# Thermal critical dynamics from equilibrium quantum fluctuations

Irénée Frérot1,2,3111Electronic address<EMAIL_ADDRESS>Adam
Rançon4222Electronic address<EMAIL_ADDRESS>and Tommaso
Roscilde5333Electronic address<EMAIL_ADDRESS>1 ICFO - Institut
de Ciencies Fotoniques, The Barcelona Institute of Science and Technology,
08860 Castelldefels (Barcelona), Spain  2 Max-Planck-Institut für
Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany  3 Univ
Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France  4
Univ. Lille, CNRS UMR 8523 - PhLAM - Laboratoire des Lasers Atomes et
Molécules, F-59000 Lille, France  5 Univ Lyon, Ens de Lyon, Univ Claude
Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France

###### Abstract

We show that quantum fluctuations display a singularity at thermal critical
points, involving the dynamical $z$ exponent. Quantum fluctuations, captured
by the quantum variance [I. Frérot and T. Roscilde, Phys. Rev. B 94, 075121
(2016)], can be expressed via purely static quantities; this in turn allows us
to extract the $z$ exponent related to the intrinsic Hamiltonian dynamics via
_equilibrium_ unbiased numerical calculations, without invoking any effective
classical model for the critical dynamics. These findings illustrate that,
unlike classical systems, in quantum systems static and dynamic properties
remain inextricably linked even at finite-temperature transitions, provided
that one focuses on static quantities that do not bear any classical analog,
namely on quantum fluctuations.

Characterizing the dynamical behavior of systems close to symmetry-breaking
phase transitions Täuber (2014) has far-reaching implications, ranging from
our understanding of the early Universe to the behavior of interacting systems
in laboratory experiments Dziarmaga (2010); del Campo and Zurek (2014);
Beugnon and Navon (2017). Indeed, close to a critical point, the fluctuations
of the order parameter become correlated over a characteristic distance (the
correlation length $\xi$) which can exceed arbitrarily the microscopic length
(namely the distance between the elementary constituents); as a consequence,
such fluctuations build up over a characteristic timescale (the relaxation
time $t_{c}$) much larger than microscopic timescales. In particular, in the
vicinity of a thermal critical point at temperature $T_{c}$, $\xi$ and $t_{c}$
are expected to be related to the temperature $T$ via

$t_{c}\sim\xi^{z}\sim|T-T_{c}|^{-\nu z}~{},$ (1)

where $\nu$ governs the divergence of $\xi$, and $z$ is the dynamical exponent
Hohenberg and Halperin (1977); Täuber (2014) governing the so-called critical
slowing down of the order-parameter dynamics. A central goal of the study of
critical phenomena is the evaluation of the various critical exponents for a
given universality class.

A fundamental paradigm in classical physics is that, in order to extract the
dynamical exponent $z$ at a thermal phase transition, it is necessary to go
beyond static thermodynamic observables, and to investigate instead the
dynamics at criticality. Indeed, all static quantities can be obtained from
the knowledge of the free energy $F=-k_{B}T\log Z$ (where $Z$ is the partition
function) and of its derivatives; and, for classical systems, $F$ is
completely independent of the dynamics. As a matter of fact, quantities such
as position and momentum enter independently in the statistical sum over phase
space, and, as an example, the partition function of a system of classical
particles is fully independent of whether these particles have a ballistic
dynamics, a stochastic dynamics, _etc._ ; therefore the dynamical exponent $z$
cannot be obtained from the knowledge of the free energy. On the other hand
the $z$ exponent governs the singular behavior of dynamics, in accordance with
dynamical scaling theory Täuber (2014), which has been confirmed either
directly by measuring the dynamical response functions in the vicinity of a
critical point Collins (1989); Fernandez-Baca et al. (1998); Tseng et al.
(2016); Tseng (2016); indirectly via signatures of the Kibble-Zurek mechanism
Navon et al. (2015); Ebadi et al. (2021); and in numerical simulations of the
dynamics of (classical) microscopic models or classical field theories Landau
and Krech (1999); Krech and Landau (1999); Tsai and Landau (2003); Berges et
al. (2010); Chesler et al. (2015); Nandi and Täuber (2019, 2020).

In the case of quantum systems, extracting ab initio the $z$ exponent at
thermal transitions for a given quantum Hamiltonian represents a notoriously
difficult task. On the theory side, the effective coarse-grained dynamics to
be studied can severely depend on the approximations made (_e.g._ how
collective modes are effectively included), see for instance Saito et al.
(2015); Morimatsu et al. (2014); Mesterházy et al. (2015); in numerical
simulations, computing the fully quantum real-time dynamics, or performing the
analytical continuation of imaginary-time correlators Jarrell and Gubernatis
(1996), are prohibitive tasks for many-body quantum systems.

Yet, in spite of the above cited difficulties to investigate their dynamics,
quantum systems differ fundamentally from classical ones in that, in quantum
mechanics, static and dynamic properties remain inextricably intertwined. As a
striking illustration, at zero-temperature (quantum) phase transitions the
corresponding dynamical exponent does govern the scaling of the (free) energy
Sachdev (2001). The link between between statics and dynamics must remain true
at finite-temperature transitions as well, given that the quantum-mechanical
partition function is the trace of the imaginary-time evolution operator. In
this work we probe this connection by considering microscopic quantum models
possessing a thermal phase transition, and whose order parameter is not a
conserved quantity, namely it does not commute with the Hamiltonian. As a
consequence, the order parameter has quantum fluctuations in addition to
thermal fluctuations Frérot and Roscilde (2016); Streltsov et al. (2017). We
show that these quantum fluctuations exhibit a weak singularity at thermal
transitions, governed by a combination of critical exponents involving the
dynamical $z$ exponent.

Our prediction for the singularity of quantum fluctuations stems from the
dynamical scaling hypothesis, the fluctuation-dissipation theorem and Kramers-
Kronig relations; and it is rigorously verified using several exactly solvable
quantum models of thermal phase transitions. As quantum fluctuations can be
obtained from the partition function, this singularity establishes a
fundamental link between the thermal critical dynamics of a quantum system and
its statistical properties. Our results are not in contradiction with the
notion that thermal criticality is fundamentally of classical origin, as the
leading singular behavior of the order-parameter fluctuations coincides with
that of the classical limit of the models of interest; yet the quantum
fluctuations of the order parameter introduce a _sub_ -leading singular
behavior (containing the exponent $z$) which can be analytically singled out,
and which disappears in the classical limit. At the practical level, this
insight allows one to extract the thermal $z$ exponent of microscopic quantum
Hamiltonians without simulating the many-body dynamics at all. We demonstrate
that our approach can be carried out successfully via numerically exact
quantum Monte-Carlo (QMC) calculations on paradigmatic quantum spin models
exhibiting thermal transitions.

