Revisiting the $^{63}$Cu NMR signature of charge order in La$_{1.875}$Ba$_{0.125}$CuO$_{4}$

We use single crystal $^{63}$Cu NMR techniques to revisit the early $^{63}$Cu NQR signature of charge order observed for La$_{1.875}$Ba$_{0.125}$CuO$_{4}$ ($T_{\text{c}} =4$~K) [A. W. Hunt et al., Phys. Rev. Lett. {\bf 82}, 4300 (1999)]. We show that the growth of spin correlations is accelerated below $\sim 80$~K, where the inverse Laplace transform (ILT) T$_{1}$ analysis of the $^{139}$La NMR spin-lattice relaxation curve recently uncovered emergence of the slow components in the lattice and/or charge fluctuations [P. M. Singer et al., {\bf 101}, 174508 (2020)]. From the accurate measurements of the $^{63}$Cu NMR signal intensity, spin echo decay $M(2\tau)$, spin-lattice relaxation rate $^{63}1/T_1$, and its density distribution function $P(^{63}1/T_{1})$, we also demonstrate that charge order at $T_{\text{charge}}\simeq 54$~K turns on strong enhancement of spin fluctuations {\it within charge ordered domains}, thereby making the CuO$_2$ planes extremely inhomogeneous. The charge ordered domains grow quickly below $T_{\text{charge}}$, and the volume fraction $F_{\text{CA}}$ of the canonical domains unaffected by charge order gradually diminishes by $\sim 35$~K. This finding agrees with our independent estimations of $F_{\text{CA}}$ based entirely on the $^{139}$La ILTT$_{1}$ analyses, but is in a stark contrast with much slower growth of charge ordered domains observed for La$_{1.885}$Sr$_{0.115}$CuO$_{4}$ from its $T_{\text{charge}}\simeq 80$~K to $T_{\text{c}}\simeq 30$~K.

A variety of phases compete or coexist in cuprate high T c superconductors, including the charge ordered phase around the magic composition at x ∼ 1/8 (see [1,2] for recent reviews). The charge ordered state was originally discovered a quarter century ago below T charge ∼ 60 K in the low temperature tetragonal (LTT) structure of La 1. 48 Nd 0.4 Sr 0.12 CuO 4 [3]. Years later, evidence for charge order based on neutron and X-ray scattering experiments also emerged in the LTT structure of La 1.875 Ba 0.125 CuO 4 (T charge 54 K) [4][5][6], followed by La 1.68 Eu 0.2 Sr 0.12 CuO 4 (T charge ∼ 80 K) [7], rather than the low temperature orthorhombic (LTO) structure of the canonical superconducting phase with much higher T c . Accordingly, many researchers continued to believe that the LTO to LTT structural transformation was the key to stabilizing the long range charge ordered state, which in turn suppresses superconductivity. However, recent advances in X-ray scattering techniques finally led to successful detection of charge order Bragg peaks even in the LTO structure of La 1.885 Sr 0.115 CuO 4 (T c 30 K) below as high as T charge 80 K [8][9][10].
Two decades have passed since our initial reports that all of these La214 type cuprates undergo charge order at comparable temperatures [11][12][13][14][15], on the ground that they all share nearly identical NMR anomalies identified at T charge of La 1. 48 Nd 0.4 Sr 0.12 CuO 4 . During these years, NMR techniques made major advances both in the instrument technologies and data analysis methods. Owing to the reduction in the signal detection dead time of the NMR spectrometers after the application of radio frequency pulses, routine NMR measurements have become possible with the pulse separation time as short as τ ∼ 2 µs between the 90 degree excitation and 180 degree refocusing pulses. This τ is an order of magnitude shorter than the typical value τ ∼ 20 µs used in the 1980's, and detection of the paramagnetic 63 Cu NMR signals with extremely fast NMR relaxation rates, which arises from the charge ordered domains (represented schematically by islands with various shades in Fig. 1(b-c)), has become feasible [16,17]. Moreover, the development of the inverse Laplace transform (ILT) T 1 analysis technique enabled us to deduce the histogram of the distribution of the nuclear spin-lattice relaxation rate 1/T 1 (i.e. the probability density distribution P (1/T 1 )) [18][19][20][21], in addition to the average value of the distributed 1/T 1 estimated from the conventional stretched exponential fit. The recent ILTT 1 analysis of 139 1/T 1 measured at the 139 La sites in La 1.875 Ba 0.125 CuO 4 [19] and La 1.885 Sr 0.115 CuO 4 [20] established the continued presence even below T charge of the canonical domains, which exhibit canonical properties expected for superconducting CuO 2 planes without anomalous enhancement of Cu spin fluctuations triggered by charge order. This new finding based entirely on 139 La NMR supports our original conjecture [11][12][13] that peculiar domain-by-domain variation emerges immediately below T charge due to the spatially growing charge ordered domains, as summarized in Fig.1.
In this paper, we revisit the earlier 63 Cu nuclear quadrupole resonance (NQR) report on the issue of charge order in La 1.875 Ba 0.125 CuO 4 [11,13] based on comprehensive single crystal 63 Cu NMR results, and compare our findings with 139 La NMR results observed for the same crystal [19]. Since La 1.875 Ba 0.125 CuO 4 has a well-defined, sharp charge order transition at T charge 54 K as determined by X-ray diffraction experiments and lacks magnetic perturbations caused by additional Nd 3+ spins, it is an ideal platform to test the NMR response that sets in precisely at T charge . We confirmed a precur-sor of enhanced spin correlations below ∼ 80 K based on the 63 Cu NMR linewidth data [22] and 1/T 1 [23], where the ILTT 1 analysis of the 139 La NMR data uncovered the presence of low frequency modes in the lattice and/or charge fluctuations [19]. These precursors are followed by dramatic, spatially inhomogeneous enhancement of low frequency spin fluctuations within charge ordered domains that begin to nucleate at T charge 54 K. The volume fraction F CO of the charge ordered domains is not 100% immediately below T charge , and grows only progressively below T charge . We estimate the volume fraction F CA = (1 − F CO ) of the canonical domains based on 63 Cu NMR spin echo decay M (2τ ) measured over a wide time range from 2τ = 4 to 100 µs. The temperature dependence of F CA (Fig.7) shows excellent agreement with the independent estimation based entirely on the ILTT 1 analysis of the 63 Cu NMR 63 1/T 1 data (Fig.5) and 139 La NMR 139 1/T 1 data (Fig.8) [19]. Moreover, we show that these canonical domains almost completely disappear by ∼ 35 K, far above T c 4 K of La 1.875 Ba 0.125 CuO 4 . This finding is in remarkable contrast with the case of La 1.885 Sr 0.115 CuO 4 , where the canonical domains still occupy nearly a half of the CuO 2 planes when superconductivity sets in at T c 30 K [14,20].
