Signatures of a pair density wave at high magnetic fields in cuprates with charge and spin orders

TL;DR

This study provides transport evidence supporting the existence of pair density wave (PDW) states in cuprates with charge and spin order, revealing how magnetic fields influence interlayer superconductivity and local pairing correlations.

Contribution

It offers experimental transport signatures of PDW states in cuprates, highlighting their behavior under magnetic fields and their competition with uniform superconductivity.

Findings

Enhanced interlayer superconducting coherence with parallel magnetic field.

Presence of local PDW correlations competing with uniform SC.

Transport signatures indicating PDW in the quantum fluctuation regime.

Abstract

In underdoped cuprates, the interplay of the pseudogap, superconductivity, and charge and spin ordering can give rise to exotic quantum states, including the pair density wave (PDW), in which the superconducting (SC) order parameter is oscillatory in space. However, the evidence for a PDW state remains inconclusive and its broader relevance to cuprate physics is an open question. To test the interlayer frustration, the crucial component of the PDW picture, we performed transport measurements on La$_{1.7}$Eu$_{0.2}$Sr$_{0.1}$CuO$_{4}$ and La$_{1.48}$Nd$_{0.4}$Sr$_{0.12}$CuO$_{4}$, cuprates with "striped" spin and charge orders, in perpendicular magnetic fields ($H_\perp$), and also with an additional field applied parallel to CuO$_2$ layers ($H_\parallel$). We detected several phenomena predicted to arise from the existence of a PDW, including an enhancement of interlayer SC phase coherence with increasing $H_\parallel$. Our findings are consistent with the presence of local, PDW pairing correlations that compete with the uniform SC order at $T_{c}^{0}< T<(2-6) T_{c}^{0}$, where $T_{c}^{0}$ is the $H=0$ SC transition temperature, and become dominant at intermediate $H_\perp$ as $T\rightarrow 0$. These data also provide much-needed transport signatures of the PDW in the regime where superconductivity is destroyed by quantum phase fluctuations.

Full text

Signatures of a pair density wave at high magnetic fields in cuprates with charge and spin orders

Zhenzhong Shi,1† P. G. Baity,1,2†† J. Terzic,1 T. Sasagawa,3 Dragana Popović1,2∗

1National High Magnetic Field Laboratory, Florida State University,

Tallahassee, Florida 32310, USA

2Department of Physics, Florida State University,

Tallahassee, Florida 32306, USA

3Materials and Structures Laboratory, Tokyo Institute of Technology,

Kanagawa 226-8503, Japan

† Present address: Department of Physics, Duke University,

Durham, North Carolina 27708, USA

†† Present address: James Watt School of Engineering, University of Glasgow,

Glasgow, G12 8QQ, Scotland, United Kingdom

∗To whom correspondence should be addressed; E-mail: dragana@magnet.fsu.edu

In underdoped cuprates, the interplay of the pseudogap, superconductivity, and charge and spin ordering can give rise to exotic quantum states, including the pair density wave (PDW), in which the superconducting (SC) order parameter is oscillatory in space. However, the evidence for a PDW state remains inconclusive and its broader relevance to cuprate physics is an open question. To test the interlayer frustration, the crucial component of the PDW picture, we performed transport measurements on La1.7Eu0.2Sr0.1CuO4 and La1.48Nd0.4Sr0.12CuO4, cuprates with “striped” spin and charge orders, in perpendicular magnetic fields ($\bm{H_{\perp}}$), and also with an additional field applied parallel to CuO2 layers ($\bm{H_{\parallel}}$). We detected several phenomena predicted to arise from the existence of a PDW, including an enhancement of interlayer SC phase coherence with increasing $\bm{H_{\parallel}}$. Our findings are consistent with the presence of local, PDW pairing correlations that compete with the uniform SC order at $\bm{T_{c}^{0}<T<(2-6)T_{c}^{0}}$, where $\bm{T_{c}^{0}}$ is the $\bm{H=0}$ SC transition temperature, and become dominant at intermediate $\bm{H_{\perp}}$ as $\bm{T\rightarrow 0}$. These data also provide much-needed transport signatures of the PDW in the regime where superconductivity is destroyed by quantum phase fluctuations.

