Pair density wave at high magnetic fields in cuprates with charge and spin orders.

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 perform transport measurements on charge- and spin-stripe-ordered La1.7Eu0.2Sr0.1CuO4 and La1.48Nd0.4Sr0.12CuO4 in perpendicular magnetic fields (H⊥), and also with an additional field applied parallel to CuO2 layers (H∥). We detect several phenomena predicted to arise from the existence of a PDW, including an enhancement of interlayer SC phase coherence with increasing H∥. These data also provide much-needed transport signatures of the PDW in the regime where superconductivity is destroyed by quantum phase fluctuations.

Introduction

The origin of the cuprate pseudogap regime has been a long-standing mystery. The richness of experimental observations[1] 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[2], have raised the possibility that putative pair density wave (PDW) correlations[3,4] may be responsible for the pseudogap regime[5,6]. In order to distinguish between different scenarios, the most intriguing open question is what happens at low (here is the H = 0 superconducting (SC) transition temperature) and high H⊥, when SC order is destroyed by quantum phase fluctuations[6] and short-range charge orders are enhanced[7-9]. However, the experimental evidence for a PDW state remains scant and largely indirect in the first place.

A PDW SC state was proposed[4,10] to explain the suppression of the interlayer (c-axis) Josephson coupling (or dynamical layer decoupling) apparent in the H = 0 anisotropic transport[11] in La1.875Ba0.125CuO4, as well as in optical measurements in La1.85−NdSr0.15CuO4 when the Nd concentration was tuned into the stripe-ordered regime[12]. The dynamical layer decoupling was observed also in the presence of an applied H⊥, in La1.905Ba0.095CuO4 (ref. [13]) and La2−SrCuO4  (ref. [14]). In La2−(Ba,Sr)(Nd,Eu)CuO4 compounds near x = 1/8, charge order appears in the form of stripes, which are separated by regions of oppositely phased antiferromagnetism (spin stripes)[5] at T < TSO < TCO; here TSO and TCO are the onsets of spin and charge stripes, respectively. In La2−SrCuO4  at x = 0.10, spin stripe order is induced[15] by applying H⊥. The dynamical layer decoupling was thus attributed[4,10] to a PDW SC state[3,10], 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° from one layer to next, antiphase superconductivity within a plane strongly frustrates the interlayer SC phase coherence[5], leading to an increase in anisotropy. This effect is reduced by doping away from x = 1/8, but H⊥ 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[5]. For example, transport measurements on La1.875Ba0.125CuO4 have employed H⊥ 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[16]; Josephson junction measurements[17] on La1.875Ba0.125CuO4 devices support the prediction of a charge-4e SC condensate, consistent with the presence of a PDW state; an additional charge order was detected[18] in Bi2Sr2CaCu2O8 by scanning tunneling microscopy (STM) at very low  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⊥. 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[6].

Therefore, we measure transport in La2−Sr(Nd,Eu)CuO4 compounds, which have the same low-temperature structure as La2−BaCuO4, over an unprecedented range of T down to and fields up to  T/K. We combine linear in-plane resistivity ρab, nonlinear in-plane transport or voltage–current (V–I) characteristics, and the anisotropy ratio ρc/ρab (here ρc 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  (see “Methods” section); the former is away from x = 1/8 and thus the stripe order is weaker[5]. We find signatures of dynamical layer decoupling in both H = 0 and with increasing H⊥, 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[5,10]. In particular, since H∥ can reorient the spin stripes in every other plane[19-21], a consequence of a PDW would be an enhancement of interplane coherence, or a reduced anisotropy. This is precisely what we test and observe.

Results

Anisotropy in H = 0

In both La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4, ρc and ρab vanish at the same within the error (see "Methods” section; see also Supplementary Note 1), indicating the onset of 3D superconductivity, similar to La2−SrCuO4  (e.g. ref. [22]). The initial drop of ρab(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 . These data look remarkably similar to those on La1.875Ba0.125CuO4 (ref. [11]) 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.