The rest of this article is organized as follows. In section 2, we provide a brief overview of the NMR response expected in the charge ordered CuO 2 planes of the La214 cuprates. In section 3, we present results and discussions, followed by summary and conclusions in section 4.

II. NMR RESPONSE IN CHARGE ORDERED LA214 CUPRATES
Until fairly recently, tremendous confusions persisted since our original publications asserting the presence of charge order in La 1.875 Ba 0.125 CuO 4 and related La214 materials. This is primarily because, except for La 1.48 Nd 0.4 Sr 0.12 CuO 4 [3], charge order Bragg peaks were not successfully detected in many La214 cuprates for years [4,[7][8][9][10]. This unfortunate circumstance misled a large number of researchers to argue that charge order was absent in all the cuprates but La 1.48 Nd 0.4 Sr 0.12 CuO 4 , and cast doubt on the link between the 63 Cu NMR anomalies and charge order (see, for example [26,27]). Moreover, many NMR experts overlooked, or failed to understand the implications of the following crucial statement in our original publication, quoted verbatim from Hunt et al. [11]:charge order turns on low frequency spin fluctuations [28], and consequently the 63 Cu nuclear spin-lattice and spin-spin relaxation rates diverge in the striped domains.
The idea outlined in this short statement about the spatial inhomogeneity of magnetic properties induced by charge order is the key to the proper understanding of the NMR data of all the La214 cuprates below T charge . Let us elaborate with the aid of Fig.2, in which we contrast the behaviors in the charge ordered state in the left  [11-13, 24, 25]. (a) Above T charge , a mild spatial modulation of the local hole concentration (represented by varying shades) exist with nm length scales, arising from the quenched disorder effects induced by random substitution of Ba 2+ ions in La 3+ sites [24,25]. (b) Small charge ordered domains nucleate immediately below T charge (islands with varying shades, in which the charge rivers are represented by vertical or horizontal lines). But the charge correlation length is still as short as several nm [5,6] and these domains are strongly disordered. As the size of the charge ordered domain grows (islands with lighter shade), Cu spin fluctuations slow down and hence enhance 1/T1. The disordered nature of the charge ordered state implies that the level of enhancement of 1/T1 varies domain by domain, resulting in a large distribution in the enhancement of 1/T1. A majority of Cu sites are not affected by charge order, and maintain the canonical properties expected for superconducting CuO2 planes. 63 Cu NMR signals in charge ordered domains are observable only with extremely short τ ∼ 2 µs. On the other hand, 63 Cu NMR signals from the canonical domains are easily observable with τ ∼ 10 µs or longer, but their volume fraction FCA gradually diminishes below T charge . (c) With decreasing temperature below T charge , charge ordered domains grow their domain size up to ∼ 20 nm [5,6], further enhancing 1/T1 in larger domains. (d) The temperature dependence of the volume fractions of the canonical domains FCA in panels (a-c). We can also estimate the volume fraction of the charge ordered domains FCO from FCO + FCA = 1. panels with those for typical antiferromagnets without any spatial inhomogeneity, such as La 2 CuO 4 [29,30], in the right panels. In Fig.2(a), we sketch the temperature dependence of the imaginary part of the dynamical local spin susceptibility Imχ(ω) at very low energy transfer ω using solid curves. See Fig. 7 in [28] for the first original data for La 1.48 Nd 0.4 Sr 0.12 CuO 4 , and Fig. 1(c) in [5] as well as Fig. 8(a) in [4] for the original data for La 1.875 Ba 0.125 CuO 4 . In the case of La 1.875 Ba 0.125 CuO 4 , Imχ(ω) at the low energy transfer of ω = 0.5 meV begins to grow dramatically precisely at T charge 54 K [5].
This inelastic glassy spin response induced by charge order below T charge 54 K should not be confused with the elastic response of Bragg scattering arising from the static spin order at T spin 40 K [4,5]. Simply put, charge order creates small, finite size domains, in which Cu spins are slowly and collectively fluctuating without entering the long range spin ordered state. The long range spin order takes place when the charge correlation length grows to ∼ 20 nm at T spin 40 K [5,6], where charge ordered domains become interconnected. This sequence of order may look similar to the nucleation of charge density wave (CDW) ordered domains in NbSe 2 [31] above the long range CDW ordered state, but the stripe charge order in La214 materials is accompanied by slow Cu spin fluctuations below T charge . NMR exhibits spectacular responses to the latter, because NMR is a local, low frequency probe, and hence we can use these NMR responses as the fingerprints of charge order as explained in the following paragraphs.
Since 1/T 1 ∝ T · Imχ(ω)/ω at the limit of ω → 0 [32,33], one would expect that 1/T 1 shows strong enhancement below T charge , as shown by a solid curve in Fig.2(b). This is somewhat different from the divergent behavior of 1/T 1 toward the Néel temperature T N in conventional antiferromagnets shown in panel (f). In the latter, Imχ(ω)/ω diverges toward T N due to spatially homogeneous critical slowing down of spin fluctuations that begins without any onset temperature, then the elastic response in neutron scattering sets in below T N as shown in Fig.2(e). In the case of charge ordered La 1.875 Ba 0.125 CuO 4 , the elastic magnetic response (Bragg peaks) sets in only below the spin ordering temperature, T spin 40 K [4,5], but strong enhancement of 1/T 1 precedes below the clear onset temperature at T charge due to the glassy nature of spin fluctuations induced by charge order.