The origin of the cuprate pseudogap regime has been a long-standing mystery. The richness of experimental observations**?** and the instability of underdoped cuprates towards a variety of ordering phenomena, such as periodic modulations of charge density discovered in all families of hole-doped cuprates**?, have raised the possibility that putative PDW correlations?, ? may be responsible for the pseudogap regime?, ?. In order to distinguish between different scenarios, the most intriguing open question is what happens at low $T\ll T_{c}^{0}$ and high $H_{\perp}$, when SC order is destroyed by quantum phase fluctuations? and short-range charge orders are enhanced?**, ?, ?. However, the experimental evidence for a PDW state remains scant and largely indirect in the first place.

A PDW SC state was proposed**?, ? to explain the suppression of the interlayer ($c$-axis) Josephson coupling (or dynamical layer decoupling) apparent in the $H=0$ anisotropic transport?** in La1.875Ba0.125CuO4, as well as in optical measurements in La1.85-yNdySr0.15CuO4 when the Nd concentration was tuned into the stripe-ordered regime**?. The dynamical layer decoupling was observed also in the presence of an applied $H_{\perp}$, in La1.905Ba0.095CuO4 (ref.  ?) and La2-xSrxCuO4  (ref.  ?). In La2-x-y(Ba,Sr)x(Nd,Eu)yCuO4 compounds near $x=1/8$, charge order appears in the form of stripes, which are separated by regions of oppositely phased antiferromagnetism (“spin stripes”)? at $T<T_{SO}<T_{CO}$; here $T_{SO}$ and $T_{CO}$ are the onsets of spin and charge stripes, respectively. In La2-xSrxCuO4  at $x=0.10$, spin stripe order is induced?** by applying $H_{\perp}$. The dynamical layer decoupling was thus attributed?, ? to a PDW SC state?, ?, such that the spatially modulated SC order parameter, with zero mean, occurs most strongly within the charge stripes, but the phases between adjacent stripes are reversed (antiphase). Since stripes are rotated by $90^{\circ}$ from one layer to next, antiphase superconductivity within a plane strongly frustrates the interlayer SC phase coherence?, leading to an increase in anisotropy. This effect is reduced by doping away from $x=1/8$, but $H_{\perp}$ can lead to dynamical layer decoupling as static stripe order is stabilized by a magnetic field.

To obtain more definitive evidence of the existence of a PDW, recent experiments have focused on testing various theoretical predictions?. For example, transport measurements on La1.875Ba0.125CuO4 have employed $H_{\perp}$ high enough to decouple the planes and then to suppress the SC order within the planes, with the results consistent with pair correlations surviving in charge stripes**?; Josephson junction measurements?** on La1.875Ba0.125CuO4 devices support the prediction of a charge-4$e$ SC condensate, consistent with the presence of a PDW state; an additional charge order was detected**?** in Bi2Sr2CaCu2O8 by scanning tunneling microscopy (STM) at very low $H_{\perp}/T_{c}^{0}\lesssim 0.1$ T/K, consistent with a PDW order that emerges within the halo region surrounding a vortex core once a uniform SC order is sufficiently suppressed by $H_{\perp}$. However, alternative explanations are still possible, and additional experiments are thus needed to search for a PDW and explore its interplay with other orders in the pseudogap regime?.