To avoid any confusion, it should be stressed that the behavior of quantum
fluctuations at thermal critical points, as studied in this paper, is
completely independent of their behavior at (zero-temperature) quantum
critical points Sachdev (2001); Hauke et al. (2016); Frérot and Roscilde
(2018, 2019), which are of no relevance to the present study. We also stress
that the thermal singularity of quantum fluctuations is different from that of
entanglement estimators, such as negativity, whose thermal critical behavior
still lacks a general understanding Sherman et al. (2016); Wald et al. (2020);
Lu and Grover (2020); Wu et al. (2020).

Dynamical scaling hypothesis, and thermal singularity of the quantum
variance.– The linear response of a system at thermal equilibrium to a weak
time-dependent perturbation is characterized by its dynamical susceptibility
Forster (1995); Täuber (2014). If $O$ denotes the order parameter of a
symmetry-breaking phase transition, and if a weak time-dependent perturbation
in the form $-f(t)O$ is added to the Hamiltonian ${\cal H}$ of the system, the
dynamical susceptibility is defined as $\chi_{O}(\omega)=\delta\langle
O\rangle(\omega)/\delta f(\omega)|_{f=0}$, where $\langle O\rangle(\omega)$
and $f(\omega)$ are the Fourier transforms of $\langle O\rangle(t)$ and
$f(t)$, respectively. Here, $\langle O\rangle(t)=Z^{-1}{\rm
Tr}[O(t)\exp{(-\beta{\cal H}})]$ denotes an average over the equilibrium
distribution at inverse temperature $\beta=(k_{B}T)^{-1}$, with $k_{B}$ the
Boltzmann constant, and $O(t)=e^{i{\cal H}t/\hbar}Oe^{-i{\cal H}t/\hbar}$ is
the time-evolved operator in the Heisenberg picture. According to the
dynamical scaling hypothesis Hohenberg and Halperin (1977); Täuber (2014),
close to the critical point, the singular part of the spectral function
$\chi^{\prime\prime}_{O}(t)=\frac{1}{2\hbar}\langle[O(t),O(0)]\rangle$ (the
imaginary part of $\chi_{O}(t)$) obeys in Fourier space the scaling form

$[\chi_{O}^{\prime\prime}(\omega)]_{\rm s}=[\chi^{(0)}_{O}]_{\rm
s}~{}g\left(\frac{\omega}{\omega_{c}}\right)~{},$ (2)

where $\omega_{c}\sim t_{c}^{-1}\sim|T-T_{c}|^{\nu z}$; $g(x)$ is a scaling
function such that $\int dx~{}g(x)/x=\pi$; and $[\chi^{(0)}_{O}]_{\rm
s}\sim|T-T_{c}|^{-\gamma}$ is the singular part of the static susceptibility
of the order parameter with critical exponent $\gamma$.

In classical systems the variance of the order parameter ${\rm Var}(O)=\langle
O^{2}\rangle-\langle O\rangle^{2}$ is related to the susceptibility via the
fluctuation-response relation ${\rm Var}(O)=k_{B}T\chi^{(0)}_{O}$, and
therefore it exhibits the same power-law singularity. In quantum systems, the
above relation holds only if $O$ is a conserved quantity, namely if $[{\cal
H},O]=0$; otherwise quantum fluctuations are responsible for an extra
contribution to the variance, the quantum variance (QV), ${\rm
Var}_{Q}(O)={\rm Var}(O)-k_{B}T\chi_{O}^{(0)}>0$ Frérot and Roscilde (2016).
Our central result is that, at thermal criticality, the QV of the order
parameter (along with a whole family of coherence measures to which it
belongs) acquires a singular part scaling as

$[{\rm Var}_{Q}(O)]_{\rm s}\approx A_{1}|T-T_{c}|^{2\nu
z-\gamma}+A_{2}|T-T_{c}|^{4\nu z-\gamma}+...$ (3)

Given that $2\nu z-\gamma\geq 0$ for all the universality classes reported in
the literature Hohenberg and Halperin (1977); Täuber (2014) this result has
the immediate implication that the QV of the order parameter does _not_
diverge at the critical point, despite being the difference between two
divergent quantities. This is consistent with the intuition that quantum
fluctuations do not alter the long wavelength behavior of the system at
thermal criticality – in fact, their characteristic length scale, the quantum
coherence length, remains finite even at $T_{c}$ Malpetti and Roscilde (2016).
At the same time, the weak singularity of the QV fully exposes the dynamical
critical exponent, despite depending only on equilibrium fluctuations and the
static response of the system.

Proof of the main result.– Eq. (3) follows directly from the dynamical scaling
hypothesis (Eq. (2)). It is indeed a basic result of linear response theory
that both ${\rm Var}(O)$ and $\chi_{O}^{(0)}$ can be expressed in terms of the
imaginary part of the dynamical susceptibility Täuber (2014); Forster (1995):

$\begin{split}{\rm Var}(O)=&\hbar\int_{0}^{\infty}\frac{d\omega}{\pi}~{}{\rm
coth}(\beta\hbar\omega/2)~{}\chi_{O}^{\prime\prime}(\omega)~{},\\\
\chi_{O}^{(0)}=&2\int_{0}^{\infty}\frac{d\omega}{\pi}~{}\frac{\chi_{O}^{\prime\prime}(\omega)}{\omega}~{}.\end{split}$
(4)

The first line is a consequence of the fluctuation-dissipation theorem Callen
and Welton (1951), and the second of causality Täuber (2014); Forster (1995).
Consequently, the QV admits the following expression

${\rm Var}_{Q}(O)=\hbar\int_{0}^{\infty}\frac{d\omega}{\pi}h_{\rm
QV}(\beta\hbar\omega)\chi_{O}^{\prime\prime}(\omega)$ (5)

where $h_{\rm QV}(x)={\rm coth}(x/2)-2/x$ filters out the low frequency modes
in the integral (namely the modes such that $\hbar\omega\ll k_{B}T$). In
contrast, the functions of which is it composed ($\coth(x/2)$ and $2/x$) both
diverge at zero frequency. The dynamical scaling hypothesis, Eq. (2), suggests
that the characteristic frequency singled out by
$\chi_{O}^{\prime\prime}(\omega)$ (corresponding _e.g._ to its maximum), moves
to zero as $T\to T_{c}$, and therefore the critical singularity of both ${\rm
Var}(O)$ and $\chi_{O}^{(0)}$ stems from the critical enhancement of the zero
frequency contribution to the integrals Eq. (4). Similarly, the singular part
of the quantum variance must also stem from the low-frequency part of the
integral Eq. (5). Assuming on other hand that
$\chi_{O}^{\prime\prime}(\omega)$ vanishes as $\omega\to\infty$ faster than
any power 444Since
$\int_{-\infty}^{\infty}d\omega\omega^{n}\chi^{\prime\prime}_{O}(\omega)\propto\partial_{t}^{n}\chi^{\prime\prime}_{O}(t)\big{|}_{t=0}$
and using the Heisenberg equation of motion $i\hbar\partial_{t}O(t)=[O(t),H]$,
we observe that these integrals are proportional to averages of nested
commutators $\langle[[[O,\ldots],H],H]\rangle$ which are expected to be finite
on physical ground. we may then Taylor expand $h_{\rm QV}(x)$ at low frequency
as $h_{\rm QV}(x)=x/6-x^{3}/360+...$, to obtain

