eprint Physical Review Letters 124, 095703 (2022)

# Critical Slowing Down at the Abrupt Mott Transition: When the First-Order
Phase Transition Becomes Zeroth-Order and Looks Like Second-Order

Satyaki Kundu  Indian Institute of Science Education and Research Kolkata,
Mohanpur, Nadia 741246, West Bengal, India  Tapas Bar  Indian Institute of
Science Education and Research Kolkata, Mohanpur, Nadia 741246, West Bengal,
India  Rajesh Kumble Nayak  Indian Institute of Science Education and Research
Kolkata, Mohanpur, Nadia 741246, West Bengal, India  Bhavtosh Bansal
<EMAIL_ADDRESS>Indian Institute of Science Education and Research
Kolkata, Mohanpur, Nadia 741246, West Bengal, India

###### Abstract

We report that the thermally-induced Mott transition in vanadium sesquioxide
shows critical-slowing-down and enhanced variance (’critical opalescence’) of
the order parameter fluctuations measured through low-frequency resistance-
noise spectroscopy. Coupled with the observed increase of also the phase-
ordering time, these features suggest that the strong abrupt transition is
controlled by a critical-like singularity in the hysteretic metastable phase.
The singularity is identified with the spinodal point and is a likely
consequence of the strain-induced long-range-interaction.

Despite their ubiquity across systems and scales [1, 2, 3, 4, 7, 5, 6, 10, 11,
9, 8, 12, 14, 13, 15, 16], first-order or abrupt phase transitions typically
get only a passing reference in a traditional statistical physics course [20].
This is because they lack many of the remarkable features of the second-order
or continuous transitions which arise from the diverging susceptibility [20,
21, 22]. The singularity leads to a power-law-divergence of the correlation
length, thereby leading to a description that is universal and largely
independent of the microscopic details [22, 20, 21]. Experimentally, this
singularity manifests in the sharp enhancement of the dynamical time scales
(critical slowing down) [23, 21] and the variance of fluctuations (critical
opalescence) [24, 25, 21].

Figure 1: (a) Temperature dependence of resistance of V2O3 shows a sharp and
hysteretic transition around $153$ K while cooling and $162$ K while heating.
(inset) A large latent heat is also measured in differential thermal analysis
[26]. (b) Return-point memory [15, 53, 65]: Although multivalued in the
hysteretic region, after any excursion out of a given hysteresis loop in the
form of a minor loop the same value of the resistance is recovered on return.
Thus the memory of the excursion is wiped out. (c) Mean field free energy as a
function of the spatially averaged order parameter $\phi$ for the compressible
Ising model (essentially the $\phi^{6}$ theory) [26] can be used to capture
transition semi-quantitatively. The points corresponding to curves (i)-(v) are
also roughly marked on the resistance curve in (a). Curve (iii) represents the
binodal where the two minima are equal and (ii) and (iv) are the two spinodals
where one minimum becomes an inflection point. $\tau_{2}$ is the relaxation
time of fluctuations measured through the autocorrelation of the resistance
noise and $\tau_{1}$ is the time associated with phase-ordering. (d) Location
of the extrema of the above free energy as a function of temperature with the
same points (i)-(v) marked. In the temperature window $\sim 153-162$ K where
hysteresis is seen, the order parameter $\phi$ has a globally stable, a
metastable and an unstable extremum.

In this letter, we experimentally demonstrate slowing down and enhancement of
fluctuations at an abrupt phase transition (APT) and argue that a certain
class of APTs are also controlled by critical-like singularities.
Specifically, we have studied the celebrated Mott transition in V2O3 [29, 3,
26, 27, 28]. As the scope of this work transcends the microscopic and material
details of the system under investigation, here we only highlight the two
essential characteristics [30]. Firstly, that this is an APT is unambiguously
inferred from the large latent heat [Fig. 1 (a) (inset)]. Second is a curious
feature that—V2O3 shares with some other vanadates [4, 7, 5, 6], nickelates
[10, 11, 9, 8], manganites [12], intermetallic alloys [14, 13], other charge-
ordered materials [15, 16], and spin-transition polymers [17, 18, 19]—the
transition is always hysteretic.

Hysteresis indicates that this ‘mixed-order’ behavior (viz. the observation of
critical slowing down as well as a latent heat at the transition) [31, 32] may
originate from the spinodal singularity [2, 33, 35, 36, 40, 34, 44, 45, 41,
37, 38, 39, 46, 47]. This would then be a manifestation of the classic mean-
field physics, already contained in van der Waal’s equation (without Maxwell
correction) [2, 34, 47]. Within equilibrium thermodynamics, an APT occurs at
the binodal, the point where the free energy minima for the two phases have
the same value [Fig. 1 (c)]. But nucleation barriers can lead to
supersaturation into a metastable phase [48]. The depth of supersaturation is
thus a function of the efficacy of the fluctuations in affecting first-passage
to the lower energy equilibrium phase [49]. It is believed that only in the
(zero-temperature or infinite-range-interaction) mean-field limit can the
hysteretic passage though the metastable phase extend up to the stability
limits, the spinodals [2, 47]. As
$\chi_{T}^{-1}={\delta^{2}F\over\delta\phi^{2}}=0$ at the spinodals (where
$F[\varphi]$ is the analytically continued free energy density,
$\chi_{{}_{T}}$ is the susceptibility [22], and $\phi$ is the spatially
averaged order parameter), the correlation length and relaxation times diverge
at the spinodals for the same reason that they do at the critical point [2,
33, 35]. Consequently, the spinodals are fixed points under the
renormalization group and should display critical behavior and universality
[44, 45, 41].

While this instability has been studied in a variety of zero-temperature
noise-free systems [35, 36, 50, 51, 40, 52], spinodals have not been
experimentally established in thermodynamic systems where fluctuations are
present [42]. From the scaling behavior of the dynamic hysteresis and the
average qualitative nature of the phase ordering, we have recently proposed
that the Mott transition in V2O3 does indeed occur around such bifurcation
points [26]. The present work, by focussing on fluctuations, transcends the
mean-field picture to establish that the transition has a genuine
thermodynamic character. Our observations of critical slowing down [23] and
enhancement [37] of the order parameter fluctuations indicate that one can at
least get close enough to the spinodal such that the singularity controls many
of the features of the transition, even if there are no divergences [33, 43].