Fig. 1

Evolution of the anisotropy in La1.7Eu0.2Sr0.1CuO4 with and ⊥.

a ρab(T) and ρc(T), and b the anisotropy ratio ρc/ρab(T), in zero field. The vertical dashed line indicates where SC correlations are established in the planes, resulting in the enhancement of the anisotropy; ρc continues to grow with decreasing T. c ρc/ρab vs. H⊥ at different T, as shown. Arrows show the positions of the anisotropy peak Hp, or the decoupling field, as well as Hb, where the anisotropy is enhanced. The method to determine Hb more precisely is described in Supplementary Fig. 1.

Evolution of the anisotropy and ρab with H⊥ and T

The evolution of ρc/ρab(T) with H⊥ is shown in Fig. 1c. The anisotropy at the highest T = 20 K is ρc/ρab ~ 6000 and practically independent of H⊥. However, as T is lowered below , ρc/ρab develops a distinctly nonmonotonic behavior as a function of H⊥. At T = 0.017 K, for example, the anisotropy increases with H⊥ by over an order of magnitude before reaching a peak (ρc/ρab > 105) at H⊥ = Hp, signifying decoupling of or the loss of phase coherence between the planes. However, strong SC correlations persist in the planes for H⊥ > Hp: here ρc/ρab decreases with H⊥ to H⊥-independent values, comparable to those at high T, for the highest H⊥ > 20 T. This is in agreement with previous evidence[23] that the H⊥ > 20 T region corresponds to the normal state. A smooth, rapid decrease of the anisotropy for H⊥ > Hp is interrupted by a bump or an enhancement in ρc/ρab, centered at Hb. Therefore, the behavior of ρc/ρab 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⊥ normal state by reducing H⊥ at a fixed T (Fig. 1c). These results thus suggest that the enhancement of the anisotropy near Hb(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 ρc/ρab, as a function of T and H⊥, with the behavior of ρab(T) for a fixed H⊥, as shown in Fig. 2 for both La1.7Eu0.2Sr0.1CuO4 and La1.48Nd0.4Sr0.12CuO4. The ρab(T) data were extracted from the in-plane magnetoresistance (MR) measurements (ref. [23], Supplementary Fig. 2a; unless stated otherwise, the results are shown for La1.7Eu0.2Sr0.1CuO4  sample B, see “Methods” section); the raw ρc(H) data are shown in Supplementary Fig. 2b, c. In Fig. 2a, b, we also include Tc(H⊥), as well as Hpeak, the position of the peak in the in-plane MR (see e.g. Supplementary Fig. 2a), which corresponds[23] to the upper critical field Hc2 in these materials (see also Supplementary Note 1). Indeed, at a fixed T, ρc/ρab starts to increase as H⊥ is reduced below Hpeak. This is followed by an enhancement of ρc/ρab near H⊥ = Hb, corresponding to the initial, metallic-like drop of ρab(T) as the SC transition is approached from the normal state for a fixed H⊥ (Fig. 2c, d). The behavior of both materials is similar, except that the layer decoupling field Hp(T) ≳ Hc(T) [or Tc(H⊥)] in La1.48Nd0.4Sr0.12CuO4, as expected[5] for a stronger stripe order and frustration of interlayer coupling for x ≈ 1/8. Therefore, practically all the data in Fig. 2c, d, i.e. for H⊥ > Hp, involve “purely” 2D physics, with no communication between the planes. The striking splitting of the ρab(T) curves in both materials (ref. [23], Fig. 2c, d), into either metallic-like (i.e. SC-like) or insulating-like, when the normal state sheet resistance R□/layer ≈ RQ, where RQ = 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[24]. In addition, as previously noted[23], the two-step ρab(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[25,26] thus suggest the formation of SC islands as H⊥ is reduced below Hb at a fixed T (e.g. Fig. 2a, b), i.e. at the initial, metallic-like drop of ρab(T) for a fixed H⊥ (Hb dashed line in Fig. 2c, d). 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 Note 2 (also, Supplementary Figs. 3–5). Therefore, at low T, the increasing H⊥ 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, metallic-like drop of ρab(T) at in H = 0 (see Hb dashed line in Fig. 2c, d, and Supplementary Figs. 3 and 4). Further increase of H⊥ 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.

