# Quantum critical fluctuations in an Fe-based superconductor

D. Jost<EMAIL_ADDRESS>Current address: SLAC National Accelerator
Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA  Walther Meissner
Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany
Fakultät für Physik, Technische Universität München, 85748 Garching, Germany
L. Peis  Walther Meissner Institut, Bayerische Akademie der Wissenschaften,
85748 Garching, Germany  Fakultät für Physik, Technische Universität München,
85748 Garching, Germany  G. He  Walther Meissner Institut, Bayerische Akademie
der Wissenschaften, 85748 Garching, Germany  A. Baum  Walther Meissner
Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany  S.
Geprägs  Walther Meissner Institut, Bayerische Akademie der Wissenschaften,
85748 Garching, Germany  J. C. Palmstrom  Stanford Institute for Materials and
Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road,
Menlo Park, CA 94025, USA  Geballe Laboratory for Advanced Materials & Dept.
of Applied Physics, Stanford University, CA 94305, USA  M. S. Ikeda  Stanford
Institute for Materials and Energy Sciences, SLAC National Accelerator
Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA  Geballe Laboratory
for Advanced Materials & Dept. of Applied Physics, Stanford University, CA
94305, USA  I. R. Fisher  Stanford Institute for Materials and Energy
Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo
Park, CA 94025, USA  Geballe Laboratory for Advanced Materials & Dept. of
Applied Physics, Stanford University, CA 94305, USA  T. Wolf  Karlsruhe
Institut für Technologie  S. Lederer  Cornell University, Ithaca, New York
14850, USA  R. Hackl<EMAIL_ADDRESS>Walther Meissner Institut, Bayerische
Akademie der Wissenschaften, 85748 Garching, Germany  Fakultät für Physik,
Technische Universität München, 85748 Garching, Germany  IFW Dresden,
Helmholtzstr. 20, 01069 Dresden, Germany

###### Abstract

The interrelation of various instabilities characterizes all unconventional
superconductors. Here, we show that the characteristic scaling of a marginal
Fermi liquid is present in inelastic light scattering data of an Fe-based
superconductor tuned through a quantum critical point (QCP) by chemical
doping. From the doping dependence of the imaginary time dynamics we are able
to distinguish regions dominated by quantum critical behavior from those
having classical critical responses. This dichotomy reveals a connection
between the marginal Fermi liquid behavior and quantum criticality. In
particular, the overlap between regions of high superconducting transition
temperatures and quantum critical scaling suggests a contribution from quantum
fluctuations to the formation of superconductivity.

##  I Introduction

The steep increase of the susceptibility upon small perturbations close to
quantum critical points (QCPs) is considered crucial for various instabilities
in solids such as the cuprates, iron-based materials, and heavy fermion
superconductors. QCPs arise from suppressing a second order thermal phase
transition to absolute zero temperature. Although this renders the direct
probing of QCPs impossible, their corresponding critical fluctuations [1, 2]
extend over a wide range of temperatures and manifest themselves under
experimentally feasible conditions, yielding direct consequences for the
macroscopic properties of materials. For instance, the $T$-linear resistivity
observed in some cuprate [3, 4, 5, 6] and Fe-based superconductors [7] tuned
to criticality has been linked to the presence of quantum critical
fluctuations [8]. Spectroscopic fingerprints strongly indicate a connection
between the $\Omega/T$ scaling of a marginal Fermi liquid [9] and quantum
criticality, as observed in a recent study of the optical conductivity [10] of
the heavy fermion superconductor YbRh2Si2 [11]. Moreover, theoretical works
have shown that quantum fluctuations may have a significant impact on
superconductivity, such as its enhancement [12] or formation [13, 14].