Such an upturn of 139 1/T 1 below T charge can be easily observed at 139 La sites by NQR [13], and more recent high precision NMR measurements confirmed that the onset of the drastic growth of 139 1/T 1 is precisely at T charge 54 K [34]. It is important to note, however, that these earlier 139 1/T 1 values were estimated based on the stretched exponential fit, and probed only the average behavior of the entire sample. In fact, based on the ILTT 1 analysis of the T 1 recovery curve at 139 La sites, we recently demonstrated that the fastest component of 139 1/T 1 indeed begins to grow precisely below T charge (the solid curve in Fig.2(b)), but the slower components exhibit no anomaly through T charge (dashed curve in Fig.2 [20]. In other words, 1/T 1 exhibits qualitatively different behaviors domain by domain. Some parts of CuO 2 planes are not immediately affected by charge order and continue to exhibit canonical behavior expected for superconducting CuO 2 planes even below T charge , as schemat- The inelastic neutron scattering intensity at low energy transfers [5]. See the main text for details. (b) 63 Cu NMR 1/T1 develops a spatial distribution below T charge with qualitatively different temperature dependences, showing an upturn for short τ1 but no anomaly for long τ2. (c) The Lorentzian (exponential) spin echo decay curve M (2τ ) observed for the Bext || ab axis geometry above T charge begins to exhibit initial fast decay in the short time regime below T charge , followed by slow decay in the long time regime. (d) The bare 63 Cu NMR signal intensity measured at a fixed 2τ1 as well as 2τ2 begins to drop at T charge . The total 63 Cu NMR intensity (gray horizontal line), arising from both the charge ordered and canonical domains, can be estimated by extrapolating M (2τ ) to 2τ = 0 in the short time regime in (c). The total intensity is proportional to the number of 63 Cu nuclear spins in the entire sample, and temperature independent. On the other hand, the volume fraction FCA of 63 Cu nuclear spins in the canonical domains, as estimated from the extrapolation of the slowly relaxing component of M (2τ ) in the long time regime of (c), begins to drop below T charge , i.e. 63 Cu NMR signal intensity wipeout effect induced by charge order [11,13].
ically shown in Fig.1(b). These findings are also consistent with the two decade old knowledge that 1/T 1 measured at 63 Cu sites of La 1.875 Ba 0.125 CuO 4 with a relatively long τ = 20 µs exhibits no anomaly at T charge [35], because it preferentially reflects the canonical domains with slow transverse relaxation times, whereas 1/T 1 measured at the 63 Cu sites with very short τ exhibits an upturn of 1/T 1 below T charge [23].
The continued presence of 139 La and 63 Cu NMR signals exhibiting the canonical behavior below T charge implies that not all Cu electron spins are involved in the low energy upturn of Imχ(ω) shown by the solid curve in Fig.2(a). Instead, some Cu electron spins continue the trend observed above T charge , as shown by the dashed curve in Fig.2(a). Since inelastic neutron scattering measures only the volume integral of the spin response, one needs to rely on a local probe such as NMR to reveal the domain by domain response schematically summarized in Fig.1.
The unusual magnetic inhomogeneity induced in the charge ordered state is also reflected on the transverse T 2 relaxation process observed for the transverse nuclear magnetization M (2τ ) at 63 Cu sites, as schematically summarized in Fig.2(c) (see Fig.6 below for the actual data). Upon entering the charge ordered state, M (2τ ) begins to exhibit initial fast decay in the short time regime, followed by slower decay in the long time regime. The transverse relaxation rate in the latter is comparable to that observed at T charge and above. The nuclear spins responsible for the fast and slow transverse relaxation in Fig.2(c) can be attributed to the 63 Cu sites located in the charge ordered and canonical domains in Fig.1, respectively.
We can estimate the volume fraction F CA of the canonical domains by extrapolating the slow decaying part of the M (2τ ) curve to 2τ = 0, as shown by dashed lines in Fig.2(c). The intercept of the extrapolated dashed line with the vertical axis at 2τ = 0 yields F CA . We emphasize that, if the charge ordered CuO 2 planes undergo uniform enhancement of spin correlations, then M (2τ ) curve would look very different, and should be similar to the case of uniform antiferromagnets shown in Fig.2(g).
In Fig.2(d), we summarize the temperature dependence of the 63 Cu NMR signal intensity at short 2τ 1 and long 2τ 2 , as expected from Fig.2(c). The total intensity in the limit of 2τ = 0 is proportional to the number of nuclear spins that does not change with temperature, and hence always conserved, as shown by the horizontal gray line. For the finite values of 2τ , the intensity exhibits an anomaly at T charge , because the signal intensity arising from the charge ordered domains is reduced by the fast transverse relaxation in the short time regime in Fig.2(c). As explained in the previous paragraph, one can estimate the volume fraction of the canonical domains F CA by extrapolating M (2τ ) in the long time regime of Fig.2(c) to 2τ = 0. The end result would be the solid curve in Fig.2(d). Recalling F CO +F CA = 1, one can also estimate F CO as shown in Fig.1  The signal intensity is normalized for the Boltzmann factor by multiplying temperature T . The peak frequency in (b) is dominated by the temperature independent chemical shift ∼ 1.2 % [36]. The double peak structure in (a) arises from the difference in the nuclear quadrupole frequency νQ [24,37]. The 63 Cu(B) sites have a larger νQ due to the presence of a Ba 2+ ion at the nearest neighbor La 3+ sites than the majority 63 Cu(A) sites. (Inset) The temperature dependence of the integrated intensity of the lineshapes in panels (a) and (b). Note that these are just raw data without any analysis, and correspond to the dashed curve in Fig.2(d) with 2τ = 24 µs. We estimated the total intensity in the limit of 2τ = 0 (defined as 1 after normalization) from the spin echo decay curves above T charge .
signal intensity wipeout effect to probe the volume fraction of the charge ordered domains, developed originally in [11][12][13].