Therefore, we measured transport in La2-x-ySrx(Nd,Eu)yCuO4 compounds, which have the same low-temperature structure as La2-xBaxCuO4, over an unprecedented range of $T$ down to $T/T_{c}^{0}\lesssim 0.003$ and fields up to $H/T_{c}^{0}\sim 10$ T/K. We combined linear in-plane resistivity $\rho_{ab}$, nonlinear in-plane transport or voltage-current ($V$–$I$) characteristics, and the anisotropy ratio $\rho_{c}/\rho_{ab}$ (here $\rho_{c}$ is the out-of-plane resistivity) to probe both charge and vortex matter on single crystals with the nominal composition La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4  (Methods); the former is away from $x=1/8$ and thus the stripe order is weaker?. We find signatures of dynamical layer decoupling in both $H=0$ and with increasing $H_{\perp}$, consistent with the presence of a PDW. However, a key proposed test of this interpretation involves relieving the interlayer frustration through the application of an in-plane magnetic field?, ?. In particular, since $H_{\parallel}$ can reorient the spin stripes in every other plane**?**, ?, ?, a consequence of a PDW would be an enhancement of interplane coherence, or a reduced anisotropy. This is precisely what we test and observe.

We first explore the anisotropy in $H=0$. In both La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4, $\rho_{c}$ and $\rho_{ab}$ vanish at the same $T_{c}^{0}$ within the error (Methods; see also Supplementary Information), indicating the onset of 3D superconductivity, similar to La2-xSrxCuO4  (e.g. ref.  ?). The initial drop of $\rho_{ab}(T)$ with decreasing $T$ (Fig. 1a) is accompanied by an enhancement of the anisotropy (Fig. 1b), which continues to increase by almost an order of magnitude as $T$ is

lowered further towards $T_{c}^{0}$. These data look remarkably similar to those on La1.875Ba0.125CuO4 (ref.  ?) that motivated theoretical proposals for a PDW SC state in striped cuprates: the initial, high-$T$ enhancement of the anisotropy is understood to reflect the establishment of SC correlations in CuO2 planes.

The evolution of $\rho_{c}/\rho_{ab}(T)$ with $H_{\perp}$ is shown in Fig. 1c. The anisotropy at the highest $T=20$ K is $\rho_{c}/\rho_{ab}\sim 6000$ and practically independent of $H_{\perp}$. However, as $T$ is lowered below $T_{c}^{0}$, $\rho_{c}/\rho_{ab}$ develops a distinctly nonmonotonic behavior as a function of $H_{\perp}$. At $T=0.017$ K, for example, the anisotropy increases with $H_{\perp}$ by over an order of magnitude before reaching a peak ($\rho_{c}/\rho_{ab}>10^{5}$) at $H_{\perp}=H_{p}$, signifying decoupling of or the loss of phase coherence between the planes. However, strong SC correlations persist in the planes for $H_{\perp}>H_{p}$: here $\rho_{c}/\rho_{ab}$ decreases with $H_{\perp}$ to $H_{\perp}$-independent values, comparable to those at high $T$, for the highest $H_{\perp}>20$ T. This is in agreement with previous evidence**?** that the $H_{\perp}>20$ T region corresponds to the normal state. A smooth, rapid decrease of the anisotropy for $H_{\perp}>H_{p}$ is interrupted by a “bump” or an enhancement in $\rho_{c}/\rho_{ab}$, centered at $H_{b}$. Therefore, the behavior of $\rho_{c}/\rho_{ab}$ is qualitatively the same whether the SC transition is approached from either (1) the high-$T$ normal state by lowering $T$ in $H=0$ (Fig. 1b) or (2) the high-$H_{\perp}$ normal state by reducing $H_{\perp}$ at a fixed $T$ (Fig. 1c). These results thus suggest that the enhancement of the anisotropy near $H_{b}(T)$ may be attributed to the establishment of SC correlations in the planes as the SC transition is approached from the high-field normal state.