Characteristic for systems with a QCP is the presence of a quantum critical
region at temperatures above a cross-over temperature
$T^{*}(x)\sim|x-x_{\mathrm{c}}|^{\nu z}$ (where $\nu$ and $z$ are the
correlation length and dynamical critical exponents, respectively, and
critical doping $x_{\mathrm{c}}$). Since this broad region of the phase
diagram manifests the behavior of the single quantum critical point at
$x=x_{c},\ T=0$, Frerot et al. [15] dubbed this region a “magnifying lens” for
quantum criticality. In the experimental work presented here, we have traced
the image produced by this lens by determining the crossover temperature
$T^{*}$ separating the high temperature quantum critical behavior from the
(effectively) classical behavior at low temperatures. This approach may prove
effective in probing the dynamics of quantum criticality without the need to
suppress superconductivity with high magnetic fields, as is typically
necessary in studies involving transport.
Due to the tunability by doping and the purity of the crystals, we have opted
to use as a testbed the compound Ba(Fe1-xCox)2As2, one of the most studied Fe-
based superconductors. Ba(Fe1-xCox)2As2 exhibits a rich phase diagram [16]
with overlapping regions of magnetism, nematicity and superconductivity, and
shows a strong enhancement of nematic fluctuations of $d_{x^{2}-y^{2}}$
symmetry (here using the 1-Fe unit cell) close to the nematic transition [17,
18, 19]. A plethora of experimental studies [17, 18, 20, 21, 22, 23] indicate
the existence of a QCP underneath the superconducting dome in this material,
and the Raman response within the superconducting phase bears an imprint of
nematic quantum criticality [24]. Most recently, Worasaran et al. [25]
provided striking evidence for nematic quantum criticality by observing power-
law behavior of the nematic/structural transition temperature as a function of
uniaxial strain. However, the influence of the QCP on the high temperature
characteristics, for instance signatures of $\Omega/T$ scaling, remain
elusive.
In this work we provide a comprehensive investigation of the Raman response of
Ba(Fe1-xCox)2As2 for doping concentrations on both sides of optimal doping
$x_{\mathrm{c}}\sim 0.061$. Specifically, we fill the gap close to
$x_{\mathrm{c}}$ where $\Omega/T$ scaling is expected to be most pronounced
and provide a solid experimental basis for an earlier proposal [26]. We use
electronic inelastic light (Raman) scattering which probes fluctuations of
various degrees of freedom [27, 28] and has led to promising results
particularly in Fe-based superconductors [17, 29, 19].

##  II Methods

###  II.1 Raman scattering

We performed inelastic light scattering experiments using an excitation
wavelength of 577 nm (Coherent GENESIS MX-SLM577-500). The samples were
attached to the cold finger of a 4He flow cryostat. Polarized photons hit the
sample at an incident angle of $68^{\circ}$. The polarized scattered photons
were collected along the surface normal of the sample and focused on the
entrance slit of a double monochromator. Polarized photons having the selected
energy were recorded with a CCD detector. The number of photons per second is
proportional to the Van-Hove function $S(\mathbf{q}\approx
0,\Omega)=1/\pi[{1+n(\Omega,T)}]\chi^{\prime\prime}(\Omega,T)$ where
$n(\Omega,T)$ is the Bose factor and $\chi^{\prime\prime}$ is the imaginary
part of Raman response function which is displayed in Fig. 1. The factor $R$
is the constant of proportionality which absorbs all experimental factors.

###  II.2 Imaginary time response

Following a method proposed in Ref. [26], we have used our measurements of the
dissipative part of the response function, $\chi^{\prime\prime}$, to
characterize the dynamics of the corresponding _imaginary time_ correlation
function $\tilde{\Lambda}(\tau)$ using the following exact identity:

$\tilde{\Lambda}(\tau)=\int\frac{\mathrm{d}\Omega}{2\pi}\chi^{\prime\prime}(\Omega,T)\frac{\exp[\Omega(\tau-\beta/2)]}{\sinh[\beta\Omega/2]},$
(1)

with the imaginary time $\tau\in(0,\beta)$ and the inverse temperature
$\beta=1/k_{\mathrm{B}}T$. Since bosonic correlations have period $\beta$ in
imaginary time, the longest time response corresponds to $\tau\approx\beta/2$.
In the following, it will be convenient to work with the specific quantity

$\beta\tilde{\Lambda}(\beta/2)=\int\frac{\mathrm{d}\Omega}{2\pi
T}\frac{\chi^{\prime\prime}(\Omega,T)}{\sinh[\beta\Omega/2]},$ (2)

which has the same universal properties as the static susceptibility
$\chi^{\prime}(\Omega=0,T)$ in the sense that the two quantities have
identical scaling with temperature on cooling to both classical and quantum
critical points [26]. A key property of $\beta\tilde{\Lambda}(\beta/2)$is that
it has a power-law dependence on temperature if $\chi^{\prime\prime}$ obeys
$\Omega/T$ scaling, and in particular is _independent of temperature_ if
$\chi^{\prime\prime}(\Omega,T)$ is of marginal Fermi liquid form [9], i.e. if
it is a function of only $\Omega/T$ and not of either variable separately.

One significant virtue of $\beta\tilde{\Lambda}(\beta/2)$in comparison to
$\chi\prime(\Omega=0)$ is that it can be unambiguously computed from the
measured Raman spectra. Since the measured $\chi^{\prime\prime}$ (in this and
other studies) fails to fall off at high frequency, the Kramers-Kronig
integral relating $\chi^{\prime\prime}$ to $\chi^{\prime}$ does not converge,
and therefore requires a manual cutoff procedure. By contrast, the denominator
in Eq. 2 grows exponentially for frequencies $\Omega\gg T$, rendering the
integral convergent so long as the range of frequencies probed extends to
several times the temperature.