$[{\rm Var}_{Q}(O)]_{\rm s}\approx[\chi_{O}^{(0)}]_{\rm
s}\left[\frac{(\hbar\omega_{c})^{2}}{k_{B}T}{\cal
I}_{1}+\frac{(\hbar\omega_{c})^{4}}{(k_{B}T)^{3}}{\cal I}_{2}+...\right]~{},$
(6)

where ${\cal I}_{1}=(6\pi)^{-1}\int_{0}^{\infty}dx~{}xg(x)$, ${\cal
I}_{2}=-(360\pi)^{-1}\int_{0}^{\infty}dx~{}x^{3}~{}g(x)$, etc. Given the
critical behavior of $[\chi_{O}^{(0)}]_{\rm s}$ and $\omega_{c}$, we obtain
Eq. (3).

Extension to asymmetry measures.– Eq. (5) is very similar to an expression
derived for the quantum Fisher information (QFI) (Hauke et al., 2016) (with
$h_{\rm QFI}(x)={\rm tanh}(x/2)$), and in fact, both the QV and the QFI belong
to a larger family of so-called quantum coherence (or “asymmetry”) estimators
Streltsov et al. (2017), all admitting an analogous expression in term of
$\chi^{\prime\prime}$ (see the Supplementary Material – SM SM  for further
details). Most importantly, for all the coherence measures of this family the
“quantum filter” $h(x)$ appearing in the frequency integrals of the dynamical
susceptibility has an odd parity, and therefore is linear at low frequency
Hauke et al. (2016); SM . The latter property, together with the dynamical
scaling hypothesis (Eq. (2)), are the only requirements leading to Eq. (3);
hence our result immediately applies to all of them. In the specific case of
the QFI, our result rectifies a statement of Ref. (Hauke et al., 2016) on the
absence of thermal singularities SM .

Two exactly-solvable models. We first illustrate our findings with two
quadratic models which belong to the same static universality class, yet
having different $z$ exponents (we also treat the case of a quantum Ising
model with infinite range interaction in SM ). The first one is the so-called
quantum spherical model (QSM) on a $d$-dimensional lattice Tu and Weichman
(1994); Nieuwenhuizen (1995); Vojta (1996), defined by the Hamiltonian ${\cal
H}_{\rm
QSM}=\frac{g}{2}\sum_{i}P_{i}^{2}+\frac{1}{2g}\sum_{i,j}U(i,j)X_{i}X_{j}+\lambda\left(\sum_{i}X_{i}^{2}-\frac{N}{4}\right)$
, $[X_{i},P_{i}]=i\hbar\delta_{ij}$, and $U(i,j)$ defines the interaction. The
second model describes a Bose-Einstein condensation (BEC) transition Gunton
and Buckingham (1968); Jakubczyk and Napiórkowski (2013); Jakubczyk and
Wojtkiewicz (2018), and is defined by ${\cal H}_{\rm BEC}=-\mu N_{B}+\sum_{\bf
k}(\epsilon_{k}+\mu)a^{\dagger}_{\bf k}a_{\bf k}$, $[a_{\bf k},a_{\bf
l}]=\delta_{\bf kl}$. The Lagrange parameter $\lambda$ (resp. $\mu$) imposes
the constraint $\sum_{i}\langle X_{i}^{2}\rangle=N/4$ (resp. $\sum_{\bf
k}\langle a^{\dagger}_{\bf k}a_{\bf k}\rangle=N_{B}$), with $N$ the number of
sites (resp. $N_{B}$ the number of bosons). In both models, we assume that at
small momenta $\epsilon_{k}\sim U_{k}\sim k^{x}$, with $d<x<2d$ ($x=2$
corresponds to short-range interactions in the QSM). Both models have a phase
transition at a critical temperature $T_{c}$, and belong to the universality
class of the spherical model Berlin and Kac (1952); Joyce (1966), with
exponents $\nu=\frac{1}{d-x}$ and $\gamma=\frac{x}{d-x}$ Joyce (1966); Vojta
(1996); Jakubczyk and Wojtkiewicz (2018), so that
$\lambda,\mu\propto|T-T_{c}|^{\frac{x}{d-x}}$. In the SM SM , we show that for
the QSM, the dynamical exponent is $z_{\rm QSM}=x/2$, and that the QV of the
order parameter ($O_{\rm QSM}=N^{-1/2}\sum_{i}X_{i}$) is ${\rm Var}_{Q}(O_{\rm
QSM})=\frac{\hbar g}{2\omega_{0}}h_{\rm QV}(\beta\hbar\omega_{0})$, with
critical frequency $\omega_{0}=\sqrt{2g\lambda}$. For the bosons ($O_{\rm
BEC}=a_{\bf 0}+a_{\bf 0}^{\dagger}$), we find $z_{\rm BEC}=x$ and ${\rm
Var}_{Q}(O_{\rm BEC})=h_{\rm QV}(\beta\mu)$. Close to criticality, we
therefore obtain:

$\begin{split}{\rm Var}_{Q}(O_{\rm
QSM})&=\frac{\hbar^{2}g}{12k_{B}T}-\frac{\hbar^{4}g^{2}\lambda}{720(k_{B}T)^{3}}+\ldots\,,\\\
{\rm Var}_{Q}(O_{\rm
BEC})&=\frac{\mu}{6k_{B}T}-\frac{\mu^{3}}{360(k_{B}T)^{3}}+\ldots\,.\end{split}$
(7)

The singular part of the QV in the QSM model stems from the (negative) ${\cal
I}_{2}$ contribution to Eq. (6), scaling as $|T-T_{c}|^{4\nu z-\gamma}$, which
is consistent with $z_{\rm QSM}=x/2$. Remarkably, the QV vanishes at the BEC
transition, as well as in the whole BEC phase 555Strictly speaking, the QV is
exponentially small in the system size.. The singular contribution scales as
$|T-T_{c}|^{2\nu z-\gamma}$ [Eq. (3)], which is consistent with $z_{\rm
BEC}=x$.

