# Competition between spin ordering and superconductivity near the pseudogap
boundary in
${\mathbf{La}}_{\mathbf{2-x}}{\mathbf{Sr}}_{\mathbf{x}}{\mathbf{CuO}}_{\bf{4}}$:
insights from NMR

I. Vinograd<EMAIL_ADDRESS>CNRS, LNCMI, Univ. Grenoble Alpes, INSA-T, UPS,
EMFL, Grenoble, France  R. Zhou  CNRS, LNCMI, Univ. Grenoble Alpes, INSA-T,
UPS, EMFL, Grenoble, France  H. Mayaffre  CNRS, LNCMI, Univ. Grenoble Alpes,
INSA-T, UPS, EMFL, Grenoble, France  S. Krämer  CNRS, LNCMI, Univ. Grenoble
Alpes, INSA-T, UPS, EMFL, Grenoble, France  S. K. Ramakrishna  National High
Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310,
USA  A. P. Reyes  National High Magnetic Field Laboratory, Florida State
University, Tallahassee, FL 32310, USA  T. Kurosawa  Department of Physics,
Hokkaido University, Sapporo 060-0810, Japan  N. Momono  Muroran Institute of
Technology, Muroran 050-8585, Japan  M. Oda  Department of Physics, Hokkaido
University, Sapporo 060-0810, Japan  S. Komiya  Central Research Institute of
Electric Power Industry, Yokosuka, 240-0196, Japan  S. Ono  Central Research
Institute of Electric Power Industry, Yokosuka, 240-0196, Japan  M. Horio
Department of Physics, University of Zürich, CH-8057 Zurich, Switzerland  J.
Chang  Department of Physics, University of Zürich, CH-8057 Zurich,
Switzerland  M.-H. Julien<EMAIL_ADDRESS>CNRS, LNCMI, Univ.
Grenoble Alpes, INSA-T, UPS, EMFL, Grenoble, France

###### Abstract

When superconductivity is suppressed by high magnetic fields in
${\mathrm{La}}_{\mathrm{2-x}}{\mathrm{Sr}}_{\mathrm{x}}{\mathrm{CuO}}_{4}$,
striped antiferromagnetic (AFM) order becomes the magnetic ground state of the
entire pseudogap regime, up to its end at the doping $p^{*}$ [M. Frachet, I.
Vinograd et al., Nat. Phys. 16, 1064 (2020)]. Glass-like freezing of this
state is detected in 139La NMR measurements of the spin-lattice relaxation
rate $T^{-1}_{1}$. Here, we present a quantitative analysis of $T^{-1}_{1}$
data in the hole-doping range $p=x=0.12-0.171$, based on the Bloembergen-
Purcell-Pound (BPP) theory, modified to include statistical distribution of
parameters arising from strong spatial inhomogeneity. We observe spin
fluctuations to slow down at temperatures $T$ near the onset of static charge
order and, overall, the effect of the field $B$ may be seen as equivalent to
strengthening stripe order by approaching $p=0.12$ doping. In details however,
our analysis reveals significant departure from usual field-induced magnetic
transitions. The continuous growth of the amplitude of the fluctuating moment
with increasing $B$ suggests a nearly-critical state in the $B\rightarrow 0$
limit, with very weak quasi-static moments possibly confined in small areas
like vortex cores. Further, the nucleation of spin order in the vortex cores
is shown to account quantitatively for both the value and the $p$ dependence
of a field scale characterizing bulk spin freezing. The correlation time of
the fluctuating moment appears to depend exponentially on $B/T$ (over the
investigated range). This explains the timescale dependence of various
experimental manifestations, including why, for transport measurements, the
AFM moments may be considered static over a considerable range of $B$ and $T$.
These results make the high-field magnetic ground state up to $p^{*}$ an
integral part of the discussion on putative quantum criticality.

##  I Introduction

There are various reasons for which high $T_{c}$ superconductivity in the
cuprates is a hard problem. It is now clear that one of these reasons is that
the superconducting state impedes the ordering of spin or charge degrees of
freedom. Because of these competing effects, ordered phases may thus remain
partially or entirely ”hidden”, which hampers full understanding of the
cuprate electronic properties. Nevertheless, tremendous progress has been
accomplished in the last two decades as experiments using magnetic fields to
quench superconductivity have played a pivotal role in exposing the spin
and/or charge orders that compete with superconductivity Lake et al. (2002);
Hoffman et al. (2002); Mitrović et al. (2003); Wu et al. (2011).

Figure 1:  Schematic phase diagram of LSCO showing that the pseudogap and
antiferromagnetic (AFM) glass phases have separate doping endpoints in zero
field ($B=0$, left) but coinciding endpoints in high fields (right), according
to the NMR and ultrasound results in ref. Frachet et al. (2020).

A recent study combining nuclear magnetic resonance (NMR) and sound velocity
measurements in
${\mathrm{La}}_{\mathrm{2-x}}{\mathrm{Sr}}_{\mathrm{x}}{\mathrm{CuO}}_{4}$
(LSCO) (Frachet et al., 2020) has provided the latest illustration of how high
fields can uncover a hidden piece of the cuprate puzzle. This work revealed a
connection between magnetism and the pseudogap phase that had been hitherto
hidden by superconductivity: when superconductivity is removed by high fields,
the striped antiferromagnetic (AFM) glass (sometimes referred to as spin-glass
or spin-stripe phase) persists well above its end-doping in zero field $p_{\rm
sg}\simeq 0.135$ (Fig. 1a), actually up to the endpoint of the pseudogap
phase, $p^{*}\simeq 0.19$ (Fig. 1b). Note that in
${\mathrm{La}}_{\mathrm{2-x}}{\mathrm{Sr}}_{\mathrm{x}}{\mathrm{CuO}}_{4}$ the
hole doping $p$ is considered to be equal to the Sr concentration $x$.

At the qualitative level, that the same frozen state of AFM moments Julien
(2003) extends from the doped Mott insulating state at $p\simeq 0.02$ up to
$p^{*}$ underlines the relevance of Mott physics throughout the pseudogap
state, even suggesting a possible connection between local-moment magnetism of
the doped Mott insulator and the pseudogap state. This work, however, raises a
number of important questions. How is the magnetic quantum phase transition
connected with the sharp changes in the electronic properties observed across
$p^{*}$ in high-field measurements Collignon et al. (2017); Michon et al.
(2019); Fang et al. (2022)? How do we describe the zero-field ground state? Is
it sharply distinct from the high-field (AFM glass) ground state? Could the
slow spin fluctuations have any impact on the transport properties?

The purpose of this paper is to perform a quantitative analysis of the NMR
results of ref. Frachet et al. (2020) in order to gain insight into these
questions. A central issue in the analysis will be the presence of strong
spatial inhomogeneity of various origins, as discussed below.

The paper is organized as follows: section II describes the fitting model,
section III discusses the results and their interpretation. The readers who
are not interested in the details may go directly to section IV, an extended
summary of the main points discussed in the previous section. Perspectives are
mentioned in section V.

##  II NMR background and model

###  II.1 Why $T_{1}$ measurements?

The most direct information on magnetic order is the magnitude of the ordered
moment, which is in principle extracted from the broadening or splitting of
the NMR lines. In La-based cuprates (La214 in short), internal fields $\langle
h_{\perp}\rangle$ of $\sim 10-40$ mT produced by ordered moments up to 0.3
$\mu_{B}$ within the CuO2 planes have been successfully detected from
measurements in zero external field Kitaoka et al. (1987); Ohsugi (1996) or
with the field applied parallel to the planes Arsenault et al. (2018).

In the specific case of field-induced order, however, measurements of NMR
spectra cannot provide information on the ordered moment. Indeed, because the
ordered moments lie within the planes, such measurement require in-plane
fields ($B\parallel ab$) while the existence of the field-induced order itself
requires out-of-plane fields ($B\parallel c$) Wu et al. (2013a); Frachet et
al. (2020). As a matter of fact, the 139La linewidth broadens only slightly
upon cooling for $B\parallel c$ and saturates when spins freeze (see Fig. 12
for an example). Therefore, magnetic ordering will be detected here through
the low-energy spin fluctuations, which are probed by measurements of the
spin-lattice relaxation rate $T_{1}^{-1}$ of 139La nuclei.

###  II.2 The standard BPP model

Our NMR experiments probe properties of the hyperfine field
$\boldsymbol{h}\propto\bar{A}\,\boldsymbol{S}$ produced at the 139La nuclear
positions by the electronic spins $\boldsymbol{S}$ in CuO2 planes ($\bar{A}$
is the hyperfine coupling tensor). In their original model Bloembergen et al.
(1948), Bloembergen, Purcell and Pound assume an auto-correlation function of
the fluctuating hyperfine field $\boldsymbol{h}(t)$ that decays exponentially
with time ($t$):

$\langle\boldsymbol{h}(t)\boldsymbol{h}(0)\rangle=\langle h^{2}\rangle
e^{-t/\tau_{c}},$ (1)

where $\tau_{c}$ is called the correlation time.

Fourier transformation of this expression and evaluation of the resulting
spectral density of fluctuations at the NMR frequency $\omega_{L}$ leads to
the following expression of the relaxation rate $T^{-1}_{1}$:

$T^{-1}_{1,\,\rm BPP}=\langle
h_{\bot}^{2}\rangle\,\gamma_{n}^{2}\frac{2\tau_{c}}{1+(\omega_{L}\tau_{c})^{2}}.$
(2)

with the nuclear gyromagnetic ratio $\gamma_{n}$ and the angular Larmor
frequency $\omega_{L}=2\pi f$, $f$ being the actual resonance frequency.

Here, $\langle h_{\bot}^{2}\rangle=\langle h_{xx}^{2}\rangle+\langle
h_{yy}^{2}\rangle$ is the time-averaged, squared hyperfine field (also called
”fluctuating field”) transverse to the direction ($z$) of the applied field
($B$). For a diagonal hyperfine tensor $\bar{A}$, $T_{1}^{-1}$ is thus related
to the correlation function $\langle S_{+}(t)S_{-}(0)\rangle$.

Eq. 2 has a maximum when $\tau_{c}=\omega_{L}^{-1}$, that is, when electronic
fluctuations are as slow as the NMR frequency. Therefore, if electronic
fluctuations slow down so much upon cooling that they eventually become slower
than the NMR frequency, there must be a temperature $T_{\rm peak}$ at which
$\omega_{L}=\frac{1}{\tau_{c}}$. Then, $T_{1}^{-1}$ reaches a maximum value:

$(T_{1}^{-1})_{\rm max}=\frac{\langle
h_{\bot}^{2}\rangle\,\gamma_{n}^{2}}{\omega_{L}}\quad.$ (3)

Since $\omega_{L}\propto B$ in NMR and since $B$ does not affect the physical
properties in general, $(T_{1}^{-1})_{\rm max}$ usually varies as $B^{-1}$. As
we shall see, this is no longer true in superconducting cuprates as $\langle
h_{\bot}^{2}\rangle$ strongly increases with $B$.

###  II.3 Basic assumptions

We follow previous studies of spin-freezing in cuprates Suh et al. (2000);
Curro et al. (2000); Simovič et al. (2003); Mitrović et al. (2008); Wu et al.
(2013a) that have shown that the peak in $T_{1}^{-1}$ could be reproduced
using the BPP formula (Eq. 2) in which all of the temperature ($T$) dependence
arises from a diverging correlation time $\tau_{c}$ upon cooling:

$\tau_{c}(T)=\tau_{\infty}\,e^{E_{0}/k_{B}T}\,.$ (4)

The activation energy $E_{0}$ for spin fluctuations is assumed to be $T$ (but
not $B$) independent over the full fitting range (and so is $\langle
h_{\bot}^{2}\rangle$ in Eq. 2). Its physical interpretation in terms of spin
stiffness will be discussed in section IV. The exponential $\tau_{c}$ has no
onset temperature: Eq. 4 describes fluctuations that continuously slow down
upon cooling, eventually becoming frozen (i.e. static) at $T=0$ without any
phase transition intervening at finite $T$. In section III.2, we also briefly
discuss fitting to a Vogel-Fulcher-Tammann dependence:

$\tau_{c}(T)=\tau_{\infty}\,e^{E_{0}/k_{B}(T-T_{\rm VF})}.$ (5)

Experimentally, however, the spin system appears to be frozen as soon as
fluctuations become slower than the time scale of the technique. The larger
the time scale, the higher the temperature of the apparent freezing. The BPP
peak temperature $T_{\rm peak}$, namely the freezing temperature at the NMR
timescale, is readily obtained by inserting Eq. 4 into the condition
$\omega_{L}=\tau_{c}^{-1}$ and $\omega_{L}=\gamma_{n}\,B$ to first
approximation (neglecting the Knight shift and the quadrupole shift that are
small corrections to $\omega_{L}$):

$T_{\rm
peak}=\frac{-E_{0}}{\ln(\omega_{L}\tau_{\infty})}\,\simeq\frac{-E_{0}}{\ln(\gamma_{n}\,\tau_{\infty}\,B)}\,.$
(6)

Notice that we shall evaluate $E_{0}$ in Kelvin and $\omega=\tau_{c}^{-1}$ in
meV, thus implicitly setting $k_{B}=1$ and $\hbar=1$.