##  III Results

Earlier studies, investigating the low frequency Raman response of
Ba(Fe1-xCox)2As2 [17, 19], showed that the contribution of nematic
fluctuations presents itself in the B1g ($d_{x^{2}-y^{2}}$) symmetry of the
1-Fe unit cell whereas the temperature dependence of the A1g response is
essentially independent of doping and follows the in-plane resistivity [19].
This suggests, that the essential critical dynamics of the system is captured
by the B1g response. We therefore limit our analysis to the B1g response, with
characteristic spectra shown in Fig. 1 for selected Ba(Fe1-xCox)2As2 samples
with $x<x_{\mathrm{c}}$, $x\sim x_{\mathrm{c}}$ and $x\gg x_{\mathrm{c}}$. All
spectra have in common that the temperature dependence is limited to low
frequencies, while no significant temperature dependence can be detected at
higher frequencies. The underdoped compound [Fig. 1 (a)] exhibits a nematic
transition at approximately $T_{\mathrm{nem}}\sim 50\,\mathrm{K}$. Similar to
results reported elsewhere for $x<x_{\mathrm{c}}$ [17, 29, 19] the strong
increase of the spectral weight between $333$ and $53\,\mathrm{K}$ in the
range below $500\,\mathrm{cm}^{-1}$ can be attributed to the presence of
(classical) critical fluctuations. The decrease below $T_{\mathrm{nem}}$,
reminiscent of a gap, is characteristic for such transitions in the Fe-based
superconductors, and consistent with earlier results [19]. Closer to optimal
doping at $x=0.059$ [Fig. 1 (b)] the response increases monotonically upon
cooling towards the superconducting transition temperature $T_{\mathrm{c}}$.
Remarkably, the response at this lowest temperature is nearly constant over
the entire frequency range. Far away from $x_{\mathrm{c}}$, the temperature
dependence is weak even at small energy transfer [Fig. 1 (c)].

Figure 1: Raman response for Ba(Fe1-xCox)2As2 at selected doping
concentrations and temperatures. (a) For the underdoped compound with
$x=0.052$, the low-frequency pile-up attributable to classical critical
fluctuations exhibits a maximum above the nematic transition and decreases
towards $T_{\mathrm{c}}$. (b) Close to optimal doping with $x=0.059$ the low
frequency response increases monotonically on cooling towards
$T_{\mathrm{c}}$. (c) For a sample with $x\gg x_{c}$, The temperature
dependence is weak at all frequencies.

To examine the possibility of quantum critical scaling, we have computed
$\beta\tilde{\Lambda}(\beta/2)$from each of our measured spectra, with the
results shown in Fig. 2 For the underdoped sample [Fig. 2 (a)] the
characteristic cusp singularity of the nematic transition is visible [26].
Notably, a similar increase observed using a sample with $x=0.057$ doping in
panel (b) towards $T_{\mathrm{c}}$ suggests a persisting nematic phase close
to the superconducting transition. Yet, from our measurements it cannot be
clearly decided whether or not nematicity sets in above $T_{\mathrm{c}}$.
Closer to optimal doping in panel (c), $\beta\tilde{\Lambda}(\beta/2)$
develops a maximum above $T_{c}$, though this cannot be attributed to a
nematic phase transition. Above $x_{\mathrm{c}}$ at $x=0.073$, the quantity
$\beta\tilde{\Lambda}(\beta/2)$ remains constant down to the lowest measured
temperature on the order of $T_{\mathrm{c}}$. Tuning further away from
$x_{\mathrm{c}}$ [Fig. 2 (e) and (f)], $\beta\tilde{\Lambda}(\beta/2)$
decreases towards low temperatures, where $\beta\tilde{\Lambda}(\beta/2)$is
generically required to vanish (see the supplementary material for a proof of
this).

Figure 2: The quantity $\beta\tilde{\Lambda}(\beta/2,T)$ derived from the
Raman spectra of Ba(Fe1-xCox)2As2. The dashed lines correspond to
$\beta\tilde{\Lambda}(\beta/2,T)=\mathrm{const}$, and the blue regions
indicate the deviation from this constant behavior, which we use to define the
crossover temperature $T^{*}$. (a) At $x=0.052$ doping a nematic transition
occurs at $T_{\mathrm{nem}}$ $\sim$ 50 K at which
$\beta\tilde{\Lambda}(\beta/2,T)$ develops a cusp and then decreases towards
$T_{\mathrm{c}}$. (b) At $x=0.057$, $\beta\tilde{\Lambda}(\beta/2,T)$
increases towards lower temperatures and develops a maximum above
$T_{\mathrm{c}}$. (c) Closer to the QCP $\beta\tilde{\Lambda}(\beta/2,T)$
develops a hump between $T^{*}$ and $T_{\mathrm{c}}$. (d) On the overdoped
side at $x=0.073$ the $\Omega/T$ scaling persists down to $T_{\mathrm{c}}$.
(e) Further away, the crossover temperature increases again. (f) The high
crossover temperature at $x=14.6$ coincides with a small superconducting
transition temperature.