III. RESULTS AND DISCUSSIONS
A. NMR lineshapes In Fig.3(a-b), we summarize the representative 63 Cu NMR lineshapes for a 51 mg single crystal [4] measured with a fixed pulse separation time τ = 12 µs in an external magnetic field B ext = 9 T. The typical radio frequency pulse width was 2 µs and 4 µs for 90 and 180 degree pulses throughout this work. We confirmed both above (60 K) and below (50 K) T charge 54 K that the lineshape hardly changes even if we use τ = 2 µs, except that the transverse relaxation process (i.e. T 2 ) reduces the overall intensity for longer τ . These lineshapes indicate that La 1.875 Ba 0.125 CuO 4 develops its charge ordered state in a fundamentally different manner from La 1.885 Sr 0.115 CuO 4 . We recall that the c-axis 63 Cu NMR lineshapes in the latter comprised of two distinct types of signals below its higher T charge 80 K: (i) a narrower, canonically behaving peak with slower relaxation rates that are typical for high T c cuprates, and (ii) a much broader wing-like signal with extremely fast relaxation rates [14]. The former is gradually wiped out below T charge , transferring the spectral weight to the latter. Accordingly, the NMR lineshapes completely change between τ = 2 µs and τ = 12 µs below T charge , since only the canonically behaving narrower peak can be detected with τ = 12 µs. That is not the case here for La 1.875 Ba 0.125 CuO 4 , and the observed c-axis lineshapes are more uniformly broadened even for τ = 12 µs.
In the inset of Fig.3, we also summarize the temperature dependence of the integral of these τ = 12 µs lineshapes. We emphasize that the intensity data are merely the integral of the lineshapes in panels (a) and (b), and have not been subjected to any data analysis. The sharp anomaly observed at ∼ 54 K in the raw integrated intensity corresponds to that in the conceptual sketch of the dashed curve in Fig.2(d) for 2τ 2 = 24 µs, and signals a lurking phase transition in La 1.875 Ba 0.125 CuO 4 .

B. Linewidth
In Fig.4(b), we summarize the temperature dependence of the half width at the half maximum (HWHM) of the c-axis lineshapes shown in Fig.3(b). For the B ext || caxis geometry, the second order nuclear quadrupole effect vanishes [36], and the temperature dependence of the linewidth is set almost entirely by magnetic effects [38]. On the other hand, since the lower frequency side of the broadened NMR lineshape nominally has negative frequency shifts, a large distribution of the chemical shift cannot account for the observed broadening, either. Therefore, the spin degrees of freedom must be playing the key role in the line broadening, but the exact mechanism of the broadening has long been an enigma.
The dashed curve overlaid on the HWHM data points above ∼ 80 K are the best empirical Curie-Weiss fit. The HWHM begins to grow more quickly below 80 K. Our single crystal result is consistent with an earlier aligned powder result [22]. This linewidth anomaly is accompanied by an analogous deviation from the Curie-Weiss growth of 1/T 1 T at 63 Cu sites [23], signaling that strong enhancement of antiferromagnetic spin correlations is playing a role in HWHM as well. Interestingly, charge order sets in for a minor volume of La 1.885 Sr 0.115 CuO 4 also at ∼ 80 K and the aforementioned wing-like 63 Cu  [35]. Also shown using grey open bullets is the center of gravity of the distributed 139 1/T1 at 139 La sites of the same single crystal, deduced from inverse Laplace transform of the nuclear spin recovery curve [19], whereas the dashed curves below 100 K represent the top or bottom 10% value of the distributed 139 1/T1. (b) 1/T1T , presented with the same symbols as in the panel (a). Also shown using the right axis is the HWHM of the c-axis lineshapes in Fig.3(b). The dashed curve represents the Curie-Weiss fit at higher temperatures with Weiss temperature −40 K. Extrapolation of the fit below ∼ 80 K underestimates the HWHM by ∼ 40% at T charge .
NMR signal emerges [17], but it may be a coincidence.
We recently showed based on the ILTT 1 analysis of the 139 La nuclear spin-lattice relaxation curve that the electric field gradient (EFG) at the 139 La sites has slowly fluctuating components (∼ MHz) below ∼ 80 K [19]. The ILT cannot distinguish the origin of the slow dynamics between the lattice and/or charge degrees of freedom. Regardless of the origin, these NMR results indicate that spin correlations begin to grow more steeply when fluctuations of the lattice and/or charge degrees of freedom slow down. It is also interesting to note that recent X-ray scattering data showed the dynamic short range charge order above T charge [6]. All pieces put together, the onset of charge order in La 1.875 Ba 0.125 CuO 4 seems to be suppressed to ∼ 54 K, until the LTO to LTT structural phase transition suddenly takes place.
C. 63 Cu spin-lattice relaxation rate 63 1/T1 We measured 63 1/T 1 at the center of the B ext || c axis peak in Fig. 3(b) using the standard inversion recovery method by applying a 180 degree pulse prior to the spin echo sequence. The goodness of the fit of the nuclear spin recovery curve M (t) with the standard formula for the central transition was similar to the case of La 1.885 Sr 0.115 CuO 4 [17], and stretching was not necessary for our purpose even below T charge owing to modest distributions, as shown in Appendix. This is simply because the signals arising from 63 Cu sites with very fast 63 1/T 1 are wiped out below T charge even for τ = 2 µs (see Fig.6 below), and hence hardly contribute to the M (t) results. In Fig.4(a), we summarize the temperature dependence of 63 1/T 1 observed for three different values of the pulse separation time τ = 2, 12, and 20 µs between the 90 and 180 degree pulses, and compare the results with 139 1/T 1 observed at the 139 La sites [19].
The gradual decrease of 63 1/T 1 with temperature observed for τ = 20 µs down to 48 K is typical for high T c cuprates [39]. The extremely broad, small NMR signal from a small single crystal made accurate measurements of 63 1/T 1 difficult below 48 K. For comparison, we show the 63 1/T 1 results measured with τ = 20 µs for an aligned powder sample [35]. The signal intensity was large and manageable even below 48 K for the large amount (∼ 300 mg) of aligned powder, and the decreasing trend of 63 1/T 1 continues below 48 K. These results for τ = 20 µs show no anomaly through T charge , and indicate that some parts of CuO 2 planes remain unaffected by charge order and the resulting enhancement of spin fluctuations even deep into the charge ordered state below T charge . That is why we initially overlooked the lurking charge order in 1990 [35].