Figure 1: Thermal singularity of the quantum variance from QMC results, and
fitted $z$ exponents. The (a1-a4) panels show the quantum variance of the
order parameter close to the transition, together with a representative fit
(black line) and the fitted regular part (blue dashed line); the vertical line
marks the transition point. The (b1-b4) panels show the resulting fitted $z$
exponent as as function of the fitting window width $w$ around $T_{c}$, along
with the reduced $\chi^{2}$ (the dashed line marks the unity threshold).
(a1-b1): 2$d$ TFIM ($\Gamma/J=1.3$ \- lattice size $L=64$); (a2-b2) 2$d$
antiferromagnetic XXZ model ($\Delta=1.2$, $L=64$); (a3-b3) 3D XX model
($L=28$); (a4-b4) 3D Heisenberg model ($L=28$).

Extraction of the dynamical exponent using QMC. The QV can be calculated for
_any_ quantum model whose thermodynamics can be calculated efficiently Frérot
and Roscilde (2016), opening the route for a systematic calculation of the $z$
exponent in a large class of quantum many-body systems. We illustrate this
possibility by focusing on four paradigmatic models exhibiting a finite-
temperature transition, defined on $d$-dimensional (hyper-)cubic lattices of
size $L^{d}$: (1) the 2$d$ ferromagnetic transverse-field Ising model (TFIM)
${\cal H}_{\rm TFIM}=-J\sum_{\langle
ij\rangle}S_{i}^{z}S_{j}^{z}-\Gamma\sum_{i}S_{i}^{x}$ with $\Gamma/J=1.3$; and
the XXZ model ${\cal H}_{\rm XXZ}=-J\sum_{\langle
ij\rangle}\left(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y}-\Delta
S_{i}^{z}S_{j}^{z}\right)$ (with $J<0$) in the three following cases: (2) 2$d$
easy-axis model ($\Delta=1.2$); (3) 3$d$ XX model ($\Delta=0$); and (4) 3$d$
Heisenberg model $\Delta=1$) 666The standard Heisenberg model has equal
(positive) sign for the couplings of all three spin components; yet on
bipartite lattices it can be mapped onto the XXZ model – considered here –
with ferromagnetic couplings for the $x$ and $y$ components.. For all models
$\langle ij\rangle$ are nearest-neighbor pairs on a $d$-dimensional lattice.
All four models possess a finite-temperature transition. For models 1 and 2,
the transition belongs to the 2$d$ Ising universality class. However in model
2 the magnetization along $z$ is conserved: it represents a diffusive mode
which could potentially couple to the order parameter and alter the $z$
exponent with respect to model 1 Täuber (2014). Model 3 is representative of
the 3$d$ XY universality class, and model 4 of the 3$d$ Heisenberg class. We
investigate all models using Stochastic Series Expansions quantum Monte Carlo
Syljuåsen and Sandvik (2002) which allows us to reconstruct the QV (as already
shown in Refs. Frérot and Roscilde (2016, 2018, 2019)). The system sizes we
analyze ($L=64$ in $d=2$; $L=28$ in $d=3$) are not the largest ones we can
simulate, but the QV exhibits very little scaling beyond these sizes SM ,
while its precision degrades significantly in the ordered phase, as the QV
(per spin) is the non-diverging difference between two divergent quantities.
For all the models, $T_{c}$ is estimated by a scaling analysis of the full
variance of the order parameter (scaling as $L^{\gamma/\nu}$ at the transition
point).

The QV per spin of the order parameter (uniform magnetization $O=J^{z}$ for
the 2$d$ TFIM, $O=J^{x}$ for the 3$d$ XX model, and staggered magnetization
$O=J^{z}_{\rm st}$ for the 2$d$ XXZ model and the 3$d$ Heisenberg model) shows
a clear anomaly at the phase transition (Fig. 1). It is then fitted as:

$\displaystyle{\rm Var}_{Q}(O)(T)$ $\displaystyle=$ $\displaystyle
a_{0}+a_{1}T+a_{2}T^{2}$ (8) $\displaystyle+$
$\displaystyle\left[A_{+}~{}\theta(\delta)+A_{-}\theta(-\delta)\right]|\delta|^{2\nu
z-\gamma}~{}.$

where $\delta=T-T_{c}$. The first line is a parabolic fit to the regular part,
while the second line is a fit to the dominant term of the singular part (the
critical exponents $\nu$ and $\gamma$ are known for each universality class
Pelissetto and Vicari (2002)). In principle we have 6 fitting parameters
($a_{0},a_{1},a_{2},A_{+},A_{-},z$), which are nonetheless further reduced: a)
for the $d=2$ models (1 and 2), we set $A_{-}=0$, as suggested by the
smallness of $A_{-}$ when treated as a free parameter; b) for the $d=3$
models, we set $A_{+}=0$ (for the same reason as above) as well as $a_{2}=0$.
A subtle aspect of the fits is the discrimination of the singular part from
the regular one, since both are non-divergent. This is particularly true for
the models 1 and 2, for which an alternative fitting analysis is presented in
the SM SM .

We perform our fits to Eq. (8) over windows of variable width
$[T_{c}-w,T_{c}+w]$ around the critical point. Fig. 1 shows the results: the
fit quality is always very good for all four models, with $(\chi^{2})_{\rm
red}$ ($\chi^{2}$ per degree of freedom) systematically reaching values around
1 upon shrinking the fitting window down to the relevant critical region. As
for our final estimates of $z$, we retain the values at which $(\chi^{2})_{\rm
red}\approx 1$ and for which the fitted value has converged upon reducing $w$
(within the error bar): (1) $z=1.95(15)$; (2) $z=1.95(10)$; (3) $z=1.61(15)$;
(4) $z=1.36(10)$. Models 1 and 2 ($2d$ Ising universality class) could
potentially be captured by Model C ($z=2$) Hohenberg and Halperin (1977);
Täuber (2014). The alternative fitting strategy presented in the SM SM  gives
$z=1.88(10)$ for model 2, compatible with $z=1.95(10)$ obtained above, while
it gives $z\simeq 1.65(5)$ for model 1, closer to experimental results on
quasi-2$d$ magnets ($z\approx 1.6-1.8$) Bucci and Guidi (1974); Hutchings et
al. (1982); Slivka et al. (1984); Tseng et al. (2016); Tseng (2016). For model
3, our estimate is compatible with that obtained by Ref. Krech and Landau
(1999) ($z=1.62$) via a dynamical simulation of the classical 3d XY model. As
for model 4, our estimate is slightly lower but compatible with $z=1.49(3)$
obtained via dynamical simulations of classical Heisenberg antiferromagnets
Bunker et al. (1996); Landau and Krech (1999); Tsai and Landau (2003), and
with the experimental estimate from neutron scattering studies of quantum
Heisenberg antiferromagnets Tucciarone et al. (1971); Coldea et al. (1998).