On account of the criticality and the essential role of hysteresis, it is
meaningful to distinguish these transitions from the usual first-order
transitions [2]. Following the Ehrenfest’s criterion, these may be called
‘zeroth-order’, as the free energy is itself discontinuous [49, 26] [Fig 1
(c)].

Figure 2: Time series of the resistance fluctuations at different
temperatures. The left column depicts the measurements done under cooling and
the right panel under heating. (a, b) Normalized resistance fluctuations after
the time series were detrended by imposing a low frequency cut-off of about
$10^{-3}$ Hz [54]. (c, d) The probability distribution of fluctuations. Solid
lines represent the Gaussian distribution with zero mean and the given
variance and the error bars roughly represent $99.7\%$ confidence for this
distribution. (e, f) Variance of the order parameter fluctuations
$\langle\delta\phi^{2}(0)\rangle$ roughly inferred from the resistivity time
series. The fluctuations are expected to follow the divergence of $\chi_{T}$
around the spinodals [Eq. 3]. Figure 3: Critical-slowing-down of fluctuations
around the spinodals. (a, b) Normalized noise power spectral density (PSD),
$S_{R}/R^{2}$ at some representative temperatures. (c, d) Temperature
dependence of the exponent $\mu$, where $S_{f}\sim f^{-\mu}$. (e, f) A surface
plot of $f\times S_{R}/R^{2}$ to visualize the non-$1/f$ behavior when the
spectral weight shifts to lower frequencies.

V2O3 is our material of choice [26] because of the sharp and definitive APT
with a large latent heat over a narrow temperature window [Fig. 1 (a)].
Although one cannot define state variables in the metastable phase, the system
does show reproducible quasistationary behavior (essential for the fluctuation
spectroscopy), as seen in the return-point memory [15, 53] of hysteresis [Fig
1 (b)].

The temperature dependence of the resistance of a typical polycrystalline
sample V2O3 used in the study, after it has been thermally cycled between
$\sim 77-250$ K a few hundred times, is shown in Fig 1 (a), with the inset
showing the latent heat peaks in the differential thermal analysis peaks
measurement [26]. The hysteresis window ($\sim 153-162$ K) can be seen to
contain the metastable phase whose properties are history-dependent, whereas
the regions outside this widow seem to represent equilibrium states. In rest
of this Letter, results for the heating and the cooling transitions are shown
in parallel [54] as the similarity of the two data sets is central to main
conclusions of the work.

Resistance-noise spectroscopy has been a powerful tool for the study of
fluctuations close to phase transitions [23, 9, 11, 14]. The experiments
reported here involved carefully recording the time series of the fluctuating
resistance $R$ with the sample temperature kept precisely fixed at the given
value [54].

Figure 2 (a, b) shows some of these time series of the normalized resistance
fluctuations $\Delta R(t)/R$ at different temperatures after the removal of
the very slowly varying smooth background [54]. It is hard to discern any
significant departure from their Gaussianity [Fig. 2 (c, d)]. To make contact
with the theory of phase transitions (see the discussion below), let us assume
that the essence of the transition may be captured by a scalar order parameter
$\varphi({\bf x},t)$, equal to the fraction of the insulating phase in the
sample [26, 58]. The time series of the sample resistance [Fig. 2 (a, b)] can
then be approximately converted to the time series of the fluctuations in the
spatial average of this order parameter $\delta\phi(t)$ using results from
percolation theory [59, 54]. Enhanced variance of the order parameter
fluctuations $\langle\delta\phi^{2}(0)\rangle$ around the transition [Fig. 2
(e, f)] is the first indication of criticality [24].

Figure 3 (a, b) shows the corresponding normalized power spectral density
(PSD) $S_{R}(f)/R^{2}$ in the frequency window $0.3-0.003$ Hz [54] at
different temperatures. The PSD obeys the characteristic $f^{-\mu}$ behavior
[9, 23], with an increase in the value to $\mu\approx 1.6$ around the
transition, indicating a shift of the spectral weight of fluctuations to lower
frequencies. Such slowing down has been previously observed in the resistance
noise data at the Mott critical point [23] and simulations of the Ising model
[25]. Noise studies at the APT have also observed qualitatively similar
features [9, 11, 14] but the combination of the large hysteresis, a very sharp
transition and the similarity in the behavior along the cooling and the
heating runs in bulk V2O3 studied here has made the identification with the
spinodal singularity more obvious. Our ongoing work on epitaxial NdNiO3 thin
films also qualitatively reproduces Fig. 3 (c, d), indicating its generality.
Indeed recent works on a ferrite [16] and VO2 [5, 6] also invoke ‘pseudo-
criticality’ [38] to explain the slowed dynamics around APT.

Figure 4:  Critical-slowing-down is reflected in both $\tau_{1}$ and
$\tau_{2}$ which, in the linear approximation, are equal to each other and
$\tau_{k=0}$ and diverge with the susceptibility exponent: (a) Temperature
dependence of $\tau_{2}$ the relaxation time for the autocorrelation of the
detrended resistance fluctuations, and (b) $\tau_{1}$ the phase ordering time
during heating and cooling. See Supplemental Material for the details of the
phase-ordering experiments. Solid lines are to guide the eye. The region
between $\sim 153-162$ K is the metastable phase.

In Fig. 4(a), this signature of slowing down is further evident from the
relaxation time $\tau_{2}$ of the autocorrelation of the fluctuations [54]. In
Fig. 4(b) we have also independently estimated the phase ordering time
$\tau_{1}$ where the sample was shock-heated and quench-cooled respectively at
$50$ K/min to the desired target temperature marked on the abscissa [54].
$\tau_{2}$ also shows a distinct enhancement around the transition [54]. Such
an increase in the lifetime with deeper supersaturation (and consequent
reduction in the energy barrier) within the metastable region would seem
counterintuitive for a thermally-activated process like nucleation, but is the
expected behavior at the spinodal [2].

The spinodal singularities.—Let $F[\varphi]$ be the free energy functional of
the system with an order parameter $\varphi({\bf x},t)$ that denotes the
fraction of the insulating phase in the material [26, 58]. The dynamical
behavior this stochastic variable $\varphi({\bf x},t)$, a non-conserved
scalar, is then given by the dissipative Model A [21]

${\partial\over\partial t}\varphi({\bf x},t)=-\lambda{\delta
F[\varphi]\over\delta\varphi({\bf x})}+\zeta({\bf x},t).$ (1)

Here $\lambda$ is a kinetic parameter (that was previously determined to be
$3.5$ s-1 [26]) and the thermal noise $\zeta({\bf x},t)$ is as usual assumed
to be $\delta$-correlated, i.e., $\langle\zeta({\bf x},t)\zeta({\bf
x^{\prime}},t^{\prime})\rangle=2\lambda k_{B}T\delta({\bf x}-{\bf
x}^{\prime})\delta(t-t^{\prime})$, with zero mean [21]. The angular brackets
denote canonical ensemble averaging.

