
variables. Finally, sociodemographic variables provide informative dimensions
for research analysis and interpretations, regardless of the sample size.
Given the great opportunities, one can expect that there will be an explosion
of highly controlled population neuroscience studies, and the collected data
can be accumulated continuously, eventually forming highly controlled large-
scale datasets.

###  3.2 Dimensions of analysis

Current neuroimaging techniques can only indirectly measure brain activity
with limited spatiotemporal resolution, inevitably losing physical details
reflecting cognitive processes. Even so, neuroimaging data are still intricate
under current research practices. Until recently, many neuroscientists
remained committed to deriving simple biomarkers from neuroimaging data, which
equates to a dimensionality reduction process. For instance, Greene et al. [1]
employed a predictive modeling protocol [2] to reduce the functional
connectivity network to a point on a 2-dimensional plane. However, they found
that points corresponding to two types of individuals with distinct behavioral
scores are often intertwined and cannot be correctly classified
simultaneously.

If a specific sociodemographic variable is used as a third dimension, the
classification accuracy will be greatly improved, which is also supported by
data presented by Greene et al. [1] The approach discussed here is not to
simply regress out sociodemographic variables but to take them as intrinsic
dimensions within the model. Similarly, the dimension of development (i.e.,
age) has frequently been regarded as a valid factor in BWAS. A typical example
is the association between cortical morphology and intelligence quotient (IQ),
which is intrinsically dynamic and encoded throughout the human lifespan.
While the pattern of the association is difficult to disentangle from cross-
sectional data, with the additional dimension from longitudinal measurements,
the differences in the age-dependent trajectories of cortical morphology are
detectable and clear across different IQ groups [19, 20]. In light of the
uniqueness of the dimension of development, developmental population
neuroscience [18, 21] takes age as a key dimension in BWAS from a lifespan
perspective [22].

Certainly, higher-dimensional information can be extracted directly and
exclusively from neuroimaging data by using multivariate models or even deep
learning methods. These approaches, however, pose at least two challenges:
interpretability and generalizability. It is difficult, first, to interpret
the implications of the high-dimensional model and, second, to determine the
model application scope without evaluating the sample representativeness.
Fortunately, population neuroscience with big data can help get neuroimaging
data analysis out of the rut in four ways. First, based on demographic
information, the distribution of the samples is straightforward, so the model
application scope is clear. Second, taking sociodemographic and genetic
features as additional dimensions beyond simple biomarkers, one can obtain
brain-wide association models with higher dimensions and better
interpretability. Third, using anchors of demographic variables to determine
the application scope and assist in interpretability analysis, sophisticated
multivariate models are more desirable. Fourth, exploiting additional
information from genetic and sociodemographic variables makes the established
models theoretically stronger and therefore more likely to yield valid
discoveries.

###  3.3 Interpretations from associations to mechanisms

With the support of interpretable high-dimensional analysis space, one can
model general patterns of brain-wide associations based on large-scale
representative samples and reliable measurements [23], i.e., deep and big
data. It is important to note that the perspective of population neuroscience
requires variables, such as genetic and sociodemographic features, that
influence or regulate the brain-behavior relationships to be treated as
intrinsic dimensions of the model. By harnessing genetic, sociodemographic,
and other population-level variables as intrinsic model dimensions, we can
gain better insights into brain-behavior relationships.

While simultaneously accounting for many dimensions is only possible based on
very large-scale samples, a simple strategy is to select relatively
independent dimensions and to model the normative variations separately [24].
The obtained normative models quantify individual differences in the brain
along diverse population information and can further facilitate innovative
small-scale BWAS [25], especially precise or personalized health care research
(e.g., psychiatry [26, 27]). Individual scores can be evaluated along the
intrinsic dimensions characterized by the normative models. In this way, the
undetectable brain-behavior associations in the raw data can be revealed with
more intrinsic dimensions and increased statistical power. The models encode
rich population information of the representative individual differences and
function as microscopes to decipher incomprehensible information into
comprehensible patterns.

Potential discoveries of the patterns, combined with knowledge from multilevel
neuromaps [28], will pave the way for researchers to develop further
hypotheses and facilitate more targeted population neuroscience research in
finding, validating, and evaluating key factors that mediate and moderate the
brain-behavior relationships. The knowledge production flywheel called
‘population neuroscience’ will be in motion, continuously generating new
insights into mechanisms and principles on how brain and behavior interact
with each other, helping to open the black box.