###  II.4 The spatial-inhomogeneity problem

Quantitative analysis of the results is a daunting task as multiple levels of
spatial inhomogeneity make the problem very intricate and they strongly affect
the results for a local probe such as NMR. Moreover, we are not aware of a
model that takes into account the full complexity of this problem. In the
superconducting mixed state, electronic properties are intrinsically
inhomogeneous at relatively short length scales as the competing magnetic
order is thought to be enhanced in and around the vortex cores Arovas et al.
(1997); Demler et al. (2001); Sachdev and Zhang (2002); Hu and Zhang (2002);
Kivelson et al. (2002); Franz et al. (2002); Zhu et al. (2002); Ghosal et al.
(2002); Andersen et al. (2003). In addition to this, there is evidence that
spatial heterogeneity characterizes the spin-freezing process, regardless of
the presence or absence of superconductivity Julien et al. (1999); Suh et al.
(2000); Curro et al. (2000); Teitel’baum et al. (2000); Julien et al. (2001);
Hunt et al. (2001); Simovič et al. (2003); Mitrović et al. (2008); Baek et al.
(2015, 2017); Arsenault et al. (2020); Singer et al. (2020); Wu et al.
(2013a). A third source of inhomogeneity lies in the possible phase separation
between magnetic and non-magnetic regions, as suggested by neutron scattering
studies Wakimoto et al. (2007). In principle, there is also significant
spatial inhomogeneity of the hole doping in LSCO Singer et al. (2002).
Nevertheless, this appears to have little influence on measurements of the
spin-lattice relaxation rate $T_{1}^{-1}$ of 139La Mitrović et al. (2008);
Arsenault et al. (2020).

Direct information on inhomogeneity is encoded in the probability density of
relaxation rates $T_{1}^{-1}$. This quantity can in principle be determined
without any assumption on the distribution by performing an inverse Laplace
transform of the NMR relaxation curve (the time evolution of the nuclear
magnetization returning back to its equilibrium value). This technique has
been put forward recently in the cuprate context Singer et al. (2020);
Arsenault et al. (2020). However, it requires levels of signal-to-noise ratio
that are difficult to achieve in time-constrained experiments at high-field
facilities and its application is also not straightforward in general Choi et
al. (2021). Furthermore, while inverse Laplace transform gives the full
probability density of relaxation rates, its result still needs to be
subsequently interpreted within a model in order to relate $T_{1}$ to physical
parameters and to characterize the evolution of these parameters with $B,p$ or
$T$. This has not been attempted yet.

Here we use a simpler approach first developed by Curro et al. Curro et al.
(2000); Suh et al. (2000); Curro (2004): the relaxation curves are fitted by
stretched exponentials (a standard way to determine a ”typical” relaxation
rate Mitrović et al. (2008); Johnston (2006)) and the temperature dependence
of the thus-determined $T^{-1}_{1}$ values is then fitted by a modified BPP
model in which the characteristic energy $E_{0}$ is distributed, e.g. by a
Gaussian. We generalize this approach by introducing ad-hoc distributions of
all relevant fitting parameters to account for the inhomogeneous relaxation
mechanism in LSCO.

###  II.5 Statistical distribution of parameters

NMR studies of spin-freezing in superconducting cuprates (Cho et al., 1992;
Julien et al., 1999, 2000; Suh et al., 2000; Curro et al., 2000; Teitel’baum
et al., 2000; Hunt et al., 2001; Julien et al., 2001; Mitrović et al., 2008;
Wu et al., 2013a; Baek et al., 2015, 2017; Arsenault et al., 2018; Frachet et
al., 2020) generally find that the relaxation curves are stretched which
indicates that there is a distribution of relaxation rates (in the absence of
any significant quadrupolar relaxation mechanism, which is true at low $T$).
In this situation, a standard procedure is to fit the relaxation curves with a
stretched exponential form, which yields a ”typical” relaxation rate
$T_{1}^{-1}$, equivalent to the median of the distribution, and a stretching
exponent $\beta$ that quantifies the breadth of the distribution (Mitrović et
al., 2008; Baek et al., 2017; Arsenault et al., 2020). Indeed, as shown by
Johnston Johnston (2006), for $0.5\leq\beta\leq 1$, $1-\beta$ is proportional
to the logarithmic full width at half maximum (FWHM) of the distribution
($\beta=0.5$ corresponding to a distribution over an order of magnitude and
$\beta=1$ to a Dirac function) and the value of $T_{1}^{-1}$ is within about
10% of the median of the distribution (i.e. the value that splits the
cumulative probability into equal halves, see Fig. 15 for an example). For
$\beta\simeq 0.3$, $T_{1}^{-1}$ is distributed over two orders of magnitude
and the value of $T_{1}^{-1}$ becomes about twice the median.

In order to fit the $T$ dependence of $T_{1}$, we convolute the BPP formula
(Eqs. 2 and 4), with positive-definite Gaussian distributions of $E_{0}$ and
$\tau_{\infty}$ (for convenience we define $a=\ln\tau_{\infty}$ and distribute
$a$) as follows:

Figure 2:  Relaxation data and fits for LSCO $p=0.148$. a) Raw $T^{\,-1}_{1}$
from stretched fits vs. $T$ at different fields. For $B=10$ T,
$T^{\,-1}_{1}(T)$ could not be measured at intermediate temperature ($T\sim 7$
K), because the signal becomes very weak due to strong superconductivity which
impedes the penetration of radio frequency pulses. However, some signal can be
recovered at lower $T$ since the nuclear magnetization is proportional to
$1/T$. Lines are calculated background relaxation rates based on linear-in-$T$
$T_{1}^{-1}$ (constant $(T_{1}T)^{-1}$) above $T_{\rm{c}}$ and exponentially
gapped relaxation below $T_{\rm{c}}(B)$ (see Appendix and ref. Frachet et al.
(2020)). b) $T^{-1}_{1,\,\rm subtr.}$ after background subtraction fitted by
$T^{-1}_{1,\,\rm BPP\,dist.}(T)$ (continuous lines). The subtraction affects
only the high $T$ range. For details about the fits, see text and Fig. 3.

$\begin{split}T^{-1}_{1,\,\rm
BPP\,dist.}(T)&=\frac{1}{\mathcal{N}}\int_{0}^{\infty}\int_{-\infty}^{\infty}T^{-1}_{1,\rm
BPP}(E_{0},a,T)\cdot
e^{-\frac{(E_{0}-E_{0,c})^{2}}{2\mathit{\Delta}\\!E_{0}^{2}}}e^{-\frac{(a-a_{c})^{2}}{2\mathit{\Delta}a^{2}}}{\rm
d}a\,{\rm d}E_{0}\\\
&=\frac{1}{\mathcal{N}}\int_{0}^{\infty}\int_{-\infty}^{\infty}\langle
h_{\bot}^{2}\rangle^{2}\gamma_{n}^{2}\frac{2\tau_{c}(E_{0},a)}{1+(\omega_{L}\tau_{c}(E_{0},a))^{2}}\cdot
e^{-\frac{(E_{0}-E_{0,c})^{2}}{2\mathit{\Delta}\\!E_{0}^{2}}}e^{-\frac{(a-a_{c})^{2}}{2\mathit{\Delta}a^{2}}}{\rm
d}a\,{\rm d}E_{0}\\\
&=\frac{1}{\mathcal{N}}\int_{0}^{\infty}\int_{-\infty}^{\infty}\langle
h_{\bot}^{2}\rangle^{2}\gamma_{n}^{2}\frac{2e^{a+\frac{E_{0}}{T}}}{1+(\omega_{L}e^{a+\frac{E_{0}}{T}})^{2}}\cdot
e^{-\frac{(E_{0}-E_{0,c})^{2}}{2\mathit{\Delta}\\!E_{0}^{2}}}e^{-\frac{(a-a_{c})^{2}}{2\mathit{\Delta}a^{2}}}{\rm
d}a\,{\rm d}E_{0}\,.\end{split}$ (7)

Here $\mathcal{N}$ is the normalisation

$\mathcal{N}=\int_{0}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{(E_{0}-E_{0,c})^{2}}{2\mathit{\Delta}\\!E_{0}^{2}}}e^{-\frac{(a-a_{c})^{2}}{2\mathit{\Delta}a^{2}}}{\rm
d}a\,{\rm d}E_{0}\,.$

Notice that, in the following, on $E_{0,c}$ and $a_{c}$ we drop the index $c$
(that indicates the center of the distribution) in order to simplify the
notations, so the fit parameters $E_{0}$ and $a$ will actually refer to the
center of the distribution of $E_{0}$ and $a$, respectively.

Introducing the distribution of $\tau_{\infty}$ improves the fit quality
significantly at $T$ above the peak (see Appendix, Fig. 14). What $\Delta a$
essentially does, and which $\Delta E_{0}$ does not, is to create a constant
$T_{1}^{-1}$ where $E_{0}$ is small. The value $a=-31.5$ (with $\tau_{\infty}$
measured in seconds), corresponding to $\tau_{\infty}=0.02$ ps, was found to
fit data well for different La-compounds (Curro et al., 2000; Suh et al.,
2000). We thus keep this value fixed. Integrating $a$ over the full numerical
range is computationally demanding, but limiting the integration range to
$a-3\Delta a\leq a\leq a+3\Delta a$ leads to equivalent results when $\Delta
a>1$ . When $\Delta a\leq 1$, we keep a fixed $\Delta a$-independent
integration range of $\pm 3$.

We point out that distributing $\langle h_{\bot}^{2}\rangle$ would have no
effect on $T^{-1}_{1,\,\rm BPP\,dist.}(T)$ because the integral over $\langle
h_{\bot}^{2}\rangle$ can be factorized out and simply gives $\langle
h_{\bot}^{2}\rangle=h_{\bot,\rm{mean}}^{2}$, as long as this distribution is
uncorrelated with those of $E_{0}$ and $\tau_{\infty}$. Nevertheless, this
does not imply that the distribution of $T^{-1}_{1}$ would be unaffected if
$\langle h_{\bot}^{2}\rangle$ were to be distributed. In fact, the width of
the $T^{-1}_{1}$ distribution, of which the stretching exponent $\beta$ is a
measure, would very well increase (see Appendix).

Figure 3:  Field and temperature dependence of fittting parameters for LSCO
$p=0.148$. (a) Root mean squared fluctuating hyperfine field $\langle
h_{\bot}^{2}\rangle^{1/2}$, (b) activation energy $E_{0}$, (c) relative width
of the distribution of $E_{0}$ and (d) width of the distribution of
$a=\ln\tau_{\infty}$, all as a function of magnetic field. For $B\geq 13.5$ T,
all fits parameters are free and unconstrained (solid symbols). For $B\leq 12$
T, $T_{1}^{-1}(T)$ is flat (Fig. 2), so the peak temperature is ill-defined
and the constraint $\Delta\\!E_{0}/E_{0}=0.42$ has been used (open symbols),
as determined from a fit of higher-field data to a constant (dashed line in
panel c). The crosses in (a) where determined from a single value of $T_{1}$
at base temperature (see text). The solid line in (b) is a linear fit through
the data while the dashes represent another possible dependence, also
consistent with the data. The distribution of $\tau_{\infty}$ increases by an
order of magnitude as $\Delta a$ increases by a factor of $2.3\approx\ln 10$.
At 10 T, $\tau_{\infty}$ is spread over 5 orders of magnitude. At the
quantitative level, $E_{0}$ appears to depend approximately linearly on $B$
while $\langle h_{\bot}^{2}\rangle^{1/2}$ varies more strongly ($\propto
B^{5/2}$).