The dashed horizontal line is a guide to the eye indicating the value of
$\beta\tilde{\Lambda}$ in the regime of $\Omega/T$ scaling. Outside this
quantum critical regime, where the response does not scale as $\Omega/T$, the
quantity $\beta\tilde{\Lambda}(\beta/2)$ deviates from this constant. The
shaded regions show the crossover temperatures $T^{*}$ from $\Omega/T$ scaling
to regions where other dynamics govern the low frequency response.

##  IV Discussion

$T^{*}(x)$ exhibits a systematic doping dependence, with a global minimum at
$x=0.073$. To further illustrate this point, $T^{*}(x)$ is plotted along with
the relevant phase transitions of Ba(Fe1-xCox)2As2 in Fig. 3. The spin-
density-wave (SDW) phase is indicated in grey, nematicity in red, and
superconductivity in blue. The open red rectangles indicate the crossover
temperatures derived from data presented in Ref. [26]. The dashed lines
represent guides to the eye.

Underdoped region: For $x<x_{\mathrm{c}}$ Ba(Fe1-xCox)2As2 has an SDW ground
state below $T_{\mathrm{SDW}}$, and fluctuations in the temperature range
$T_{\mathrm{SDW}}<T<T^{*}$. Above $T^{*}$ the $\Omega/T$ scaling
characteristic for a marginal Fermi liquid is recovered. Moving closer to
$x_{\mathrm{c}}$, the magnetic and nematic phases become unstable as signified
by the decreasing transition temperatures. In the picture of quantum
criticality, this results from a continously increasing superposition of
entangled states which manifests itself as a divergence of the coherence
length of the quantum fluctuations directly at the QCP. Simultaneously, the
region of marginal Fermi liquid behavior with $\Omega/T$ scaling begins to
persist over wider ranges of temperature with $T^{*}$ falling more rapidly
after superconductivity sets in.

Overdoped region: On the overdoped side for $x>x_{\mathrm{c}}$, the crossover
line re-emerges from the superconducting dome and continues to rise as a
function of doping, with a significantly smaller slope than for
$x<x_{\mathrm{c}}$. In the absence of a competing magnetic ground state, the
quantum fluctuations may support superconductivity over a wider region of
doping concentrations.

Figure 3: Phase diagram of Ba(Fe1-xCox)2As2 with the spin-density-wave (SDW)
in grey, nematicity in red and superconductivity (SC) in blue. The rectangular
data points indicate the cross-over temperatures $T^{*}$ from the region of
$\Omega/T$ scaling. The dashed lines are guides to the eye. Open red squares
represent the data points from Ref. [26]. The grey, red and blue phase
separation lines are based on Ref. [16].

The endpoints of the dashed lines coincide with the region of optimal doping
$x_{\mathrm{c}}$ consistent with a scenario in which quantum fluctuations
increasingly destroy the long range order of the ground state for
$x<x_{\mathrm{c}}$ at temperatures $T>0$. Thus, $T_{\mathrm{c}}$ is high where
the quantum fluctuations dominate the normal state, and low where ”classical”
dynamics take over. Therefore the quantum fluctuations appear to have a
twofold effect: on the classically ordered side below $x_{\mathrm{c}}$, the
competing magnetic ground state is destabilised. On the overdoped side, the
quantum fluctuations protect superconductivity at least partially from
increasing disorder. This scenario is consistent with quantum criticality
being a driving force behind the superconducting state in Ba(Fe1-xCox)2As2.

##  V Conclusion

In conclusion, we tracked the B1g Raman response of Ba(Fe1-xCox)2As2 as a
function of doping $x$. $x$ is tuned from $x<x_{\mathrm{c}}$ through a quantum
critical point far into the overdoped regime of the phase diagram. From the
data we calculated the temperature dependence of the imaginary time ordered
correlation function $\tilde{\Lambda}(\tau)$ at $\tau=\beta/2$ and extracted
cross-over temperatures $T^{*}(x)$ above which $\Omega/T$ scaling prevails.
These temperatures define the boundaries of the quantum critical fan
corroborating the existence of a QCP around optimal doping. Thereby we show
that the determination of the imaginary time dynamics from spectroscopic data
is a straightforward way to reveal the effects of quantum criticality in the
normal state of Ba(Fe1-xCox)2As2. The most important distinction is drawn
between this analysis and any type of Kramers-Kronig analysis where an upper
limit needs to be set. Here, the integral is naturally cut off at
$k_{\mathrm{B}}T$ by an exponential factor. The analysis is not limited to
inelastic light scattering, but can be applied to a wide range of experimental
probes as already outlined in Ref. [26]. We hope that this transformation may
enable the comparison of results from different spectroscopic probes directly
and on the same footing, allowing a more unified analysis of quantum
criticality.

Competing financial interests: The authors declare that there are no competing
financial interests.

