Magnetic field reveals vanishing Hall response in the normal state of stripe-ordered cuprates

The origin of the weak insulating behavior of the resistivity, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho }_{xx}\propto {\mathrm{ln}}\,(1/T)$$\end{document}ρxx∝ln(1/T), revealed when magnetic fields (H) suppress superconductivity in underdoped cuprates has been a longtime mystery. Surprisingly, the high-field behavior of the resistivity observed recently in charge- and spin-stripe-ordered La-214 cuprates suggests a metallic, as opposed to insulating, high-field normal state. Here we report the vanishing of the Hall coefficient in this field-revealed normal state for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\ <\ (2-6){T}_{{\rm{c}}}^{0}$$\end{document}T<(2−6)Tc0, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{{\rm{c}}}^{0}$$\end{document}Tc0 is the zero-field superconducting transition temperature. Our measurements demonstrate that this is a robust fundamental property of the normal state of cuprates with intertwined orders, exhibited in the previously unexplored regime of T and H. The behavior of the high-field Hall coefficient is fundamentally different from that in other cuprates such as YBa2Cu3O6+x and YBa2Cu4O8, and may imply an approximate particle-hole symmetry that is unique to stripe-ordered cuprates. Our results highlight the important role of the competing orders in determining the normal state of cuprates.


Introduction
The central issue for understanding the high-temperature superconductivity in cuprates is the nature of the ground state that would have appeared had superconductivity not intervened.Therefore, magnetic fields have been commonly used to suppress superconductivity and expose the properties of the normal state, but the nature of the high-H normal state may be further complicated by the interplay of charge and spin orders with superconductivity.La 2−x−y Sr x (Nd,Eu) y CuO 4 compounds are ideal candidates for probing the nature of the field-revealed ground state 1 of underdoped cuprates in the presence of intertwined orders because, for doping levels near x = 1/8, they exhibit both spin and charge orders with the strongest correlations and lowest T 0 c already at H = 0.In particular, in each CuO 2 plane, charge order appears in the form of static stripes that are separated by charge-poor regions of oppositely phased antiferromagnetism 2 , i.e. spin stripes, with the onset temperatures T CO > T SO > T 0 c ; stripes are rotated by 90 • from one layer to next.The low values of T 0 c have made it possible to determine the in-plane T -H vortex phase diagram 3 using both linear and nonlinear transport over the relatively largest range of T and perpendicular H (i.e.H ⊥ CuO 2 layers), and to probe deep into the high-field normal state.The most intriguing question, indeed, is what happens after the superconductivity is suppressed by H, i.e. for fields greater than the quantum melting field of the vortex solid where T c (H) → 0. It turns out that a wide regime of vortex liquid-like behavior, i.e. strong superconducting (SC) phase fluctuations, persists in two-dimensional (2D) CuO 2 layers, all the way up to the upper critical field H c2 .It is in this regime that recent electrical transport measurements have also revealed 4 the signatures of a spatially modulated SC state referred to as a pair density wave 5 (PDW).
The normal state, found at H > H c2 , is highly anomalous 3 : it is characterized by a weak, insulating T -dependence of the in-plane longitudinal resistivity, ρ xx ∝ ln(1/T ), without any sign of saturation down to at least T /T 0 c ∼ 10 −2 , and the negative magnetoresistance (MR).In contrast to the H-independent ln(1/T ) reported 6,7 for the case where there is no clear evidence of charge order 8 in H = 0 and where the high-H normal state appears to be an insulator 6,9 , here the ln(1/T ) behavior is suppressed by H, strongly suggesting that ρ xx becomes independent of T , i.e. metallic, at high enough magnetic field (H > 70 T).
In either case, the origin of such a weak, insulating behavior is not understood 7,[10][11][12][13] , but it is clear that the presence of stripes seems to affect the nature of the normal state.
Therefore, additional experiments are needed to probe the highest-H regime.