###  II.6 Analysis in the absence of a peak in $T_{1}^{-1}$

At low fields ($B\lesssim 10$ T), it is impossible to determine the BPP
parameters from regular fits as no $T_{1}^{-1}$ peak is discernible in the
data. As explained in Appendix (Fig. 14b), a single $T_{1}^{-1}$ value at low
$T$ is sufficient to determine $\langle h_{\bot}^{2}\rangle$ because
$T_{1}^{-1}$ becomes effectively $T$ independent as $\Delta a$ values become
large. In doing so, we of course implicitly assume that freezing described by
the BPP model still applies in this field range, which is justified by the
fact that the $T_{1}^{-1}$ values are still considerably larger than the
estimated background (Fig. 2a).

###  II.7 Assumptions, limitations, caveats

$\bullet$ Following Curro and coworkers Curro et al. (2000); Suh et al.
(2000); Curro (2004), the present analysis posits that the width of the peak
in $T_{1}^{-1}(T)$ arises from a distribution of $E_{0}$ values, or
equivalently from a distribution of $T_{\rm peak}$ values (Eq. 6), and that
$\tau_{c}$ varies exponentially with $T$.

$\bullet$ Our approach assumes unimodal distributions of the parameters. This
could be an oversimplification if the distribution is bimodal or multimodal,
for example if the sample separates into regions that either undergo spin
freezing or do not show it at all. Accurate analysis of $T_{1}^{-1}$ data in
La1.885Sr0.115CuO4 and La1.875Ba0.125CuO4 has actually uncovered a bimodal
distribution of the relaxation rate. However, both modes show qualitatively
similar behavior, namely slowing down of spin fluctuations Arsenault et al.
(2020); Singer et al. (2020), so unimodal distribution is still a reasonable
approximation in that case. In fact, while our approach determines properties
of the distribution indirectly and is in principle less powerful than the
direct use of the Inverse Laplace Transform (ILT), its value lies in the
capability to obtain a consistent parametrization of 139La datasets in LSCO
Frachet et al. (2020); Mitrović et al. (2008); Frachet et al. (2021) and to
characterize how the parameters evolve as a function of field and doping.
Furthermore, we show in Appendix an example of a $T_{1}^{-1}$ distribution
that is not unimodal even though the distributions of BPP parameters are
unimodal.

$\bullet$ We are fitting $T^{-1}_{1}$ data from stretched fits, that
correspond to the median of the $T^{-1}_{1}$ distribution, by an expression
$T^{-1}_{1,\,\rm BPP\,dist.}(T)$ (Eq. 7) that gives the mean $T^{-1}_{1}$.
While this may in principle be incorrect because the mean and the median may
be very different, we explain in Appendix that our model is justified by the
fact that $T^{-1}_{1,\,\rm mean}\approx T^{-1}_{1,\,\rm median}$ for a
realistic distribution of relaxation rates that cannot have a tail extending
to infinity.

Figure 4:  Doping evolution of magnetism in high fields. a) $T$ dependence of
the relaxation rate $T_{1}^{-1}$ divided by $T$, at different hole doping
levels. The data for $p=0.21$ has been lowered by 0.002 s-1K-1 corresponding
to the approximated contribution from electric field gradient fluctuations
Frachet et al. (2020). Field values for each doping are: 28 T ($p=0.12$), 34 T
($p=0.148$), 28 T ($p=0.155$), 44.7 T ($p=0.171$) and 40 and 45 T ($p=0.21$).
Complete datasets and fits of doping levels other than $p=0.148$ are presented
in Appendix, Fig. 13). b) Root mean squared fluctuating hyperfine field
$\langle h_{\bot}\rangle^{1/2}$ determined either from from fits to
$T^{-1}_{1,\,\rm BPP\,dist.}(T)$ for fields at which a peak is visible in
$T_{1}^{-1}(T)$ (i.e. $B\geq 12$ T for $p=0.148$, $B\geq 19$ T for $p=0.155$
and all fields for $p=0.12$) or from a single $T_{1}^{-1}(B)$ value at low $T$
otherwise (see text). In this latter case, temperatures are in the range
$1.5-2$ K, depending on sample. For $p=0.12$, the value of $\langle
h_{\bot}^{2}\rangle^{1/2}$ at $B=0$ (open symbol) is taken from a fit of
$T_{1}^{-1}$ peak measured at 20 T with the field oriented in the
$\rm{CuO}_{2}$ plane, i.e. a situation for which the field weakly affects
superconductivity. The horizontal dashed line marks the 1.5 mT value used as a
criterion to define the onset of quasi-static spin fluctuations (see text).

$\bullet$ We fix $\tau_{\infty}=0.02$ ps for all dopings but add a
distribution of $\tau_{\infty}$. Good BPP fits would be possible without this
distribution, provided $\tau_{\infty}$ is a free, i.e. field and material
dependent parameter Hammerath et al. (2013). However, $\tau_{\infty}$ is the
correlation time at $T\gg T_{\rm peak}$, so it is unphysical to assume that it
is strongly field dependent. $\tau_{\infty}$ is strongly cross-correlated with
other fitting parameters such as $\langle h_{\bot}^{2}\rangle$ and $E_{0}$, so
fixing $\tau_{\infty}$ helps to identify the intrinsic $B$ dependence of these
parameters across the investigated hole doping range. In principle, a field
dependence of the distribution $\Delta a$ ($a=\ln\tau_{\infty}$) is also
unphysical but we consider that this betrays an actual, but nonessential,
shortcoming of the model (see discussion in section IV).

$\bullet$ No correlation between the distributions of $E_{0}$, $\tau_{\infty}$
and $h_{\bot}$ is introduced.

$\bullet$ Part of the nuclear relaxation is due to processes other than the
spin-freezing: there are other contributions to the spin fluctuation spectrum
as well as electric-field gradient fluctuations Frachet et al. (2020). Since
we are only interested in fluctuations from the local moments, these
contributions should thus be subtracted out before fitting with the BPP model.
We take a semi-phenomenological approach that assumes uncorrelated relaxation
mechanisms, namely a background $T_{1}^{-1}$ on top of which the enhanced
relaxation due to glassy magnetic ordering develops. As explained in ref.
Frachet et al. (2020) (supplementary information), this background is taken to
be $T$ and $B$ independent above $T_{c}$ (in agreement with available data
below $\sim$100 K) and with the $B$ and $T$ dependence expected for a
superconductor below $T_{c}$. We thus fit the $T^{\,-1}_{1}$ from stretched
fits after subtracting the background. It is important to remark that the
field dependence of the assumed background makes a negligible difference for
fields where a clear peak in the relaxation rate is visible, but in the
following we subtract the background at all fields for consistency.

$\bullet$ We reiterate that our analysis is an ad-hoc parametrization of the
data that aims at effectively accounting for spatial inhomogeneity in a
tractable model but that does not pretend to be a realistic representation of
the actual physical parameters. Our goal with this ”first-order approach” (a
zeroth-order approach would assume no distribution at all) is to give an idea
of the physical quantities that determine $T_{1}$, to gain physical insight
into the glassy freezing and its competition with superconductivity as well as
to motivate more sophisticated approaches.

##  III Results and interpretations

###  III.1 Organization of the discussion

Having defined the distributed BPP model we can now show that it captures the
full $T$ and $B$ dependence of $T^{-1}_{1}$ (see Appendix for details about
samples and experimental methods). We shall first provide a brief survey of
the fitting results and then proceed with a more detailed discussion. The
discussion will revolve around three main physical parameters that
characterize glassy antiferromagnetism: the fluctuating field $\langle
h_{\bot}^{2}\rangle$ and the activation energy $E_{0}$ (both of which are $B$,
but not $T$, dependent in our model) as well the correlation time $\tau_{c}$
that depends exponentially on $E_{0}$ (thus on $B$) and on $T$ (Eq. 4).

Since we are presenting data as a function of temperature ($T$), field ($B$)
and hole doping ($p$), we have organized the discussion in the following way:
we first discuss aspects that are more related to the $T$ dependence and then
we discuss aspects that are more related to the $B$ dependence. Since our most
comprehensive set of data is for the crystal with $p=0.148$ doping, we
systematically describe results for this sample first, and then for other
doping levels when relevant.

###  III.2 Brief survey of the fit results

Fig. 2 shows $T^{\,-1}_{1}$ before and after the background subtraction,
together with the fits described above for La1.852Sr0.148CuO4. Fig. 3 shows
the field dependence of the fitting parameters. The results of the fits
rationalize three characteristic behaviors of the data:

$\bullet$ The decreasing amplitude of the peak in $T^{\,-1}_{1}(T)$ upon
decreasing field is rooted in the strong decrease of the fluctuating field
$\langle h_{\bot}^{2}\rangle$ (Fig. 3a).

$\bullet$ The shift of the peak towards low temperatures upon decreasing field
arises from both the term $-1/\ln(B)$ in Eq. 6 (a property of the BPP model,
irrespective of the exact physics at play) but also from the field dependence
of $E_{0}$ (Fig. 3b) that is specific of LSCO.

$\bullet$ The broadening of the peak upon decreasing field is accounted for by
a rapid increase in the width of the distribution of $\tau_{\infty}$ (Fig. 3d)
whereas the distribution of $E_{0}$ relative to $E_{0}$ is constant vs. field
(Fig. 3c). Recall that $a=\ln\tau_{\infty}$ is a logarithmic quantity, so an
order of magnitude change in $\Delta\tau_{\infty}$ increases or decreases
$\Delta a$ by $\ln 10^{\pm 1}=\pm 2.3$. At 10 T, $\tau_{\infty}$ is thus
distributed over five orders of magnitude.

Figure 5:  (a) Correlation time $\tau_{c}$ vs. temperature in the $T$ range
where the BPP fits are actually performed (continuous traces) and extrapolated
over the whole $T$ range (dashes). The typical time scales of a $T_{1}$
experiment in NMR (for a $\sim$100 MHz resonance frequency) and of elastic
neutron scattering (for an an energy resolution of $\sim$1 meV) are indicated.
The quasiparticle (QP) lifetime of 0.1 ps is taken from ref. Fang et al.
(2022). (b) $\tau_{c}$ vs. magnetic field at $T=2$ K. Lines are linear fits to
the data. Dashes represent linear and nonlinear extrapolation of the data at
the low- and high-field ends. The horizontal dotted line marks the value
$\tau_{\infty}=0.02$ ps (Eq. 4) used in our analysis. In both (a) and (b),
$\tau_{c}$ values are deduced from the $E_{0}$ values in Fig. 3b and Eq. 4.

We point out that Vogel-Fulcher-Tammann fits (Eq. 5) did not bring significant
improvement: for fixed $a=\ln\tau_{\infty}$, the introduction of a
distribution of $T_{\rm VF}$ values was completely unable to achieve the
fitting goodness that $\Delta a$ does and $T_{\rm VF}$ values were negligibly
small.

In order to visualize the weakening of slow spin fluctuations upon increasing
hole doping, Fig. 4a shows the $(T_{1}T)^{-1}$ data vs. $T$ at the highest-
measured field for different doping levels (dividing $T_{1}^{-1}$ by $T$ helps
to visualize the difference with the nearly-constant behavior of the $p=0.21$
sample). The steep peak in $T_{1}^{-1}$ vs. $T$ for $p=0.12$ shifts to lower
$T$, decreases in amplitude and broadens upon increasing $p$ and then entirely
disappears for $p=0.21>p^{*}$. The weakening of quasi-static magnetism upon
decreasing $B$ or increasing $p$ is also reflected in the $B$ and $p$
dependence of $\langle h_{\bot}^{2}\rangle^{1/2}$ shown in Fig. 4 b.

###  III.3 Probe-frequency dependence

The activated behavior of $\tau_{c}$ for $p=0.148$, as deduced from Eq. 4 and
the fitted $E_{0}$ values (Fig. 3b), is shown in Fig. 5a at two different
fields. $\tau_{c}$ is seen to cross the typical time scale of various
experimental techniques at different temperatures. This means that the
freezing temperatures $T_{g}$ depends on the time scale of the measurement.

In neutron scattering (NS), the timescale is defined by the energy resolution,
which is rather coarse in standard experiments ($\sim$meV). This results in a
quasi-elastic, rather than purely elastic, signal over a substantial range of
$T$ above $T_{g}^{\rm NMR}$ Haug et al. (2010). In the data at 34 T for
example, $T_{g}^{\rm NS}\sim$13 K whereas $T_{g}^{\rm NMR}=4$ K.