###### Acknowledgements.

We thank E. Berg, S. Kivelson and B. Moritz for fruitful discussions. The work
was supported by the Deutsche Forschungsgemeinschaft (DFG, Projekt-IDs
107745057 – TRR 80, HA 2071/8-1 and HA2071/12-1)), the Friedrich-Ebert
foundation (D.J.) and the Alexander-von-Humboldt foundation (G.H.). Part of
the analysis was performed at the Stanford Institute for Materials and Energy
Sciences (SIMES) funded by the U.S. Department of Energy, Office of Basic
Energy Sciences, Materials Sciences and Engineering Division (D.J.). J. C. P.
was supported by a Gabilan Stanford Graduate Fellowship and a Stanford
Lieberman Fellowship.

##  Appendix A Raman data

Fig. 2 of the main text shows the correlation function
$\beta\tilde{\Lambda}(\beta/2)$ resulting from the transformation of
$R\chi^{\prime\prime}(\Omega,T)$ according to Eq. (1). Fig. 4 and Fig. 5
compile the spectra which entered this analysis.

For reasons of clarity, we plot the results for 5.2 % Co content above
$T_{\mathrm{nem}}$ in panel (a) and below $T_{\mathrm{nem}}$ in panel (b) of
Fig. 4. Here, the pile-up of spectral weight from nematic fluctuations [19]
and subsequent collapse for $T<T_{\mathrm{nem}}$ yield the characteristic cusp
[26] of Fig. 2 (a) in the main text.

Fig. 5 shows the remaining results for $x>0.052$. Here, too, the spectral
weight at small frequencies increases as temperature is decreased for all
doping concentrations. Yet, this increase is weaker for $x>x_{\mathrm{c}}$
than for $x<x_{\mathrm{c}}$ for which the presence of nematic fluctuations
yields a significant contribution to the response.

Figure 4: Inelastic light (Raman) scattering data for $x=0.052$ of
Ba(Fe1-xCox)2As2 in B1g symmetry. The temperature dependence is limited to
frequencies $\Omega<750\,\mathrm{cm}^{-1}$. (a) Spectra taken for temperatures
$333\,\mathrm{K}>T>T_{\mathrm{nem}}~{}\sim 50\,\mathrm{K}$. At small
frequencies, the spectral weight increases sharply towards the nematic
transition. (b) Upon entering the nematic phase, the spectral weight decreases
down to the lowest measured temperature at $26\,\mathrm{K}>T_{\mathrm{c}}$.
Figure 5: Inelastic light scattering (Raman) data in B1g symmetry for doping
concentrations as indicated. For all doping concentrations, the small
frequency contribution increases with decreasing temperature, but varies
qualitatively as the doping $x$ is tuned through and beyond $x_{\mathrm{c}}$
to the overdoped side of the phase diagram.

##  Appendix B Scaling

In the main text, we have identified quantum critical scaling in the vicinity
of optimal doping using the imaginary time correlation function
$\tilde{\Lambda}(\beta/2)$. More precisely, we have analyzed the temperature
dependence of $\beta\tilde{\Lambda}(\beta/2)$ to extract, for each doping
value, a crossover temperature, $T^{*}$, below which this function begins to
deviate from the constant behavior indicative of a marginal Fermi liquid.
Another, more traditional approach is to analyze the real frequency data
directly using energy and intensity scaling. In this section we describe this
method and its theoretical background, and apply it to the experimental data
presented in Fig. 4 and Fig. 5. The scaling method yields results consistent
with the imaginary time analysis, but involves a number of arbitrary
methodological choices that hamper the interpretation of the results.

###  B.1 Scaling of response functions

Let us consider a system with a quantum tuning parameter $x$ (such as doping)
that is near a quantum critical point at $x=x_{c}$, and let us assume that the
system obeys scaling. At each value of $x$, there is a characteristic energy
scale $T^{*}$, which will vanish as $x$ approaches $x_{c}$. Approaching from
the ordered side of the phase diagram, $T^{*}$ will be (proportional to) the
critical temperature, while on the disordered side it will represent a
crossover111In both cases, $T^{*}$ should vanish with critical exponent $\nu
z$, where $\nu$ is the correlation length critical exponent and $z$ is the
dynamical critical exponent, but this is not relevant to the current
discussion. From here on, we will assume that $T,\ T^{*}$, and the frequencies
($\Omega$) at which we probe the system are much less than “UV” scales such as
the bandwidth and Fermi energy, so that we are in the low energy regime where
quantum criticality can be observed.

Within this regime there are two asymptotic limits of temperature. When $T\ll
T^{*}$, the relevant quantum fluctuations (for instance phonons near a
structural transition, or paramagnons near a magnetic transition) have
energies much greater than the temperature and are thus frozen out. Therefore,
these quantum fluctuations produce only quantitative changes to the properties
of the system, in comparison to its behavior far from the QCP. As such, this
part of the phase diagram is sometimes called the _renormalized classical_
regime. By contrast, when $T\gg T^{*}$, the quantum fluctuations are
effectively gapless and therefore produce their most spectacular consequences,
such as divergences of thermodynamic quantities with cooling. This is the
_quantum critical_ regime, typically having a “fan” shape, as shown in Fig. 3
of the main text.