##  4 Proof-of-Concept: A Paradigm with Brain Charts

From the perspective of population neuroscience, human lifespan brain charts
[15], which encode population information along key dimensions such as age and
sex, can immediately enable a paradigm shift in BWAS. First, brain charts can
help distinguish individual differences originating from subtle developmental
and genetic variables by harnessing the most robust and fundamental age- and
sex-rooted variations. Second, interpretable normative scores (e.g., centile
scores) derived from brain charts enable the transformation of feature/trait
space from absolute to relative values, allow for heuristic explorations such
as direct comparisons between different brain features/traits, and enhance the
validity and generalizability of findings by referencing external big or deep
data information. Finally, lifespan brain charts provide precise longitudinal
references that can be used to assess within-individual longitudinal changes
in centile scores, which can greatly empower innovative longitudinal studies.

A simple proof-of-concept for the research paradigm with brain charts is
illustrated in Figure 1. Measurements of the brain are represented by the two
features, and therefore samples are represented by points on the 2-dimensional
plane with the two features as axes, while different scores of a certain
behavioral trait are distinguished by different colors. We can see two groups
of differently colored points (indicating markedly different behavioral
scores) in panel a. However, the distribution areas of the two groups largely
overlap, preventing us from effectively capturing the brain-behavior
associations. The tricky situation cannot be overcome simply by a larger
sample size (panel b).

With age as the third dimension, the two groups are effectively separated in
the 3-dimensional space (panels c, d), demonstrating the great power of
additional intrinsic dimensions in helping us gain better insights. A larger
sample size (panel d) allows us to more clearly see the different distribution
patterns of the groups around the normative trajectory (solid line) than a
smaller sample size (panel c). The raw feature scores can be transformed into
the normative feature scores by leveraging the human lifespan brain charts to
accurately interpret the age-rooted variations (panels e, f). On the brain
chart-derived 2-dimensional plane, the distinction between the previously
indistinguishable groups is clearly revealed even with a small sample size
(panel e), presenting a proof-of-concept of how small-scale studies can
reliably discover hidden brain-wide association patterns. Animations that show
the distribution of samples in panel c and the process of transforming raw
feature scores into normative feature scores are provided online
(https://github.com/zuoxinian/CCS/tree/master/projects/chartdemo) to improve
understanding of the chart guided BWAS.

##  5 Conclusions

Reconsidering the experimental design, data analysis strategies and
interpretations in cognitive neuroscience research from the perspective of
population science with big data exhibits great potential to break through the
limitations of current research practices [29, 30]. The research paradigm of
population neuroscience driven by big data not only outlines a tantalizing
vision that ultimately reveals the mechanisms of interactions between brain,
behavior, gene, and environment, but also immediately informs small-scale BWAS
with the additional intrinsic dimensions and the population information
encoded in the normative brain models, allowing us to uncover previously
invisible patterns and effects. We believe that population neuroscience will
reform the research practices of BWAS and advance cognitive neuroscience.

## Acknowledgements

This study has been supported by the STI 2030 - the major project of the Brain
Science and Brain-Inspired Intelligence Technology (2021ZD0200500). We thank
the Research Program on Discipline Direction Prediction and Technology Roadmap
of China Association for Science and Technology for bibliometric resources,
the Chinese Data-sharing Warehouse for In-vivo Imaging Brain at National Basic
Science Data Center and the Lifespan Brain Chart Consortium
(https://github.com/brainchart/lifespan) for big data resources.

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Figure 1: Gain better insights with additional intrinsic dimensions and
normative brain charts. (a) The two groups of sample points distinguished by
different colors are non-separable on the 2-dimensional plane with the two
features as axes. (b) When the sample size increases, the respective
distribution areas of the two groups appear clearer but still overlap. (c)
With age as the third dimension, the two groups are effectively separated, and
their different distribution patterns around the normative trajectory (solid
line) of the two features are revealed. (d) Increasing the sample size helps
to more clearly obtain the different distribution patterns of the groups.
(e)&(f) By obtaining normative features with reference to the normative
trajectory, the hidden distinction between the two groups is revealed on the
2-dimensional plane with the two normative axes, which is clear even with a
small sample size as in panel e. Animations that show the distribution of
samples in panel c and the process of transforming raw feature scores into
normative feature scores are provided online
(https://github.com/zuoxinian/CCS/tree/master/projects/chartdemo).