If $\varphi=\varphi_{0}$ for the metastable phase in the mean field
approximation, we can Taylor-expand $F[\varphi]$ around this saddle point
keeping only Gaussian fluctuations, viz. $F[\varphi]\approx
F[\varphi_{0}]+{1\over 2}\int d{\bf x}d{\bf x}^{\prime}\delta\varphi({\bf
x})[\beta G({\bf x}-{\bf x}^{\prime})]^{-1}\delta\varphi({\bf x}^{\prime})$,
where $[\beta G({\bf x}-{\bf
x}^{\prime})]^{-1}={\delta^{2}F[\varphi_{0}]\over\delta\varphi({\bf
x})\delta\varphi({\bf x}^{\prime})}$ and $\beta^{-1}=k_{B}T$. $G({\bf x}-{\bf
x}^{\prime})$ is correlation function evaluated at $\varphi_{0}$ and let
$\tilde{G}_{\bf k}$ be its Fourier transform [21]. If
$\delta\tilde{\varphi}_{\bf k}(t)$ is the Fourier transform of
$\delta\varphi({\bf x},t)\equiv\varphi({\bf x},t)-\varphi_{0}$, the first two
moments of $\delta\tilde{\varphi}_{\bf k}(t)$ evolve as

$\langle\delta\tilde{\varphi_{\bf
k}}(t)\rangle=\langle\delta\tilde{\varphi}_{\bf
k}(0)\rangle\,\exp(-t/\tau_{\bf k}),$ (2) $\langle\delta\tilde{\varphi}_{\bf
k}(t)\delta\tilde{\varphi}_{{\bf k}}(0)\rangle=V\tilde{G}_{\bf
k}\,\exp(-t/\tau_{\bf k}).$ (3)

$\tau_{\bf k}=\beta\tilde{G}_{\bf k}/\lambda$ is the relaxation time for the
${\bf k}^{th}$ mode.

The sample resistance, after correcting for percolation effects [54, 59],
tracks the spatial average of the order parameter [58], i.e., essentially the
zero-wave vector Fourier mode $\tilde{\varphi}_{k=0}(t)[=\int d{\bf
x}\,\varphi({\bf x},t)]\equiv\phi(t)$. Thus our measurements reflect Eq. 2 and
3 with $k=0$. By the sum rule [22], the isothermal susceptibility
$\chi_{{}_{T}}=\beta\tilde{G}_{k=0}$. It follows that
$\chi_{T}=[\delta^{2}F/\delta\phi^{2}]^{-1}\rightarrow\infty$ at the
spinodals, in strict analogy with its behavior at the critical point [21].
Phase-ordering monitored through resistance should thus evolve as

$\langle\delta\phi(t)\rangle=\langle\delta\phi(0)\rangle\,\exp(-t/\tau_{0});\;\tau_{0}=\chi_{T}/\lambda,$
(4)

and the autocorrelation of the resistance noise should be related to

$\langle\delta\phi(t)\delta\phi(0)\rangle=Vk_{B}T\chi_{T}\,\exp(-t/\tau_{0});\;\tau_{0}=\chi_{T}/\lambda.$
(5)

The phase ordering time $\tau_{1}$ and the autocorrelation time $\tau_{2}$
[Fig. 4] are equal to $\tau_{0}$ and diverge proportional to $\chi_{{}_{T}}$.
Eq. 4 further shows that the variance $\langle\delta{\phi}^{2}(0)\rangle$
should also diverge as $\chi_{T}$ (‘critical opalescence’), in agreement with
Fig. 2 (e, f) [37]. Regarding the slopes of the PSD [Fig. 3], the linear
theory would predict Brownian dynamics ($\mu=2$) at the spinodals. Theoretical
estimates for the critical point of the two-dimensional Ising model under
Glauber dynamics suggest $\mu\sim 1.8$ [25].

Discussion.— It had long been an open question whether the spinodal
singularity can be seen in finite temperature experiments. While for short-
ranged interactions, the spinodals are physically meaningless [2], in the
opposite limit of infinite-ranged interactions, the mean field theory becomes
exact [47]. Thus the point of contention is whether the spinodal may be a
meaningful concept for long but finite-ranged interactions [42].
Theoretically, the issue is addressed by constructing, as usual [21], a
relationship (Ginzburg criterion) between the relative order parameter
fluctuations, system dimensionality, and the range of interactions as one
approaches the spinodal singularity [2, 43, 60]. Provided the interactions are
sufficiently long-ranged, one can get close enough to the instability such
that finite temperature effects broaden the transition but may not completely
mask it [43, 33, 64], mimicking a finite-size effect. This is indeed what we
see; the singularity still manifests as a very discernable (but finite) growth
in the susceptibility. That we have long-range interactions in the system [61]
is independently indicated by the observations of broken ergodicity [Fig. 1
(b)] and very slow relaxation [Fig. 4], which are otherwise unusual for hard
condensed matter systems. Furthermore the robust presence of hysteresis itself
indicates long-range forces on rigorous theoretical grounds [62].

The electronic phase transitions in a number of correlated electron systems
are often also accompanied by abrupt structural transitions on account of the
large polaronic coupling [10, 28]. The resulting strain fields are the source
of long-range interactions [63, 64, 18]. Indeed a structural transition does
seem like the feature shared across disparate systems undergoing APT with
pronounced hysteresis [3, 26, 27, 28, 4, 7, 5, 6, 10, 11, 9, 8, 12, 14, 13,
15, 16, 17]. These are all candidates for the zeroth-order transition
belonging to the mean-field ‘universality class’ [56, 19].

We have not discussed disorder. Though the phase coexistence and ramified
fractal-like structures in the transition regions [7] are also explained by
the non-classical nucleation expected at the spinodal [34, 64] without
invoking disorder, it is perhaps essential for understanding avalanches [40,
65] or even the return-point memory [65, 53] seen in Fig. 1 (b). While one
would naturally expect the smooth evolution suggested by Eq. 1 to be not quite
valid [66], the spinodal singularity itself is expected to be robust to
disorder for sufficiently long range interactions [69]. But this is clearly an
issue that requires further systematic investigation.