We will discuss these regimes separately below, but first describe the highly
constrained form of response functions that we obtain throughout the full
range of (small) $T$ and $T^{*}$ under the assumption of quantum critical
scaling. Let $\chi(\Omega)$ be a response function that diverges with
characteristic exponent $y$ at the QCP 222If $\chi$ is the order parameter
susceptibility, this exponent is $\gamma/\nu z$, where $\gamma$ is the
susceptibility critical exponent., and write $\chi=\chi_{0}+\chi_{s}$, where
$\chi_{0}$ is the analytic (smooth) part of the response and $\chi_{s}$ is the
singular part. If the quantum critical point obeys scaling, then $\chi_{s}$
has the form

$\displaystyle\chi_{s}(\Omega;x,T)=|\Omega|^{-y}\cdot{\cal
F}\left(\frac{\Omega}{T},\frac{\Omega}{T^{*}}\right),$ (3)

where we have suppressed the $x$ dependence of $T^{*}$ for compactness. In
this expression, ${\cal F}$ is a _scaling function_ , which is identical for
transitions within a given universality class. Except for the characteristic
divergence as $\Omega^{-y}$, the variable $\Omega$ and the parameters $T$ and
$T^{*}$ appear only as the dimensionless, scaled combinations shown in Eq. 3
333We could write a variety of equivalent expressions involving different
scaled variables (for instance, $T/T^{*}$ instead of $\Omega/T^{*}$). An
example would be $\chi_{s}=T^{-y}{\cal G}(\frac{\Omega}{T},\frac{T}{T^{*}})$,
with $\cal G$ a different universal scaling function. We have chosen the form
above for convenience.. The limits of $\cal{F}$ when its arguments go to zero
and infinity correspond to the various asymptotic regimes of the phase diagram
and of the dynamical response.

###  B.2 Asymptotic regimes

We can approach the QCP from the side, with $T=0$ and $T^{*}>0$, or from the
side, with $T>0$ and $T^{*}=0$. The former case corresponds to the so-called
renormalized classical regime, and the latter to the quantum critical regime.
In the renormalized classical regime, the first argument of $\cal F$ is
infinite, and the response obeys $\Omega/T^{*}$ scaling:

$\displaystyle\chi_{s}(\Omega;T=0)=$ $\displaystyle|\Omega|^{-y}\cdot
f_{h}\left(\frac{\Omega}{T^{*}}\right),\text{ where}$ (4) $\displaystyle
f_{h}(u)\equiv$ $\displaystyle\lim_{v\to\infty}{\cal F}(v,u)$ (5)

Where the subscript “h” is for “horizontal.” More precisely:

$\displaystyle\chi_{s}(\Omega;T,T^{*})\cdot|\Omega|^{y}\approx
f_{h}\left(\frac{\Omega}{T^{*}}\right),\quad\text{ for }\Omega,T^{*}\gg T$ (6)

So, for a given value of $T^{*}$, the spectra at $T\ll T^{*}$ depend only on
$\Omega$.

In the quantum critical regime, the second argument of $\cal F$ is infinite,
so the response obeys $\Omega/T$ scaling:

$\displaystyle\chi_{s}(\Omega;T^{*}=0)=$ $\displaystyle|\Omega|^{-y}\cdot
f_{v}\left(\frac{\Omega}{T}\right),\text{ where}$ (7) $\displaystyle
f_{v}(u)\equiv$ $\displaystyle\lim_{v\to\infty}{\cal F}(u,v),$ (8)

and the subscript “v” is for “vertical.” More precisely:

$\displaystyle\chi_{s}(\Omega;T,T^{*})\cdot|\Omega|^{y}\approx
f_{v}\left(\frac{\Omega}{T}\right),\quad\text{ for }\Omega,T\gg T^{*}$ (9)

So, for a given value of $T^{*}$, spectra for $T\gg T^{*}$ depend only on the
scale-invariant combination $\Omega/T$.

###  B.3 Data collapse

The expressions above suggest a method to estimate the scaling exponent $y$ as
well as the crossover temperature $T^{*}$. By Eq. 9 above,
$\chi_{s}\cdot|\Omega|^{y}$ is a function only of $\Omega/T$ for the high
temperature data. Therefore, to the extent that the total response
$\chi=\chi_{0}+\chi_{s}$ is dominated by the singular part $\chi_{s}$, the
spectra for $T\gg T^{*}$ and $T\ll T^{*}$ can be distinguished as follows.
First, plot $\chi\cdot\Omega^{y}$ versus $\Omega/T$ for several values of $y$,
and choose the value of $y$ that yields the best collapse of the high
temperature spectra. Specializing to this optimal value of $y$, the spectrum
measured at the crossover temperature $T^{*}$ will form a separatrix between
the collapsing and non-collapsing spectra. This procedure is shown in Fig. 6
for an idealized data set in which $\chi^{\prime\prime}$ grows as $\Omega/T$
for $T>T^{*}$ and as $\alpha(T)\cdot\Omega/T$ for $T<T^{*}$.