# Taming spin susceptibilities in frustrated quantum magnets:
Mean-field form and approximate nature of the quantum-to-classical
correspondence

Benedikt Schneider  Department of Physics and Arnold Sommerfeld Center for
Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstr. 37,
80333 Munich, Germany  Munich Center for Quantum Science and Technology
(MCQST), 80799 Munich, Germany  Björn Sbierski  Institut für Theoretische
Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany

###### Abstract

In frustrated quantum magnets the empirically found quantum-to-classical
correspondence (QCC) matches the real-space static susceptibility pattern of a
quantum spin-$1/2$ model with its classical counterpart computed at a certain
elevated temperature. This puzzling relation was observed via bold line
diagrammatic Monte Carlo simulations in dimensions two and three, where the
matching was within error bars and seemed valid down to the lowest accessible
temperatures $T$ about an order of magnitude smaller than the exchange
coupling $J$. Here we employ resummed spin diagrammatic perturbation theory to
show analytically that the QCC breaks at fourth order in $J/T$ and provide the
approximate mapping between classical and quantum temperatures. Our treatment
further reveals that QCC is an indication of the surprising accuracy with
which static correlators can be approximated by a simple renormalized mean-
field form. We illustrate this for all models discussed in the context of QCC
so far, including a recent example of the $S=1$ material
$\mathrm{K}_{2}\mathrm{Ni}_{2}(\mathrm{SO}_{4})_{3}$. The success of the mean-
field form is traced back to partial diagrammatic cancellations.

_Introduction_.—Frustrated quantum magnets remain at the forefront of current
research in condensed matter physics [1]. In this arena, enhanced spin
fluctuations suppress magnetic ordering and conspire to stabilize delicate
highly-entangled quantum states characterized by long-range entanglement and
fractional excitations at low temperature $T$ [2, 3]. But how low is “low”?
And which experimental observables reveal the sought-after quantum spin liquid
properties unambiguously?

While complete answers to these questions, even in theory, remain elusive,
impressive progress has been made on the numerical front [4, 5, 6, 7]. In
2013, a particularly puzzling empirical observation has been made for the
triangular lattice quantum $S=1/2$ Heisenberg anti-ferromagnet (AFM): In Ref.
[8] the bond-resolved spin susceptibility $\chi(\mathbf{r})$ [also referred to
as the static spin correlation function, see Eq. (2)] was simulated via bold
line diagrammatic Monte Carlo (BDMC). For all temperatures $T\\!\geq\\!0.375J$
attainable, the intricate and highly featured normalized pattern
$\chi(\mathbf{r})/\chi(\mathbf{0})$ (see Fig. 3 left) can be matched by
correlation data obtained from the classical ($S=\infty$) vector-spin version
of the same Heisenberg model at an empirically obtained elevated temperature
$T^{(c)}>T$! This was dubbed the quantum-to-classical correspondence (QCC)
111QCC is not to be confused with the fact that quantum phase transitions in
$d$ spatial dimensions can share a universality class with a classical
counterpart in $d+1$ spatial dimensions [27]..

The BDMC [10] is one of the few numerical methods which remains operative for
frustrated quantum spin models in high spatial dimensions ($d=2,3$) and
moderately low $T/J\gtrsim 0.1$. It builds on a complex fermionic
representation of spin $S=1/2$ operators and stochastically samples millions
of skeleton Feynman diagrams [11]. Importantly, results for $\chi(\mathbf{r})$
from BDMC are not exact but come with error bars of $\simeq 1\%$. The matching
of QCC above is to be understood within these error bars.

Figure 1: QCC for systems with a single nearest-neighbor AFM Heisenberg
coupling $J$, left for triangular and pyrochlore lattice with coordination
number $z=6$ and right for kagome and square lattice ($z=4$). The normalized
susceptibility pattern for the quantum $S=1/2$ system at temperature $T$ was
found empirically to be reproduced by the classical (vector-spin) counterpart
at $T^{(c)}$ in the BD(MC) studies of Refs. [8, 12, 13] from where the markers
are reproduced. The solid lines show the analytical prediction for the
_approximate_ QCC obtained by numerically inverting Eq. (8), the dashed line
denotes the high-$T$ limit $T^{(c)}=4T/3$.