Finally, catastrophes [49] in natural and social systems—earthquakes,
ecosystem collapse, climate change, onset of depression, epilepsy, market
crash—are sometimes modeled after such zeroth-order transitions, with the
spinodal singularity signifying the loss of ‘resilience’ [70]. Anticipating
them is of course an important objective of complex systems science. The idea
that fluctuations can carry precursory signatures of such an impending event,
in the form of critical-slowing down accompanied by an enhanced variance of
fluctuations [70], is perfectly vindicated by our work.

It is a pleasure to thank Arup Raychaudhuri for critical advise on the noise
measurements, Amit Ghosal for comments on the manuscript, and Satyabrata Raj
for technical advice on sample preparation.

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Supplemental Material

In this Supplemental Material we discuss the details of the resistance noise
experiments including a discussion on the temperature stability during the
time series measurements and the method employed for detrending the time
series. Raw data of the autocorrelation of the resistance fluctuations [Fig. 4
(a) (Main text)] and of the phase-ordering experiments [Fig. 4 (b) (Main
text)] are discussed. The algorithm to convert the resistance noise data to
the order parameter fluctuation using the approximate percolation model [Fig.
2 (e, f) (Main text)] is discussed. Finally we show that a rough but
independent thermal analysis measurement also indicates sharp maxima in the
phase ordering times around the spinodals.

###### Contents

  1. I Sample
  2. II Noise Measurements
    1. II.1 Experimental setup
    2. II.2 Temperature stability
  3. III Analysis of the voltage time series
  4. IV Robustness of PSD slope $\mu$
  5. V Order parameter from the resistance
  6. VI Autocorrelation of fluctuations
  7. VII Phase-ordering time
  8. VIII Time constant measurement using DTA technique
  9. IX Ruling out the effect of temperature fluctuations on the resistance noise
    1. IX.1 Cross-correlation
    2. IX.2 Temperature coefficient of resistance under different conditions
  10. X Spinodals as bifurcation points and the role of fluctuations

##  I Sample

Single phase polycrystalline V2O3 samples were used in the measurements. The
details can be found in the Supplementary Material of our previous work [26].
As already mentioned, the samples were thermally cycled hundreds of times such
the measurements were reproducible.

##  II Noise Measurements

Transition metal oxides like V2O3, VO2, or NdNiO3 are known to have a large
Hooge parameter [9]. A conventional 4-probe geometry is thus sufficient and
the experimental constraints on reducing the background noise are less
stringent. The measurements nevertheless require extreme caution and multiple
crosschecks. Unlike semiconductors, it can sometimes be hard to get
reproducible data and the samples require a fair amount of training.

Previous work on noise measurements around the phase transition have often
involved ramping the temperature at a given rate ($\sim 0.1$-$10$ K/min) and
constantly recording the resistance during the temperature ramp [23, 14]. This
is them broken into segments within temperature windows where the time series
is detrended and analyzed. While we attempted to do this, even for the lowest
temperature ramp rate of $0.1$ K/min (below which the measurements become
untenably long), the change in the resistance around the transition was too
abrupt and it was difficult to reliably detrend the time series. Therefore the
time series were recorded with the temperature kept fixed a given value.

###  II.1 Experimental setup

The setup consists of a liquid nitrogen-cooled cryostat with a temperature
range from $77$-$300$ K. Gold wire contacts made with silver epoxy paste in
van der Pauw geometry were used through the study.

Given that the characteristic relaxation time, previously inferred from the
time-dependent Landau dynamics [26], is about $3.5$ s-1, we focussed on much
lower frequencies ($<1$ Hz) than are typically used ($<10-100$ Hz). This also
proved helpful because the sample resistance could be $>10^{5}\,\Omega$ in the
insulating phase and capacitive effects begin to be prominent at higher
frequencies. Note that it is necessary that the lockin excitation frequency be
about $50-100$ times larger than the high frequency cut-off in the power
spectral density to avoid any spurious settling time effects associated with
the lowpass filter required for the homodyne detection, a safe settling time
being $>5$, say $10$, lockin time constants.

The measurement circuit is shown in Fig. 1 (supplement). The measurements
involved exciting the sample with an oscillatory voltage at the frequency of
about $23$ Hz, with a large $1$ M$\Omega$ low noise reference resistance
connected in series. The voltage across the other two contacts was fed into a
high input impedance (100 M$\Omega$) voltage preamplifier (Stanford Research
Systems SR560) equipped with bandpass filters which also helped in cutting
down the aliasing effects. The output of the preamplifier was fed into the
lockin amplifier (Stanford Research Systems SR830) whose time constant was
chosen to be $300$ milliseconds. The output of the lockin amplifier was read
out the rate of $10$ Hz (time stamp from the computer clock was made on each
data point). For the purpose of analysis of the power spectral density (PSD)
of the time series, the upper frequency cut off was chosen to be $0.3$ Hz
(i.e, a very safe $10$ time constants). The sample current was also
independently monitored by another locking amplifier (Stanford Research
Systems SR830). The sample temperature was controlled with a proportional-
integral-derivative (PID) based digital temperature controller (Lakeshore 336)
with a temperature resolution of 1 millikelvin. The measurement electronics
was electrically isolated from the measurement computer and proper shielding
and grounding of the set up was ensured.

Figure 1 (supplement): Circuit for the resistance noise measurements.

###  II.2 Temperature stability

As the sample response in the metastable phase is highly history dependent,
the sample was always cooled (heated) to well outside the metastable phase
before each time series measurement and then the temperature was ramped up
(down) to the desired set point. Then the voltage fluctuation of the sample
with time were recorded. For each temperature set point, the best PID control
parameter values were determined. Typically, the voltage time series was
recorded for many hours and then an appropriate segment (of duration $>60$
minutes) was picked out by manual inspection, where the fluctuation in the
temperature was less than $10$ mK, usually better and limited by the
digitization noise [Fig. 2 (supplement)].

Figure 2 (supplement): Some typical temperature profiles in the time windows
where the resistance time series data were analyzed. (a) Insulating region
(132 K), (b) cooling spinodal (153K), (c) heating spinodal (162K), and (d)
metallic region (190K). Note that the temperature stability is essentially
limited by the resolution set by digitization.