Unfortunately, choosing the best value of $y$ and identifying this separatrix
for experimental data must involve some arbitrarily chosen qualitative and/or
quantitative criteria, which makes the data collapse procedure somewhat ad
hoc. This is in addition to the challenges posed by the regular part
$\chi_{0}$.

Figure 6: Data collapse for synthetic data with a temperature evolution as
indicated. The black curves correspond to $T^{*}$. (a) Down to $T^{*}$, the
response $\chi^{\prime\prime}(\Omega,T)$ grows with $\Omega/T$. Not
immediately apparent is the deviation from this scaling for $T<T^{*}$. (b)-(d)
For this idealized data set, data collapse of
$\chi^{\prime\prime}\cdot\Omega^{y}$ plotted against $\Omega/T$ is observed
for the high temperature data at $T>T^{*}$ and $y=0$.

We illustrate the data collapse method using experimental data in Fig. 7. Upon
attempting the collapse for various doping concentrations and various values
of $y$, we find that only small (or $0$) values of $y$ show reasonable
collapse of the high temperature spectra. This is consistent with the nearly
constant behavior of $\beta\tilde{\Lambda}(\beta/s)$ at high temperatures. For
each doping $x$, we highlight in black the spectrum at the value of $T^{*}(x)$
determined from the imaginary time analysis of the main text. With the
exception of $x=7.3\%$, where $T^{*}$ is below the base temperature of the
experiment, the black curves do indeed roughly separate the collapsing and
non-collapsing spectra. Therefore, the two methods are consistent, but the
imaginary time method provides a much more straightforward way of identifying
$T^{*}$.

Figure 7: Data collapse method illustrated in the text for varying values of
the exponent $y$.

###  B.4 Asymptotics of $\beta\tilde{\Lambda}(\beta/2)$

Here we prove two claims mentioned in the text: The first is that
$\beta\tilde{\Lambda}(\beta/2)$has the same singular temperature dependence as
the static susceptibility upon on cooling towards a quantum critical point
exhibiting $\Omega/T$ scaling. To first look at the singularity of
$\chi^{\prime}$ as $T\to 0$ with $T^{*}=0$, write its singular part as in Eq.
8:

$\displaystyle\chi_{s}(\Omega)=$ $\displaystyle|\Omega|^{-y}\cdot
f_{v}\left(\frac{\Omega}{T}\right),$ (10)

At nonzero temperature, the static susceptibility must be finite, so the
divergence of the factor $|\Omega|^{-y}$ at low frequencies must be
compensated by the asymptotic behavior of $f_{v}$. In particular, cancelling
the divergence requires $f_{v}(u)\sim u^{y}$ as $u\to 0$, so that the static
susceptibility diverges with temperature as

$\displaystyle\chi_{s}(\Omega=0)\sim$
$\displaystyle|\Omega|^{-y}\cdot\left(\frac{\Omega}{T}\right)^{y}$ (11)
$\displaystyle=T^{-y}$ (12)

Now let us substitute Eq. 8 into the expression for
$\beta\tilde{\Lambda}(\beta/2)$:

$\displaystyle\beta\tilde{\Lambda_{s}}(\beta/2)=$
$\displaystyle\int\frac{\mathrm{d}\Omega}{2\pi
T}\frac{\chi^{\prime\prime}(\Omega)}{\sinh[\beta\Omega/2]}$ (13)
$\displaystyle=$ $\displaystyle\int\frac{\mathrm{d}\Omega}{2\pi
T}\frac{|\Omega|^{-y}\cdot Im[f_{v}(\Omega/T)]}{\sinh[\beta\Omega/2]},$ (14)

where $\tilde{\Lambda}_{s}$ is the singular part of $\tilde{\Lambda}$. Now we
rewrite the integral in terms of the scaled variable $x=\Omega/T$, so that
$\Omega=xT$:

$\displaystyle\beta\tilde{\Lambda_{s}}(\beta/2)=$
$\displaystyle\int\frac{T\mathrm{d}x}{2\pi T}\frac{|Tx|^{-y}\cdot
Im[f_{v}(x)]}{\sinh[x/2]}$ $\displaystyle=$ $\displaystyle
T^{-y}\cdot\int\frac{\mathrm{d}x}{2\pi}\frac{|x|^{-y}\cdot
f_{v}(x)}{\sinh[x/2]},$

The integral is now independent of T and is some order 1 number dependent on
the scaling function $f_{v}$. Therefore, as promised,
$\chi^{\prime}(\Omega=0)$ and $\beta\tilde{\Lambda}(\beta/2)$have the same
$T^{-y}$ divergence upon cooling towards the QCP.