In the years following the initial observation, QCC was found wherever the
BDMC was aimed at: The square- and kagome lattice in $d=2$ [8, 13], the
pyrochlore lattice in $d=3$ [12] as well as $J_{1}\\!-\\!J_{2}$ models on the
square and anisotropic triangular lattices [13]. The corresponding
$T^{(c)}(T)$ for various nearest-neighbor quantum models are reproduced from
Ref. [8] in Fig. 1 (markers). Recently, QCC was also observed in a $S=1$ model
for the material $\mathrm{K}_{2}\mathrm{Ni}_{2}(\mathrm{SO}_{4})_{3}$
consisting of two interconnected $d=3$ trillium lattices [14], here the
quantum data was obtained from the pseudo-fermion functional renormalization
group (PFFRG) [7].

As of today, the QCC remains a puzzle with lots of hard-to-ignore anecdotal
evidence but no explanation. Why do the celebrated quantum fluctuations
maximized for the smallest spin length $S\\!=\\!1/2$ merely seem to be
accounted for classically by an effective heating effect? And would the QCC
break down and reveal an approximate nature if a more powerful successor of
BDMC could reach lower $T$? If not, Ref. [8] speculated, could the empirical
$T^{(c)}(T)$ curve be extrapolated to $T=0$ such that a classical simulation
would reveal properties of the highly sought-after quantum ground state?

In this letter we shed analytical light on the origin of QCC building on a
recent advance in spin-diagrammatics to be discussed in detail in an upcoming
work [15]. The QCC turns out to be only approximate for all finite spin
lengths $S$ and dimensions $d$. We quantify QCC’s failure at order
$[J/T]^{4}$, provide a closed-form expression of the (approximate)
$T^{(c)}(T)$ curve, and reveal a close connection between QCC and the
surprising success of a renormalized mean-field (MF) ansatz for
$\chi(\mathbf{r})$. We also quantitatively explain the failure of QCC for the
$d=1$ Heisenberg chain at intermediate temperature, as empirically pointed out
already in Ref. [8].

_Perturbation theory_.—We present a resummed perturbative expansion of the
susceptibility in spin-$S$ Heisenberg models. For ease of presentation, we
here restrict to nearest-neighbor models on $N$-site Bravais lattices with
single atomic bases (e.g. triangular or square lattices),

$\displaystyle
H=J\sum_{\langle\mathbf{r},\mathbf{r^{\prime}}\rangle}\mathbf{S}_{\mathbf{r}}\cdot\mathbf{S}_{\mathbf{r^{\prime}}},$
(1)

for generalizations see the supplemental material (SM)[16]. In momentum space,
the susceptibility or static spin correlator (at zero Matsubara frequency) is

$\displaystyle\chi(\mathbf{k})=\frac{1}{N}\sum_{\mathbf{r},\mathbf{r^{\prime}}}e^{-i\mathbf{k}\cdot(\mathbf{r}-\mathbf{r^{\prime}})}\\!\int^{\beta}_{0}\\!\mathrm{d}\tau\langle
S_{\mathbf{r}}^{z}(\tau)S_{\mathbf{r^{\prime}}}^{z}(0)\rangle,$ (2)

where $\beta=1/T$ and $S_{\mathbf{r}}^{z}(\tau)$ is the Heisenberg spin
operator at imaginary time $\tau$. The Larkin equation [17, 18] expresses (2)
via the set of one-$J$-irreducible (static) two-legged spin-correlator
diagrams $\Sigma(\mathbf{k})$,

$\displaystyle\chi(\mathbf{k})^{-1}=\Sigma(\mathbf{k})^{-1}+J\gamma(\mathbf{k}),$
(3)

where
$\gamma(\mathbf{k})=\frac{1}{N}\sum_{\langle\mathbf{r},\mathbf{r^{\prime}}\rangle}e^{-i\mathbf{k}\cdot(\mathbf{r}-\mathbf{r^{\prime}})}$
is the spatial Fourier transform of the real-space coupling pattern normalized
to unit strength. A diagrammatic representation of Eq. (3) is shown in Fig.
2(a) and a few low-order in $J$ contributions to $\Sigma$ are depicted in Fig.
2(b).