The 63 1/T 1 results for τ = 2 µs are consistently larger by ∼ 6% than those for τ = 20 µs down to ∼ 80 K. This is merely because the quenched disorder caused by Ba 2+ substitution into the La 3+ induces a nanoscale inhomogeneity in the local hole concentration of the CuO 2 planes [24,25], as represented schematically by different shades in Fig.1(a). 63 1/T 1 is generally smaller for larger values of the hole concentration x [24,29], but the 63 Cu NMR peak frequency for the B ext || c axis geometry is set entirely by the chemical shift that is independent of x. Moreover, the 63 Cu B-sites located at the nearest neighbor of La 3+ sites are superposed in this field geometry [40], and their 63 1/T 1 is somewhat slower than those at the main 63 Cu A-sites [24]. Accordingly, a mild distribution of 63 1/T 1 is always present. As shown below in Fig.6(a), the transverse relaxation does not reduce the spin echo intensity significantly for τ = 2 µs above T charge , and hence nearly 100% of the 63 Cu nuclear spins contribute to the observed value of 63 1/T 1 . 63 1/T 1 for τ = 2 µs levels off towards T charge , and shows qualitatively different behavior from the τ = 20 µs results. This is consistent with the increased distribution of 139 1/T 1 observed at 139 La sites in the same temperature range [19,34]. 63 1/T 1 measured with τ = 2 and 12 µs begins to increase precisely below T charge , in agreement with the earlier NQR report by Tou et al. [23]. Pelc et al. observed greater values of 63 1/T 1 below T charge with NQR than Tou et al., because they used τ = 2 µs and captured more nuclear spins with faster relaxation rates [16]. Our 63 1/T 1 results for τ = 2 µs is slower below T charge than Pelc et al.'s, probably because the fastest nuclear spins are pushed aside to the tail sections of the magnetically broadened NMR lineshape below T charge due to locally stronger spin correlations.
We present 63 1/T 1 T in Fig.4(b) in comparison to the HWHM. 63 1/T 1 T probes the wave vector q integral of Imχ(q, ω n ). 63 1/T 1 T obeys the Curie-Weiss behavior analogous to that observed for the HWHM. The 63 1/T 1 T results for τ = 2 µs as well as the HWHM begin to deviate from the Curie-Weiss behavior somewhat above T charge , in agreement with the earlier report by Tou et al. as noted above [23].
The enhancement of 63 1/T 1 observed at 63 Cu sites below T charge is only modest, compared with the steep divergent behavior observed at 139 La sites for the same crystal (grey bullets) [41]. This apparent discrepancy arises from the fact that 139 1/T 1 plotted for the 139 La sites represents the spatially averaged (center of gravity) value of the widely distributed 139 1/T 1 below T charge . In contrast, 63 1/T 1 at 63 Cu sites reflects only the nuclear spins that are still observable below T charge owing to their slower NMR relaxation rates. For example, the observable 63 Cu NMR signal intensity at 48 K is only about a half of the total intensity even for τ = 2 µs, because a majority of 63 Cu NMR signals is already suppressed by their extremely fast transverse relaxation rates (see the data points at 2τ = 4 µs in Fig.6(b) below).
To underscore this point, we used dashed lines in Fig.4(a) to mark the top and bottom 10% values of the distributed 139 1/T 1 at 139 La sites estimated from the ILTT 1 analysis [19]. The comparison indicates that 1/T 1 measured at the observable 63 Cu sites below T charge , especially for the longer values of τ , reflects only the bottom end of the spatial distribution in spin fluctuations.
These findings can be corroborated by the density distribution function P ( 63 1/T 1 ) at 63 Cu sites deduced by ILT. In Fig.5, we summarize P ( 63 1/T 1 ) obtained from the M (t) curves measured with τ = 2 µs in Fig. 9(a). The integrated area underneath the P ( 63 1/T 1 ) curve for 50 K is set to 0.63 to reflect the suppressed signal intensity observed with τ = 2 µs at 50 K, as shown in Fig. 6(a) in the next section. In general, the stretched fit value of 1/T 1 is merely a crude approximation of the center of gravity of the distribution P (1/T 1 ) [19]. In fact, we found that the center of gravity 63 1/T 1 = 1840 s −1 of P ( 63 1/T 1 ) at 50 K is close to the stretched fit value (1920 s −1 ) plotted in Fig.4. It is important to notice, however, that 63 1/T 1 at 50 K has both the slower and faster components than 100 K. The fast components reach as large as ∼ 10 4 s −1 , in agreement with the expectations from the divergent growth of 139 1/T 1 below T charge . On the other hand, the slow components extends to below 10 3 s −1 , again in agreement with the expectations from 63 1/T 1 measured with τ = 20 µs.

D. Spin echo decay
In Fig.6, we summarize the representative spin echo decay curves M (2τ ) observed at the peak of the NMR lineshapes in Fig.3. We normalized the magnitude of M (2τ ) with the integral of the lineshape in Fig.3 as well as the Boltzmann factor. Thus M (2τ ) in the limit of 2τ = 0 represents the temperature independent total intensity as represented schematically by the grey horizontal line in Fig.2(d). For clarity, we normalize the total intensity to M (2τ = 0) = 1.
The B ext || c results in Fig.6(a) show the typical Gaussian-Lorentzian decay form for the long time regime above 2τ ∼ 20 µs both above and below T charge ; the indirect nuclear spin-spin coupling causes the Gaussian curvature associated with the real part Reχ(q, ω) of the dynamical electron spin susceptibility of Cu [42,43]. For the B ext || ab geometry shown in Fig.6(b), this Gaussian contribution is motionally narrowed to Lorentzian [42]. In addition, Redfield's T 1 process associated with the imaginary part Imχ(q, ω) of the dynamical electron spin susceptibility of Cu leads to the exponential, Lorentzian process in both field geometries. Since Reχ(q, ω) and Imχ(q, ω) are related to each other, 1/T 1 T and 1/T 2 in cuprates generally exhibit analogous temperature dependences [30,44,45], unless the pseudo-gap strongly suppresses only 1/T 1 T [44].