In cuprates, the Hall effect has been a powerful probe of the T = 0 field-revealed normal state (e.g.5][16][17][18][19] and refs.therein).In the high-field limit as T → 0, the Hall coefficient R H , obtained from the Hall resistivity ρ yx (H) = R H H, can be used to determine the sign and the density (n) of charge carriers.In a single-band metal, for example, n = n H , where the Hall number n H = 1/(eR H ) and e is the electron charge (R H > 0 for holes, R H < 0 for electrons).In general, the magnitude of R H reflects the degree of particle-hole asymmetry and, thus, understanding the Hall coefficient provides deep insight into the microscopic properties.However, the interpretation of the Hall effect in cuprates has been a challenge, because R H can depend on both T and H, and it can be affected by various factors, such as the presence of SC correlations and the topological structure of the Fermi surface.For example, a drop of R H from positive to negative values with decreasing T , observed in underdoped cuprates for dopings where charge orders are present 20 at high H, has been attributed 14,19,21,22 to the Fermi surface reconstruction, which includes the appearance of electron pockets in the Fermi surface of a hole-doped cuprate.A drop in the normal state, positive R H (T ) is, in fact, observed in all hole-doped cuprates near x = 1/8 (see ref. 21 and refs.therein).Other studies of the Hall effect in cuprates have focused on the effects of SC fluctuations (refs. 23,24  refs.therein), and on the pronounced change in the Hall number across the charge order and the pseudogap quantum critical points 16-18, 25, 26 .However, the Hall behavior in the T → 0, H > H c2 regime has remained mostly unexplored.In particular, recent studies of the La 2−x−y Sr x (Nd,Eu) y CuO 4 compounds have demonstrated 3,4 that reliable extrapolations to the T → 0 normal state can be made only by tracking the evolution of SC correlations down to T ≪ T 0 c and H/T 0 c [T/K] ≫ 1, but there have been no studies of the Hall effect in stripe-ordered cuprates that extend to that regime of T and H and, specifically, to the anomalous normal state at H > H c2 .Therefore, we measure the Hall effect on La 1.7 Eu 0.2 Sr 0.1 CuO 4 and La 1.48 Nd 0.4 Sr 0.12 CuO 4 (see Methods) over the entire in-plane T -H vortex phase diagram previously established 3,4 for T down to T /T 0 c 0.003 and fields up to H/T 0 c ∼ 10 T/K, and deep into the normal state.Combining the results of several techniques allows us to achieve an unambiguous interpretation of the Hall data for H < H c2 , and reveal novel properties of the normal state for H > H c2 .Our main results are summarized in the T -H phase diagrams shown in Fig. 1.The key finding is that, in the high-field limit, the positive R H decreases to zero at T = T 0 (H) upon cooling, and it remains zero (see Methods) all the way down to the lowest measured T , despite the absence of any observable signs of superconductivity.Here, (5.7 ± 0.3) ∼ 15 (ref. 27) ∼ 40 (ref. 27) ∼ 175 (ref. 28) La 1.48 Nd 0.4 Sr 0.12 CuO 4 (3.6 ± 0.4) ∼ 50 (ref. 29) ∼ 70 (ref. 29) ∼ 150 (ref. 28) , where the linear resistivity ρ xx becomes zero, and other characteristic temperatures, such as the pseudogap T PG , are summarized in Table 1.Therefore, the vanishing Hall coefficient appears well below T PG , in the temperature region where both charge and spin orders (i.e.stripes) have fully developed.Meanwhile, we note that the drop of R H at T > T 0 does not depend on H, while T 0 (H) is very weakly dependent on H (Fig. 1), almost constant, suggesting that R H ≈ 0 is characteristic of the zero-field (normal) ground state in the presence of stripes.

Hall coefficient
Our main results are shown in Fig. 1.From the Hall measurements, we are able to identify regions in (T, H) phase space with different signs of R H (Fig. 1a,b) and, in particular, we find R H ≈ 0 over a wide range of T and H in both materials.Further insight is obtained by comparing the Hall results with the phase diagram obtained by other transport techniques, as shown in Fig. 1c,d.The measurements of the in-plane magnetoresistance ρ xx (H) at different T were used to determine 3,4 T c (H), the melting temperature of the vortex solid in which ρ xx = 0.Although the quantum melting fields of the vortex solid are relatively low (∼ 5.5 T and ∼ 4 T, respectively, for La 1.7 Eu 0.2 Sr 0.1 CuO 4 and La 1.48 Nd 0.4 Sr 0.12 CuO 4 ), the regime of strong 2D SC phase fluctuations (vortex liquid) extends up to much higher fields 3,4 H peak (T ) ∼ H c2 (T ), where H peak (T ) is the position of the peak in ρ xx (H).For T → 0, H c2 ∼ 20 T for La 1.7 Eu 0.2 Sr 0.1 CuO 4 and H c2 ∼ 25 T for La 1.48 Nd 0.4 Sr 0.12 CuO 4 .