This picture is supported by the comparison of $\rho_{c}/\rho_{ab}$, as a function of $T$ and $H_{\perp}$, with the behavior of $\rho_{ab}(T)$ for a fixed $H_{\perp}$, as shown in Fig. 2 for both La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4. The $\rho_{ab}(T)$ data were extracted from the in-plane magnetoresistance (MR) measurements

(ref.  ?, Supplementary Fig. 2a; unless stated otherwise, the results are shown for La1.7Eu0.2Sr0.1CuO4  sample “B”, see Methods); the raw $\rho_{c}(H)$ data are shown in Supplementary Figs. 2b and 2c. In Figs. 2a and 2b, we also include $T_{c}(H_{\perp})$, as well as $H_{peak}$, the position of the peak in the in-plane MR (see, e.g., Supplementary Fig. 2a), which corresponds? to the upper critical field $H_{c2}$ in these materials (see also Supplementary Information). Indeed, at a fixed $T$, $\rho_{c}/\rho_{ab}$ starts to increase as $H_{\perp}$ is reduced below $H_{peak}$. This is followed by an enhancement of $\rho_{c}/\rho_{ab}$ near $H_{\perp}=H_{b}$, corresponding to the initial, metalliclike drop of $\rho_{ab}(T)$ as the SC transition is approached from the normal state for a fixed $H_{\perp}$ (Figs. 2c and 2d). The behavior of both materials is similar, except that the layer decoupling field $H_{p}(T)\gtrsim H_{c}(T)$ [or $T_{c}(H_{\perp})$] in La1.48Nd0.4Sr0.12CuO4, as expected? for a stronger stripe order and frustration of interlayer coupling for $x\approx 1/8$. Therefore, practically all the data in Figs. 2c and 2d, i.e. for $H_{\perp}>H_{p}$, involve “purely” 2D physics, with no communication between the planes. The striking splitting of the $\rho_{ab}(T)$ curves in both materials (ref.  ?, Figs. 2c and 2d), into either metalliclike (i.e. SClike) or insulatinglike, when the normal state sheet resistance $R_{\square/{\mathrm{layer}}}\approx R_{Q}$, where $R_{Q}=h/(2e)^{2}$ is the quantum resistance for Cooper pairs, further supports this conclusion: it agrees with the expectations for a 2D superconductor-insulator transition (SIT) driven by quantum fluctuations of the SC phase**?. In addition, as previously noted?, the two-step $\rho_{ab}(T)$ is reminiscent of that in granular films of conventional superconductors and systems with nanoscale phase separation, including engineered Josephson junction arrays, where they are generally attributed to the onset of local (e.g. in islands or puddles) and global, 2D superconductivity. Similarities to the behavior of various SC 2D systems?**, ? thus suggest the formation of SC “islands” as $H_{\perp}$ is reduced below $H_{b}$ at a fixed $T$ (e.g. Figs. 2a and 2b), i.e. at the initial, metalliclike drop of $\rho_{ab}(T)$ for a fixed $H_{\perp}$ ($H_{b}$ dashed line in Figs. 2c and 2d). Additional evidence in support of this interpretation, such as the $V$–$I$ that is characteristic of a viscous vortex liquid in the “puddle” regime, is discussed in Supplementary Information (also, Supplementary Figs. 3-5). Therefore, at low $T$, the increasing $H_{\perp}$ destroys the superconductivity in the planes by quantum phase fluctuations of Josephson-coupled SC puddles. The evolution of this “puddle” region with $T$ can be traced to the initial, metalliclike drop of $\rho_{ab}(T)$ at $T>T_{c}^{0}$ in $H=0$ (see $H_{b}$ dashed line in Figs. 2c and 2d, and Supplementary Figs. 3 and 4). Further increase of $H_{\perp}$ at low $T$ then leads to the loss of SC phase coherence in individual puddles and, eventually, transition to the high-field normal state. These results are summarized in the sketch of the phase diagram, shown in Fig. 3a.