Conclusions.– We have shown that the dynamical exponent $z$, governing the
critical slowing down of the dynamics close to a second-order thermal phase
transition of a quantum system, manifests itself in a weak singularity of the
quantum fluctuations of the order parameter. This general result was
illustrated by two exactly solvable models with the same thermodynamic
criticality but different critical dynamics; by a quantum spin model with
infinite-range interactions amenable to exact diagonalization for large system
sizes (as discussed in the Supplemental Material SM ); and exploited to
extract the exponent $z$ in four quantum spin models in $d=2$ and $d=3$ from
unbiased QMC data. Our scheme gives access to the $z$ exponent associated with
the _intrinsic_ Hamiltonian dynamics (in the absence of any external bath)
without the need to simulate the real-time quantum dynamics itself (which is a
prohibitive numerical task); and the $z$ exponent can be extracted from
numerical (_e.g._ QMC) data without the need for analytic continuation (which
is also a very demanding task). Here we have focused on the singularity of
quantum fluctuations of the order parameter, but other quantities are expected
to display similar singularities exposing the $z$ exponent, offering
alternative strategies for its numerical evaluation. Therefore our approach
opens a way to the calculation of dynamical critical exponents for the
Hamiltonian dynamics of a large class of quantum many-body models – something
which is of extreme importance in the light of the recent generation of
experiments addressing the dynamics of closed quantum systems close to
critical points Chomaz et al. (2015); Braun et al. (2015); Navon et al.
(2015); Beugnon and Navon (2017). Moreover, the ability of inelastic neutron
scattering experiments to reconstruct quantum coherence estimators Mathew et
al. (2020); Scheie et al. (2021a); Laurell et al. (2021); Scheie et al.
(2021b) along with dynamical critical scaling Collins (1989) could lead to a
direct test of our predictions within the same experimental platform. Finally,
at a fundamental level, the precise connection between microscopic quantum
models (as studied in this work) and effective classical theories Hohenberg
and Halperin (1977); Täuber (2014); Krech and Landau (1999); Tsai and Landau
(2003); Berges et al. (2010) is a fascinating topic for future investigations.

###### Acknowledgements.

Acknowledgments. We are grateful to V. Alba, J. Dalibard, N. Laflorencie, N.
Dupuis, and M. Lewenstein for stimulating discussions. IF and TR acknowledge
support by ANR (“ArtiQ” project, “EELS” project) and QuantERA (“MAQS”
project). IF acknowledges support by the Government of Spain (FIS2020-TRANQI
and Severo Ochoa CEX2019-000910-S), Fundació Cellex, Fundació Mir-Puig,
Generalitat de Catalunya (CERCA, AGAUR SGR 1381 and QuantumCAT) the Agence
Nationale de la Recherche (Qu-DICE project ANR-PRC-CES47), the John Templeton
Foundation (Grant No. 61835). AR is supported by the Research Grants QRITiC
I-SITE ULNE/ ANR-16-IDEX-0004 ULNE. All authors acknowledge support of CNRS
(PEPS project ”EnIQMa”).

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Supplemental Material

##  Appendix A Singularity of a family of quantum coherence estimators.

In this appendix, we show that a general family of quantum coherence
estimators, to which both the quantum variance (QV) and the quantum Fisher
information (QFI) belong, have an analogous expression in terms of the
dynamical susceptibility. As a consequence, they all display a singularity of
the same form at critical points. This appendix is a summary of a longer
discussion presented in ref. (Frérot, 2017, Sections 1.5-1.6). The relevant
family of coherence estimators, quantifying the coherence of a quantum state
$\rho$ with respect to an observable ${\cal O}$, can be written as:

$C_{f}(\rho,{\cal O})=\frac{f(0)}{2}\sum_{i\neq
j}\frac{(p_{i}-p_{j})^{2}}{p_{i}f(p_{j}/p_{i})}|\langle i|{\cal
O}|j\rangle|^{2}~{},$ (9)

where $|i\rangle$ is the eigenvector of $\rho$ corresponding to eigenvalue
$p_{i}$. Here, $f$ is a function obeying certain mathematical properties Petz
(1996) ensuring that $C_{f}$ is a valid coherence estimator, among which
$f(x)\geq 0$; $f(x)=xf(1/x)$ and $f(1)=1$. For pure states, it can be verified
that $C_{f}(|\psi\rangle\langle\psi|,{\cal O})=\langle\psi|{\cal
O}^{2}|\psi\rangle-\langle\psi|{\cal O}|\psi\rangle^{2}$ is the variance; in
general $C_{f}$ is smaller than the variance, and obey several mathematical
properties which make it a well-defined estimator of quantum coherence Frérot
(2017) (or, according to an alternative terminology, of “asymmetry” Streltsov
et al. (2017)). To this general family of coherence estimators belong the QFI
(for which $f(x)=F(x)=\frac{1+x}{2}$), the QV ($f_{\rm
QV}(x)=\frac{(1-x)^{2}/6}{1+x+2(1-x)/\log x}$), and the Wigner-Yanase-Dyson
skew information, defined as $I_{\alpha}={\rm Tr}(\rho^{1-\alpha}{\cal
O}\rho^{\alpha}{\cal O})$
($f_{\alpha}(x)=\frac{\alpha(1-\alpha)(1-x)^{2}}{1+x-x^{\alpha}-x^{1-\alpha}}$).
The fact that all coherence estimators $C_{f}$ behave in a qualitatively
similar fashion across the phase diagram of a many-body system descends from
the fundamental inequality Gibilisco et al. (2009): $C_{f}\leq
C_{F}\leq\frac{1}{2f(0)}C_{f}$, implying in particular that all $C_{f}$ have
the same scaling behavior. Most importantly, for thermal states,
straightforward manipulations and basic results of linear response theory
allow to establish the following expression Frérot (2017):

$C_{f}(\rho,{\cal
O})=\hbar\int_{0}^{\infty}\frac{d\omega}{\pi}h_{f}(\beta\hbar\omega)\chi^{\prime\prime}_{\cal
O}(\omega)~{},$ (10)

where $\beta$ is the inverse temperature, and with the “quantum filter”:

$h_{f}(x)=f(0)\frac{1-e^{-x}}{f(e^{-x})}~{}.$ (11)

Eqs. (10)-(11) extend the result for the QFI Hauke et al. (2016) and the QV
(main text of this paper) to the complete family of coherence estimators
defined via Eq. (9). At small frequency ($\hbar\omega\ll k_{B}T$), one finds
$h_{f}(x)\approx xf(0)$, namely $h_{f}$ is linear. Furthermore, the property
$f(x)=xf(1/x)$ implies that $h_{f}(x)$ is an odd function, leading to an
expression similar to Eq. (6) of the main text. In conclusion: for all allowed
$f$ functions Petz (1996), $C_{f}(\rho,{\cal O})$ displays a singularity of
the same form as that of the QV at critical transitions.