Figure 2 shows the field dependence of R H = ρ yx (H)/H for various T in both materials (see Supplementary Figs. 1 and 2 for the ρ yx (H) data at different T ).At relatively high T > T 0 > T 0 c in the pseudogap regime, the positive R H is independent of H (Fig. 2a,c), as observed in conventional metals, although the in-plane transport is already insulatinglike, i.e. dρ xx /dT < 0 (Fig. 3a,c, also Supplementary Fig. 5).Upon cooling, R H decreases to zero at T = T 0 (H), and then becomes negative in the regime of lower fields.The field dependence remains weak at all T , similar to the observations 30 25 ), where the charge order is at best very weak.The most striking finding is that, at the highest fields (H > H peak ∼ H c2 ), R H remains immeasurably small for T < T 0 , down to the lowest measured T (Fig. 2b,d).In other words, for a fixed T < T 0 (H), R H < 0 at low H, but it becomes zero and remains zero (see Methods) with increasing field.
In Fig. 3, we compare R H (T ) and ρ xx (T ) for various fields.The drop of R H observed at T > T 0 does not depend on H (Fig. 3b,d), similar to earlier studies of the striped La-214 family [30][31][32][33][34] and other cuprates 21 .The independence of the drop of R H on field implies that this is a property of the zero-field state, as opposed to some field-induced phase.In YBCO, the drop in R H was attributed 18,21 to the Fermi surface reconstruction by charge order.In striped cuprates, however, the onset of the drop in R H seems closer to the structural phase transition temperature T d2 (Fig. 3), where T SO < T CO < T d2 < T PG (ref. 3 ), but its origin is still under debate [30][31][32][33][34] .We define T 0 (H) as the temperature at which R H becomes zero or negative, and it is apparent that it has a very weak, almost negligible field dependence.
T 0 [∼ (2−3)T 0 c for La 1.7 Eu 0.2 Sr 0.1 CuO 4 ; ∼ 6T 0 c for La 1.48 Nd 0.4 Sr 0.12 CuO 4 ] is comparable to the temperature at which ρ xx (T ) curves in both materials split into either metalliclike (i.e.SClike) or insulatinglike, a correlation that seems to be manifested only in the presence of stripes 34 .We find that, interestingly, this occurs (Fig. 3a,c) when the normal state sheet resistance R /layer ≈ R Q , where R Q = h/(2e) 2 is the quantum resistance for Cooper pairs.Transport in the high-T , H < H c2 regime Previous studies have identified 3,4 the H < H peak regime as the vortex liquid.The Hall resistivity due to mobile vortex cores is expected 35,36 to obey the relation ρ xx 2 /ρ yx ∝ H, which is indeed observed in this regime in our samples (Supplementary Fig. 6), thus confirming its identification as the vortex liquid.We also find that, in this field range, R H is negative for T ′ 0 < T < T 0 (dark beige areas in Fig. 1c,d) and it exhibits a minimum, which is suppressed by increasing H (Fig. 3b,d).Such behavior is generally understood 14,33,34 to result from the vortex contribution to ρ yx .The minimum is less pronounced for x ≈ 1/8 (Fig. 3d) than for x = 0.10 (Fig. 3b), consistent with prior observations 33,34 , as well as with the recent evidence 4 of a more robust SC PDW state at x ≈ 1/8.Therefore, the agreement of the results of different techniques allows an unambiguous interpretation of the negative R H as being dominated by the motion of vortices, even if other effects might, in principle, also contribute to R H .For example, in contrast to stripe-ordered La-214, in YBa 2 Cu 3 O y the negative R H increases with increasing H (ref. 14 ), suggesting that other effects dominate over the vortex contribution.Our results, however, show that the observation of a field-independent T 0 , at which R H changes sign, does not necessarily imply that R H < 0 is not caused by vortices.