##  III Analysis of the voltage time series

On reaching the set point (especially around the spinodal regions), the sample
displays phase ordering where a substantial change in the resistance occurs
within a few tens seconds. The temperature also stabilizes during this time.
The lockin amplifier was then offset to zero and the gain enhanced to get the
best possible voltage resolution. As mentioned above, from the many hours long
time series, we only take the part where the temperature is most stable. It
was cross-checked that even at the spinodals (where the change of the
resistance with temperature is the sharpest), the possible resistance
fluctuations due to the temperature fluctuations were at least one order of
magnitude smaller than the measured noise. To ensure that the measured noise
was (i) indeed from the sample and not the associated electronics and (ii) the
bias on the sample was small enough to not cause heating, we verified that the
power spectral density remain unchanged (i.e., the power spectrum scaled
proportional to $\langle V^{2}\rangle$) when the bias across the sample was
doubled.

Figure 3 (supplement): (a) The raw time series of the voltage fluctuations at
162 K, which is close to the heating spinodal. Note that there is a small but
significant ($<1\%$ in about $10^{5}$ seconds) monotonic drift due to the slow
phase ordering. Solid line shows a straight line fit. (b) Detrending by
subtracting this straight line. This detrended time series is still dominated
by large low frequency fluctuations. These are real but it is hard to get good
statistics on these. As a result, the assessment of the autocorrelation,
variance and the probability distribution function of the fluctuations is not
reliable. (c) A simple alternative is to directly highpass filter the data in
shown in (a). This removes both the linear trend and puts a systematic low
frequency cut-off on the bandwidth too. Note that filtering has the (well-
known) artifact of introducing spurious oscillations in the beginning of the
time series. This part of the time series is ignored. (d, e, f) The same for
the cooling spinodal time series at $153$ K.

Around the spinodals, as the phase ordering is a slow process, there is always
a small monotonic drift (about $<1\%$ in $>3$ hours) in the value of the
sample resistance [Fig. 3 (a, d) (supplement)]. The time series thus need to
be detrended.

The second problem, again increased at the spinodals due to the slowing down,
is that there is a large spectral weight in the very low frequency components
of the power spectral density (PSD), which can be seen as large fluctuations
occurring on the time scale of tens of minutes and modulate the whole time
series. While these are real, clearly it is not possible to get sufficient
statistics on such large time scales. While the determination of PSD was
robust to the method of detrending (as the lowest frequencies which are most
affected by detrending could be thrown away), quantities like the variance,
autocorrelations and the probability density functions of the fluctuations are
very hard to determine precisely.

A simple and robust solution was found in passing the raw recorded time series
through a digital high pass filter and limiting the low frequency to about $3$
millihertz. This was implemented using the designfilt function in MATLAB
software, with the following parameters: Stopband frequency: $0.0008$ Hz;
Passband frequency: $0.003$ Hz, Stopband attenuation: 50, Passband ripple: 1
and Sample rate: 10 Hz.

Figure 4 (supplement): Power spectrum of the voltage time series (a) around
the cooling spinodal at $153$ K and (b) close to the heating spinodal $162$ K
calculated by three different methods of detrending, namely via a linear
subtraction, by subtracted a best fit quadratic and via high pass filtering.
In case of linear and quadratic fits, the difference is only at the lowest
frequency. We have band-width limited the measurement to a lower frequency of
$0.003$ Hz. The high frequency cut-off ($0.3$ Hz) is imposed by the low pass
filter of the lockin amplifier.

##  IV Robustness of PSD slope $\mu$

Fig. 4 (supplement) and Fig. 5 (supplement) shows that the inferred slope
$\mu$, $S_{V}/\langle V\rangle^{2}\sim f^{-\mu}$ of the PSD between
$0.3-0.003$ Hz is essentially robust to the method used for detrending the
time series. Since the number of data points in a frequency octave vary
drastically from octave to octave, a naive least square fitting (on the log-
log scale) to determine $\mu$ puts a disproportionate weight on the high
frequency region of the PSD. To circumvent this problem, the PSD were smoothed
and the data between $0.003-0.3$ Hz were interpolated, equispaced on the log
scale. These are the PSDs shown in Fig. 2 (a, b) (Main text).

Figure 5 (supplement): Inferred values of the noise PSD exponent $\mu$ for the
time series detrended using a linear subtraction, substraction of the best
fitting quadratic, and high pass filtering at different temperatures for the
measurements done under (a) heating and (b) cooling protocols showing that the
conclusions of the slowing down are robust to the method of detrending.

##  V Order parameter from the resistance

It is interesting to attempt to convert the sample’s resistance to a more
microscopic and theoretically interesting quantity, the order parameter. As is
argued in our previous work [26] and also in previous work on V2O3 around the
critical point [58], the fraction of the insulating phase in the material
could reasonably be taken as the order parameter. But the relationship between
the measured resistance and the insulating fraction is a monotonic but a
complicated non-linear function. We shall use the equation given by McLachlan,
which is based on a general effective medium (GEM) theory for percolative
transitions. If $\sigma_{E}$ is the effective electrical conductivity of a
binary metal-insulator mixture, the GEM equation is given by [59]

$(1-f)\frac{\sigma_{I}^{1/t}-\sigma_{E}^{1/t}}{\sigma_{I}^{1/t}+A\sigma_{E}^{1/t}}+f\frac{\sigma_{M}^{1/t}-\sigma_{E}^{1/t}}{\sigma_{M}^{1/t}+A\sigma_{E}^{1/t}}=0$

where f is the volume fraction of the metallic phases and $A=(1-f_{c})/f_{c}$,
$f_{c}$ being the volume fraction of metallic phases at the percolation
threshold, and t is a critical exponent which is close to 2 in three
dimensions.The constant $f_{c}$ depends on the lattice dimensionality, and for
3D its value is $0.16$.

Here we are considering the insulator fraction ($\varphi({\bf x},t)$) as order
parameter and the conductivity is related to the spatial average of
$\int\varphi d^{x}\sim\phi(t)$. Hence the relationship between $\phi$ and the
$f$ defined above is simply $\phi=1-f$. By using the relation between
conductivity and resistance ($\sigma=\frac{l}{AR}$ where $l$ and $A$ are
constants) we can rewrite the equation in terms of the sample resistance, viz.

$\varphi\frac{R_{I}^{-1/t}-R_{E}^{-1/t}}{R_{I}^{-1/t}+AR_{E}^{-1/t}}+(1-\varphi)\frac{R_{M}^{-1/t}-R_{E}^{-1/t}}{R_{M}^{-1/t}+AR_{E}^{-1/t}}=0$
(S1)

The data in Fig. 8 (supplement) are inferred using the above equation.