Figure 2: (a) Diagrammatic representation of the Larkin equation (3). Lines
represent the coupling $-J$. (b) Some diagrams contributing to $\Sigma$. More
details will be presented in [15]. Diagram $[4h]$ and a two-chain $[2b]^{2}$
contribute to $\Sigma(\mathbf{k})^{-1}$ as $\sim J^{4}\gamma(\mathbf{k})^{2}$.

Replacing the exact $\Sigma$ in Eq. (3) with its contribution at order
$J^{0}$, i.e. with the free-spin (Curie) susceptibility
$\Sigma^{(0)}\\!=\\!\beta b_{1}$ [where $b_{1}\\!=\\!S(S\\!+\\!1)/3$] yields
the MF approximation [18]. Consequently, terms of higher order capture
corrections beyond MF. It is convenient to parameterize the _exact_ (inverse)
susceptibility $\chi(\mathbf{k})^{-1}$ of Eq. (3) in terms of a renormalized
MF part and a correction $\epsilon(\mathbf{k})$ characterized via its beyond-
MF momentum dependence,

$[T\chi(\mathbf{k})]^{-1}=f+g\gamma(\mathbf{k})+\epsilon(\mathbf{k}).$ (4)

In MF approximation, $f=1/b_{1}$, $g=\beta J$ and $\epsilon(\mathbf{k})=0$. In
that sense, the exact $f$ is the (dimensionless) renormalized on-site inverse
susceptibility, while $g$ describes the renormalized coupling. Therefore, we
will refer to $\chi(\mathbf{k})$ of the form (4) with $\epsilon(\mathbf{k})=0$
as renormalized MF form.

Figure 3: Absolute values of normalized real space spin susceptibility for AFM
$S=1/2$ Heisenberg models with $J_{1}=1$ and temperature $T$ (dots, data taken
from BDMC calculation of Refs. [8, 12, 13]) compared to the renormalized MF
form (circles) parameterized with temperature $T_{MF}$ (see text) optimized
for best agreement. The almost perfect match for a wide range of models and
for rather low $T$ shows the accuracy of the renormalized MF form in $d>1$
dimensions. For pyrochlore, the mean value of $|\chi(\mathbf{r})|$ with the
same $r$ is plotted to match the presentation in Fig. 4 of Ref. [12].

Recent methodological progress to be detailed in [15] provides the expansion
of $T\Sigma$ in the dimensionless coupling $X=\beta J$ complete to $O(X^{4})$
for various spin models and arbitrary lattice geometry. Focusing on the static
correlator for the Heisenberg case, the right-hand side of Eq. (4) is expanded
as

$\displaystyle f=$
$\displaystyle\frac{1}{b_{1}}+z\left(\frac{1}{6}(6b_{1}+1)X^{2}+\frac{1}{24}(4b_{1}+1)X^{3}\right)$
$\displaystyle-\frac{I^{(3)}}{6}b_{1}(6b_{1}+1)X^{3}+{O}(X^{4}),$ (5)
$\displaystyle g=$ $\displaystyle
X+\frac{X^{2}}{12}+\frac{X^{3}}{120}\left(48b_{1}^{2}+16b_{1}+3\right)+{O}(X^{4}),$
(6) $\displaystyle\epsilon(\mathbf{k})\\!=$
$\displaystyle-\frac{X^{4}b_{1}}{720}\\!\biggl{[}\gamma^{2}(\mathbf{k})-z-\frac{I^{(3)}}{z}\gamma(\mathbf{k})\biggr{]}+{O}(X^{5}).$
(7)

We defined $I^{(n)}=\int_{\mathbf{k}}\gamma^{n}(\mathbf{k})$. The special case
$I^{(2)}=z$ counts the neighbors per site, the coordination number. For
example, the nearest-neighbor Heisenberg model on the $d$-dimensional hyper-
cubic lattice is characterized by $\gamma(\mathbf{k})=2\sum_{l=1}^{d}\cos
k_{l}$, $z=2d$ and $I^{(3)}=0$. The second and third term in brackets of Eq.
(7) subtract contributions $\sim\gamma(\mathbf{k})^{2}$ that can be associated
to $O(X^{4})$ terms in $f$ and $g$, respectively.