Fig. 2

Anisotropy and the in-plane resistivity for different and ⊥.

The color map in a and b shows ρc/ρab in La1.8−Eu0.2SrCuO4 (LESCO) with x = 0.10 (data from Fig. 1c) and La1.6−Nd0.4SrCuO4 (LNSCO) with x = 0.12, respectively. Black squares: Tc(H⊥); ρab = 0 for all T < Tc(H⊥). Green dots: Hpeak(T), i.e. fields above which the in-plane MR changes from positive to negative; it has been established[23] that Hpeak(T) ~ Hc2(T), i.e. the upper critical field. The error bars reflect the uncertainty in defining the MR peak within our experimental resolution (see inset of Supplementary Fig. 2a for an example; also see Supplementary Fig. 6a and ref. [23] for the raw MR data). Pink dots: Hp(T), the layer decoupling field; red triangles: Hb(T), where SC correlations are established in the planes as the SC transition is approached from the normal state. ρab(T) of c La1.7Eu0.2Sr0.1CuO4  and d La1.48Nd0.4Sr0.12CuO4  for several H⊥, as shown. Open symbols in c show the data from another run. Short-dashed lines guide the eye. The Hb(T) values obtained from the anisotropy are represented by the black dashed lines, as shown. The lower black dashed line in c corresponds to the layer decoupling field, Hp(T). In d, Hp(T) ≳ Hc(T) [or Tc(H⊥)]. Black arrows in c and d show that the splitting of the ρab(T) curves for different H⊥ becomes pronounced when R□/layer ≈ RQ = h/(2e)2.

Fig. 3

Evidence for a PDW from anisotropic transport.

a Schematic T–H⊥ phase diagram. H⊥ suppresses the 3D superconductivity (gray) and decouples (dotted line) the CuO2 layers at H⊥ = Hp(T). Strong SC phase fluctuations persist in the planes up to Hc2(T) (short-dashed line). The behavior in the pink region, the precursors of which appear already in H = 0 at (see dashed lines), is consistent with the presence of SC puddles in CuO2 planes. An additional, in-plane field enhances the interlayer coupling for Hp(T) < H⊥ < Hc2(T), consistent with the presence of PDW correlations (thin hatched lines). Except for the thick solid line, other lines do not represent phase boundaries, but correspond to finite-temperature crossovers. b ρc/ρab (for La1.7Eu0.2Sr0.1CuO4  in-plane sample B1) vs. H⊥ for different H∥, as shown, at T = 0.070 K. Larger inset: Enlarged view of the same data shows the suppression of the anisotropy by H∥ for Hp < H⊥ < Hc2. Smaller inset: ρc/ρab is reduced by  ~10% near Hp by H∥ up to 10 T. c The corresponding [ρab(H∥)/ρab(H∥ = 0) − 1] (top, sample B1) and [ρc(H∥)/ρc(H∥ = 0) − 1] (bottom) vs. H⊥ at T = 0.028 K for different H∥, as shown. In all panels, solid lines guide the eye.

Our experiments are thus consistent with the presence of local PDW correlations (in puddles) at in H = 0, which are overtaken by the uniform d-wave superconductivity at low . In transport, the PDW SC order becomes apparent when the uniform d-wave order is sufficiently weakened by H⊥: 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 Hp 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 → 0 limit and for even higher H⊥ (5].

Effects of H∥ on the anisotropy

We have performed angle-dependent measurements of both ρab(H) and ρc(H), where the angle θ is between H and the crystalline c-axis. This has allowed us to explore the effect of in-plane fields at different , i.e. fields parallel to the c-axis, discussed above. The angle-dependent ρab(H) was measured also on another La1.7Eu0.2Sr0.1CuO4  sample (sample B1, see “Methods” section; Supplementary Fig. 8); the results are qualitatively the same on both samples. Figure 3b illustrates the effect of H∥ on ρc/ρab at low T = 0.070 K on sample B1 (see Supplementary Fig. 9a–d for the raw ρc and ρab data at different T). Clearly, there is no effect of H∥ for H⊥ > Hc2 (T = 0.070 K) ≈ 17.5 T. Since H∥ should break up Cooper pairs through the Zeeman effect, this confirms the absence of any observable remnants of superconductivity above the previously identified[23] Hc2 (along c-axis). In contrast, for Hp ≤ H⊥ < Hc2, H∥ reduces the anisotropy, which is precisely what is expected in the presence of a PDW SC state if the dominant effect of H∥ is to reorient the spin stripes[10].