As explained above using Fig.2(c), (d), (g), and (h), M (2τ = 0) in the limit of 2τ = 0 is proportional to the number of nuclear spins in our sample, and is a conserved quantity. Above T charge , the extrapolation of M (2τ ) curves to 2τ = 0 based on the Gaussian-Lorentzian and Lorentzian fit for the B ext || c and B ext || ab geometry, respectively, is consistent with such expectations.
Notice, however, that the situation completely changes once charge order sets in at T charge . M (2τ ) begins to exhibit a very fast initial decay from 2τ = 0 up to 2τ ∼ 15 µs for both field geometries. For example, as shown by the green solid line in Fig.6(a), the M (2τ ) measured at 48 K in B ext || c decays quickly from M (2τ = 0) = 1 to M (2τ = 10 µs) ∼ 0.2 with positive curvature, followed by much slower Gaussian-Lorentzian decay above 2τ = 15 µs with the relaxation times comparable to those observed above T charge . The crossover of M (2τ ) from the short to long time regime is observed only below T charge , and indicate emergence of the distributed fast transverse T 2 relaxation processes in the charge ordered state. This corresponds to the analogous crossover depicted in Fig.2(c). That is, the CuO 2 planes develop strong inhomogeneous spin correlations as soon as charge order sets in.
The M (2τ ) results in Fig.6 are also consistent with the reduction of the signal intensity below T charge observed at a fixed 2τ = 24 µs in the inset of Fig.3. That is, the signal loss reflects the emergence of glassy spin state in the charge ordered domains, where the NMR relaxation rates 1/T 1 and 1/T 2 become divergently large, resulting in the initial quick decay in the spin echo intensity M (2τ ).
We note that our original publications two decades ago probed M (2τ ) only in the long time regime above 2τ 20 µs [11] because the spectrometer dead time prevented us from accessing the short time regime in Fig.6 and Fig.2(c). Our original M (2τ ) data barely missed the crossover from the short to long time regime [11]. On the other hand, a recent work by Pelc et al. presented M (2τ ) data only in the short time regime, and did not demonstrate the crossover to the long time regime, either [16] (see their Fig. 3(a)). Pelc et al.'s limited data set might inadvertently leave unsuspecting readers with a false impression that CuO 2 planes in charge ordered La 1.875 Ba 0.125 CuO 4 is spatially homogeneous, and exhibit uniformly fast transverse relaxation, similar to the case of homogeneous antiferromagnet shown in Fig.2(g). But our new data presented in Fig.6 firmly establish that is not the case. We also emphasize that what matters in understanding the inhomogeneous glassy state induced by charge order based on 63 Cu NMR intensity is the extrapolation of the slowly decaying part of M (2τ ) observed in the long time regime above 2τ = 20 µs to 2τ = 0 µs, as explained in section 2 and shown with the dashed curves in both Fig.2(c) and Fig.6. Instead, Pelc et al. examined the extrapolation of the fast decaying part of their M (2τ ) data in the short time regime to 2τ = 0 (solid curves extrapolated to 2τ = 0 in their Fig. 3(a)), only to confirm that the total intensity arising from both the charge ordered and canonical domains is conserved. Their finding below T charge summarized in the inset of their Fig.  2 corresponds to the trivial conservation law of the total intensity F CO + F CA = 1, represented schematically by the gray horizontal line in Fig.2(d). It does not provide any useful insight into the nature of the glassy, charge ordered state.
E. 63 Cu NMR signal intensity wipeout and estimation of FCA based on ILT Finally but not the least, let us return to the issue of the integrated intensity of the 63 Cu NMR lineshapes in the inset of Fig.3. To eliminate the minor effects of the transverse relaxation on the bare integrated intensity measured at a fixed τ = 12 µs, we extrapolated the M (2τ ) curves to 2τ = 0 as shown by the dashed lines in Fig.6(b), and estimated the volume fraction F CA of the canonical domains. We summarize the B || ab-axis results in Fig.7 using bullets [46]. For comparison, we also plot our original signal intensity wipeout data measured with NQR for a 63 Cu isotope enriched powder sample (open triangles) [13]. The agreement between the new NMR and older NQR results is satisfactory, in view of the greater uncertainties in the latter arising from the extra Gaussian T 2G term in the spin echo decay.
As explained in detail in section 2 using Fig.2(d), the temperature dependence of the 63 Cu NMR signal intensity in Fig.7 indicates that charge order does not set in homogeneously in the CuO 2 planes. The finite value of F CA below T charge implies that a significant fraction of the volume is hardly affected by charge order even below T charge , and exhibits the canonical behavior expected for CuO 2 planes that seem destined to undergo superconducting transition at T c 30 K. But the residual volume fraction of such canonically behaving CuO 2 planes almost vanishes by ∼ 35 K, where the charge correlation length saturates at ∼20 nm [5]. In addition, spin stripe order sets in at ∼ 35 K at the time scale of µSR experiments [47,48] and the volume-averaged value of the distributed 139 1/T 1 is peaked at 139 La sites [19,34]. In contrast, the volume fraction of the canonical domains exceeds 40 % at its T c = 30 K in La 1.885 Sr 0.115 CuO 4 [17,19,20].
We can achieve more quantitative understanding of the 63 Cu NMR intensity anomaly and its relation with the unconventional nature of charge order with the aid of the ILTT 1 analysis of the 139 La nuclear spin recovery curve [19]. For convenience, we reproduce the key results of the probability density distribution function, P ( 139 1/T 1 ) of 139 1/T 1 in Fig.8. The main peak of P ( 139 1/T 1 ) has finite values only below 139 1/T 1 1 s −1 from 100 K down to 60 K. In other words, the upper bound of the distributed values of 139 1/T 1 is 1 s −1 . Notice, however, that the main peak gradually broadens below 77 K, accompanied by a small split-off peak centered around 139 1/T 1 5 s −1 . Analogous anomalies of P ( 139 1/T 1 ) are observed also around 240 K near the high temperature tetragonal to low temperature orthorhombic structural phase transition [19]. Since the charge order transition is accompanied by a first order structural transition from low temperature orthorhombic to low temperature tetragonal phase [4], we can attribute these anomalies slightly above T charge to the contributions of fluctuating electric field gradient precursor to the structural phase transition and/or fluctuating charges [19].