##  Appendix B Quantum variance of the quantum spherical model

The quantum spherical model (QSM) Tu and Weichman (1994); Nieuwenhuizen
(1995); Vojta (1996) is a quantum generalization of the spherical model Berlin
and Kac (1952); Joyce (1966), and is defined in terms quantum operators
$X_{i}$ on a $d$-dimensional lattice with $N$ sites, with a mean-field
constraint $\sum_{i}\langle X_{i}^{2}\rangle=\frac{N}{4}$. The Hamiltonian is

$H_{\rm
QSM}=\frac{g}{2}\sum_{i}P_{i}^{2}+\frac{1}{2g}\sum_{i,j}U(i,j)X_{i}X_{j}+\lambda\left(\sum_{i}X_{i}^{2}-\frac{N}{4}\right),$
(12)

with $P_{i}$ the canonical momentum $[X_{i},P_{j}]=i\hbar\delta_{ij}$ (all
other commutators vanish), $\lambda$ is a Lagrange multiplier, and $U(i,j)$ is
the two-body interaction. The order parameter is $O:=N^{-1/2}\sum_{i}X_{i}$.
The QSM is essentially equivalent to the quantum non-linear sigma model with
$O(N)$ symmetry in the large $N$ limit Sachdev (2001).

Assuming that $U(i,j)$ is invariant under translations, and that its Fourier
transform $U_{k}\sim k^{x}$ at small momenta ($x<d<2x$, and $x=2$ for short-
range interaction), the free energy is given by Vojta (1996)

$f_{\rm QSM}=-\frac{\lambda}{4}+\frac{k_{B}T}{N}\sum_{\bf
k}\ln\left(2\sinh\left(\frac{\hbar\omega_{k}}{2k_{B}T}\right)\right),$ (13)

with $\omega_{k}^{2}=U_{k}+2g\lambda$. The mean-field constraint is imposed by

$\frac{\partial f_{R}}{\partial\lambda}=0.$ (14)

Solving this equation for the Lagrange multiplier gives $\lambda(g,T)$. For a
given $g$, there is a finite-temperature phase transition at a critical
temperature $T_{c}(g)$, defined by $\lambda(g,T_{c}(g))=0^{+}$. One finds
$\lambda\propto\delta^{\frac{x}{d-x}}$ and $\omega_{k}^{2}\simeq
k^{x}+2g\lambda$ at small momenta, with $\delta=|T-T_{c}|\ll 1$, which implies
that the static critical exponents are those of the spherical model Joyce
(1966), namely $\nu=\frac{1}{d-x}$ and $\eta=0$ (and in particular
$\gamma=\frac{x}{d-x}$) Vojta (1996).

The finite-momentum imaginary part of the dynamical susceptibility is computed
from

$\chi_{O}^{\prime\prime}(k,t)=\frac{1}{2\hbar
N}\left\langle[X_{k}(t),X_{-k}(0)]\right\rangle,$ (15)

which gives

$\chi_{O}^{\prime\prime}(k,\omega)=\frac{\pi
g}{2\omega_{k}}\left(\delta(\omega-\omega_{k})-\delta(\omega+\omega_{k})\right).$
(16)

At zero momentum, we get

$\chi_{O}^{\prime\prime}(\omega)=\frac{\pi}{4\lambda}\left(\delta\left(\frac{\omega}{\sqrt{2g\lambda}}-1\right)-\delta\left(\frac{\omega}{\sqrt{2g\lambda}}+1\right)\right),$
(17)

which obeys the scaling law
$\chi_{O}^{\prime\prime}(\omega,\delta)\propto\delta^{-\gamma}f(\omega\delta^{-\frac{x}{2(d-x)}})$,
from which we read $z_{\rm QSM}=x/2$.

A straightforward calculation gives the variance as ${\rm
Var}(O)=N^{-1}\lim_{k\to 0}\langle X_{k}X_{-k}\rangle=2g\lim_{k\to
0}\frac{\partial f_{R}}{\partial U_{k}}$:

${\rm Var}(O)=\frac{\hbar
g}{2\omega_{0}}\coth\left(\frac{\hbar\omega_{0}}{2k_{B}T}\right),$ (18)

and the static susceptibility:

$\chi^{(0)}_{O}=\frac{g}{\omega_{0}^{2}}.$ (19)

The quantum variance thus reads

$\begin{split}{\rm Var}_{Q}(O)&=\frac{\hbar g}{2\omega_{0}}h_{\rm
QV}(\beta\hbar\omega_{0}),\\\
&\simeq\frac{\hbar^{2}g}{12k_{B}T}-\frac{\hbar^{4}g^{2}\lambda}{720(k_{B}T)^{3}}+\ldots\,.\end{split}$
(20)

Comparing with Eq. (6) of the main text, we see that the scaling of the
quantum variance is consistent with $z_{\rm QSM}=x/2$, using $2\nu z_{\rm
QSM}-\gamma=0$ and $4\nu z_{\rm QSM}-\gamma=x/(d-x)$.

##  Appendix C Quantum variance at a Bose-Einstein condensation with mean-
field interaction

A simple description of Bose-Einstein condensation (BEC) is obtained from a
model of bosons with mean-field (all-to-all) interaction Gunton and Buckingham
(1968); Jakubczyk and Napiórkowski (2013). Alternatively, one can study free
bosons with a constraint on the mean density $\frac{1}{N}\sum_{i}\langle
n_{i}\rangle=n_{B}$, with $n_{i}$ the number operator on the site $i$ and
$n_{B}$ is the density of bosons. In momentum space, the Hamiltonian is given
by

$H_{B}=-N\mu n_{B}+\sum_{\bf k}E_{k}a^{\dagger}_{\bf k}a_{\bf k},$ (21)

where $E_{k}=\epsilon_{k}+\mu$, $\mu$ is an effective chemical potential that
plays the role of a Lagrange multiplier to enforce the constraint. We assume
that the bosons have a modified dispersion $\epsilon_{k}\propto k^{x}$. The
creation and annihilation operator satisfy the canonical commutation relations
$[a_{\bf k},a_{\bf l}]=\delta_{\bf kl}$.

The free energy of the bosons is given by

$f_{B}=-\mu n_{B}-\frac{1}{\beta N}\sum_{\bf k}\ln\left(1-e^{-\beta
E_{k}}\right),$ (22)

while the constraint reads

$\frac{1}{N}\sum_{\bf k}\frac{1}{e^{\beta E_{k}}-1}=n_{B}.$ (23)

The transition happens at a critical temperature $T_{c}(n_{B})$ such that
$\mu=0^{+}$. A standard analysis gives $\mu\propto\delta^{\frac{x}{d-x}}$
Jakubczyk and Wojtkiewicz (2018), and thus the static critical exponent are
those of the spherical model, namely $\nu=\frac{1}{d-x}$ and $\eta=0$.

