Signatures of a pair density wave at high magnetic fields in cuprates with charge and spin orders
Zhenzhong Shi P. G. Baity J. Terzic T. Sasagawa Dragana Popović
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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 Popović^1,2∗



^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

    
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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_1.7Eu_0.2Sr_0.1CuO_4 and La_1.48Nd_0.4Sr_0.12CuO_4, cuprates with “striped” spin and charge orders, in perpendicular magnetic fields (H_⊥), and also with an additional field applied parallel to CuO_2 layers (H_∥).  We detected several phenomena predicted to arise from the existence of a PDW, including an enhancement of interlayer SC phase coherence with increasing H_∥.  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_⊥ as T→ 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<cit.> 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<cit.>, have raised the possibility that putative PDW correlations<cit.> may be responsible for the pseudogap regime<cit.>.  In order to distinguish between different scenarios, the most intriguing open question is what happens at low T≪ T_c^0 and high H_⊥, when SC order is destroyed by quantum phase fluctuations<cit.> and short-range charge orders are enhanced<cit.>.  However, the experimental evidence for a PDW state remains scant and largely indirect in the first place.

A PDW SC state was proposed<cit.> to explain the suppression of the interlayer (c-axis) Josephson coupling (or dynamical layer decoupling) apparent in the H=0 anisotropic transport<cit.> in La_1.875Ba_0.125CuO_4, as well as in optical measurements in La_1.85-yNd_ySr_0.15CuO_4 when the Nd concentration was tuned into the stripe-ordered regime<cit.>.  The dynamical layer decoupling was observed also  in the presence of an applied H_⊥, in La_1.905Ba_0.095CuO_4 (ref. Stegen2013) and    (ref. SchafgansPRL2010).  In La_2-x-y(Ba,Sr)_x(Nd,Eu)_yCuO_4 compounds near x=1/8, charge order appears in the form of stripes, which are separated by regions of oppositely phased antiferromagnetism (“spin stripes”)<cit.>  at T<T_SO<T_CO; here T_SO and T_CO are the onsets of spin and charge stripes, respectively.  In   at x=0.10, spin stripe order is induced<cit.> by applying H_⊥.  The dynamical layer decoupling was thus attributed<cit.> to a PDW SC state<cit.>, 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<cit.>, 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<cit.>.  
For example, transport measurements on La_1.875Ba_0.125CuO_4 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<cit.>; Josephson junction measurements<cit.> on La_1.875Ba_0.125CuO_4 devices support the prediction of a charge-4e SC condensate, consistent with the presence of a PDW state; an additional charge order was detected<cit.> in Bi_2Sr_2CaCu_2O_8 by scanning tunneling microscopy (STM) at very low H_⊥/T_c^0≲ 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_⊥.  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<cit.>.

Therefore, we measured transport in La_2-x-ySr_x(Nd,Eu)_yCuO_4 compounds, which have the same low-temperature structure as , over an unprecedented range of T down to T/T_c^0≲ 0.003 and fields up to H/T_c^0∼ 10 T/K.  We combined linear in-plane resistivity ρ_ab, nonlinear in-plane transport or voltage-current (V–I) characteristics, and the anisotropy ratio ρ_c/ρ_ab (here ρ_c is the out-of-plane resistivity) to probe both charge and vortex matter on single crystals with the nominal composition   and   (Methods); the former is away from x=1/8 and thus the stripe order is weaker<cit.>.  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<cit.>.  In particular, since H_∥ can reorient the spin stripes in every other plane<cit.>, 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   and , ρ_c and  ρ_ab vanish at the same T_c^0 within the error (Methods; see also Supplementary Information), indicating the onset of 3D superconductivity, similar to   (e.g. ref. Ando1995).  The initial drop of ρ_ab(T) with decreasing T (Fig. <ref>a) is accompanied by an enhancement of the anisotropy (Fig. <ref>b), 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 La_1.875Ba_0.125CuO_4 (ref. Li2007) 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 CuO_2 planes.  