To understand exactly how H∥ affects the anisotropy, we also investigate Δρab = ρab(H∥) − ρab(H∥ = 0) and Δρc = ρc(H∥) − ρc(H∥ = 0) at different H⊥ (Fig. 3c and Supplementary Fig. 8d for sample B1; Supplementary Fig. 9e–h for sample B). It is obvious that ρab is reduced by H∥ for all H⊥, which is the opposite of what would be expected if pair-breaking was dominant. The suppression of ρab is weaker for those H⊥ where the superconductivity is stronger, e.g. near Hb ~ 15 T in Fig. 3c, and conversely, it is most pronounced above Hc2, indicating that the dominant effect of H∥ is not related to superconductivity. In fact, it occurs most strongly in the two regimes where ρab(H⊥) 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 Note 2 and Supplementary Fig. 7). This observation, therefore, further supports the conclusion that the main effect of H∥ is the reorientation of spin stripes in every other plane[19-21] (see also Supplementary Note 3). The suppression of ρab by H∥ seems to vanish at experimentally inaccessible H⊥, where the anomalous, insulating-like dependence observed in the field-induced normal state also appears to vanish[23], suggesting that the origin of the behavior might be related to the presence of short-range spin stripes. As the spin stripes in every other plane are rotated by H∥, in the PDW picture the interlayer frustration should be suppressed, leading to a decrease in ρc. This is precisely what is observed (Fig. 3c). The anisotropy ratio ρc/ρab is reduced (Fig. 3b) because the effect of H∥ on ρc is relatively stronger than on ρab. Similar results are obtained in La1.48Nd0.4Sr0.12CuO4  (Supplementary Fig. 10): here the reduction in ρc is weaker than in La1.7Eu0.2Sr0.1CuO4  and ρab is not affected within the experimental resolution, both consistent with the stronger pinning of stripe order at x = 1/8 (see also Supplementary Note 3). Nevertheless, the reduction of ρc/ρab by H∥ is comparable to that in La1.7Eu0.2Sr0.1CuO4  (Fig. 3b). Therefore, by applying an in-plane magnetic field, as proposed theoretically[5,10], our measurements confirm the presence of a PDW in both La1.7Eu0.2Sr0.1CuO4  and La1.48Nd0.4Sr0.12CuO4. The effects of H∥ are observable up to (i.e. T ~ TSO in La1.7Eu0.2Sr0.1CuO4: Supplementary Fig. 9), providing additional evidence for the PDW correlations in H = 0 at , as sketched in Fig. 3a.

Discussion

Our findings are thus consistent with the presence of local, PDW pairing correlations that compete with the uniform SC order at , and become dominant at intermediate H⊥ as T → 0. Our results also provide an explanation for the surprising, and a priori counterintuitive, observation[23] that Hc2 in La1.48Nd0.4Sr0.12CuO4  (Hc2 ~ 25 T) is higher than that in La1.7Eu0.2Sr0.1CuO4  (Hc2 ~ 20 T), even though its zero-field 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 ≈ 1/8 that the superconductivity persists to higher fields as T → 0.