An observable playing the role of an order parameter for the BEC transition is
the field quadrature $q_{\bf k}=a_{\bf k}+a^{\dagger}_{\bf k}$ at zero
momentum. The imaginary part of its dynamical susceptibility is given by

$\chi_{q}^{\prime\prime}(k,\omega)=\frac{\pi}{\hbar}\left(\delta(\omega-
E_{k})-\delta(\omega+E_{k})\right),$ (24)

from which we read $\omega_{c}=\mu/\hbar$ and thus $z_{\rm BEC}=x$, which is
twice that of the quantum spherical model. This is in agreement with the
critical behavior of the quantum variance of $q_{0}$,

$\begin{split}{\rm Var}_{Q}(q)&=h_{\rm QV}(\beta\mu),\\\
&\simeq\frac{\mu}{6k_{B}T}-\frac{\mu^{3}}{360(k_{B}T)^{3}}+\ldots,\end{split}$
(25)

since $\mu\propto\delta^{\frac{x}{d-x}}=\delta^{2\nu z-\gamma}$ and
$\mu^{3}\propto\delta^{3\frac{x}{d-x}}=\delta^{4\nu z-\gamma}$.

##  Appendix D Ising model with infinite-range interactions

In this section, we compute the quantum variance (QV) and quantum Fisher
information (QFI) at the thermal phase transition of the $s=1/2$ transverse-
field Ising model with infinite-range interactions:

${\cal H}=-\frac{1}{N}(J^{z})^{2}-gJ^{x}~{},$ (26)

where $J^{a}=\sum_{i=1}^{N}\sigma_{i}^{a}/2$, with $\sigma_{i}^{a}$ the Pauli
matrices ($a=x,y,z$). Upon varying $g$ from $0$ to $1$, this model displays a
continuous line of second-order phase transitions at
$T_{c}=g/\ln\left(\frac{1+g}{1-g}\right)$ Das et al. (2006) with mean-field
exponents ($\nu=1/2,\gamma=1$). This represents another example of a quantum
model for which the quantum coherence measures of interest (QV and QFI) can be
computed exactly at any temperature. As we shall see in this section, these
computations confirm our general result concerning the singular behavior of
these quantities at a thermal transition, involving the dynamical $z$
exponent. Regarding the QFI, the present computation rectifies the results of
Ref. Hauke et al. (2016) which reported the absence of any singularity at the
transition (Fig. 3 of Ref. Hauke et al. (2016)).

First, we show that $z=1$ at the thermal transition; then, we investigate the
singularity of the QV and the QFI at the thermal transition and relate it to
the $z$ exponent.

Dynamical exponent. As the total spin $(\bm{J})^{2}$ commutes with the
Hamiltonian, the latter can be diagonalized sector by sector, taking into
account the degeneracy of each sector. The largest dimension ($N+1$)
corresponds to a total spin $J=N/2$, which sets the maximal number of spins
reachable in a numerical exact diagonalization. The dynamical susceptibility
$\chi^{\prime\prime}(\omega)$ of the order parameter $J^{z}$ is given by
$\chi^{\prime\prime}(\omega)=(1-e^{-\beta\hbar\omega})S(\omega)/2$, with
$S(\omega)$ the dynamical structure factor Callen and Welton (1951):

$\displaystyle S(\omega)=Z^{-1}\sum_{J=0}^{N/2}D(J)$
$\displaystyle\sum_{k,k^{\prime}=-J}^{J}e^{-\beta E(J,k)}|\langle
J,k|J^{z}|J,k^{\prime}\rangle|^{2}\times$ (27)
$\displaystyle\delta(\omega-[E(J,k^{\prime})-E(J,k)]/\hbar)$

where $E(J,k)$ and $|J,k\rangle$ are the energy and eigenstates in a sector of
fixed $J$. The factor $D(J)={N\choose N/2-J}-{N\choose N/2-J-1}$ accounts for
the degeneracy of each $J$ sector (Arecchi et al., 1972). In Fig. 2, we plot
$\chi^{\prime\prime}(\omega)/\chi^{(0)}$ at the critical temperature for
$g=0.8$, for different system sizes ($N=500,1000,1500,2000$). We computed
$S(\omega)$ using Eq. (27), with frequency bins of width $100/N^{5/4}$.

Figure 2: Dynamical susceptibility for the Ising model with infinite-range
interactions ($g=0.8,T=T_{c}\approx 0.3641$). The imaginary part of the
dynamical susceptibility $\chi^{\prime\prime}(\omega)$ of $J^{z}$ is rescaled
to the static susceptibility $\chi^{(0)}$. It is plotted as a function of
$\omega N^{z/4}$ with $z=1$.

For a $d$-dimensional system of linear size $L$, the characteristic frequency
$\omega_{c}$ scales according to $\omega_{c}(L)\sim L^{-z}$ with $z$ the
dynamical exponent. In the case of infinite-range systems, $L$ is replaced by
$N^{1/d}$ with $d$ the upper critical dimension ($d=4$ in the present case)
Botet et al. (1982). In Fig. 2, the frequency is therefore rescaled to
$\omega_{c}(N)\sim N^{-z/d}$, with the choice $z=1$ providing excellent
collapse of the data for various system sizes. In summary, Fig. 2 shows that
the dynamical exponent is $z=1$ at the thermal transition. Interestingly, the
value $z=1$ differs from the mean-field prediction $z=2$ for the critical
dynamics of a non-conserved order parameter (model A) Hohenberg and Halperin
(1977); Täuber (2014). However, $z=1$ is in agreement with both the quantum
O(1) model in a gaussian approximation Frérot (2017); and with the quantum
spherical model, which has the same symmetries as the Ising model, and which
for $x=2$ has the same (mean-field) static critical exponents.

Critical behavior of quantum fluctuations. Fig. 3 shows the QV (panel a1) and
QFI (a2) per spin across the thermal phase transition [respectively $f_{\rm
QV}=N^{-1}{\rm Var}_{Q}(J^{z})$ and $f_{\rm QFI}=N^{-1}F_{Q}(J^{z})$], clearly
exhibiting the buildup of a singularity for $N\to\infty$.

Figure 3: Weak singularity of the QV and QFI around the thermal critical point
of the TFIM with infinite-range interactions ($g=0.8.T_{c}\approx 0.3641J$).
(a1) QV per spin $f_{\rm QV}=N^{-1}{\rm Var}_{Q}(J^{z})$ as a function of the
reduced temperature $\delta=T/T_{c}-1$ for various system sizes
($N=200,500,1000,2000$); (b1) same for the QFI. (b1)-(b2) Linearized first
derivative, showing a discontinuity building up with system size; (c1)-(c2)
second derivative at the critical point, exhibiting a $\sqrt{N}$ scaling.