For all the following arguments, it is crucial that corrections to the
renormalized MF form in $\epsilon(\mathbf{k})$ appear only from order $X^{4}$
on, meaning they are suppressed at high temperatures. Further, the small
prefactor in Eq. (7) results from partial diagrammatic cancellation,
$\epsilon(\mathbf{k})\sim(b_{1}^{-1}[\mathrm{2b}]^{2}-[\mathrm{4h}])+{O}(X^{5})$,
see Fig. 2.

_Approximate QCC_.—To address the QCC puzzle [8] we consider classical (unit-
vector) spins via the $S\rightarrow\infty$ limit. This is straightforward in
our general-$S$ formalism after spin-operators in Eqns. (1) and (2) are
rescaled by $1/S$. The resulting classical coupling and susceptibility are
denoted with superscript (c), hence
$\chi^{(c)}(\mathbf{k})^{-1}=\underset{S\rightarrow\infty}{\mathrm{lim}}S^{2}\chi(\mathbf{k})^{-1}$
with $X\\!\xrightarrow{}\\!X^{(c)}/S^{2}$. Note that $b_{1}$ also depends on
$S$, hence the right-hand sides of Eqns. (5) to (7) simplify by the erasure of
quantum effects.

However, as shown above, the susceptibility $\chi(\mathbf{k})$ remains a
function of two parameters $f,g$ for all $S$ with $O(X^{4})$ corrections. The
QCC was discussed for the normalized susceptibility
$\chi(\mathbf{k})/\chi(\mathbf{r}=\mathbf{0})$ where
$\chi(\mathbf{r}=\mathbf{0})=\int_{\mathbf{q}}\chi(\mathbf{q})$. The (inverse
of the) normalized susceptibility can be computed straightforwardly for both
finite $S$ and the classical case. We find an approximate analytic mapping
between the two (up to and including order $(X^{3})\sim[X^{(c)}]^{3}$) where
$f$ is fixed by the normalization and $g$ can be fixed by relating $X$ to
$X^{(c)}$,

$X\\!=\\!\frac{X^{(c)}}{3b_{1}}-\frac{(X^{(c)})^{2}}{108b_{1}^{2}}+(X^{(c)})^{3}\frac{15b_{1}z\\!-\\!12b_{1}\\!-\\!1}{2430b_{1}^{3}}+{O}(X^{4}).$
(8)

Equivalently, this approximately relates the temperature $T/J=1/X$ of the
quantum spin-$S$ system to that of the classical system,
$T^{(c)}/J=1/X^{(c)}$. In the SM [16] we generalize Eq. (8) to
$J_{1}\\!-\\!J_{2}\\!-\\!\ldots$ Heisenberg models with multiple (equivalent)
basis sites. For $J_{1}$-models on lattices with non-trivial basis (e.g.
kagome or pyrochlore), Eq. (8) is not changed. This is our first main result.

In Fig. 1, we plot the numerical inverse of Eq. (8) [without $O(X^{4})$] for
various $S=1/2$ systems (lines). Note that the lattice only enters via the
coordination number $z$. Results are in good agreement with the empirical data
points reproduced from the (BD)MC studies of various $z=4$ and $z=6$ models in
Refs. [8, 12, 13].

For all lattices considered here, the corresponding classical temperature is
always higher than the quantum temperature. Intuitively, quantum fluctuations
heat up the system [8]. However, this is not generally the case with the spin-
dimer as a counter-example [16].

_Renormalized MF form of susceptibility_.—The QCC is only approximate and
breaks down at order $X^{4}$ where first corrections $\epsilon(\mathbf{k})$ to
the renormalized MF form appear, see Eq. (7). However, this correction
vanishes in the classical limit,
$\lim_{S\xrightarrow{}\infty}S^{2}\epsilon(\mathbf{k})=0+{O}((X^{(c)})^{6})$.
Therefore, from fourth order on, the momentum dependence of the quantum and
classical $\chi(\mathbf{k})$ is inherently different, making it fundamentally
impossible to extend the mapping $X(X^{(c)})$ in Eq. (8) for the full
$\chi(\mathbf{k})$ to order $X^{4}$ in an exact way. The QCC is therefore
always of approximate nature and, contrary to the speculations in Ref. [8],
fails for $T\rightarrow 0$.