The evolution of ρ_c/ρ_ab(T) with H_⊥ is shown in Fig. <ref>c.   The anisotropy at the highest T=20 K is ρ_c/ρ_ab∼ 6000 and practically  independent of H_⊥.  However, as T is lowered below T_c^0, ρ_c/ρ_ab 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 (ρ_c/ρ_ab>10^5) at H_⊥=H_p, signifying decoupling of or the loss of phase coherence between the planes.  However, strong SC correlations persist in the planes for H_⊥>H_p: here ρ_c/ρ_ab 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<cit.> that the H_⊥>20 T region corresponds to the normal state.  A smooth, rapid decrease of the anisotropy for H_⊥>H_p is interrupted by a “bump” or an enhancement in ρ_c/ρ_ab, centered at H_b.  Therefore, the behavior of ρ_c/ρ_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. <ref>b) or (2) the high-H_⊥ normal state by reducing H_⊥ at a fixed T (Fig. <ref>c).  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 ρ_c/ρ_ab, as a function of T and H_⊥, with the behavior of ρ_ab(T) for a fixed H_⊥, as shown in Fig. <ref> for both   and .  The ρ_ab(T) data were extracted from the in-plane magnetoresistance (MR) measurements 





(ref. ZShi2019-Hc2, Supplementary Fig. 2a; unless stated otherwise, the results are shown for   sample “B”, see Methods); the raw ρ_c(H) data are shown in Supplementary Figs. 2b and 2c.  In Figs. <ref>a and <ref>b, we also include T_c(H_⊥), as well as H_peak, the position of the peak in the in-plane MR (see, e.g., Supplementary Fig. 2a), which corresponds<cit.> to the upper critical field H_c2 in these materials (see also Supplementary Information).  Indeed, at a fixed T, ρ_c/ρ_ab starts to increase as H_⊥ is reduced below H_peak.  This is followed by an enhancement of ρ_c/ρ_ab near H_⊥=H_b, corresponding to the initial, metalliclike drop of ρ_ab(T) as the SC transition is approached from the normal state for a fixed H_⊥ (Figs. <ref>c and <ref>d).  The behavior of both materials is similar, except that the layer decoupling field H_p(T)≳ H_c(T) [or T_c(H_⊥)] in , as expected<cit.> for a stronger stripe order and frustration of interlayer coupling for x≈ 1/8.  Therefore, practically all the data in Figs. <ref>c and <ref>d, i.e. for H_⊥ >H_p, involve “purely” 2D physics, with no communication between the planes.  The striking splitting of the ρ_ab(T) curves in both materials (ref. ZShi2019-Hc2, Figs. <ref>c and <ref>d), into either metalliclike (i.e. SClike) or insulatinglike, when the normal state sheet resistance R_□/layer≈ 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<cit.>.  In addition, as previously noted<cit.>, the two-step ρ_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<cit.> thus suggest the formation of SC “islands” as H_⊥ is reduced below H_b at a fixed T (e.g. Figs. <ref>a and <ref>b), i.e. at the initial, metalliclike drop of ρ_ab(T) for a fixed H_⊥ (H_b dashed line in Figs. <ref>c and <ref>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 Information (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, metalliclike drop of ρ_ab(T) at T>T_c^0 in H=0 (see H_b dashed line in Figs. <ref>c and <ref>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. <ref>a.  







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_⊥: 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   than in , since it is farther away from x=1/8.  In the T→ 0 limit and for even higher H_⊥ (<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_∥, to reduce the interlayer frustration and decrease the anisotropy<cit.>.

Therefore, we have performed angle-dependent measurements of both ρ_ab(H) and ρ_c(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 H_∥=Hsinθ at different H_⊥=Hcosθ, i.e. fields parallel to the c axis, discussed above.  The angle-dependent ρ_ab(H) was measured also on another   sample (sample “B1”, Methods; Supplementary Fig. 8); the results are qualitatively the same on both samples.  Figure <ref>b illustrates the effect of H_∥ on ρ_c/ρ_ab at low T=0.070 K on sample B1 (see Supplementary Figs. 9a-d for the raw ρ_c and ρ_ab data at different T).  Clearly, there is no effect of H_∥ for H_⊥>H_c2(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<cit.> H_c2 (∥ c).  In contrast, for H_p⩽ H_⊥<H_c2, 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<cit.>.  