According to the discussion in the main text, the dominant contribution to the
singularity is of the form $[f_{\rm QV}]_{s},[f_{\rm
QFI}]_{s}\sim|\delta|^{2\nu z-\gamma}$ with $\delta=(T-T_{c})/T_{c}$. In the
present case ($\nu=1/2,z=1,\gamma=1$), we have $2\nu z-\gamma=0$. Since a jump
singularity (compatible with a 0 exponent) is not observed in the QV and QFI
[Fig. 3(a1-a2)], the critical singularity must come from the second term in
Eq. 3 of the main text, that is: $[f_{\rm QV}]_{s},[f_{\rm
QFI}]_{s}\sim|\delta|^{4\nu z-\gamma}=|\delta|$. We expect therefore a
discontinuous change of slope of the QV and QFI at $T=T_{c}$. More precisely,
a finite-size scaling analysis shows that the first derivative of the QV and
QFI with respect to the temperature admits the scaling form
$\partial_{T}[f_{\rm QV}]_{s},\partial_{T}[f_{\rm QFI}]_{s}\sim
f(\delta\sqrt{N})$. Fig. 3(b1-b2) indeed confirms this scaling behavior,
although the finite-size effects are still important for the largest sizes we
investigate ($N=2000$), leading to an imperfect data collapse. Finally, this
scaling form implies that the second derivative at the critical point diverges
as $\sqrt{N}$, as confirmed on Fig. 3(c1-c2).

In summary, our numerical and theoretical findings are consistent with a
dynamical exponent $z=1$ for the infinite-range transverse-field Ising model
[Hamiltonian in Eq. (26)] at the finite-temperature critical point,
responsible for a weak singularity of both the QV and QFI of the form
$|T-T_{c}|$ in the thermodynamic limit. The likely explanation for the absence
of such a singularity in the QFI as computed in Ref. Hauke et al. (2016) is an
incorrect account for the sectors $J<N/2$ in Eq. (27). While the ground state
only lies in the $J=N/2$ sector, all sectors with their exponential degeneracy
$D(J)$ contribute to the thermal properties.

Figure 4: Singularity of the quantum variance in the vicinity of the
transition. The four panels refer to the four models discussed in the main
text, and referenced to in the panel labels. In all panels the points are QMC
data for two different $L^{d}$ lattices, the black line is an example of a fit
to the function given in the main text, and the dashed line shows the regular
part. The vertical blue line indicates the position of the transition.

##  Appendix E Discussion of the Quantum Monte Carlo results and their fits

_Quantum Monte Carlo results._ Fig. 4 shows a zoom of the quantum variance of
the order parameter close to the thermal phase transition for the four models
studied via QMC – compare Fig. 1 of the main text for an extended view. In
each figure two system sizes are compared ($L=64$ and $L=72$ for the 2$d$
models; $L=28$ and $L=32$ for the 3$d$ models), showing that size effects are
very small for the system sizes considered; and that the numerical precision
on the quantum variance degrades with system size, especially below the phase
transition, because of the divergence of the two quantities whose difference
defines the quantum variance. The figure also shows the fitting function,
visually highlighting the consistency between the singularity contained in the
fitting function and the (weakly) singular behavior exhibited by the quantum
variance.

In particular, in order to achieve the numerical precision on the quantum
variance that allowed us to perform meaningful fits, we had to resort
systematically to choosing a computational basis _orthogonal_ to the order
parameter, so that the quantum fluctuations of the order parameter are
estimated as off-diagonal observables. This aspect is motivated by the fact
that in Stochastic Series Expansion, as well as in any finite-temperature
quantum Monte Carlo schemes, the estimate of the off-diagonal correlations is
made during the (directed-loop or worm) update Syljuåsen and Sandvik (2002),
and it therefore enjoys a much richer statistics than diagonal observables
which are sampled instead only in between updates.

_Alternative fitting strategies for the two-dimensional models._ As stated in
the main text, a fundamental difficulty in fitting the QV data with a singular
and a regular part is that both parts are non-diverging, so that
discriminating between them is generally non-trivial. For the fits of the QMC
data presented in the main text, we have chosen the regular part to fully
account for the QV on one side of the transition; while the singular part adds
up to the regular part on the opposite side. This choice is justified in the
case of the 3$d$ models we considered, where one clearly observes a slope
discontinuity on the two sides of the transition (Fig. 4(c-d)), which can be
accounted for by a linear (and regular) QV above the transition and a weakly
non-linear QV below; we have not found an alternative hypothesis that could
lead to good fits.

On the other hand, for the 2$d$ models, the curvature of the QV captured by
the regular part on one side of the transition could as well be accounted for
by the singular part being finite on both sides. Therefore we have analyzed
the 2$d$ data with an alternative hypothesis: 1) for the 2$d$ TFI model, we
have chosen to take the regular part as a constant, and to allow for a
singular part on both sides of the transition, fitting therefore with the
parameters $a_{0},A_{+},A_{-},z$; 2) for the 2$d$ XXZ model, we have also
turned on a linear term (with coefficient $a_{1}$) in the regular part, due to
the manifestly finite slope at the critical point. The resulting fits are
shown in Fig. 5, exhibiting a quality which is fully comparable to that of the
fits shown in the main text. In the case of the 2$d$ XXZ model, the $z$
exponent resulting from the alternative fitting scheme, $z=1.88(10)$, is
compatible with the one we have obtained in the main text ($z=1.95(10)$), so
that the estimate of the $z$ exponent appears to be robust. As for the 2$d$
TFI model, on the other hand, the alternative fitting scheme gives
systematically lower values for the $z$ exponent, with a final estimate –
drawn using the same criteria as the one declared in the main text – of
$1.65(5)$. This is to be compared to $z=1.95(15)$ obtained with the fitting
procedure of the main text. Interestingly, the values of $z$ obtained with the
alternative fitting scheme for both models (2$d$ quantum Ising and XXZ) are
closer to the experimental values obtained by various methods on $2d$ quantum
magnets, see Refs. Bucci and Guidi (1974); Hutchings et al. (1982); Slivka et
al. (1984); Tseng et al. (2016); Tseng (2016). Our current conclusion is that
theoretical data of higher quality, or a fundamental justification on the
choice of the fitting function, would be needed to discriminate between the
two fitting schemes, and draw a conclusive value of $z$ for the model in
question.

Figure 5: Results of the alternative fitting scheme for the 2$d$ TFI and XXZ
models. All symbols in panels (a1) and (b1) are as in Fig. 4, except that the
solid black curve now indicates the fit to the modified fitting function (see
text). Panels (a2) and (b2) show the fitted $z$ exponent and the reduced
$\chi^{2}$ of the fit as a function of the fitting window. In the case of the
2$d$ XXZ model, the fits have been performed on an asymmetric window
$[T-w,T+1.5*w]$ due to the higher quality of the data above the transition.