More interestingly, using this relation we can approximately convert
resistance fluctuations to the fluctuations of the order parameter, plotted
Fig. 2 (e, f) (Main text). The result is in accordance with the theoretical
expectation [Eq. 3 (Main text) with $k=0$ and $t=0$] of observing a
‘divergence’ at the two spinodals.

##  VI Autocorrelation of fluctuations

In Fig 4 (a) (main text), we have plotted the relaxation time $\tau_{2}$ for
the autocorrelation of fluctuations. $\tau_{2}$ at different temperatures were
inferred from the numerically calculated autocorrelation function $A(t)$ of
the detrended time series, where $A(t)$ is defined as

$A(t)\equiv\langle\delta R(t^{\prime})\cdot\delta
R(t+t^{\prime})\rangle_{t^{\prime}}.$ (S1)

$\langle\cdot\cdot\cdot\rangle_{t^{\prime}}$ denotes the time average. Figure
6 (supplement) shows $A(t)$ at a few temperatures. It can clearly be seen that
the decay is significantly slower around the transition regions. Note that
since the measurements were done with a lockin time constant of $300$
milliseconds, we would expect to have a low-pass-filter-circuit-induced
correlations which may persist up to a few ($\lesssim 5$) time constants,
which is about $1-2$ seconds.

Figure 6 (supplement): Autocorrelation. The reason for the oscillatory
behavior of the autocorrelation in the insulating phase is not understood.
Note however that the envelope shows a much faster decay compared to the
spinodals.

The relaxation time $\tau_{2}$ is the characteristic decay time of the
autocorrelation $A(t)$ and there is some arbitrariness in the definition
unless one forces a model. It is only within the linear theory that the
autocorrelation function is expected to be of a simple exponential form $[$Eq.
5 (main text)$]$, namely

$A(t)\sim\exp[-t/\tau_{0}]\;\;\;\;\;\;\textrm{(linear \, theory\, only)}.$
(S1)

and the real $A(t)$ never fits this. Luckily, we are not interested in the
exact numerical value but only wish to (unambiguously) show that the
characteristic scale of the relaxation time rapidly grows around the
spinodals. This fact is not sensitive to the details of the definition.

Figure 7 (supplement): The characteristic autocorrelation time scale estimated
for different values of $\kappa$ shows the same characteristic trend. For a
perfect monoexponential decay, all the curves would have coincided.

But one can always extract a characteristic decay time $\tau_{2}$ assuming
such a monoexponential decay and by finding out the value of the time $t_{1}$
when the the autocorrelation function has a value $\kappa$ with $0<\kappa<1$,
e.g. if one chooses $\kappa=0.8$, $0.8=e^{-t_{1}/\tau_{2}}$. So by reading off
the time $t_{1}$ from the autocorrelation graph, one can estimate $\tau$

$\tau_{2}=\frac{t_{1}}{\ln(1/\kappa)}$ (S1)

While there is an arbitrariness about which value of $\kappa$ to take, it was
reassuring to find that the trends in the temperature dependence of the
characteristic time $\tau$ and the observation of its enhancement around the
transition does not depend on the value of $\kappa$ chosen (up to about
$\kappa=0.7$ below which $A(t)$ in the insulating phase begins to oscillate
with a decaying envelope). We considered $\kappa=0.9$ for Fig 4 (main text).

##  VII Phase-ordering time

In addition to the measurement of the relaxation times of the autocorrelation
of the fluctuations, Fig. 4 (a) (Main text) also shows the characteristic
phase ordering time $\tau_{1}$. Figure 8 (supplement) shows the data used to
infer $\tau_{1}$ at different temperatures around the two spinodals. The
measurements involved shock-heating (cooling) the sample to a given set point
at the rate of 50 K/min, starting deep into the insulating (metallic) phase.
On reaching the set-point, the temperature was quickly stabilized and the
change in the resistance with time was recorded. The resistance data were then
converted to the order parameter using the transformation discussed above and
the phase ordering time $\tau_{1}$ was inferred by fitting the equation

$\varphi(t)=[1-\varphi_{\infty}]\exp(-t/\tau_{1})+\varphi_{\infty}$ (S2a)

to the observed order parameter evolution after the shock heating quench and

$\varphi(t)=\varphi_{\infty}[1-\exp(-t/\tau_{1})]$ (S2b)

for the evolution after the cooling quench [Fig. 8 (supplement)].

Figure 8 (supplement): Phase ordering after a shock-heating (left) and cooling
(right) quench. Solid lines are fits to Eq. (20).

##  VIII Time constant measurement using DTA technique

In this section, we will show that the observed slowing down can also be
qualitatively inferred from a completely independent thermodynamic measurement
using differential thermal analysis (DTA). While this novel technique has some
limitations and is only qualitative, it has the advantage that the extent of
phase ordering is directly estimated from the amount of the latent heat
measured. The method is an extension of the well-known first-order-reversal-
curve (FORC) measurements of the hysteresis loops. The details of the DTA
apparatus can be found in Ref. [26].

Figure 9 (supplement): (a) Temperature profile for the heating FORC
measurement. The waiting time was fixed (1000 s) for this set of measurements
(Left-Right arrow). The Leftwards arrow corresponds to the first reversal
temperature (Tw). (b) DTA signal for different wait temperatures. The area of
the DTA peak at $T\sim 152$K is a measure of the amount of material
transformed due to the quench and wait at reversal temperature (Tw). (c) The
insulator (phase) fraction versus reversal temperature (Tw) for different
waiting time.

The experiment involves, starting from a temperature $T_{s}$ (say) deep in the
insulating phase and then shock-heating the sample at the rate of 50 K/minute
to a given target temperature Tw where Tw’s are around the heating spinodal
($\sim 162$ K). Then the sample temperature is kept fixed at this Tw for a
given wait time tw, after which the sample is rapidly quench-cooled at the
rate of 50 K/minute to the starting temperature $T_{s}$. As the sample is
cooled, it will cross the cooling spinodal ($\sim 152$ K) and latent heat will
be released. The main idea of the experiment is that this latent heat is
proportional to the extent of the phase ordering corresponding to (Tw, tw).
Figure 9 (a) (supplement) shows the temperature protocol for a given wait time
tw and Fig. 9 (b) (supplement) shows that different amount of latent heats are
measurement depending on Tw. The complete measurement of the phase ordering
[Fig. 9 (c) (supplement)] involves repeating this for different wait times,
$\\{t_{w}=1,5,10,30,50,100,200,500,1000\\}$ seconds. t${}_{w}=1000$ s
corresponds to (almost) complete phase transformation and would yield the
temperature dependence of the order parameter under quasistatic conditions, as
was already discussed previously [26].