To understand exactly how H_∥ affects the anisotropy, we also investigate Δρ_ab=ρ_ab(H_∥)-ρ_ab(H_∥=0) and Δρ_c=ρ_c(H_∥)-ρ_c(H_∥=0) at different H_⊥ (Fig. <ref>c and Supplementary Fig. 8d for sample B1; Supplementary Figs. 9e-h for sample B).  It is obvious that ρ_ab is reduced by H_∥ for all H_⊥, which is the opposite of what would be expected if pair-breaking was dominant.  The suppression of ρ_ab is weaker for those H_⊥ where the superconductivity is stronger, e.g. near H_b∼ 15 T in Fig. <ref>c, and conversely, it is most pronounced above H_c2, indicating that the dominant effect of H_∥ is not related to superconductivity.  In fact, it occurs most strongly in the two regimes where ρ_ab(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 Information 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<cit.> (see also Supplementary Information).  The suppression of ρ_ab by H_∥ seems to vanish at experimentally inaccessible H_⊥, where the anomalous, insulatinglike ln(1/T) dependence observed in the field-induced normal state also appears to vanish<cit.>, 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_∥, in the PDW picture the interlayer frustration should be suppressed, leading to a decrease in ρ_c.  This is precisely what is observed (Fig. <ref>c).  
The anisotropy ratio ρ_c/ρ_ab is reduced (Fig. <ref>b) because the effect of H_∥ on ρ_c is relatively stronger than on ρ_ab.  Similar results are obtained in   (Supplementary Fig. 10): here the reduction in ρ_c is weaker than in   and ρ_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 ρ_c/ρ_ab by H_∥ is comparable to that in   (Fig. <ref>b).  Therefore, by applying an      in-plane magnetic field, as proposed theoretically<cit.>, our measurements confirm the presence of a PDW in both   and .  The effects of H_∥ are observable up to T>T_c^0 (i.e. T∼ T_SO in : Supplementary Fig. 9), providing additional evidence for the PDW correlations in H=0 at T>T_c^0, as sketched in Fig. <ref>a.

Finally, our results provide an explanation for the surprising, and a priori counterintuitive, observation<cit.> that H_c2 in   (H_c2∼ 25 T) is higher than in   (H_c2∼ 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≈ 1/8 that the superconductivity persists to higher fields as T→ 0.

In summary, by probing the previously inaccessible high H_⊥/T_c^0 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<cit.> for a PDW order that emerges in vortex halos.  Since the observed PDW correlations extend only up to T≪ 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.



Samples.  Several single crystal samples of La_1.8-xEu_0.2Sr_xCuO_4 with a nominal x=0.10 and La_1.6-xNd_0.4Sr_xCuO_4 with a nominal x=0.12 were grown by the traveling-solvent floating-zone technique<cit.>.  The high homogeneity of the crystals was confirmed by several techniques, as discussed in detail elsewhere<cit.>.  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   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. <ref>c and <ref>d for   and , 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 dρ_ab/dT with H_⊥ (Supplementary Figs. 3a, 4, 8b).  In particular, both materials exhibit a reentrant metalliclike behavior at high H_⊥, below H_c2 (e.g. see the reentrant darker blue color band for ). 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_⊥, 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 , 
detailed measurements of ρ_ab were performed on sample “B” with dimensions 3.06× 0.53× 0.37 mm^3 (a× b× c); ρ_c was measured on a bar with 0.34× 0.41× 1.67 mm^3.  The in-plane   crystal with dimensions 3.82× 1.19× 0.49 mm^3 was cut along the crystallographic [110] and [11̅0] axes, i.e. at a 45^∘ angle with respect to a and b.  A bar with 0.21× 0.49× 3.9 mm^3 (a× b× c) was used to measure ρ_c in .  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 ∼ 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   (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   and 2.00 mm for   in-plane samples; 0.47 mm for   and 1.26 mm for   out-of-plane samples.  Gold leads (≈ 25 μm 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 Ω for   (0.5 Ω for ) 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± 0.3) K for   and T_c^0=(3.6± 0.4) K for ; the out-of-plane resistivity ρ_c vanishes at (5.5± 0.3) K for   and (3.4± 0.5) K for .  In , T_SO∼ 15 K, T_CO∼ 40 K (ref. Fink2011), and the pseudogap temperature T_pseudogap∼ 175 K (ref. Cyr2018); in , T_SO∼ 50 K, T_CO∼ 70 K (ref. Tranquada1996), and T_pseudogap∼ 150 K (ref. Cyr2018).  





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 A cm^-2 and ∼ 2 × 10^-3 A cm^-2 for   and , respectively) for the in-plane samples  and 10 nA (∼ 7× 10^-6 A cm^-2 and ≲ 10^-5 A cm^-2 for   and , 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 A cm^-2 and ∼ 4 × 10^-4 A cm^-2 for   and   in-plane samples, respectively) and a small ac current excitation I_ac≈ 1 μA (∼ 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∼ 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 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 sweep rates; a portable dilution refrigerator (0.02 K ⩽ T ⩽ 0.7 K) in a 35 T resistive magnet, using 1 T/min sweep rate; and a ^3He system (0.3 K ⩽ T ⩽ 20 K) in a 31 T resistive magnet, using 1 – 2 T/min 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 ρ_ab(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<cit.>.




The resistance per square per CuO_2 layer R_□/layer=ρ_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.