For $p=0.12$, $T_{g}^{\rm NMR}=10$ K while a similar analysis (not shown)
gives $T_{g}\simeq 23$ K at the neutron timescale. This is in reasonable
agreement with the experimental value $T_{g}^{\rm NS}\simeq 30$ K Lake et al.
(2002); Chang et al. (2008); Rømer et al. (2013) that (fortuitously) coincides
with the zero-field $T_{c}$. The slight difference between our model’s
prediction and the neutron onset temperature may result from $\tau_{c}(T)$ not
exactly diverging exponentially and/or from a distribution of $T_{\rm peak}$
values (Eq. 6): indeed, the distribution does not contribute to the value
predicted from the median $T_{1}^{-1}$ but it tends to increase the
temperature onset of the signal detected in neutron scattering (or muon spin
rotation) experiments.

The probe-frequency dependence of the freezing temperature arises from the
divergence of $\tau_{c}$ being slower than the critical slowing down at a
second-order phase transition. Such a gradual slowing down is typical of
glassy systems and glassiness may indeed be expected from quenched disorder
induced by the dopant atoms and by lattice inhomogeneity Božin et al. (1999);
Horibe et al. (2000); Pelc et al. (2021). Disorder is likely to induce
frustration in the spin system, especially as long as AFM order is intertwined
with charge-stripe order, the latter being more sensitive to disorder than the
former, as witnessed by its shorter correlation length. Furthermore,
characteristic properties of spin glasses (irreversibility, scaling behavior
and remanent magnetization) are observed in non-superconducting LSCO at low
doping $p\simeq 0.04-0.05$ Chou et al. (1995); Wakimoto et al. (2000).

###  III.4 Physical interpretation of $E_{0}$

An alternative explanation for the relatively slow divergence of $\tau_{c}$ is
that it is an intrinsic consequence of the quasi two-dimensional, nearly
isotropic (Heisenberg) nature of the spin system that orders only at $T=0$
Haug et al. (2010). While this latter view appears to be at odds with three
sets of experimental observations (traditional glass-like properties at low
doping Chou et al. (1995); Wakimoto et al. (2000), evidence that disorder
plays a role Sasagawa et al. (2002); Mendels et al. (1994) and the three-
dimensionality of spin correlations for $p\simeq 0.12$ Lake et al. (2005);
Rømer et al. (2015)), it is interesting to note the similarity with the
renormalized classical regime of the 2D Heisenberg AFM already pointed out in
several NMR works Cho et al. (1992); Julien et al. (1999, 2000); Curro et al.
(2000); Teitel’baum et al. (2000); Julien et al. (2001); Hunt et al. (2001):
in the limit $T>T_{\rm peak}$, $\omega_{L}\tau_{c}\ll 1$ and thus Eq. 2
reduces to
$T_{1}^{-1}\propto\tau_{c}=\tau_{\infty}\exp{\frac{E_{0}}{k_{B}T}}$, which
ressembles the expression derived by Chakravarty and Orbach (Chakravarty and
Orbach, 1990) for the renormalized classical regime:

$\frac{1}{T_{1}}\propto\xi(T)\propto\exp(\frac{2\pi\rho_{s}}{T})\,,$ (8)

where the spin stiffness $\rho_{s}$ is proportional to the nearest-neighbor
AFM coupling $J$. Eq. 8 is found to describe the undoped parent compound LCO
very accurately (Birgeneau et al., 1995). Even if the correlation length in
optimally doped LSCO is probably limited at low temperature by quenched
disorder and by competing effects from superconductivity, it is plausible that
the dynamics of freezing moments retain characteristics of the renormalized
classical regime, consistent with the idea that spin fluctuations are ”nearly
singular” above $T_{c}$ Aeppli et al. (1997). The activation energy $E_{0}$
characterizing the low-frequency dynamics (Eq. 4) may thus be interpreted as a
measure of the spin stiffness.

###  III.5 Slow spin fluctuations vs. CDW

Figure 6:  $T$ dependence of the stretching exponent $\beta$ above the zero-
field $T_{c}$. The departure from $\beta=1$ signals spatial heterogeneity of
$T_{1}$ values (due to slowing down of spin dynamics developing
inhomogeneously across the sample) and appears to be mostly correlated with
the presence of static CDW correlations detected in x-ray scattering. $T_{\rm
CDW}$ data are from refs. Wen et al. (2019); Miao et al. (2021). For
$p=0.171$, $\beta$ decreases again above 80 K (and thus does not reach 1)
because of quadrupole fluctuations related to the structural transition at 141
K Frachet et al. (2020). Figure 7:  (a) Doping ($p$) dependence of the spin
stiffness $E_{0}$ that tracks the doping dependence of the freezing
temperature at the NMR timescale $T_{g}$ ($E_{0}$ and $T_{g}$ values are from
our highest-field data, see text). (b) Doping dependence of the onset
temperature of slow spin fluctuations (from our highest-field data)
represented as vertical bars: the upper value corresponds to the deviation of
$\beta$ from 1 (i.e. apparition of the first regions with slow fluctuations in
the sample) and the lower value to the temperature at which $T_{1}^{-1}$ shows
a minimum (regions with slow fluctuations become large enough to affect the
median $T_{1}^{-1}$). The CDW phase is drawn according to x-ray results in
refs. Croft et al. (2014); Wen et al. (2019); Miao et al. (2021); von Arx et
al. preprint (2022). The dots indicate that the boundaries of the phase are
uncertain in zero field (there are issues related to whether the x-ray signal
is elastic or inelastic) and even more so in high fields (no experiment yet).
NMR data for $p=0.06$, 0.07 and 0.10 are from refs. Julien et al. (1999), Baek
et al. (2012) and Julien et al. (2001), respectively.

Eq. 4 defines an exponential slowing down that has no onset temperature. Slow
fluctuations may thus, in principle, be present up to arbitrarily high
temperatures. However, in La1.875Ba0.125CuO4 that has the longest-range CDW
order of all cuprates, the upturn of (the median) $T_{1}^{-1}$ of 139La turns
out to be very sharp and exactly coinciding with the CDW transition
temperature $T_{\rm CDW}\simeq 54$ K Baek et al. (2015); Singer et al. (2020).
The stretching exponent $\beta$ (that relates to spatial inhomogeneity as
explained in section II.5) also appears to deviate from 1 at $T_{\rm CDW}$
Baek et al. (2015) (in a recent report on the same material, $\beta$ deviates
from 1 below a slightly higher temperature of $\sim$80 K Singer et al. (2020),
which might arise from slightly different doping or homogeneity of the
sample). Clearly, it is principally the CDW transition that abruptly triggers
the slow spin fluctuations in La1.875Ba0.125CuO4 (notice that the concomitance
of the structural transition might also play a role in the abruptness). If
present above $T_{\rm CDW}$, slow spin fluctuations must have an amplitude
lower than the NMR detection threshold (determined by the ”background”
$T_{1}^{-1}$ due to other components of the spin fluctuation spectrum and to
electric-field-gradient fluctuations). This includes the possibility that slow
fluctuations are present in a volume fraction of the sample that is too small
to be detected.

Consistent with earlier works Julien et al. (2001); Curro et al. (2000); Hunt
et al. (2001), the link between slow spin fluctuations and charge order is
visible in our data for LSCO $p=0.12$. Nonetheless, due to shorter-range
charge order in LSCO, things are expectedly a bit less clear-cut than in
La1.875Ba0.125CuO4: $\beta$ still deviates from 1 at the CDW onset $T_{\rm
CDW}\simeq 75$ K (i.e. spins already start to slow down in parts of the sample
at that temperature) but the median $T_{1}^{-1}$ increases less steeply than
for La2-xBaxCuO4 and only below $\sim$55 K Mitrović et al. (2008). That the
median $T_{1}^{-1}$ does not increase immediately at $T_{\rm CDW}$ indicates
that the transition is more gradual because of stronger spatial inhomogeneity
in LSCO. This is consistent with conclusions reached in ref. Singer et al.
(2020).

For $0.14\lesssim p\lesssim 0.17$, the upturn in $T_{1}^{-1}(T)$ occurs
entirely below the zero-field $T_{c}$. Its onset is thus field dependent due
to the competing effect of superconductivity. However, $\beta$ deviates from 1
already above $T_{c}$, actually $\sim 50-70$ K (Fig. 6), consistent with the
onset temperature of static CDW order (see Fig. 7b). Therefore, the quasi-
static spin fluctuations are still intertwined with CDW correlations in this
doping range but the weakening of both orders and the greater strength of
superconductivity now tip the balance in favor of superconductivity: spin
freezing is hampered by superconductivity, which results in a nonmagnetic
ground state in zero field (Fig. 1a).

The correlation between CDW order and quasi-static spin fluctuations is also
manifested in the doping dependence of the spin stiffness $E_{0}$ from our
highest field data: like the NMR freezing temperature $T_{g}$, $E_{0}$ is
strongly enhanced around $p=0.12$ (Fig. 7a), that is, where charge-density
wave (CDW) order is the strongest.

It is important to point out that the $T=0$ phase boundaries for static CDW
order are not settled yet. On the high doping side in particular, a recent
report Miao et al. (2021) of CDW order in zero field at $p=0.21$ seemingly
points to a disconnection from quasi-static spin fluctuations that are absent
at this doping, at any field Frachet et al. (2020). However, the CDW signal
might be dynamic rather than static Miao et al. (2021); von Arx et al.
preprint (2022). Therefore, where truly static CDW order exactly ends in the
phase diagram is at present unclear in zero field, let alone in high fields
where no experiments have been performed yet. Quite evidently, it is of
paramount interest to investigate the field dependence of the static CDW in
the vicinity of $p^{*}$ and to determine the doping endpoint of static CDW
order, both in zero and high field. Is there any connection with $p^{*}$ and
with our results concerning spin order?

Disconnection between spin freezing and CDW may be more evident on the low
doping side: for $p\leq 0.10$, CDW order, if any, is weak but the onset of
slow fluctuations tends to occur at increasingly higher $T$ as $p$ decreases
and the slowing down now takes place over a much larger $T$ range (see refs.
Julien et al. (1999); Baek et al. (2017); Hunt et al. (1999) and Fig. 7).
Glassy spin freezing is visibly less and less CDW driven as $p$ is decreased
towards the non-superconducting boundary at $p\simeq 0.055$.Fang2022

The AFM spin-glass arises at low doping from strong two-dimensional AFM
correlations in the presence of disorder. Per se, its strength would
monotonically decrease upon increasing doping but, in La214 cuprates, it is
actually reinforced over a range of doping around $p=0.12$ because it
intertwines with charge-stripe order. This is unlike YBa2Cu3Oy for which
static magnetism does not extend beyond $p\sim 0.08$ where charge order starts
to emerge (Coneri et al., 2010; Wu et al., 2011).

###  III.6 Overall field dependence

The increase of the mean squared fluctuating field $\langle
h_{\bot}^{2}\rangle$ upon increasing $B$ (Fig. 4b) is reminiscent the increase
of the ordered moment measured in neutron scattering Chang et al. (2008). The
concomitant increase of the activation energy $E_{0}$ (Fig. 3b) signifies that
the spin stiffness increases in high fields, which results in a steeper
divergence of $\tau_{c}(T)$ (Fig. 5a). A consequence of the approximately
linear increase of $E_{0}(B)$ in the investigated range is that $\tau_{c}$
grows exponentially with $B$ (Fig. 5b): at 2 K, $\tau_{c}$ increases by eleven
orders of magnitude between 10 T and 34 T. Therefore, the correlation time of
spin fluctuations grows exponentially upon both decreasing $T$ and increasing
$B$: $\tau_{c}\propto\exp(B/T)$ in the explored range of $B$ and $T$.

As already mentioned, we also see a considerable increase of the distribution
$\Delta a$ at low fields (Fig. 3d). At the qualitative level, this parallels
the growing spatial inhomogeneity at low fields indicated by the values of the
stretching exponent $\beta$ at low $T$: for $p=0.148$, $\beta\simeq 0.5$ at
$\sim$30 T and $\beta\simeq 0.3$ at $\sim$4 T Frachet et al. (2020). The
distribution of $\Delta a$, however, seems more extreme than the $T_{1}$
distribution quantified by $\beta$. It is then possible that the increase in
$\Delta a$ does not only reflect a growing spatial inhomogeneity in the
superconducting state but also a deviation from the assumed exponential $T$
dependence of $\tau_{c}$ that becomes more significant at low fields as
superconductivity progressively quenches spin freezing.

In summary, the field dependence suggests that superconductivity ”dissolves”
magnetic ordering by reducing the moment amplitude and by undermining both the
stiffness of the spin system and the strong divergence of the correlation
time.