Transport in the low-T , H < H c2 regime
Similarly, at lower temperatures for H < H peak , in the viscous VL region 3 , the negative R H is suppressed by decreasing T , resulting in R H = 0 at T < T ′ 0 (light violet area in Fig. 1c,d) down to the lowest measured T = 0.019 K (Fig. 3b,d insets).Here, R H = 0 is thus attributed to the slowing down and freezing of the vortex motion with decreasing T in the presence of disorder.This observation is reminiscent of the zero Hall resistivity observed within the VL regime (H < H c2 ) in some conventional disordered 2D superconductors 37,38 and oxide interfaces 39 .Indeed, it has been proposed 36 that the vanishing of R H in such socalled "failed superconductors" can be also explained by the strong pinning of the vortex motion.

Transport in the H > H c2 regime
The remaining, most intriguing question is the origin of R H = 0 observed beyond the VL regime, at all T < T 0 and H > H peak ≈ H c2 (blue areas in Figs.1c,d).This anomalous normal state is also characterized 3 by ρ xx ∝ ln(1/T ).In addition, here the out-of-plane resistivity has the same T -dependence 4 , ρ c ∝ ln(1/T ), implying that the transport mechanism is the same for both in-plane and c directions.We discuss several potential scenarios for the origin of R H = 0 in this regime.

Discussion
For H > H c2 , the first possibility to consider is whether there are any remnants of superconductivity, such as SC fluctuations that may no longer be detectable in the ρ xx measurement.In cuprates (ref. 23and refs.therein), as well as in conventional superconductors 23,45 , the effect of SC fluctuations on the Hall signal has been extensively studied in the high-T normal state, at low fields and above T 0 c , within the conventional, weak-pairing fluctuation formalism built upon the Ginzburg-Landau (GL) theory of the BCS regime.The qualitative picture of SC fluctuations at low temperatures and high fields (H > H c2 ), however, drastically differs from the GL one 23,46 , but in either case, existing models predict nonzero R H with particle-hole asymmetry terms 23,46 .Recently, a strong-pairing fluctuation theory that also incorporates pseudogap effects has been proposed 24 for R H in cuprates, but only for the low-field, T > T 0 c regime.However, it does not describe the H-independence of the drop in R H with decreasing T observed for T > T 0 (Fig. 3b,d).
Extensive transport studies 3,4 , including those of the anisotropy ratio ρ c /ρ xx , have not found any observable signs of superconductivity, including the PDW, for H > H c2 .
For example, here ρ c /ρ xx no longer depends on a magnetic field, neither H c nor H ⊥ c, and it reaches its high-T , normal-state value.As discussed elsewhere and La 1.48 Nd 0.4 Sr 0.12 CuO 4 , given also that R H = 0 spans a ∼10 T-wide range of fields in Fig. 1.Therefore, models that rely on the existence of preformed pairs 41,42 , strong SC correlations such as those in "failed superconductors" 37,38 , or conventional Gaussian SC fluctuations 23,46 do not seem relevant for the H > H c2 regime.Hence, we consider other possible scenarios.
The drop of the positive R H (T ) to zero, observed at T > T 0 , has been attributed 14,19,21,22 to the Fermi surface reconstruction, implying the presence of both hole and electron pockets in the Fermi surface.Although this issue is not fully settled 47 , partly because of the disagreement with photoemission experiments, a similar drop of R H (T ) seen in La 1.7 Eu 0.2 Sr 0.1 CuO 4 and La 1.48 Nd 0.4 Sr 0.12 CuO 4 at T > T 0 (Fig. 3b,d) suggests the possibility that the same mechanism might be responsible for the normal-state behavior of R H at T > T 0 in these stripe-ordered cuprates, and even in their T < T 0 , high-field regime, which is the focus of our study.We note, however, that there is no consensus on how the Fermi surface is affected by the presence of spin stripes, including in La and unique measurement challenges).However, our results, in fact, place stringent constraints on any realistic models for the Hall effect in this regime: R H < 0.05 mm 3 /C, but this condition also needs to be satisfied over a wide range of H and T for two different materials and doping levels (Fig. 1).In a multiband picture, this would require that a subtle balance, or a near-cancellation, of contributions from hole and electron pockets is maintained over a huge range of parameters T and H, as well as change in x and the rare-earth composition y.Therefore, a multiband picture seems unlikely considering the robustness of our results.