Our experiments are thus consistent with the presence of local PDW correlations (in “puddles”) at $T>T_{c}^{0}$ in $H=0$, which are overtaken by the uniform $d$-wave superconductivity at low $T<T_{c}^{0}$. In transport, the PDW SC order becomes apparent when the uniform $d$-wave order is sufficiently weakened by $H_{\perp}$: it appears beyond the melting field of the vortex solid, within the vortex liquid regime, i.e. in the regime of strong 2D phase fluctuations. Higher fields $H_{p}$ are needed to decouple the layers in La1.7Eu0.2Sr0.1CuO4  than in La1.48Nd0.4Sr0.12CuO4, since it is farther away from $x=1/8$. In the $T\rightarrow 0$ limit and for even higher $H_{\perp}$ ($<H_{c2}$), the system seems to break up into SC puddles with the PDW order. However, the final and key test of the presence of a PDW requires the application of a suitable perturbation, in particular $H_{\parallel}$, to reduce the interlayer frustration and decrease the anisotropy?.

Therefore, we have performed angle-dependent measurements of both $\rho_{ab}(H)$ and $\rho_{c}(H)$, where the angle $\theta$ is between $H$ and the crystalline $c$ axis. This has allowed us to explore the effect of in-plane fields $H_{\parallel}=H\sin\theta$ at different $H_{\perp}=H\cos\theta$, i.e. fields parallel to the $c$ axis, discussed above. The angle-dependent $\rho_{ab}(H)$ was measured also on another La1.7Eu0.2Sr0.1CuO4  sample (sample “B1”, Methods; Supplementary Fig. 8); the results are qualitatively the same on both samples. Figure 3b illustrates the effect of $H_{\parallel}$ on $\rho_{c}/\rho_{ab}$ at low $T=0.070$ K on sample B1 (see Supplementary Figs. 9 a-d for the raw $\rho_{c}$ and $\rho_{ab}$ data at different $T$). Clearly, there is no effect of $H_{\parallel}$ for $H_{\perp}>H_{c2}(T=0.070$ K)$\approx 17.5$ T. Since $H_{\parallel}$ should break up Cooper pairs through the Zeeman effect, this confirms the absence of any observable remnants of superconductivity above the previously identified? $H_{c2}$ ($\parallel c$). In contrast, for $H_{p}\leqslant H_{\perp}<H_{c2}$, $H_{\parallel}$ reduces the anisotropy, which is precisely what is expected in the presence of a PDW SC state if the dominant effect of $H_{\parallel}$ is to reorient the spin stripes?.