Figure 8:  Regions of the field $B-T$ phase diagram where spin fluctuations
are frozen at the timescale of transport (10-13 s), neutron scattering
($4\times 10^{-12}$ s) and NMR (10-8 s) measurements (the corresponding
frequency values $\omega=\tau_{c}^{-1}$ are also indicated). The open symbols
correspond to the intersections between experimental $\tau_{c}$ values
determined from BPP fits and the horizontal grey bars in Fig. 5a (see the
symbol correspondance). Solid lines are linear fits to the data. Filled point:
experimental onset (7.5 T at 2 K) of quasi-elastic intensity in neutron
scattering Chang et al. (2008), that nicely matches the low-field
extrapolation of the estimate based on NMR fits. Linear extrapolation of the
upper line ($\tau_{c}=10^{-13}$ s) up to 55 T yields a temperature of 70 K,
approximately matching the onset of the resistivity upturn at this field
Bourgeois-Hope et al. (2019). Magenta and black dashes represent linear and
nonlinear extrapolation of the data at low fields, with and without an onset
field. Figure 9:  Comparison of the field dependence of the fluctuating
hyperfine field $\langle h_{\bot}^{2}\rangle^{1/2}$ measured in NMR and of the
ordered moment measured in elastic neutron scattering for $p=x=0.12$ (a) and
$p=x=0.148$ (b). Neutron data are from ref. Chang et al. (2008). The $p=0.148$
crystal studied in NMR has a precisely-determined doping Frachet et al. (2020)
and is from the same batch as the neutron crystal initially labelled
$p=0.145$. Notice that a slight vertical offset of neutron data in panel (b)
(implying a tiny, rather than strictly null, ordered moment at low fields)
would improve the scaling with $\langle h_{\bot}^{2}\rangle^{1/2}$ below 7 T.
This is not impossible given the $\mu$SR detection threshold of 0.005
$\mu_{B}$ in ref. Chang et al. (2008).

###  III.7 High-field puzzle

A most striking evidence that the field-dependent freezing arises primarily
from the quenching of superconductivity is the absence of spin freezing when
the field is applied parallel to the CuO2 planes, which affects
superconductivity much less than perpendicular fields Frachet et al. (2020).
Within a scenario of competition between superconductivity and spin order, one
would also expect the results in perpendicular fields to become field
independent above the upper critical field $B_{c2}$, as indeed observed for
the CDW state in YBa2Cu3Oy Wu et al. (2013a); Zhou et al. (2017); Kačmarčík et
al. (2018).

Our data in LSCO, however, does not show any saturation as a function of $B$
(Figs. 3a,b and 4b). This implies either that superconductivity is still
present in some form and competing with spin ordering at these fields or that
the field has an additional, direct effect on the moments, irrespective of the
presence of superconductivity (see ref. Katanin and Sushkov (2011) for one
such theoretical proposal). It is impossible to answer this question here
since our highest field values (30 - 45 T) are not significantly larger than
estimates of $B_{c2}$ based on an extrapolation of the vortex melting line at
$T=0$ (see Frachet et al. (2020); Girod et al. (2021) and refs. therein).
Still, the absence of saturation up to at least 28 T for $p=0.12$ with
$B_{c2}\simeq 20$ T is puzzling and we notice that there is also no saturation
in the sound velocity results up to at least 80 T at any doping level below
$p^{*}$ Frachet et al. (2020, 2021).

This lack of saturation in the field dependence contrasts with neutron
scattering measurements: for $p=0.12$ (Fig. 9a), $\langle
h_{\bot}^{2}\rangle^{1/2}$ varies much more with $B$ than the ordered moment
does. This is also possibly true for $p=0.148$ (Fig. 9b): for this doping, the
NMR and neutron datasets are not inconsistent within error bars but the strong
field dependence of $\langle h_{\bot}^{2}\rangle^{1/2}$ up to at least 34 T is
inconsistent with the presumed saturation of the ordered moment above $\sim$12
T Chang et al. (2008). Neutron data at higher fields would definitely be
helpful here. Had the field dependence of $\langle h_{\bot}^{2}\rangle^{1/2}$
clearly matched that of the ordered moment, we could have scaled the two
quantities to predict the ordered moment in putative neutron high field
experiments close to $p^{*}$. In the present situation, we refrain from doing
so.

###  III.8 Defining a magnetic-field scale

Inasmuch as spin fluctuations continuously slow down upon decreasing $T$ and
freeze at a temperature $T_{g}$ that depends on the frequency of the
experimental probe, one may expect similar behavior as a function of field,
namely a freezing field that depends on the experimental time scale. This is
indeed the case for LSCO $p=0.148$: from the $\tau_{c}(B,T)$ results in Fig.
5a (and data at other fields, not shown in this figure), we deduce the
$\tau_{c}=4\times 10^{-12}$ s boundary in $(B,T)$ space, i.e. where
fluctuations become as slow as $\tau_{c}^{-1}=\omega=1$ meV and thus
contribute an elastic signal in neutron scattering (ignoring inhomogeneity
aspects that complicate the problem). As Fig. 8 shows, linear extrapolation of
this data below 10 T matches the neutron scattering finding that elastic
scattering at 2 K onsets at $\sim$7.5 T Chang et al. (2008). In other words,
for this $p=0.148$ doping, both the transition field at the neutron timescale
and the NMR data are consistently accounted for by the same correlation time
of spin fluctuations $\tau_{c}\propto\exp(B/T)$.

At $T=2$ K, the fluctuations reach the NMR frequency at a higher field of
$\sim$14 T (Fig. 8) for $p=0.148$, which corresponds to the field at which a
peak becomes discernible in the data of Fig. 2. For other doping levels,
however, the same amount of information is unfortunately not available,
especially as a field of 45 T was insufficient to produce a measurable peak in
$T_{1}^{-1}$ for the $p=0.171$ sample. Therefore, we have no information on
$E_{0}$ and $\tau_{c}$ for this doping. The only available data, $\langle
h_{\bot}^{2}\rangle$, shows a smooth $B$ dependence from which no
characteristic field scale naturally emerges.

Therefore, in order to determine a field scale for any doping level, we define
the onset field $B_{\rm slow}$ as the field above which an ”ordered moment”
should be detected in neutron scattering (quotes are because of the quasi-
elastic, rather than purely elastic, nature of the signal as discussed above).
We use as a criterion the value of $\langle h_{\bot}^{2}\rangle$ at the
transition field of 7 T as determined from neutron scattering for $p=0.148$
Chang et al. (2008) (horizontal line $\langle h_{\bot}^{2}\rangle^{1/2}=1.5$
mT at 7 T in Fig. 4b). There is thus no independently-measured value of
$B_{\rm slow}$ from NMR at $p=0.148$. A merit of this definition is that it
agrees with the onset of slow fluctuations detected in sound velocity Frachet
et al. (2020) (notice that a criterion based on the value of $T_{1}^{-1}$,
rather than on $\langle h_{\bot}^{2}\rangle$, was used in ref. Frachet et al.
(2020) and gives similar values of $B_{\rm slow}$). As Fig. 10 shows, $B_{\rm
slow}$ increases with $p$ up to $p^{*}\simeq 0.19$, meaning that, as doping
increases, larger fields are necessary to induce spin freezing. We now discuss
the interpretation of this observation.

###  III.9 Evidence for competing order in vortex cores

There is a fundamental difference between decreasing temperature and
increasing field: in the latter case, the AFM glass emerges from the
electronically inhomogeneous vortex state. This situation, in which the
competing order is enhanced in and around the vortex cores, is analogous to
the emergence of long range CDW order upon increasing field in YBa2Cu3Oy Wu et
al. (2013b). To first approximation, the field at which a bulk frozen state is
reached corresponds to the critical vortex density for which the halos of
charge and/or spin order start to overlap, just as the superconducting upper
critical field $B_{c2}$ corresponds to overlap of vortex cores. By analogy,
$B_{\rm slow}$ should then scale with the inverse of the halo radius squared
Wu et al. (2013b):

$B_{\rm slow}\,\simeq\,\frac{\Phi_{0}}{2\pi\xi_{\rm AF}^{2}}$ (9)

where the halo radius is taken to be set by the AFM correlation length
$\xi_{\rm AF}$. In this picture, the characteristic field scale associated
with AFM ordering increases with increasing doping (as indeed observed)
because $\xi_{\rm AF}$ decreases as one goes away from the Néel phase.

According to Eq. 9, $B_{\rm slow}$ values of 2.3 and 7.5 T (taken from neutron
scattering results in two different samples with $p\simeq 0.145$ Khaykovich et
al. (2005); Chang et al. (2008)) correspond to $\xi_{\rm AF}$ values of 111 Å
and 69 Å. The experimental values, measured in fields of 13–14 T for these
samples are: $\xi_{\rm AF}\geq~{}120$ Å Khaykovich et al. (2005) and $\xi_{\rm
AF}=75$ Å Chang et al. (2008). The agreement is remarkable, especially as the
length scale for the AFM halos to overlap should be proportional but not
necessarily equal to $\xi_{\rm AF}$ (it depends on how the staggered
magnetization exactly decays with distance from the vortex core).

In principle, the $\xi_{\rm AF}$ values satisfying Eq. 9 are those measured in
an elastic ($\omega\simeq 0$) neutron scattering experiment at $B=B_{\rm
slow}$. These are unknown for $p\geq 0.15$. Nonetheless, $\xi_{\rm AF}$ values
from inelastic neutron scattering studies Aeppli et al. (1997); Vignolle et
al. (2007); Lipscombe et al. (2007) at low energy ($\omega\sim$ few meV) and
in zero field are sufficient for the purpose of comparing the doping
dependence (notice that, strictly speaking, diffraction peaks being at
incommensurate positions, their width defines a spin-stripe, rather than
purely AFM, correlation length). As Fig. 10 shows (notice that the $\xi_{\rm
AF}^{-2}$ scale does not start from zero, reflecting the fact that the
measured $\xi_{\rm AF}$ is not infinite even when magnetic order is present in
zero field), $B_{\rm slow}$ values at different doping levels are found to
scale with $\xi_{\rm AF}^{-2}$, just as expected from Eq. 9. This thus
provides quantitative support to the picture of a competing order emerging
from the vortex cores.

In passing, we note that, based on the scaling between $B_{\rm slow}$ and
$\xi_{\rm AF}^{-2}$, one would extrapolate $B_{\rm slow}\simeq 50$ T for
$p=0.22>p^{*}$ (Fig. 10). Since slow spin fluctuations have not been seen in
ultrasound measurements performed up to 90 T for $p=0.21$ Frachet et al.
(2020), spin freezing is surely absent at any field at this doping and by
extension above $p^{*}$.

Figure 10:  Doping dependence of the characteristic field $B_{\rm slow}$
defined from NMR data (this work) so as to match the transition field $B_{\rm
neutron}$ at $p=0.148$ Chang et al. (2008) and compared to $B_{\rm slow}$ from
ultrasound measurements (defined as the field above which a softening of the
shear elastic constant is observed) Frachet et al. (2020). The $B_{\rm
neutron}$ value for $p=0.144$ is from ref. Khaykovich et al. (2005). $B_{\rm
slow}$ appears to vary with $p$ as $\xi_{\rm AF}^{-2}$, with $\xi_{\rm AF}$
the AFM correlation length measured by inelastic neutron scattering at low
energy and zero field for $p=0.14$ Aeppli et al. (1997), $p=0.16$ Vignolle et
al. (2007) and $p=0.22$ Lipscombe et al. (2007). This scaling between $B_{\rm
slow}$ and $\xi_{\rm AF}^{-2}$ supports the notion of AFM order emerging from
the superconducting vortex cores (see text). The dashes guide the eye through
$\xi_{\rm AF}$ data.

###  III.10 Gap filling vs. gap closing

At doping levels $p\geq 0.15$ for which magnetic order is absent in zero
field, a spin gap $\Delta_{\rm spin}$ is observed in neutron scattering
studies (Tranquada (2020) and references therein). If the excitations at
$\Delta_{\rm spin}$ have a triplet character, increasing the field should
linearly decrease the energy of the lower branch of the $S=1$ state which then
ultimately crosses the singlet ground state at a critical field
$B_{c}=\Delta_{\rm spin}/g\mu_{B}$, thus giving rise to a magnetic ground
state as observed in low-dimensional quantum magnets Chaboussant, G. et al.
(1998); Giamarchi et al. (2008). The field scale for magnetic order in that
case is thus set by the spin gap $\Delta_{\rm spin}$.

Here in LSCO $p=0.148$, $\Delta_{\rm spin}\simeq 4$ to 9 meV (depending on the
definition used for the spin gap Chang et al. (2009); Li et al. (2018))
whereas the Zeeman energy associated with the field-induced freezing is
$g\mu_{B}\,B\sim$ 1 meV for a typical field scale of $\sim$10 T. Therefore,
the field-induced order arises before $\Delta_{\rm spin}$ is closed.