Figure 10 (supplement): Approximate time constants close to the transition
temperature extracted from DTA data during cooling and heating.

The same idea is used to determine the phase ordering during the cooling
spinodals using cooling FORCs.

For a particular reversal temperature Tw, the time-dependent insulator
fraction can be extracted from Fig. 9 (supplement) (c) (dashed line in Fig. 9
(supplement) (c). The phase ordering curves, similar to Fig. 8 (supplement),
can be constructed from inferring how the insulator fraction varies with the
wait time tw for a given temperature and are plotted in Fig. 10 (supplement).
However we found that the inferred time constants are almost an order of
magnitude larger than those in Fig. 4 (Main text). There are two reasons for
this. Firstly the quench rates are themselves are finite and these
measurements were much more difficult than those carried out to generate Fig.
8 (supplement) because they not only required reaching a target temperature
and rapidly settling down at that temperature, but also a quench back to Ts
after a wait time tw. As can be seen in Fig. 8 (supplement), there is a rapid
phase-ordering in the initial $5-10$ seconds. It is especially this initial
data which is very imprecise or lost in these thermal analysis measurements as
the temperature stabilization time is of the same order. A larger phase
ordering time is thus inferred. Secondly, the calculation of the fraction of
the latent heat as a function of (T$w$, tw) requires background subtraction,
which can be slightly ambiguous when the latent heat is small. Moreover, the
goodness of fit in Fig. 9 (supplement) (c) (supplement) is directly related to
the goodness of the decay curve.

For these reasons, these latent heat measurements must be considered very
rough. Nevertheless, the peaks around the spinodals are in striking
qualitative agreement with our other measurements.

##  IX Ruling out the effect of temperature fluctuations on the resistance
noise

While the temperature stability [Fig 2 (supplement)] during the resistance
time series measurements at the two spinodals was essentially limited by the
resolution of the temperature controller, it is still worth ruling out that
the observed noise is not simply due to the temperature fluctuations.

###  IX.1 Cross-correlation

The most direct way to probe this question is to simply compute the cross-
correlation between the two time series and quantify the extent to which the
measured resistance noise is affected by the temperature fluctuations. The
Pearson correlation coefficient $\rho_{{}_{T,R}}$ provides one such measure.
$\rho_{{}_{T,R}}$ is simply the covariance of the two variables normalized by
their standard deviations and in our context may be defined as

$\rho_{{}_{T,R}}={\int_{t_{i}}^{t_{f}}dt\,\delta S(t)\cdot\delta
T(t)\over\sqrt{\int_{t_{i}}^{t_{f}}dt\,\delta
T^{2}(t)}\sqrt{\int_{t_{i}}^{t_{f}}dt\,\delta S^{2}(t)}}$ (S3)

Here $\delta S(t)$ is the measured signal and $\delta T(t)$ is the time series
of the temperature fluctuations. $|\rho_{T,R}|\in[0,1]$ with $1$ indicating
perfect (linear) correlation and $0$ indicating no correlation. We can
equivalently define this correlation in the frequency domain—by the Parseval’s
theorem the power contained in each of these three integrals is the same in
the time and frequency domains, i.e.,

$\rho_{{}_{T,R}}={{1\over
2}\int_{\nu_{l}}^{\nu_{u}}d\nu\left[\delta\tilde{T}^{*}(\nu)\cdot\delta\tilde{S}(\nu)+\delta\tilde{T}(\nu)\cdot\delta\tilde{S}^{*}(\nu)\right]\over\sqrt{\int_{\nu_{l}}^{\nu_{u}}d\nu\left|\delta\tilde{T}(\nu))\right|^{2}}\sqrt{\int_{\nu_{l}}^{\nu_{u}}d\nu\left|\delta\tilde{S}(\nu))\right|^{2}}}$
(S4)

$\delta\tilde{S}(\nu)$ and $\delta\tilde{T}(\nu)$ denote the Fourier
transforms at frequency $\nu$. Taking $\nu_{l}=0.003$ Hz and $\nu_{u}=0.3$ Hz,
which is the frequency window for which the PSD slopes were calculated in Fig
2 (main text) and what approximately corresponds to the bandwidth of the
detrended time series used in the rest of the paper, we find that
$\rho_{{}_{T,R}}\simeq 0.035$ for the measurement at 162 K (heating) in a time
series with 120,000 points and $\rho_{{}_{T,R}}\simeq 0.001$ for the
measurement at 153 K (cooling) in the time series with 100,000 points. Thus
the correlation in both cases is below 5% and safely ignored.

What this roughly means is that if the measured signal $\delta S(t)$ is
affected by the normalized temperature fluctuations $\delta T(t)$ in the
following fashion

$\delta S(t)\simeq\alpha\cdot\delta S_{T}(t)+(1-\alpha)\cdot\delta S_{R}(t),$
(S5)

where $S_{R}(t)$ is the true signal and $S_{T}(t)$ is the spurious component
arising out temperature fluctuations $\delta T(t)$, then the fraction $\alpha$
of the measured signal arising from the temperature fluctuations is below 5%.

###  IX.2 Temperature coefficient of resistance under different conditions

While the correlation calculation settles the issue, it is still interesting
to understand why this correlation is so small. Physically, we would require

${1\over R}\left|{dR\over dT}\right|\delta T<<{|\delta R|\over R}$ (S6)

Here $R$ is the average sample resistance at the temperature of interest,
${dR\over dT}$ temperature coefficient of resistance at the given temperature,
$\delta T$ the magnitude of the temperature fluctuations and $|\delta R|$ the
magnitude of the noise.

While the change in the sample resistance with temperature at the transition
points is very large under a monotonic temperature ramp ($|{1\over R}{dR\over
dT}|\sim 2$ K-1), luckily the slope is dramatically reduced when one makes
small amplitude cyclic excursions around a given point in the metastable
phase. The temperature coefficient of the change in resistance ($|{1\over
R}{dR\over dT}|$) is related to the magnitude of the excursion around a given
set point as well as the thermal history.