To understand exactly how $H_{\parallel}$ affects the anisotropy, we also investigate $\Delta\rho_{ab}=\rho_{ab}(H_{\parallel})-\rho_{ab}(H_{\parallel}=0)$ and $\Delta\rho_{c}=\rho_{c}(H_{\parallel})-\rho_{c}(H_{\parallel}=0)$ at different $H_{\perp}$ (Fig. 3c and Supplementary Fig. 8d for sample B1; Supplementary Figs. 9 e-h for sample B). It is obvious that $\rho_{ab}$ is reduced by $H_{\parallel}$ for all $H_{\perp}$, which is the opposite of what would be expected if pair-breaking was dominant. The suppression of $\rho_{ab}$ is weaker for those $H_{\perp}$ where the superconductivity is stronger, e.g. near $H_{b}\sim 15$ T in Fig. 3c, and conversely, it is most pronounced above $H_{c2}$, indicating that the dominant effect of $H_{\parallel}$ is not related to superconductivity. In fact, it occurs most strongly in the two regimes where $\rho_{ab}(H_{\perp})$ exhibits hysteretic behavior at low $T$ (Supplementary Figs. 3 and 6); the latter is attributed to the presence of domains with spin stripes (see also Supplementary Information and Supplementary Fig. 7). This observation, therefore, further supports the conclusion that the main effect of $H_{\parallel}$ is the reorientation of spin stripes in every other plane?, ?, ? (see also Supplementary Information). The suppression of $\rho_{ab}$ by $H_{\parallel}$ seems to vanish at experimentally inaccessible $H_{\perp}$, where the anomalous, insulatinglike $\ln(1/T)$ dependence observed in the field-induced normal state also appears to vanish?, suggesting that the origin of the $\ln(1/T)$ behavior might be related to the presence of short-range spin stripes. As the spin stripes in every other plane are rotated by $H_{\parallel}$, in the PDW picture the interlayer frustration should be suppressed, leading to a decrease in $\rho_{c}$. This is precisely what is observed (Fig. 3c). The anisotropy ratio $\rho_{c}/\rho_{ab}$ is reduced (Fig. 3b) because the effect of $H_{\parallel}$ on $\rho_{c}$ is relatively stronger than on $\rho_{ab}$. Similar results are obtained in La1.48Nd0.4Sr0.12CuO4  (Supplementary Fig. 10): here the reduction in $\rho_{c}$ is weaker than in La1.7Eu0.2Sr0.1CuO4  and $\rho_{ab}$ is not affected within the experimental resolution, both consistent with the stronger pinning of stripe order at $x=1/8$ (see also Supplementary Information). Nevertheless, the reduction of $\rho_{c}/\rho_{ab}$ by $H_{\parallel}$ is comparable to that in La1.7Eu0.2Sr0.1CuO4  (Fig. 3b). Therefore, by applying an in-plane magnetic field, as proposed theoretically?, ?, our measurements confirm the presence of a PDW in both La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4. The effects of $H_{\parallel}$ are observable up to $T>T_{c}^{0}$ (i.e. $T\sim T_{SO}$ in La1.7Eu0.2Sr0.1CuO4: Supplementary Fig. 9), providing additional evidence for the PDW correlations in $H=0$ at $T>T_{c}^{0}$, as sketched in Fig. 3a.

Finally, our results provide an explanation for the surprising, and a priori counterintuitive, observation? that $H_{c2}$ in La1.48Nd0.4Sr0.12CuO4  ($H_{c2}\sim 25$ T) is higher than in La1.7Eu0.2Sr0.1CuO4  ($H_{c2}\sim 20$ T), even though its zero-field $T_{c}^{0}$ is lower because of stronger stripe correlations. It is clear, though, that it is precisely because of the stronger stripe order and the presence of a more robust PDW SC state at $x\approx 1/8$ that the superconductivity persists to higher fields as $T\rightarrow 0$.

In summary, by probing the previously inaccessible high $H_{\perp}/T_{c}^{0}$ and $T\rightarrow 0$ regime dominated by quantum phase fluctuations and by testing a theoretical prediction, we have obtained evidence consistent with the existence of a PDW state in the La-214 family of cuprates with stripes. Our observation of several signatures of a PDW in the regime with many vortices (i.e. a vortex liquid) is also consistent with the STM evidence? for a PDW order that emerges in vortex halos. Since the observed PDW correlations extend only up to $T\ll T_{pseudogap}$ and not beyond $H_{c2}(T)$, our results do not support a scenario in which the PDW correlations are responsible for the pseudogap.

Methods

Samples. Several single crystal samples of La1.8-xEu0.2SrxCuO4 with a nominal $x=0.10$ and La1.6-xNd0.4SrxCuO4 with a nominal $x=0.12$ were grown by the traveling-solvent floating-zone technique**?**. The high homogeneity of the crystals was confirmed by several techniques, as discussed in detail elsewhere?. It was established that the samples were at least as homogeneous as those previously reported in the literature and, in fact, the disorder in our La1.7Eu0.2Sr0.1CuO4  crystals was significantly lower than in other studies. We note that the trivial possibility that the two-step SC transition observed at $H=0$ (e.g. Figs. 2c and 2d for La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4, respectively) may be due to an extrinsic inhomogeneity, e.g. the presence of two regions with different values of $T_{c}^{0}$, is clearly ruled out also by the behavior of $d\rho_{ab}/dT$ with $H_{\perp}$ (Supplementary Figs. 3a, 4, 8b). In particular, both materials exhibit a reentrant metalliclike behavior at high $H_{\perp}$, below $H_{c2}$ (e.g. see the reentrant darker blue color band for La1.48Nd0.4Sr0.12CuO4). This is the opposite of what is expected in case of two different $T_{c}^{0}$ values corresponding to different doping levels, where one would expect a gradual suppression of superconductivity with $H_{\perp}$, i.e. no reentrance.