Instead, the slow fluctuations appear as in-gap states Chang et al. (2009),
which is consistent with the phase coexistence expected for a competing order
emerging from the vortex cores. Further, the similarity of field-induced and
Zn-induced magnetism Panagopoulos et al. (2003) is also in favor of in-gap
states, since these latter are well documented from neutron scattering studies
of Zn-doped LSCO Kimura et al. (2003a). This picture is likely to remain true
for $p>0.148$ since $\Delta_{\rm spin}$ is weakly $p$ independent in the
doping range $p\simeq 0.15-0.20$ Li et al. (2018) whereas $B_{\rm slow}$ has a
strong $p$ dependence (Fig. 10).

###  III.11 Relevance to transport measurements

Most discussions on ”quantum criticality” in cuprates (in the loose sense of
ground-state properties changing across $p^{*}$) have so far overlooked the
possible effect of the magnetic field. Our results, however, reveal that there
is some difference in magnetic properties between superconducting and
nonsuperconducting ground states. As pointed out in ref. Frachet et al.
(2020), the field-dependent nature of magnetism up to $p^{*}$ may be relevant
for the interpretation of high-field experiments in La214, and particularly
for the question of quantum criticality inferred from specific heat
measurements Michon et al. (2019). In the present section, we would like to
discuss a different but complementary aspect, which is related to experimental
time scales.

Looking again at the $T$ dependence of the correlation time $\tau_{c}$ for
$p=0.148$ (Fig. 5), one sees that the AFM moments can be considered static on
the scale of the quasiparticle lifetime of $\sim 0.1$ ps (from ref. Fang et
al. (2022) in Nd-LSCO) below a temperature as high as $\sim 40$ K at 34 T,
even though the moments become frozen at the NMR timescale only at $\sim 5$ K.
Fig. 8 provides an overview of how fluctuations evolve in the $B-T$ plane. The
$T$ range of slow fluctuations visibly expands in high fields.

The same analysis shows that the 0.1 ps timescale is reached far above
$T_{c}$, at $\sim 80$ K, for $p=0.12$ but only at $\sim$10 – 20 K (at 30
Tesla) for $p=0.155$.

We thus see that, up to $p^{*}$, the slow spin fluctuations are likely to
impact transport properties measured in high fields, not only at the lowest
temperatures but also significantly above the temperature of freezing at the
NMR timescale. This can even be above $T_{c}$ when stripe order is already
well developed in zero-field, namely close to $p=0.12$ in LSCO and presumably
up to $p\sim 0.2$ in LTT (low temperature tetragonal) variants of La214.
Therefore, the presence of quasi-static AFM fluctuations is potentially
relevant to important optical Post et al. (2021) and transport (Boebinger et
al., 1996; Collignon et al., 2017; Bourgeois-Hope et al., 2019; Fang et al.,
2022) experiments performed in a similar range of $T$, $B$ and $p$ in La214.

In particular, a correlation between resistivity upturn and magnetic freezing
has been recently discussed in the context of field-induced spin order in LSCO
$p=0.143$ (Bourgeois-Hope et al., 2019), thereby extending previous evidence
of such correlation at lower doping and zero field Harshman et al. (1988);
Hayden et al. (1991); Julien et al. (1999); Hunt et al. (2001); Panagopoulos
and Dobrosavljević (2005). In our $p=0.148$ sample (from the same batch as the
$p=0.143$ sample of ref. (Bourgeois-Hope et al., 2019)), the temperature at
which fluctuations reach the 0.1 ps timescale increases linearly with $B$
(squares in Fig. 8). Extrapolating this linear dependence up to 55 T, we find
that the 0.1 ps timescale should be reached at $\sim$70 K, which is about the
onset of resistivity upturn at this field Bourgeois-Hope et al. (2019). Of
course, such correlation is only approximate: the onset temperature of the
upturn is not very precisely defined, the quasiparticle lifetime of 0.1 ps
that was determined for Nd-LSCO $p=0.20$ may be different in LSCO $p=0.148$
and the linear extrapolation at high fields may not be justified. Still, that
the numbers match shows that our analysis of the NMR data is at least semi-
quantitatively consistent with a link between resistivity upturns and quasi-
static spin fluctuations (we note that a somewhat weaker field dependence of
$E_{0}$ above 34 T, as suggested by ultrasound measurements Frachet (2019),
leads to an extrapolated temperature of $\sim$55 K at 55 T for the 0.1 ps
timescale, which rather corresponds to the resistivity minimum).

Correlation, however, does not imply causation. Therefore, while the
coincidence of $p^{*}$ with a transition in the high-field magnetic ground
state is unlikely to be accidental, we are not claiming that the
transformation of the Fermi surface below $p^{*}$ (Collignon et al., 2017;
Fang et al., 2022) necessarily results from a reconstruction by frozen
antiferromagnetism in La214. The credibility of this scenario depends, for
instance, on whether the AFM correlation length $\xi_{\rm AF}$ is large enough
in high fields up to $p^{*}$, which is unknown (we nonetheless note that
$\xi_{\rm AF}\geq 120$ Å at 14.5 T for $x=0.144$ in ref. Khaykovich et al.
(2005)). Furthermore, it is possible that the Fermi surface of the pseudogap
state (i.e. in zero field and up to $T\sim T^{*}$) consists of pockets that
are insensitive to a reconstruction by the low-$T$ AFM order observed here.
Our point here is simply that, in a striped cuprate such as Nd-LSCO with
$p\simeq 0.2$, spin fluctuations should be sufficiently slow at $T\sim T_{c}$
(i.e. at temperatures relevant to most transport experiments) for a scenario
of reconstruction by quasi-static AFM order to be considered. Our results
actually beg the question: why would this state not reconstruct the Fermi
surface?

###  III.12 Nature of the low-field ground state

We now come back to the discussion of the field dependence at low fields,
which is directly connected to the question of the nature of the ground state
in zero or low field.

As already alluded to in the above, the values of $T_{1}^{-1}$ at low $B$ and
$T$ remain well above the values expected if the field had only closed the
superconducting gap (dashed lines in Fig. 2a for $p=0.148$), even though the
peak in $T_{1}^{-1}(T)$ has disappeared below 10 T. Within our analysis, this
is reflected in the finite and smoothly $B$ dependent values of $\langle
h_{\bot}^{2}\rangle$ (Fig. 4b).

In principle, vestiges of frozen magnetism may arise from the least-doped
regions of the sample Singer et al. (2002). However, as already noted above,
139La NMR measurements do not seem to be affected by this type of spatial
inhomogeneity in general Mitrović et al. (2008); Arsenault et al. (2020), so
it is unlikely that this explains the enhanced $T_{1}^{-1}$ at low fields.

Actually, we have seen from the consistency between neutron scattering and NMR
data (Fig. 8) that the frozen state arises gradually upon increasing the
field, in the same way as it arises from a gradual slowing down upon cooling.
As Fig. 8 also shows, the $\tau_{c}=10^{-13}$ s, $\tau_{c}=4\times 10^{-12}$ s
and $\tau_{c}=10^{-8}$ s lines all extrapolate linearly to a field as small as
3.6 T at $T=0$ (the common field value regardless of the $\tau_{c}$ value
being a consequence of the linear dependence of $E_{0}$ on $B$). This already
points to quasi-static magnetism very close to the zero-field ground state.
Furthermore, it is possible that $\tau_{c}$ no longer depends linearly on $B$
at low $B$ values: Fig. 3b shows a possible S-shape dependence - saturating on
both high and low field ends - that is as consistent with the data as the
purely linear dependence). This thus suggests that there may be no sharp
difference between the low-field and high-field ground states: quasi-static
moments are probably present at $B\ll B_{\rm slow}$ but of very weak magnitude
and/or occupying a relatively small volume fraction of the sample.

It is then possible that the ground state in zero field can be considered as
nearly critical Chubukov et al. (1994) between $p_{\rm sg}\simeq 0.135$ (the
end of the frozen ground state in zero-field) and $p=p^{*}$ (the end of the
frozen ground state in high fields): the system ”knows” that it is destined to
become critical once superconductivity is locally suppressed in the vortex
cores. This is different from a situation in which superconductivity simply
shifts the end doping of the magnetic phase from $p^{*}$ down to $p_{\rm sg}$
with no change in the zero-field ground state across $p^{*}$.

In this regard, it is interesting to note that theory Kivelson et al. (2002)
predicts the possibility that the order appears at arbitrarily low fields
around (and along) each single 3D vortex, before crossing over to a more
spatially-homogenous order at higher fields (corresponding to the above-
described overlap of AFM halos in each plane). Evidence of such vortex-core
magnetism has been claimed from muon spin rotation measurements in fields as
low as 0.5 T Sonier et al. (2007).

###  III.13 Broad picture

Our work sheds light on how superconductivity coexists and competes with spin
order, a question of obvious topical interest in the cuprates Fradkin et al.
(2015); Yamase et al. (2016); Cui et al. (2020). In combination with the
ultrasound results Frachet et al. (2020) that are complementary to the NMR
results, this work also sheds new light on the magnetism of
${\mathrm{La}}_{\mathrm{2-x}}{\mathrm{Sr}}_{\mathrm{x}}{\mathrm{CuO}}_{4}$ by
revealing that an AFM glass constitutes the non-superconducting ground state
from the weakly-doped insulator at $p=0.02$ all the way up to $p^{*}=0.19$. As
argued in ref. Frachet et al. (2020), this suggests that the same local-moment
antiferromagnetism as found in the doped Mott insulator survives throughout
the pseudogap regime. This observation adds to previous experimental evidence
that local-moment magnetism prevails in a large part of the cuprate phase
diagram Tranquada (2013), thus making the doped Mott insulator the most
natural starting point for describing the pseudogap state.

The most striking aspect of the NMR and ultrasound results is that the
competition between superconductivity and spin order is confined to the
pseudogap state: as shown in ref. Frachet et al. (2020), no signature of
competition is observed for $p=0.21>p^{*}$. It may be worth recalling here
that short-range AFM correlations have been found to survive above $p^{*}$
from NMR Ohsugi et al. (1994), neutron scattering Wakimoto et al. (2007);
Lipscombe et al. (2007) and resonant x-ray scattering Dean et al. (2013);
Wakimoto et al. (2015); Monney et al. (2016) experiments in LSCO so our
results are not explained by the disappearance of AFM correlations. What
disappears at $p^{*}$ is the ability of the moments to freeze out and to
compete with superconductivity.

These results establish an unexpected connection between the pseudogap phase,
defined by a relatively high temperature scale $T^{*}$, and a much lower
temperature phenomenon, spin freezing, that may be taken as a signature of the
doped Mott insulator. It is even possible that the transition at $p^{*}$ is
precisely associated with the loss of Mott physics. This would be in line with
theoretical works using dynamical-mean-field-theory Sordi et al. (2011) and
with a recent pump-probe study of Bi2201 Peli et al. (2017). Consistent with
this view, a resonant x-ray scattering study of Bi2201 and Tl2201 observed a
change in the nature of (high-energy) magnetism around $p^{*}$ Minola et al.
(2017). An alternative, albeit not necessarily incompatible explanation views
the transition at $p^{*}$ as a percolation phenomenon in a phase-separated
system with coexisting magnetic and nonmagnetic patches whose relative areas
change with doping Tranquada (2020).

The unique ability to sustain local-moment magnetism would then generically
distinguish the pseudogap phase from the correlated metal at $p>p^{*}$. In the
specific case of La214 cuprates, this would ultimately result in an ordered
magnetic ground state up to $p^{*}$, provided competing effects from
superconductivity have been removed. However, we stress that this is not
necessarily so in other cuprates: while the frozen AFM state is a symptom of
local-moment magnetism, not all cuprates are symptomatic from this standpoint.
The intertwined spin and charge stripes are known to be specificity of the
La214 system while a cuprate like Bi2201 does not seem to feature any
resurgence of magnetism in high fields Kawasaki et al. (2010).

Finally, we underline the relevance of our results to recent theoretical work
exploring the nature of the doping-driven quantum phase transition from a
metallic spin-glass with small Fermi surface to a Fermi liquid with large
Fermi surface in the $t-J$ model with random all-to-all hopping and exchange
interactions (refs. Joshi et al. (2020); Shackleton et al. (2021) and
references therein). In this respect, the connection, if any, between the
small Fermi surface observed below $p^{*}$ in several cuprate families Fang et
al. (2022); Badoux et al. (2016); Collignon et al. (2017); Putzke et al.
(2021) and magnetic properties (such as the local-moment nature of the Cu
$3d^{9}$ states or the incipient ordering of the AFM moments) is a fascinating
issue.