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Acknowledgements

We acknowledge helpful discussions with L. Benfatto, E. Berg, V. Dobrosavljević, E. Fradkin, S. A. Kivelson, J. M. Tranquada, K. Yang, and J. Zaanen.  This work was supported by NSF Grants Nos. DMR-1307075 and DMR-1707785, and the National High Magnetic Field Laboratory (NHMFL) through the NSF Cooperative Agreements Nos. DMR-1157490, DMR-1644779, and the State of Florida. 




Author contributions

Single crystals were grown and prepared by T.S.; Z.S., P.G.B., and J.T. performed the measurements and analysed the data; Z.S., P.G.B., J.T. and D.P. wrote the manuscript, with input from all authors; D.P. planned and supervised the investigation.




Additional information

Supplementary information accompanies 
this paper.  Correspondence and requests for materials should be addressed to D.P. (email: dragana@magnet.fsu.edu).




Competing financial interests

The authors declare no competing interests.






Supplementary Information









=24pt

Superconducting transition temperature and vortex phase diagram 

T_c(H_⊥) were determined as the temperatures at which the linear resistance R≡lim_I_dc→ 0 V/I (or resistivity) becomes zero  (V – voltage, I – current).  In both   and , ρ_c and ρ_ab vanish at the same temperature within the error, indicating the onset of 3D superconductivity.  In contrast, in striped La_1.875Ba_0.125CuO_4 in H=0, a 2D superconductivity was reported^11 to appear at a T higher than the onset of 3D superconductivity, although it has been suggested^4 that higher precision measurements might reveal the same T_c^0 for both ρ_c and ρ_ab. 

The position of the peak in ρ_ab(H_⊥) (see, e.g., Fig. S2a),  H_⊥=H_peak(T), was found^23 to be of the order of the upper critical field (H_c2), i.e. the field scale corresponding to the closing of the SC gap.  Therefore, the superconductor with T_c(H)>0 (i.e. a ρ_ab=0 state) is separated from the normal state at H_⊥>H_peak by a wide regime of SC phase fluctuations arising from the motion of vortices.  At low T, this regime exhibits non-Ohmic transport^23 consistent with the motion of vortices in the presence of disorder: it was thus identified^23 as a viscous vortex liquid with the zero freezing temperature, i.e. T_c=0.  



In-plane transport in perpendicular magnetic fields

The in-plane T–H_⊥ phase diagrams of   and   are shown in Figs. S<ref>a and S<ref>, respectively.  In order to display ρ_ab(T) for all H_⊥, we use the color maps.  The metalliclike, dρ_ab/dT>0 regions where ρ_ab(T>0)≠ 0 (blue regions II  in Figs. S<ref>a and S<ref>), which exhibit non-Ohmic transport at low T, were identified^23 as a viscous vortex liquid with the zero freezing temperature, i.e. T_c=0. 

In , within the phase-fluctuations regime, there is clearly a region of pronounced insulatinglike (dρ_ab/dT<0) behavior at low T (region IV in Fig. S<ref>a), with its precursors, i.e. the weakening of the metalliclike T dependence, becoming visible already at T≲ T_c^0 (see also Fig. S<ref> for ), i.e. at H_b(T) (see also Fig. 2).  The insulatinglike dρ_ab/dT that develops at low T, in region IV, is at least as strong as the one observed in the field-revealed normal state, i.e. for H_⊥>H_c2∼ 20 T (Fig. S<ref>b).  By tracking the “h/4e^2” line where R_□/layer changes from R_□/layer<R_Q at lower H_⊥ and higher T, to R_□/layer>R_Q at higher H_⊥ and lower T, we find that it has two branches (Figs. S<ref>a and S<ref>): while the upper one seems to form an upper limit for the presence of vortices^23, the lower one extrapolates roughly to the onset of region IV as T→ 0, suggesting that region IV may be related to the localization of Cooper pairs.  Indeed, although the range of T and H_⊥ is limited, the scaling behavior of ρ_ab(T,H_⊥)  near the onset of region IV (Fig. S<ref>) seems consistent with the presence of a T=0 SIT driven by quantum phase fluctuations in a disordered 2D system^23.  However, the V–I measurements in region IV reveal a non-Ohmic increase of dV/dI with I_dc (Fig. S<ref>c), in contrast to the observations [S1] on the insulating side of the 2D SIT where dV/dI decreases with I_dc.  On the other hand, a non-Ohmic increase of dV/dI with I_dc is consistent with the motion of vortices in the presence of disorder (i.e. a viscous vortex liquid) (refs. 23, [S2]).  The increase of dV/dI with I_dc is precisely the opposite of what would be expected in the case of simple Joule heating, confirming the presence of SC correlations, characteristic of a vortex liquid, in region IV.  Our results thus strongly suggest that region IV consists of SC puddles, with no inter-puddle phase coupling, in an insulatinglike, high-field normal-state background: 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 region with T can be traced to the initial, metalliclike drop of ρ_ab(T) at T>T_c^0 in H=0 (see also H_b dashed line in Figs. 2c and 2d).  In , the lower branch of the “h/4e^2” line, together with H_b(T), practically outlines the region of the weakened, metalliclike ρ_ab(T).  Although the insulatinglike behavior is not observed, these results strongly suggest that it would ultimately emerge at even lower, experimentally inaccessible T, roughly in the ∼10-20 T field range, similar to .  