However, it turns out that corrections $\epsilon(\mathbf{k})$ to renormalized
MF form of $[T\chi(\mathbf{k})]^{-1}$ are relatively small even for small
$T/J$ (large $X$). This can be anticipated from the partial cancellation of
diagrams as mentioned above, but it can also be inferred empirically by the
high quality of the QCC mapping even for very low $T/J$, e.g. in the
pyrochlore case down to $T/J=0.1667$ [12]. Formally, the renormalized MF form
of $[T\chi(\mathbf{k})]^{-1}$ is a good approximation to the exact value as
long as the minimum of the first two terms $\Delta=f-gz$ (the MF gap) is large
compared to $\epsilon(\mathbf{k})$, such that $\|\epsilon\|_{1}/\Delta\ll 1$.
Here we use a $L_{1}$ norm in real-space,
$\|\epsilon\|_{1}=\sum_{j}|\epsilon_{i,j}|$.

In the following we show that for the $d=2,3$ models considered in the context
of QCC the susceptibility patterns are very accurately approximated by
$\epsilon(\mathbf{k})\approx 0$ well into the cooperative paramagnetic regime
$T/J\gtrsim 0.1$. In Fig. 3 we match the $S=1/2$ BDMC data from various models
treated in Refs. [8, 12, 13] to the renormalized MF form with the empirically
optimized MF temperature $T_{MF}$ proportional to the ratio of $f$ and $g$,
$T_{MF}=\frac{fb_{1}}{g}J$. For the $J_{1}\\!-\\!J_{2}$ models, where the
renormalized MF form reads $[T\chi(\mathbf{k})]^{-1}\simeq
f+g_{1}\gamma_{1}(\mathbf{k})+g_{2}\gamma_{2}(\mathbf{k})$ [16], also $J_{2}$
is adjusted, see the labels in Fig. 3. In all instances, the renormalized MF
and BDMC data fit very well, even at temperatures below which the analytical
mapping based on the truncated version of Eq. (8) would yield reasonable
results. This is our second main result. Importantly, this goes beyond the
empirical QCC [8] since the exact susceptibilities in the classical case, to
which the quantum susceptibilities were matched so far, also feature
$O([X^{(c)}]^{6})$ corrections to the renormalized MF form, c.f. the
discussion above.

As stated by Kulagin et al. [8] the QCC does not hold in the $d=1$ Heisenberg
chain. The susceptibility of this model’s classical counterpart is exactly
described by the renormalized MF form with $f=\frac{3u^{2}+3}{1-u^{2}}$,
$g=\frac{3u}{1-u^{2}}$ and $\epsilon(\mathbf{k})=0$ where $u=\coth(X)-1/X$
[19]: Therefore, QCC can be observed as long as
$\lVert\epsilon(\mathbf{k})\rVert_{1}/\Delta\ll 1$ for the quantum correlator.
We calculate the susceptibility for the $S=1/2$ AFM Heisenberg chain with
quantum MC [20]. In Fig. 4 we show that $|\Delta|$ vanishes around $T/J\simeq
0.3$ making a renormalized MF approximation of $\chi(k)$ impossible at this
temperature. Already for $T/J=0.5$, the corrections to renormalized MF form
are sizable, $\lVert\epsilon(\mathbf{k})\rVert/\Delta\approx 20\%$. This
explains why the range of $T/J$ where QCC could be observed is significantly
smaller in $d=1$ than in the $d=2,3$ models, where no violation was found at
the temperatures available to BDMC.