Figure 11 (supplement): The normalized temperature coefficient of resistance
$|{1\over R}{dR\over dT}|\approx 2$ K-1 under monotonic temperature ramp. As
the next figure shows, this value is greatly reduced when the temperature has
a small amplitude fluctuation around a set point.  Figure 12 (supplement): (a)
Sample resistance under cyclic temperature variation with the cycle of
different amplitudes. (c) $|{1\over R}{dR\over dT}|$ monotonically decreases
with the amplitude of temperature oscillation. When $\Delta T$ is $4$ K, the
amplitude is reduced to about $0.04$ K-1 which is a factor of about 30 lower
than the value under a monotonic ramp (marked as ‘0’ in (a)) where $|{1\over
R}{dR\over dT}|\approx 1$-$2$ K-1

Since our noise measurements involve stabilizing the temperature over tens of
minutes, it is reasonable to consider the $dR/dT$ under conditions when the
temperature goes back and forth around the given set point, rather than what
it is under the condition of a monotonic ramp. While we had initially started
our measurements by making slow temperature ramps, it was soon realized that
this method was untenable for V2O3 due to the large $dR/dT$ and the very sharp
transition.

Figure 12 (supplement) shows variation of the normalized slope $|{1\over
R}{dR\over dT}|$ in the return-point-memory experiment of Fig. 1 (b) (main
text). It can be seen that the slope is a strong function of the amplitude of
the excursion in the metastable phase. When the temperature fluctuation is
about $4$ K, the slope is only $0.04$ k-1. Note that the amplitude of
fluctuations in our experiments is nearly three orders of magnitude smaller at
$\sim 5$ mK [Fig. 2 (supplement)] and thus even if we take $|{1\over
R}{dR\over dT}|\approx 0.01$ K-1, a temperature fluctuation of $\delta T\sim
5\times 10^{-3}$ K yields

$\displaystyle{1\over R}\left|{dR\over dT}\right|\delta T\sim 5\times 10^{-5}$

Note that ${|\delta R|\over R}$ is already depicted in Fig. 2 (main text) and
${|\delta R|\over R}\sim 1\times 10^{-3}$.

Thus the condition

$\displaystyle{1\over R}\left|{dR\over dT}\right|\delta T<<{|\delta R|\over
R}$

is satisfied.

We thus have a qualitative understanding for why the degree of temperature
control in our experiments can be considered good enough to not affect the
resistance noise.

##  X Spinodals as bifurcation points and the role of fluctuations

It is perhaps worth pointing out for completeness that, in the language of
nonlinear dynamics, phase transitions correspond to bifurcations [52, 49]. The
thermodynamic spinodals that we discuss here also appear in the more general
setting of catastrophe theory [52, 49, 70].

The dynamical systems description of bifurcations [52, 55] closely parallels
the thermodynamic mean-field theory in terms of the Landau free energy [21].
The continuous or second-order transitions are identified with the
supercritical bifurcations [55] and the Landau $\phi^{6}$ potential used to
describe an abrupt phase transition [shown in Fig. 1 (c) (main text)] is a
standard manifestation of the pitchfork subcritical bifurcation in the context
of nonlinear dynamical systems [52, 49]. As a consequence, systems undergoing
the subcritical bifurcation also display hysteresis and metastability.
Subcritical bifurcations have been extensively studied in a variety of
(nonthermo-)dynamical systems—e.g., Ref. [51] discusses some dynamical aspects
of the lasing instability, Ref. [55] discusses plasma dynamos relevant in
astrophysical contexts, and Ref. [70] discuss such bifurcations or ‘tipping
points’ in context of various complex systems.

Furthermore, it not just the potential but the similarity extends also to the
dynamical equation. The dissipative Model A (time-dependent Landau-Ginzbug)
dynamics [21] [Eq. 1 (main text)] is also often used to describe the
associated dynamics (with and without noise) [52, 49] for bifurcations in non-
thermodynamic systems. It thus follows that such gradient dynamical systems
(where the restoring force is proportional to the gradient of the generalized
potential/free energy and there is no inertial term corresponding to the
second time derivative) should also exhibit critical slowing down [51] in
phase-ordering as the system’s resilience (which is the inverse of the order
parameter susceptibility) [70, 49] vanishes around the spinodals.

Going beyond deterministic dynamics, we have argued in the text that it seems
reasonable to expect that the fluctuations in the relevant coordinate (order
parameter) should also exhibit slowing-down and anomalously large variance as
one approaches the spinodal bifurcation. This fact has been also been much
celebrated in the catastrophe theory literature [49, 70]. While the arguments
presented are exactly along the lines of the theory critical point
fluctuations, the question of fluctuations in the context of the spinodals is
a delicate matter that had not so far been experimentally resolved.

Spinodals can only be defined in the limit where the mean field theory is
exact and [2] and the mean field description of course is arrived at by
ignoring fluctuations. The relationship between the spinodal points and
fluctuations is thus antithetical and has been doubtful if the manifestations
of the spinodal singularity could ever seen in a real thermodynamic system
[2].

Theoretical work over especially over the last two decades has carefully
addressed this issue in context of various thermodynamic models using both
field-theoretic and computational techniques [45, 60, 43, 64, 57, 56]. It has
been anticipated that even for systems with (long but) finite-ranged
interaction, even though the spinodal should be technically inaccessible,
fluctuations may be suppressed enough such that the one characteristic aspect
of the mean-field physics of abrupt phase transitions, namely hysteresis, may
always be manifest. The effect of the finite range of the interaction is that
the divergence is broadened and the susceptibility remains finite at the
transition. Correspondingly the lifetime of the metastable phase close to the
spinodal is also finite [57]. Interestingly, it has been shown that the
absence of a true divergence can be thought of as the magnetic field (or
temperature) at the point of transition picking up an imaginary component,
with the finiteness of the range of the interactions being akin to a finite
size effect [33, 43]. Such finite-size effects for models with long-range
interactions have also been studied in the context of the relaxation dynamics
[56]. In summary, despite the critical features only approximately manifesting
at the transitions, such transitions are expected to be qualitatively very
different from the usual abrupt (‘first-order’) transitions and closer to the
critical transitions because one should still observe a rapid growth in
fluctuations and the relaxation time scales [64].

Our experimental observations do indeed seem to support this general idea of
such mean-field ‘universality class’ [43, 18] (or a zeroth order transition
[49]). Although previously spin crossover [18] and martensite materials [63]
have been identified as candidates for this mean-field universality class,
this is the first experimental study where the critical slowing down and
critical opalescence has been sought and so clearly observed in an abrupt
phase transition.