##  IV Summary

Figure 11:  False-color representation of the doping ($p$) and field ($B$)
dependence of the mean-squared fluctuating hyperfine field $\langle
h_{\bot}^{2}\rangle^{1/2}$ from data in Fig. 4b. $\langle h_{\bot}^{2}\rangle$
represents the amplitude of the fluctuating moments. The dashed line ($B_{\rm
slow}$) is from Fig. 10. The vertical line shows the end doping of the
pseudogap phase at $p^{*}=0.19$ in LSCO.

In the doping range $p=x=0.14-0.18$, superconducting
${\mathrm{La}}_{\mathrm{2-x}}{\mathrm{Sr}}_{\mathrm{x}}{\mathrm{CuO}}_{4}$ has
a nonmagnetic ground state protected by a spin-gap. However, the application
of magnetic fields $B$ perpendicular to CuO2 planes promotes a ground state
with frozen antiferromagnetic moments, thus providing one of the clearest
cases of competition between superconductivity and electronic ordering.

The slow fluctuations of the moments are detected in NMR relaxation rate
measurements. In this work, we have described a semi-quantitative BPP model
for the relaxation rate of inhomogeneously-freezing electronic spins. The
model assumes that the characteristic timescale of the field-induced
fluctuations becomes exponentially large upon cooling down to $T=0$. There are
also two $T$ independent parameters: the fluctuation amplitude $\langle
h_{\bot}^{2}\rangle$ and the spin stiffness $E_{0}$, in addition to
distributions of the parameters. We have used this model to fit the relaxation
data Frachet et al. (2020) as a function of $T$, for different values of $B$
and $p$. Our main findings may be summarized as follows:

  * •

A correlation time that depends exponentially on $B/T$ accounts for the
differences in the freezing temperature $T_{g}$ and in the onset field $B_{\rm
slow}$ between NMR and neutron scattering.

  * •

Above $T_{g}$, the spin fluctuations are already slow enough to potentially
impact on transport properties. Whether this leads to a reconstruction of the
Fermi surface or not is unsettled. Nonetheless, that quasi-static spins are
present up to $p^{*}$ and not beyond makes magnetism obviously relevant to the
question of quantum criticality at $p^{*}$ and more generally to the
interpretation of those experiments that find a sharp change in the electronic
properties across $p^{*}$, especially if these experiments are performed in
high fields.

  * •

Both the fluctuation moment $\langle h_{\bot}^{2}\rangle$ and the spin
stiffness $E_{0}$ are monotonously enhanced upon increasing the magnetic
field. The relative strength of quasi-static magnetism, as quantified by the
fluctuating hyperfine field $\langle h_{\bot}^{2}\rangle$ (proportional to the
mean-squared or fluctuating moment), is represented as a function of field and
doping in Fig. 11.

  * •

We defined a characteristic field scale $B_{\rm slow}$ that marks the onset of
slow spin fluctuations in NMR and ultrasound measurements as well as the onset
of quasi-static scattering in neutron experiments. We find that $B_{\rm slow}$
increases with doping in a way that is quantitatively consistent with the
notion that bulk freezing occurs when the vortex density is large enough that
halos of local AFM order around each vortex core start to overlap.

  * •

It is possible that the difference between the high-field and low-field ground
states is more quantitative than qualitative. Indeed, while the emergence of
quasi-static magnetism upon increasing field is reflected in the strong
increase of both $\langle h_{\bot}^{2}\rangle$ and $E_{0}$, no sharp
transition is observed as a function of field. The data suggests that there
may be quasi-static moments of extremely weak amplitude already below $B_{\rm
slow}$. These could correspond to quasi-ordering within a single vortex, which
would be consistent with the idea of field-induced low-energy fluctuations
filling the spin-gap.

  * •

Spin order and charge-stripe order are known to be intertwined in
superconducting LSCO but, because the 139La probe lacks sensitivity to CDW
order in CuO2 planes, we could investigate the relationship between the two
orders only indirectly. We observe that the onset of slow spin fluctuations is
essentially triggered by CDW order over the doping range $p=0.12-0.18$.
Moreover, both $E_{0}$ and $T_{g}$ are sharply enhanced at $p=0.12$ where CDW
order is the strongest and they both decrease upon increasing $p$ up to
$p^{*}$. It is thus possible that the primary competitor of superconductivity
up to $p^{*}$ is charge-stripe order. However, an entirely open and
fundamental question is whether spin order would remain in the high-field
ground state up to $p^{*}$ if CDW order could be suppressed by any means.
Results in Zn-doped LSCO suggest that the answer to this question might be in
the affirmative Panagopoulos et al. (2003); Kimura et al. (2003b); Hücker et
al. (2011) but better characterization of the CDW in Zn-doped samples is
needed for a firmer answer.

  * •

The more stable is the stripe phase (as a function of either doping or field),
the stiffer is the spin system and the sharper the transition as a function of
$T$. In La1.875Ba0.125CuO4 that shows the sharpest magnetic transition of all
striped cuprates, the difference in $T_{g}$ values between neutron scattering
and NMR does not exceed a few Kelvin Baek et al. (2015); Singer et al. (2020).

  * •

There is a puzzling absence of saturation in the high-field data that is
worthy of further investigation. Possible interpretations include the
possibility that AFM order competes with superconducting fluctuations well
above the bulk $B_{c2}$ determined by specific heat Girod et al. (2021), as
well as a direct effect of the field on low-energy spin fluctuations (i.e.
irrespective of the presence of superconductivity).

##  V Perspectives

  * •

The described analysis could allow to identify signatures of spin freezing in
cuprates where the relaxation rate is enhanced but a clear peak in
$T^{-1}_{1}(T)$ is not visible.

  * •

Our results provide motivation and guidance for further high-field experiments
in LSCO in the vicinity of $p^{*}$. Neutron scattering in high (most likely
pulsed) fields should observe the AFM glass above $B_{\rm slow}$ and allow a
much-needed measurement of the AFM correlation length in high fields. Optical
measurements could probe whether the field-induced state is a putative 2D
superconductor with interlayer decoupling as observed at much lower
field/doping values Schafgans et al. (2010). Direct probes of CDW order in
high fields across $p^{*}$ would also be highly desirable. Finally, another
natural extension of our work is the NMR study of LTT La214 cuprates across
$p^{*}$. A recent neutron scattering study of Nd-doped LSCO finds spin-stripe
order above $p^{*}$ Ma et al. (2021), seemingly contradicting our NMR results
in LSCO. However, as noted by the authors, timescale issues might need to be
considered here. Furthermore, the Nd moment might complicate the analysis in
this system, thus calling for further investigations in La2-xBaxCuO4 or Eu-
doped LSCO.

  * •

Spatial inhomogeneity, due to both disorder and the vortex state, is clearly a
key issue. Progress in our understanding will require more sophisticated
analysis of the NMR data as well as quantitative predictions from theoretical
models that account realistically for this inhomogeneity.

  * •

It is possible that quantum simulations using cold atoms Mazurenko et al.
(2017); Koepsell et al. (2021) provide in a near future an appropriate
settings for elucidating the particularly intricate interplay between spin,
charge and superconducting orders, with or without quenched disorder.

## Acknowledgements

We thank David LeBoeuf, Mehdi Frachet, Adam Dioguardi, Mladen Horvatić, Jeff
Sonier, John Tranquada and Shangfei Wu for useful discussions.

Work in Grenoble was supported by the Laboratoire d’Excellence LANEF
(ANR-10-LABX-51-01) and by the French Agence Nationale de la Recherche (ANR)
under reference ANR-19-CE30-0019 (Neptun). Part of this work was performed at
the LNCMI, a member of the European Magnetic Field Laboratory (EMFL). A
portion of this work was performed at the National High Magnetic Field
Laboratory, which is supported by the US National Science Foundation
Cooperative Agreement no. DMR-1644779 and the State of Florida. Work in Zürich
was supported by the Swiss National Science Foundation.

##  Appendix A Additional datasets

Fig. 12 presents the $T$ dependence of the 139La linewidth.

Fig. 13 presents additional (background subtracted) $T_{1}$ datasets and the
corresponding fits to the distributed BPP model.

Figure 12:  Width at half maximum of the 139La central line for LSCO
$p=0.148$. The field is applied perpendicular to the CuO2 planes. The width
saturate below $\sim$20 K, the temperature at which $T_{1}^{-1}$ shows a
maximum (Fig. 2). Figure 13:  $T_{1}^{-1}$ data (symbols) from which a
superconducting background has been subtracted as described in ref. Frachet et
al. (2020) for different doping levels and the corresponding fits (continuous
lines) with the distributed BPP model described in the text.

##  Appendix B Experimental methods

The samples were grown by travelling solvent floating zone and the sample
information is summarized in table 1. Where available, the hole doping $p$ is
estimated from the High Temperature Tetragonal (HTT) to Low Temperature
Orthorhombic (LTO) structural phase transition at $T_{s}$.

Table 1:  Sample information hole doping $p$ |  $T_{\textrm{c}}$ (K) |  $T_{\textrm{s}}$ (K) | Ref.
---|---|---|---
$0.122\pm 0.002$ | $29.3\pm 1.5$ | $252\pm 3$ | Frachet et al. (2021)
$0.135\pm 0.002$ | $35.5\pm 1.5$ | $225\pm 3$ |
$0.148\pm 0.001$ | $36.2\pm 1$ | $194\pm 2$ | Frachet et al. (2020); Chang et al. (2008)
$0.155\pm 0.001$ | $38.1\pm 1$ | $177.5\pm 2$ | Frachet et al. (2020)
$0.171\pm 0.002$ | $37.5\pm 1$ | $141.5\pm 3$ | Frachet et al. (2020)
$0.210\pm 0.005$ | $25.6\pm 1$ | $6\pm 10$ | Frachet et al. (2020)

We used home-built heterodyne NMR probes and spectrometers, superconducting
magnets for fields up to 20 T, the LNCMI M10 resistive magnet for fields up to
30 T and the NHMFL hybrid magnet for fields up to 45 T. Fields were applied
along the $c$-axis, i.e. perpendicular to the $\rm{CuO}_{2}$ planes.

The relaxation rate $T_{1}^{-1}$ was measured on the central transition of
139La (nuclear spin $I=7/2$) without any contamination from satellite
transitions as the quadrupole splitting of $\sim$6 MHz greatly exceeds the
excitation width $\Delta f\simeq 50$ kHz. The recoveries were defined by the
time dependence of the nuclear magnetization $M(t)$ following a saturating
pulse yielding $M(t=0)\simeq 0$. $T_{1}$ values were determined by fitting
these recoveries to a stretched version of the theoretical law for magnetic
relaxation between $m_{I}=\pm 1/2$ levels of a nuclear spin 7/2 (Mitrović et
al., 2008; Arsenault et al., 2020; Frachet et al., 2020).

$\begin{split}M(t)=M_{t\rightarrow\,\infty}-\left(M_{t\rightarrow\,\infty}-M_{t=0}\right)\left(0.714\,e^{-\left(\frac{28\,t}{T_{1}}\right)^{\beta}}+0.206\,e^{-\left(\frac{15\,t}{T_{1}}\right)^{\beta}}+0.068\,e^{-\left(\frac{6\,t}{T_{1}}\right)^{\beta}}+0.012\,e^{-\left(\frac{t}{T_{1}}\right)^{\beta}}\right),\end{split}$
(10)

Figure 14:  (a) Relaxation rate after background subtraction fitted by
$T^{-1}_{1,\,\rm BPP\,dist.}(T)$ without distribution of $a=\ln\tau_{\infty}$
(dashed line) and with $\Delta a=3.8$ (where $\tau_{\infty}$ is in s-1, black
line). Notice that the distribution of $a$ is assumed to be uncorrelated from
that of $E_{0}$ (see Eq. 7). (b) Simulated $\Delta a$ dependence of $\langle
h_{\bot}^{2}\rangle^{1/2}$ based on Eq. 7 for an experimentally determined
value $T^{\,-1}_{1}=0.018$ s-1 at $T=1.7$ K and $B=3.7$ T. For small $\Delta
a$, $E_{0}$ affects $\langle h_{\bot}^{2}\rangle$ strongly but above $\Delta
a\sim 9$, the $\langle h_{\bot}^{2}\rangle$ value is essentially independent
of $E_{0}$. This allows to determine $\langle h_{\bot}^{2}\rangle$ from a
single $T^{\,-1}_{1}$ value at low $T$.