Figure 4: Parameters of the inverse susceptibility in Eq. (4) for the $S=1/2$
AFM Heisenberg chain of length $L=48$. Lines are a guide to the eye. For
better visibility $f,g,\epsilon$ where all multiplied by the normalization
constant: $\chi(r=0)=\int_{k}\chi(k)$. It can be observed, that the mean field
gap $\Delta$ closes, rendering a description by the renormalized MF form
impossible. The inaccuracy of the renormalized MF form (and thus of QCC) can
be seen in the strong increase of $\|\epsilon\|_{1}/\Delta$. Figure 5: Static
spin structure factor
$S(\mathbf{k})=\sum_{\nu,{\nu^{\prime}}}\chi_{\nu,{\nu^{\prime}}}(\mathbf{k})f_{\mathrm{Ni}^{2+}}(\mathbf{k})$
along different planes in reciprocal space, where
$f_{\mathrm{Ni}^{2+}}(\mathbf{k})$ is the form factor of the
$\mathrm{Ni}^{2+}$ ion. Panels (a)-(c) show classical MC results at
$T=0.35J_{4}$ (top left part of each panel), PFFRG results at flow parameter
$\Lambda=0.58J_{4}$ (top right), and a fit to the renormalized MF form (rMF,
lower half) at $T_{\mathrm{MF}}=0.787J_{4}$. Panels (e)-(g) display line cuts
along three principal directions. The classical MC and PFFRG data was taken
from Ref. [14].

Finally, we test the renormalized MF form for a model of a complex realistic
material $\mathrm{K}_{2}\mathrm{Ni}_{2}(\mathrm{SO}_{4})_{3}$ where QCC was
empirically observed by Gonzalez et al. [14]. According to _ab-initio_
calculations [14], this $d\\!=\\!3$ material realizes a $S\\!=\\!1$ Heisenberg
model on two interconnected trillium lattices with
$\\{J_{1},J_{2},J_{3},J_{4},J_{5}\\}=\\{0.066,-0.026,0.144,1,0.479\\}$ where
$J_{4}=1$ is dominant. This model was treated with the ground state PFFRG [7]
with results extracted at finite Matsubara frequency cutoff parameter
$\Lambda=0.67J_{4}$. The latter was argued to act (at least qualitatively) as
a finite temperature [21]. Unlike BDMC, the error of PFFRG is uncontrolled.
However, empirical confirmation of QCC with $T^{(c)}=0.35J_{4}$ [14] suggests
that the susceptibility is well described by a renormalized MF form. In Fig. 5
we match the classical MC and PFFRG susceptibilities [14] along multiple
planes and cuts in $\mathbf{k}$-space to a renormalized MF form at
$T_{\mathrm{MF}}=0.787J_{4}$ and all $J_{m}$ unchanged. The quality could be
further improved by also renormalizing the latter.

_Conclusion_.—The approximate description of quantum spin systems with
classical theories is a recurring theme [22, 23, 24, 25]. However, the so far
empirical QCC for the spin susceptibility [8] revisited in this letter stands
out by its surprising universality and accuracy even for low temperatures $T$
down to about an order of magnitude below $J$. Via resummed spin diagrammatic
perturbation theory, we showed rigorously that the QCC is only approximate and
fails at order $(J/T)^{4}$. At this order, the spatial dependence of the exact
susceptibility deviates from a simple renormalized MF form for $S<\infty$.

In this sense, the QCC can be understood as a symptom of a more fundamental
insight put forward in this work: The renormalized MF form almost perfectly
accounts for a plethora of spin susceptibility patterns in $d=2$ and $d=3$
reported in the literature on the basis of numerically expensive computations.
We rationalize this from the partial diagrammatic cancellations of corrections
observed at order $(J/T)^{4}$ but likely also operative at higher orders.

The success of the renormalized MF form is surprising, since ordinary MF
theory is only valid for high temperatures or lattices with large coordination
numbers. For future work, it is thus an interesting question if the
renormalized MF ansatz could inform the development of novel theoretical
(renormalization-)methods or experimental data analysis. The latter could be
especially relevant for atom-tweezer array quantum simulators where
$\chi(\mathbf{r})$ should be directly accessibly due to single-site control
and measurement capabilities [26]. As a first step, it would be useful to
extend the analytical (and approximate) QCC mapping of Eq. (8) to higher
orders in $(J/T)$ for better analytical estimation of the renormalized MF
parameters.

_Acknowledgment_.—We thank Matías G. Gonzalez, Philip Osterholz, Johannes
Reuther, Nepomuk Ritz and Manuel Weber for helpful discussions. We acknowledge
support from DFG grant no. 524270816. B. Sb. acknowledges support from DFG
through the Research Unit FOR 5413/1 (grant no. 465199066). B.Sch.
acknowledges funding from the Munich Quantum Valley, supported by the Bavarian
state government with funds from the Hightech Agenda Bayern Plus.

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Supplement.pdf