The second statement in the text is that
$\beta\tilde{\Lambda}(\beta/2)$generically vanishes as $T\to 0$. By
generically we mean that the response function is analytic in frequency, i.e.
non-critical. In this case, we can expand $\chi^{\prime\prime}$ as a
convergent Taylor series in $\Omega$:

$\chi^{\prime\prime}(\Omega)=\sum_{n=0}^{\infty}a_{n}\Omega^{2n+1},$ (15)

where $a_{n}$ are real coefficients, and only odd powers of $\Omega$ are
included because $\chi^{\prime\prime}$ is odd in frequency. Substitute this
into the expression for $\beta\tilde{\Lambda}(\beta/2)$:

$\displaystyle\beta\tilde{\Lambda_{s}}(\beta/2)=$
$\displaystyle\int\frac{\mathrm{d}\Omega}{2\pi
T\sinh[\beta\Omega/2]}\sum_{n}a_{n}\Omega^{2n+1}$

Now once again substitue $\Omega=xT$

$\displaystyle\beta\tilde{\Lambda_{s}}(\beta/2)=$
$\displaystyle\int\frac{T\mathrm{d}x}{2\pi
T\sinh[x/2]}\sum_{n}a_{n}T^{2n+1}x^{2n+1}$ $\displaystyle=$
$\displaystyle\sum_{n}\alpha_{n}T^{2n+1},$

where

$\displaystyle\alpha_{n}=$ $\displaystyle
a_{n}\int\frac{\mathrm{d}x}{2\pi\sinh[x/2]}x^{2n+1}$

And each of the $\alpha_{n}$ are convergent because of the exponential cutoff.
Accordingly, $\beta\tilde{\Lambda}(\beta/2)$can be written as a series in odd
powers of $T$, and therefore vanishes as $T\to 0$. The only way this can fail
to occur is if the response diverges sufficiently rapidly at low frequencies.

##  Appendix C Sample characterization

The samples were cleaved ex-situ prior to mounting the sample holder into the
cryostat. Characterization with a superconducting quantum interference device
(SQUID) was performed after completion of the Raman measurements. The results
are presented in Fig. 8 (a). Assuming an S-shaped function for the
magnetization curves of the form

$M[T]\sim\frac{1}{1+\mathrm{exp}[-(T-T_{\mathrm{c}})/\Gamma^{(1)}]}\cdot\frac{1}{1+\mathrm{exp}[-(T-T_{\mathrm{c}})/\Gamma^{(2)}]}$
(16)

we derive the critical temperatures $T_{\mathrm{c}}$ of the samples from the
peak positions of $\partial\mathrm{M}(T)/\partial T$ from a standard
asymmetric double sigmoidal function as a fit model (black curves in Fig. 8
(b)). The results are compiled in Table 1.

Figure 8: (a) Normalized magnetization as a function of temperature for doping concentrations as indicated. The increase from the high temperature side signifies the onset of the superconducting transition which saturates in the superconducting state. (b) First derivative of the magnetization $\partial\mathrm{M}(T)/\partial T$ normalized to the peak maxima alongside the fits in black described in the text.  Table 1: Sample characterization. Doping concentrations $x$ and their variation within each sample $\Delta x$ derived from the width $\pm\Delta T_{\mathrm{c}}$ of $\partial\mathrm{M}(T)/\partial T$. $T_{\mathrm{c}}$ corresponds to the peak position. Doping $x$ (%) | Variance $\Delta x$ (%) |  $T_{\mathrm{c}}$ (K) |  $\pm\Delta T_{\mathrm{c}}$ (K)
---|---|---|---
5.2 | 0.15 | 20.7 | 0.2
5.7 | 0.05 | 23.9 | 0.2
5.9 | 0.05 | 24.2 | 0.05
7.3 | 0.13 | 23.0 | 0.13
9.6 | 0.05 | 18.4 | 0.10
14.6 | 0.16 | 6.4 | 0.25

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  * Note [1] Note1.  In both cases, $T^{*}$ should vanish with critical exponent $\nu z$, where $\nu$ is the correlation length critical exponent and $z$ is the dynamical critical exponent, but this is not relevant to the current discussion.
  * Note [2] Note2.  If $\chi$ is the order parameter susceptibility, this exponent is $\gamma/\nu z$, where $\gamma$ is the susceptibility critical exponent.
  * Note [3] Note3.  We could write a variety of equivalent expressions involving different scaled variables (for instance, $T/T^{*}$ instead of $\Omega/T^{*}$). An example would be $\chi_{s}=T^{-y}{\cal G}(\frac{\Omega}{T},\frac{T}{T^{*}})$, with $\cal G$ a different universal scaling function. We have chosen the form above for convenience.