As temperature is lowered through T charge , P ( 139 1/T 1 ) gradually transfers spectral weight to larger values of 139 1/T 1 while broadening asymmetrically. This corresponds to the fact that a sharp divergent behavior sets in precisely at T charge for 139 1/T 1 estimated from the stretched fit, which tends to be close to the center of gravity of the distributed 139 1/T 1 [19]. We emphasize that a half of the spectral weight of P ( 139 1/T 1 ) still remains below 1 s −1 even at 50 K. This implies that the corresponding sample volume is still unaffected by charge For comparison, we also reproduce the powder NQR intensity (black triangles, adopted from [13]). The increase in the NQR intensity below ∼ 15 K is due to the freezing of the fluctuations of the hyperfine magnetic fields from Cu electron spins; we multiplied a factor of 1.63 [13] for the Zeeman perturbed NQR results below 15 K to account for the missing contribution below 25 MHz [13]. order, and 139 1/T 1 is as slow as at 77 K. This is consistent with our findings for 63 1/T 1 at 63 Cu sites in Fig.4 and 5 . 63 Cu nuclear spins that are still easily observable below T charge owing to slow NMR relaxation rates are located in the same domains as these 139 La sites with slower relaxation rates.
In view of the fact that the ILT curve P ( 63 1/T 1 ) observed at 50 K in Fig. 5 has a well-defined peak associated with the canonical domains with slower 63 1/T 1 , perhaps it may be somewhat surprising to find that P ( 139 1/T 1 ), which should also encompass the canonical component centered around 139 1/T 1 0.5 s −1 , is broader and increasingly featureless below T charge . But this is simply because the transverse relaxation does not suppress the faster components of P ( 139 1/T 1 ) arising from charge ordered domains. In this context, we recall that the charge correlation length in the charge ordered state is known to be as short as several nm immediately below T charge , and the spin correlation length cannot exceed it. This means that the extent of enhancement of 1/T 1 in each charge ordered domain is set by the domain size. The highly disordered nature of the charge ordered state with varying domain sizes naturally explains the very broad distribution of 139 1/T 1 below T charge , ranging from the small canonical value to the upper bound set by the largest charge ordered domains. It is also worth noting that P ( 139 1/T 1 ) curve exhibits somewhat more distinct features in La 1.885 Sr 0.115 CuO 4 for the canonical and charge ordered domains [20]. That is probably because the canonical domains are more robust below T charge in a wider temperature range in La 1.885 Sr 0.115 CuO 4 , and in agreement with the fact that T c is as high as ∼ 30 K.
The featureless, continuous distribution of P ( 139 1/T 1 ) makes it difficult to de-convolute P ( 139 1/T 1 ) and estimate F CA from P ( 139 1/T 1 ). We therefore introduce a cut off in Fig.8 at 2 s −1 , at the upper end of the distributed values of 139 1/T 1 observed at 56 K, as represented by the upward vertical arrow in Fig.8. Then we can estimate the F CA of the canonical 63 Cu nuclear spins as the integrated area of P ( 139 1/T 1 ) below the cut-off. We summarize the temperature dependence of thus estimated F CA in Fig.7 using × symbols, in comparison to F CA estimated from the 63 Cu NMR intensity. Despite the simplicity of this analysis, the estimation based entirely on the 139 La NMR results reproduces the 63 Cu NMR signal intensity anomaly very well.
We can also test the consistency of F CA with the 63 Cu ILT result of P ( 63 1/T 1 ) at 50 K. From the integral of the light dashed curve in Fig.5 arising from the canonical contribution with slower relaxation rate, we estimate F CA ∼ 0.63 at 50 K. We plot the result in Fig.7 with a diamond, in comparison to F CA estimated from two other methods, the extrapolation of M (2τ ) (bullets) and cut-offs introduced for P ( 139 1/T 1 ) (×). Despite the completely different methodologies between the three approaches, agreement is good.
Turning our attention to the low temperature side below 30 K, Zeeman perturbed NQR signal is known to La nuclear spin-lattice relaxation curve (adopted from [19]). The black dashed vertical arrows represent the cut-off 1 and 2 set at 139 1/T1 = 2 s −1 and at 139 1/T1 = 0.6 s −1 , respectively, used for estimating FCA above and below 30 K. A small split-off peak observed at 240 K (marked by EFG), accompanied by the broadening of the main peak, is caused by slow fluctuating electric field gradient due to the structural phase transition. Analogous features manifest slightly above T charge as well.
reemerge when the hyperfine magnetic field from frozen Cu electron spins become static at the NMR measurement time scale below ∼ 15 K [13,49]. In general, 1/T 1 is proportional to the dynamical spin susceptibility Imχ(ω) multiplied by temperature T , and hence the cut-off of 139 1/T 1 = 2 s −1 for the same magnitude of Imχ(ω) needs to be scaled down to 139 1/T 1 = 0.6 s −1 by the ratio between 15 K and T charge . This second cut-off is shown with a downward vertical arrow in Fig.8. We can estimate the fraction of frozen Cu electron spins as the area integral below this cut-off. The results, also shown in Fig.7 using ×, reproduce the qualitative aspects of the signal intensity recovery observed by Hunt et al. [13]. The agreement can be improved if we estimate F CA based on P ( 139 1/T 1 T ) (i.e. the distribution of 139 1/T 1 divided by T ) by introducing a single cut-off at 139 1/T 1 T = 0.036 s −1 K −1 for both above and below 30 K; this cut-off value corresponds to 139 1/T 1 = 2 s −1 divided by T charge . We present these estimations using + symbols also in Fig.7.