The samples were shaped as rectangular bars suitable for direct measurements of the in-plane and out-of-plane resistance. In La1.7Eu0.2Sr0.1CuO4, detailed measurements of $\rho_{ab}$ were performed on sample “B” with dimensions $3.06\times 0.53\times 0.37$ mm3 ($a\times b\times c$); $\rho_{c}$ was measured on a bar with $0.34\times 0.41\times 1.67$ mm3. The in-plane La1.48Nd0.4Sr0.12CuO4  crystal with dimensions $3.82\times 1.19\times 0.49$ mm3 was cut along the crystallographic [110] and [1$\bar{1}$0] axes, i.e. at a $45^{\circ}$ angle with respect to $a$ and $b$. A bar with $0.21\times 0.49\times 3.9$ mm3 ($a\times b\times c$) was used to measure $\rho_{c}$ in La1.48Nd0.4Sr0.12CuO4. The behavior of these samples remained astonishingly stable with time, without which it would have not been possible to conduct such an extensive and systematic study that required matching data obtained using different cryostats and magnets (see below) over the period of 2-3 years during which most of this study was performed, thus further attesting to the high quality of the crystals. After $\sim 3$ years, the low-$T$ properties of sample B changed, resulting in a quantitatively different $T$–$H_{\perp}$ phase diagram (Supplementary Fig. 8b); this is why we consider it a different sample (“B1”). The phase diagram of sample B1 seems to be intermediate to those of sample B (Supplementary Fig. 3a) and La1.48Nd0.4Sr0.12CuO4  (Supplementary Fig. 4). Gold contacts were evaporated on polished crystal surfaces, and annealed in air at 700 ∘C. The current contacts were made by covering the whole area of the two opposing sides with gold to ensure uniform current flow, and the voltage contacts were made narrow to minimize the uncertainty in the absolute values of the resistance. Multiple voltage contacts on opposite sides of the crystals were prepared. The distance between the voltage contacts for which the data are shown is 1.53 mm for La1.7Eu0.2Sr0.1CuO4  and 2.00 mm for La1.48Nd0.4Sr0.12CuO4  in-plane samples; 0.47 mm for La1.7Eu0.2Sr0.1CuO4  and 1.26 mm for La1.48Nd0.4Sr0.12CuO4  out-of-plane samples. Gold leads ($\approx 25~{}\mu$m thick) were attached to the samples using the Dupont 6838 silver paste, followed by the heat treatment at 450 ∘C in the flow of oxygen for 15 minutes. The resulting contact resistances were less than 0.1 $\Omega$ for La1.7Eu0.2Sr0.1CuO4  (0.5 $\Omega$ for La1.48Nd0.4Sr0.12CuO4) at room temperature. The values of $T_{c}^{0}$ and the behavior of the samples did not depend on which voltage contacts were used in the measurements, reflecting the absence of extrinsic (i.e. compositional) inhomogeneity in these crystals.

$T_{c}^{0}$ was defined as the temperature at which the linear resistivity becomes zero. For the in-plane samples, $T_{c}^{0}=(5.7\pm 0.3)$ K for La1.7Eu0.2Sr0.1CuO4  and $T_{c}^{0}=(3.6\pm 0.4)$ K for La1.48Nd0.4Sr0.12CuO4; the out-of-plane resistivity $\rho_{c}$ vanishes at $(5.5\pm 0.3)$ K for La1.7Eu0.2Sr0.1CuO4  and $(3.4\pm 0.5)$ K for La1.48Nd0.4Sr0.12CuO4. In La1.7Eu0.2Sr0.1CuO4, $T_{SO}\sim 15$ K, $T_{CO}\sim 40$ K (ref.  ?), and the pseudogap temperature $T_{pseudogap}\sim 175$ K (ref.  ?); in La1.48Nd0.4Sr0.12CuO4, $T_{SO}\sim 50$ K, $T_{CO}\sim 70$ K (ref.  ?), and $T_{pseudogap}\sim 150$ K (ref.  ?).