The weakening of the metalliclike T dependence at intermediate H_⊥, which leads to the insulatinglike behavior in   at low T (region IV in Fig. S<ref>), is manifested by the appearance of a “shoulder” in the in-plane MR curves for H<H_peak (Figs. S<ref>a and S<ref>a).  The shoulder in the MR becomes more noticeable with decreasing T and, at very low T≲ 0.05 K, the MR in this range of fields becomes hysteretic (Fig. S<ref>a).  The size of the hysteresis grows with decreasing T (Fig. S<ref>b), and the range of fields where it is observed, independent of the sweep rate, outlines the boundary of region IV (Fig. S<ref>a) where dρ_ab/dT<0.  In other words, the hysteretic MR is not observed at even higher H_⊥, where dρ_ab/dT>0 (blue sliver in Fig. S<ref>a).  Another hysteretic regime appears as the system enters the normal state (Fig. S<ref>a), but it is much less robust: its width in H_⊥ is reduced with decreasing sweep rate (Fig. S<ref>a inset; Fig. S<ref>a shows the boundaries corresponding to 1 T/min).  In general, a hysteresis is a manifestation of the coexistence of phases, i.e. it indicates the presence of domains of different phases in the system.  Typical signatures of such systems include slow, nonexponential relaxations and memory effects, which are indeed observed here (Fig. S<ref>).  The hysteretic response to H_⊥, observed when the superconductivity is suppressed, is attributed to the presence of domains with spin stripes.



Effects of parallel magnetic fields on the spin structure

In contrast to La_2-xBa_xCuO_4, the magnetization of La_1.8-xEu_0.2Sr_xCuO_4 and La_1.6-xNd_0.4Sr_xCuO_4
is dominated by the rare-earth ion (e.g. refs. 19, [S3]), so that studies of the Cu spin magnetism are difficult and scarce in these compounds.  Nevertheless, it is known that, in the low-temperature tetragonal phase of antiferromagnetic La_1.8Eu_0.2CuO_4, an in-plane magnetic field (H∥ b) of ∼ 6 T leads to a spin-flop transition^19 of the Cu spin moments in every other plane.  Moreover, this spin-flop transition field is roughly the same as that in La_1.875Ba_0.125CuO_4 for T<T_SO (ref. 20).  Since the structure of these three materials is similar [S4], it is thus likely that the spin-flop transition occurs also in La_1.8-xEu_0.2Sr_xCuO_4 and La_1.6-xNd_0.4Sr_xCuO_4 near x=1/8 at comparable fields.  La_2-xSr_xCuO_4 with x=0.115, for example, also exhibits a spin-flop transition at a similar field ≈ 7.5 T [S5].
We note that, in case of H∥ [110], the transition is broader: spins in all planes continuously rotate until staggered moment is again perpendicular to the field, so there is no sharp spin-flop transition^20,21.  In all cases, however, the reorientation of spins under the influence of H∥ ab should enhance Josephson coupling between layers, within the PDW picture.

In the H∥ [110] configuration, Josephson coupling between layers may be enhanced also by another mechanism, namely by the field partially compensating for the momentum mismatch between the layers; that mechanism would reduce ρ_c, but it would have no effect on ρ_ab  [S6].  In our in-plane   crystal, which was cut at a 45^∘ angle with respect to a and b axes (Methods), we do not see, indeed, any observable effect of H∥ [110] on ρ_ab (Fig. S<ref>a).  This also implies that, in contrast to , (Fig. 3c, top), the effect of spin reorientation on ρ_ab is too weak to be observed within the experimental resolution.  We note that, at the same time, in our out-of-plane   crystal for which H∥ [100] (Methods) and in which the mechanism of ref. [S6] thus cannot play a role, the reduction in ρ_c (Fig. S<ref>b) is also weaker than in .  The weaker spin reorientation effect of H_∥ on ρ_c and ρ_ab in   than in   is, therefore, attributed to the stronger pinning of stripe order at x=1/8.  














































































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