Strictly speaking, the cut-off for dividing the canonical and charge ordered domains at 2 s −1 for 139 1/T 1 should be slightly temperature dependent, because 139 1/T 1 in canonically superconducting compositions with x ∼ 0.15 decreases slightly below 54 K toward T c = 38 K [40,50]. But the observed decrease is weak, and the varying cutoff hardly affects our estimation of F CA . In fact, the cut-off 139 1/T 1 T = 0.036 s −1 K −1 effectively incorporates such temperature dependent shift of the cut-off in 139 1/T 1 , but the F CA results in Fig. 7 show no significant changes.

IV. SUMMARY AND CONCLUSIONS
We reported new comprehensive single crystal 63 Cu NMR results for La 1.875 Ba 0.125 CuO 4 , and compared the results with our recent report on 139 La NMR. We confirmed the precursors of enhanced growth in spin correlations below ∼ 80 K based on HWHM and 1/T 1 T , in agreement with earlier reports [22,23]. This is the same temperature range, where our recent ILTT 1 analysis of 139 1/T 1 at the 139 La NMR sites identified the presence of slow lattice and/or charge fluctuations [19]. We demonstrated that the apparently contradictory reports of 1/T 1 and spin echo decay curves M (2τ ) at the 63 Cu sites near and below T charge , as well as the apparently different behavior between 63 Cu and 139 La sites below T charge , are the consequence of a large spatial distribution in the enhancement of spin fluctuations.
Our findings of the sudden onset of NMR anomalies precisely at T charge for M (2τ ), F CA , 63 1/T 1 , and 139 1/T 1 and its asymmetric distribution are consistent with the earlier inelastic neutron scattering experiments with very small energy transfer, i.e. charge order turns on glassy spin dynamics precisely at T charge before the static magnetic order sets in at T spin 40 K, and hence the low frequency Cu spin fluctuations begin to undergo a dramatic enhancement at T charge [4,5,28].
We revisited our earlier report of the intensity anomaly of 63 Cu NMR below T charge of La 1.875 Ba 0.125 CuO 4 . We reproduced our original discovery of the intensity anomaly at T charge [11] with much higher precision, by taking advantage of the convenient magnetic field geometry of B ext || ab axis. We demonstrated once and for all that glassy spin dynamics induced within charge ordered domains begins to suppress the 63 Cu NMR intensity exactly at T charge , where T charge has already been determined independently by diffraction experiments. We also explained in detail why the observable fraction F CA of the 63 Cu NMR signal intensity provides a good measure of the canonical domains that have not been affected significantly by charge order. Our finding was corroborated by a completely different approach based on the ILTT 1 analysis of the distributed 63 1/T 1 and 139 1/T 1 [19]. We recall that we recently achieved the same for La 1.885 Sr 0.115 CuO 4 based on single crystal 63 Cu NMR and the ILTT 1 analysis of 139 La NMR [17,20].
We identified a key difference between La 1.875 Ba 0.125 CuO 4 with T c = 4 K and La 1.885 Sr 0.115 CuO 4 with much higher T c = 31 K; charge order enhances spin fluctuations in nearly 100% volume of the CuO 2 planes in the former by ∼ 35 K, while nearly a half of the volume fraction is still hardly affected when superconductivity sets in at higher T c in the latter [17,20]. On the other hand, in view of the fact that the aforementioned anomalous enhancement of spin correlations are commonly observed below ∼ 80 K for both La 1.875 Ba 0.125 CuO 4 [23] and La 1.885 Sr 0.115 CuO 4 [15,51], it is not clear why T charge 54 K is much lower in La 1.875 Ba 0.125 CuO 4 than T charge 80 K in La 1.885 Sr 0.115 CuO 4 . The only signature of charge order for La 1.875 Ba 0.125 CuO 4 observed to date above T charge is dynamic in nature [6]. It seems as if charge order in La 1.875 Ba 0.125 CuO 4 is suppressed from 80 K to 54 K, until the first-order low temperature tetragonal structural transition sets in.
It may be worthwhile to caution that the 63 Cu NMR intensity anomaly is not always entirely related to static charge order. In the case of La 2−x Sr x CuO 4 with [12] and without [11] Nd co-doping, we initially attributed the onset of the 63 Cu NQR intensity anomaly to charge order not only for the optimal charge order composition of x ∼ 1/8, but also for above and below x ∼ 1/8. Subsequent X-ray diffraction experiments for La 1.6−x Na 0.4 Sr x CuO 4 [52] showed that our estimation of T charge was accurate for x = 1/8 and above, but we overestimated T charge for x = 0.10 and below. Our overestimation for x = 0.10 resulted from the fact that the gradual intensity loss can arise also from localization of doped holes that precedes charge order. It turned out that the inflection point in the temperature dependence of F CO at a lower temperature corresponds to T charge for x < 1/8. We refer readers to Fig. 15(a), Fig. 18 and related discussions in [13] for details.

Appendix. The recovery curve M (t)
We measured 63 1/T 1 at the 63 Cu sites using the central transition. The standard formula for the relaxation recovery M (t) calculated from the coupled rate equations is where M o , A, and 63 1/T 1 are the free parameters, and the stretched exponent β should be normally set to 1.
We present examples of the normalized M (t) curves measured with fixed τ = 2 µs in Fig. 9(a), together with the best fit with Eq.(1) for fixed β = 1. The standard fit with β = 1 is sufficiently good even at 50 K below T charge to illustrate the key aspects summarized in Fig. 4.
Also summarized in Fig. 9(b) is the τ dependence of M (t) curves observed at 50 K with the phenomenological stretched fit. Lifting the constraint of β = 1 only marginally improves the fit, and the fitted value of 63 1/T 1 hardly changes. The overall relaxation becomes faster for shorter τ , accompanied by greater distribution, as evidenced by the smaller value of β. But the deviation of β from the non-distributed case of 1 is not very significant. Since the signal intensity becomes very small below T charge especially for longer τ , we fixed β = 1 for 63 Cu sites to reduce the number of free fitting parameters in Eq. (1), and thereby reducing the scattering in the 63 1/T 1 results.