##  Appendix C $\langle h_{\bot}^{2}\rangle$ in the limit of large $\Delta a$

The fact that for large $\Delta a$ and small $E_{0}$ the BPP peak becomes
essentially $T$ independent implies that measuring the relaxation rate at a
single temperature, e.g. $T=1.7$ K is sufficient to determine the fluctuating
field $\langle h_{\bot}^{2}\rangle$. It is the non-zero $\langle
h_{\bot}^{2}\rangle$ which leads to relaxation rates $(T_{1}T)^{-1}$ which are
enhanced with respect to the normal state above $T_{\rm{c}}$ and the upturn in
$T_{1}^{-1}(B)$ at the lowest fields. Fig. 14b illustrates that $\langle
h_{\bot}^{2}\rangle$ no longer depends on $E_{0}$ for $\Delta a\gtrsim 9$.
According to Fig. 3d, for the $p=0.148$ sample this limit is reached for
$B\lesssim 12$ T. For samples of higher doping it follows that even for fields
somewhat larger than $B_{\rm{slow}}$ this limit is still valid.

##  Appendix D Mean and median values of probability distribution functions

As mentioned in the main text, we are fitting the $T^{-1}_{1}$ values that
correspond to the median of the experimental $T^{-1}_{1}$ distribution as
shown in Fig. 15a (because these values are obtained from stretched fits of
the recoveries Johnston (2006)) by an expression $T^{-1}_{1,\,\rm
BPP\,dist.}(T)$ (Eq. 7) that calculates the mean $T^{-1}_{1}$ of a model
distribution, depicted in Fig. 15b.

$T^{-1}_{1,\,\rm BPP\,dist.}(T)$ (Eq. 7) consists of a convolution of a
function $f(x)$ with a probability density function PDF$(x-x_{0})$ (in our
case we convolute $T^{-1}_{1,\rm BPP}$ with a Gaussian where
$x-x_{0}=E_{0}-E_{0,c}$) which by construction gives the mean value, not the
median, of that function with respect to the probability density function
(PDF):

$f_{\rm mean}=\int f(x)\cdot P\\!D\\!F(x-x_{0})\,\rm d\textit{x}\,.$

As mentioned, the mean relaxation rate calculated from the model distribution
depends only on the average fluctuating field, so an advantage of the mean
$T^{-1}_{1}$ is that we do not need to assume a specific distribution of
fluctuating fields. This would be necessary if we needed to calculate the
histogram of the model distribution of $T^{-1}_{1}$.

Figure 15:  Situation where the model distribution of relaxation rates (panel
b) has a smaller asymmetry than the experimental distribution (panel a) such
that $\frac{1}{T_{1}}_{\rm mean,\,model}(T)=\frac{1}{T_{1}}_{\rm
median,\,experimental}(T)$ (thick vertical line). The model distribution is
”wrong” since its median relaxation rate differs from the experimental median.
(c) Regular (untruncated) and truncated log-normal probability distribution
functions of the relaxation rate $\frac{1}{T_{1}}$. The regular log-normal
distribution has a median value of 30 s-1 and a much larger mean. For the
truncated distribution ($1/T_{1}$(max) = 100 s-1) the mean decreases strongly,
so one finds $\rm{median_{trunc}}\sim\rm{mean_{trunc}}$. After truncation the
distribution is normalized again, so the integrated area is conserved.

In general, the median and mean of an asymmetric distribution with a long tail
can differ significantly. In a realistic situation, however, the mean
$T^{-1}_{1}$ does not differ greatly from the median because the distribution
of $T^{-1}_{1}$ cannot extend to infinity and must be truncated (Fig. 15c).
Within the BPP model, according to Eq. 3, an infinite relaxation rate would
imply an infinite fluctuating moment.

For the log-normal distribution there are analytical expressions available
from Zaninetti (Zaninetti, 2017) for the relation of the mean and the median
with which one can verify the effect of the truncation: Fig. 15c displays that
the mean of the truncated log-normal distribution, $\rm{mean_{trunc}}$,
becomes much more comparable to the distribution median,
$\rm{median_{trunc}}$. We thus see that, although the distribution of
relaxation rates is asymmetric, the mean and median relaxation rates are
similar, which justifies the fitting by Eq. 7.

If we wanted to correctly model the experimental $T^{-1}_{1}$ by the median of
a model distribution, this would require a specific distribution of
fluctuating fields to numerically evaluate the histogram of the distributed
relaxation rates. While we could make assumptions about the distribution of
the fluctuating fields (see for example the inset of Fig. 16), we do not have
an efficient implementation of the process embedded in an automated fitting
routine which finds the best set of model parameters.

##  Appendix E Comparison of distributed BPP method with ILT distributions

Figure 16: Comparison of probability distributions PDF$(T_{1}^{-1})$ for
La1.885Sr0.115CuO4 at different temperatures based on the Inverse Laplace
Transform (ILT) (colored curves, reproduced from Arsenault et al. (Arsenault
et al., 2020)) with calculated distributions based on parameters $E_{0}=90.5$
K, $\Delta E_{0}=54.4$ K and $h_{\bot}=41.3$ mT from a distributed BPP fit of
$T_{1}(T)$ data of Arsenault et al. (grey histograms). The dashed trace
follows the background relaxation rate discussed in the text (see also Fig. 2)
and in ref. Frachet et al. (2020). The inset depicts the assumed half-Gaussian
distribution of the fluctuating field, $\tilde{h}=\langle
h_{\bot}^{2}\rangle^{1/2}$, whose width $\Delta\tilde{h}$ is taken to be equal
to the 41.3 mT found in the BPP fit.

To assess the quality of the distributed BPP model, we would like to compare
the distribution of $T_{1}^{-1}$ values that corresponds to the fitted BPP
parameters with the distribution obtained from the inverse Laplace transform
(ILT). For this purpose, we extract the parameters ($\langle
h_{\bot}^{2}\rangle$, $E_{0}$, $\Delta E_{0}$) from a distributed BPP fit of
the $T_{1}^{-1}~{}vs.~{}T$ data of Arsenault et al. (Arsenault et al., 2020)
in La1.885Sr0.115CuO4 and we calculate the corresponding $T_{1}^{-1}$
distribution, which we then compare with the calculated ILT of Arsenault et
al.

In our model, the maximum possible $T_{1}^{-1}$ value is $(T_{1}^{-1})_{\rm
max}=\langle h_{\bot}^{2}\rangle\,\gamma_{n}^{2}/\omega_{L}$. (Eq. 3). As we
did not introduce any distribution of $\langle h_{\bot}^{2}\rangle$,
$T_{1}^{-1}$ must reach this exact value at some temperature for every
nucleus. This means that we have introduced an artificial cutoff in the
distribution (no higher value is allowed). Consequently, the probability
density function (PDF) of $T_{1}^{-1}$ calculated from the BPP parameters
diverges at $(T_{1}^{-1})_{\rm max}$ for temperatures close to the peak. Since
this is unphysical, we introduce a distribution of $\langle
h_{\bot}^{2}\rangle$ values that smoothes this singularity. Defining
$\tilde{h}=\langle h_{\bot}^{2}\rangle^{1/2}$ to simplify notations, we model
$\tilde{h}$ by a half Gaussian (see inset of Fig. 16) with a width
$\Delta\tilde{h}$. For this particular distribution the mean value of
$\tilde{h}^{2}$ is equal to $\Delta\tilde{h}^{2}$:
$\frac{\sqrt{2}}{\sqrt{\pi}\Delta\tilde{h}}\int_{0}^{\infty}\tilde{h}^{2}\exp(-\frac{\tilde{h}^{2}}{2\Delta\tilde{h}^{2}})\mathrm{d}\tilde{h}=\Delta\tilde{h}^{2}$,
so this distribution is consistent with the fluctuating fields determined from
BPP fits provided that we set $\Delta\tilde{h}=\tilde{h}_{\rm{BPP}}$. Such a
distribution, having its most probable value at $\tilde{h}=0$, is consistent
with the distribution of static local fields $\langle h_{\bot}\rangle$
inferred by Hunt et al. from 63Cu nuclear quadrupole resonance lineshapes in
La2-xBaxCuO4 (Hunt et al., 2001).

There is a similar issue on the low $T_{1}^{-1}$ side of the PDF. While we
have modeled the BPP relaxation after subtracting a background $T_{1}^{-1}$
(see Fig. 2 and ref. (Frachet et al., 2020)), the ILT is performed on
recoveries that contain all contributions to the relaxation. These include the
background $T_{1}^{-1}$ that actually determines the smallest possible value
of $T^{-1}_{1}$ at $T\ll T_{\rm{peak}}$. Indeed, based solely on the BPP
mechanism, $T_{1}^{-1}$ would drop to arbitrarily low values as $T\rightarrow
0$. We thus need to reestablish the subtracted background ”by hand” in order
to avoid unphysically small values when we reverse-engineer the BPP parameters
to get PDF$(T^{-1}_{1})$. Again, if all nuclei had the same $T_{1}^{-1}$
background, there would be a singularity in PDF$(T^{-1}_{1})$, so we
distribute this background relaxation rate by a Gaussian, whose width
increases as $1/T$ with cooling. The $1/T$ dependence mimics the increased
width of PDF$(T^{-1}_{1})$ from high to low temperature and is chosen for the
sake of simplicity (a more realistic choice would probably require introducing
an onset temperature near $T_{\rm{CDW}}=80$ K, as discussed in the main text).

The calculation of PDF$(T^{-1}_{1})$ histograms shown in Fig. 16 takes place
in two steps. First, we calculate $T^{-1}$ for each point ($E_{0}$,
$\tilde{h}$, $T^{-1}_{1,\mathrm{backgr.}}$) in parameter space and its
probability $P(E_{0},\tilde{h},T^{-1}_{1,\mathrm{backgr.}})$ considering the
(half-)Gaussian distributions with widths $\Delta E_{0}$, $\Delta\tilde{h}$
and $\Delta T^{-1}_{1,\mathrm{backgr.}}$. Second, as we want to compare to ILT
distributions that are plotted on logarithmic scale, the probabilities are
sorted and binned together in 50 bins from $\log_{10}T^{-1}_{1,\mathrm{min}}$
to $\log_{10}T^{-1}_{1,\mathrm{max}}$ and each binned probability is weighted
with its corresponding $T^{-1}_{1}$. We restrict the comparison to
temperatures below 100 K, for which the data is correctly described by the BPP
peak and/or the approximated background $T^{-1}_{1,\mathrm{backgr.}}\propto
T$.

The agreement between the calculated PDF$(T^{-1}_{1})$ and the ILT is not
perfect but some features of the ILT distributions are reproduced (Fig. 16):
i) the peak position initially shifts to lower values with cooling below 100
K, tracking $T^{-1}_{1,\mathrm{backgr.}}\propto T$. ii) the distribution
eventually broadens, leading to the shoulder at intermediate temperatures.
Asymmetric histograms, however, systematically underestimate the median value
compared to the ILT distributions. iii) At base temperature 4.2 K, the two
distributions are similar: two peaks appear, reminiscent of the bimodal ILT
distribution, although the peak positions do not match. In the calculated
histogram, the peak on the left hand side is rooted in
$T^{-1}_{1,\mathrm{backgr.}}$ and is not centered at the dashed line but is
shifted due to the weighting by $T^{-1}_{1}$ in combination with a large
$\Delta T^{-1}_{1,\mathrm{backgr.}}$. Possibly, the left peak of the ILT
distribution corresponds to frozen spins that relax at a faster background
relaxation rate than assumed in our model.

We are of course aware that our distributed BPP analysis is based on modeling
the data with over-simplified parametrization and various assumptions but it
is a bold attempt to understand a very complex problem within a physically
reasonable model. For instance, it might be argued that assuming additive
contributions (the BPP freezing and the residual ”background” relaxation) is
itself suggestive of some form of phase separation. Nevertheless, any nucleus
must always be affected by residual relaxation, for instance, relaxation by
paramagnetic impurities at low $T$. Therefore, despite the shortcomings of the
analysis, the findings suggest that a unimodal distribution of the BPP
parameters can reasonably describe the inhomogeneous relaxation process and
that the bimodal nature of the $T_{1}^{-1}$ distribution found in the ILT
analysis could arise from residual relaxation processes yielding a lower bound
to $T_{1}^{-1}$ values at low $T$, rather than from spatial differentiation
Arsenault et al. (2020). More work is certainly needed to clarify the issues
related to inhomogeneous nuclear relaxation.

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