In summary, by probing the previously inaccessible high and T → 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[18] for a PDW order that emerges in vortex halos. Since the observed PDW correlations extend only up to T ≪ Tpseudogap and not beyond Hc2(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−Eu0.2SrCuO4 with a nominal x = 0.10 and La1.6−Nd0.4SrCuO4 with a nominal x = 0.12 were grown by the traveling-solvent floating-zone technique[27]. The high homogeneity of the crystals was confirmed by several techniques, as discussed in detail elsewhere[23]. 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 that in other studies. We note that the trivial possibility that the two-step SC transition observed at H = 0 (e.g. Fig. 2c, d 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 , is clearly ruled out also by the behavior of dρab/dT with H⊥ (Supplementary Figs. 3a, 4, and 8b). In particular, both materials exhibit a reentrant metallic-like behavior at high H⊥, below Hc2 (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 values corresponding to different doping levels, where one would expect a gradual suppression of superconductivity with H⊥, 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 ρab were performed on sample B with dimensions 3.06 × 0.53 × 0.37 mm3 (a × b × c); ρc was measured on a bar with 0.34 × 0.41 × 1.67 mm3. The in-plane La1.48Nd0.4Sr0.12CuO4  crystal with dimensions 3.82 × 1.19 × 0.49 mm3 was cut along the crystallographic [110] and [10] axes, i.e. at a 45° angle with respect to a and b. A bar with 0.21 × 0.49 × 3.9 mm3 (a × b × c) was used to measure ρc in La1.48Nd0.4Sr0.12CuO4. The behavior of these samples remained stable for the duration of numerous experimental runs carried out in different cryostats and magnets (see below) that were needed for this study. After  ~3 years, the low-T properties of sample B changed, resulting in a quantitatively different T–H⊥ 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). Electrical contacts were made by evaporating Au on polished crystal surfaces such that, for current contacts, the two opposing faces were fully covered with Au to ensure a uniform current flow, while multiple voltage contacts made on the side faces were narrow enough to minimize the error in the absolute values of the resistance. This was followed by annealing in air at 700 °C. The data are shown for the voltage contacts separated by 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. Dupont 6838 Ag paste was used to attach gold leads (≈25 μm thick) to the samples, with a subsequent heat treatment at 450 °C in the flow of oxygen for 15 min. The room T contact resistances were <0.1 Ω for La1.7Eu0.2Sr0.1CuO4, i.e. <0.5 Ω for La1.48Nd0.4Sr0.12CuO4. The properties of the samples, including the values of , did not depend on the choice of voltage contacts used in the measurements, as expected in the absence of extrinsic (i.e. compositional) inhomogeneity.

was defined as the temperature at which the linear resistivity becomes zero, i.e. falls below the experimental noise floor (~0.5 mΩ). For the in-plane samples,  K for La1.7Eu0.2Sr0.1CuO4  and  K for La1.48Nd0.4Sr0.12CuO4; the out-of-plane resistivity ρc vanishes at (5.5 ± 0.3) K for La1.7Eu0.2Sr0.1CuO4  and (3.4 ± 0.5) K for La1.48Nd0.4Sr0.12CuO4. In La1.7Eu0.2Sr0.1CuO4, TSO ~ 15 K, TCO ~ 40 K (ref. [28]), and the pseudogap temperature Tpseudogap ~ 175 K (ref. [29]); in La1.48Nd0.4Sr0.12CuO4, TSO ~ 50 K, TCO ~ 70 K (ref. [30]), and Tpseudogap ~ 150 K (ref. [29]).

Measurements

The standard four-probe ac method (~13 Hz) was used for measurements of the sample resistance, with the excitation current (density) of 10 μA (~5 × 10−3 and  ~2 × 10−3 A cm−2 for La1.7Eu0.2Sr0.1CuO4 and La1.48Nd0.4Sr0.12CuO4, respectively) for the in-plane samples and 10 nA (~7 × 10−6 and  ≲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 μA (~1 × 10−3 and  ~4 × 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 Iac ≈ 1 μA (~13 Hz) through the sample and measuring the ac voltage across the sample. For each value of Idc, 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 ~ 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 Idc. 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 kΩ resistor in series with a π 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 ≤ T ≤ 0.7 K) and a 3He system (0.3 K ≤ T ≤ 35 K) in superconducting magnets (H up to 18 T), using 0.1–0.2 T min−1 sweep rates; a portable dilution refrigerator (0.02 K ≤ T ≤ 0.7 K) in a 35 T resistive magnet, using 1 T min−1 sweep rate; and a 3He system (0.3 K ≤ T ≤ 20 K) in a 31 T resistive magnet, using 1–2 T min−1 sweep rates. Below  ~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 ρab(T) curves below  ~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∥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[31].

The resistance per square per CuO2 layer R□/layer = ρab/l, where l = 6.6 Å  is the thickness of each layer.