Measurements. The standard four-probe ac method ($\sim 13$ Hz) was used for measurements of the sample resistance, with the excitation current (density) of 10 $\mu$A ($\sim 5\times 10^{-3}$ A cm*-2* and $\sim 2\times 10^{-3}$ A cm*-2* for La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4, respectively) for the in-plane samples and 10 nA ($\sim 7\times 10^{-6}$ A cm*-2* and $\lesssim 10^{-5}$ A cm*-2* for La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4, respectively) for the out-of-plane samples. $dV/dI$ measurements were performed by applying a dc current bias (density) down to 2 $\mu$A ($\sim 1\times 10^{-3}$ A cm*-2* and $\sim 4\times 10^{-4}$ A cm*-2* for La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4  in-plane samples, respectively) and a small ac current excitation $I_{ac}\approx 1~{}\mu$A ($\sim$ 13 Hz) through the sample and measuring the ac voltage across the sample. For each value of $I_{dc}$, the ac voltage was monitored for 300 s and the average value recorded. The relaxations of $dV/dI$ with time, similar to that in Supplementary Fig. 7, were observed only at the lowest $T\sim 0.016$ K. Even then, the change of $dV/dI$ during the relaxation, reflected in the error bars for the $T=0.017$ K data in Supplementary Fig. 3c, was much smaller than the change of $dV/dI$ with $I_{dc}$. The data that were affected by Joule heating at large dc bias were not considered. To reduce the noise and heating by radiation in all measurements, a 1 k$\Omega$ resistor in series with a $\pi$ filter [5 dB (60 dB) noise reduction at 10 MHz (1 GHz)] was placed in each wire at the room temperature end of the cryostat.

The experiments were conducted in several different magnets at the National High Magnetic Field Laboratory: a dilution refrigerator (0.016 K $\leqslant$ T $\leqslant$ 0.7 K) and a 3He system (0.3 K $\leqslant$ T $\leqslant$ 35 K) in superconducting magnets ($H$ up to 18 T), using 0.1 – 0.2 T/min sweep rates; a portable dilution refrigerator (0.02 K $\leqslant$ T $\leqslant$ 0.7 K) in a 35 T resistive magnet, using 1 T/min sweep rate; and a 3He system (0.3 K $\leqslant$ T $\leqslant$ 20 K) in a 31 T resistive magnet, using 1 – 2 T/min sweep rates. Below $\sim 0.06$ K, it was not possible to achieve sufficient cooling of the electronic degrees of freedom to the bath temperature, a common difficulty with electrical measurements in the mK range. This results in a slight weakening of the $\rho_{ab}(T)$ curves below $\sim 0.06$ K for all fields. We note that this does not make any qualitative difference to the phase diagram (Supplementary Fig. 3a). The fields were swept at constant temperatures, and the sweep rates were low enough to avoid eddy current heating of the samples. The MR measurements with $H\parallel c$ were performed also by reversing the direction of $H$ to eliminate by summation any Hall effect contribution to the resistivity. Moreover, since Hall effect had not been explored in these materials in large parts of the phase diagrams studied here, we have also carried out detailed measurements of the Hall effect; the results of that study will be presented elsewhere**?**.

The resistance per square per CuO2 layer $R_{\square/\mathrm{layer}}=\rho_{ab}/l$, where $l=6.6$ Å  is the thickness of each layer.

Data availability. The data that support the findings of this study are available from the corresponding author upon reasonable request.