Since dρ xx /dT < 0 in the normal state, one could speculate whether R H vanishes (i.e.ρ yx = 0, or conductivity σ xy = 0) because of some kind of localization.Strong, exponential localization does not describe the data because the T -dependence of the resistivity is very weak, it becomes even weaker with increasing H, and at the same time, the absolute value of ρ xx remains relatively low and comparable to that at T > T 0 (Fig. 3a,c).Similarly, as the system goes from the VL to the normal state with H at a fixed, relatively high T < T 0 , the H-dependence of ρ xx is negligible 3 (e.g. at ∼ 4 K in Fig. 3a), while R H changes qualitatively from a finite negative value to zero (Fig. 3b).Our results for R H are indeed the opposite of those in lightly-doped 17 , i.e. insulating cuprates with a diverging ρ xx (T → 0), or in highly underdoped 16 cuprates, both of which seem to show a diverging R H at low T .If n = n H = 1/(eR H ) holds, this is indeed consistent with a depletion of carriers, whereas in our case it would indicate a diverging number of carriers.
Likewise, weak localization in 2D is not consistent with the data, since the same ln(1/T ) behavior is observed also along the c axis, just like in underdoped La x = 1/8 at low H = 5 T and high T , i.e.T > T 0 in Fig. 3b,d.Although, in contrast, our central result is R H = 0 in the high-field (H > H c2 ), T < T 0 regime, models based on the quasi-1D picture seem to be a plausible description of stripe-ordered cuprates also when the applied H suppresses the interstripe Josephson coupling.One such model, for example, predicts 49 , both in the presence and the absence of a spin gap, a non-Fermi-liquid smectic metal phase, in which the transport across the stripes is incoherent, whereas it is coherent inside each stripe.Importantly, a smectic metal has an approximate particlehole symmetry 49 for x < 1/8, which implies ρ yx ≈ 0, as observed in our experiment.
Incidentally, the same model had been proposed as the origin of the drop of R H in the early studies 50 of YBa 2 Cu 3 O y at T > T 0 .Other, more general scenarios include holographic models for doped Mott insulators 51 , which also feature emergent particle-hole symmetry 52 .
Our study of the Hall effect across the entire in-plane T -H phase diagram has clarified and further confirmed that the origin of R H = 0 reported in earlier studies 31,33,34,40 of stripe-ordered cuprates is associated with the presence of SC fluctuations.In contrast, our central result is that, at much higher fields, such that H > H c2 , the field-revealed normal state of La 2−x−y Sr x (Nd,Eu) y CuO 4 cuprates with static spin and charge stripes is characterized by a zero, i.e. immeasurably small, Hall coefficient.Indeed, since the vanishing of R H is pronounced over a wider range of H and T for x = 0.12 (Fig. 1b,d) than for x = 0.10 (Fig. 1a,c), this strongly suggests that R H ≈ 0 is crucially related to the presence of static stripe order.Further insight into this issue might come from other experiments at high fields, such as optical conductivity, Raman scattering, and thermal transport, to determine whether R H ≈ 0 results from a fortuitous near-cancellation of contributions from multiple bands or it signals an approximate particle-hole symmetry, as expected for a smectic metal in a stripe-ordered cuprate 49 and in more general models of correlated matter 51,52 .

Samples
Several single crystal samples of La  Depending on the temperature, the excitation current (density) of 10 µA to 316 µA to ∼ 6.3 × 10 −2 A cm −2 for La 1.48 Nd 0.4 Sr 0.12 CuO 4 ) was used: 10 µA for 0.019 K (Supplementary Fig. 1d); 100 µA for all measurements in fields up to 12 T (Supplementary Fig. 1a), and for the 0.3 K data in Supplementary Figs. 2 and 3; 316 µA for all other measurements.These excitation currents were low enough to avoid Joule heating 3 .Traces with different excitation currents were also compared to ensure that the reported results are in the linear response regime.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 to reduce the noise and heating by radiation in all measurements.
Several different cryostats at the National High Magnetic Field Laboratory were used, including a dilution refrigerator (0.016 K T 0.7 K) and a 3 He system (0.R yx d ∼ 0.005 mΩ cm, while ρ xx 0.5 mΩ cm (see Fig. 3 for H = 0, but at T > 15 K, the magnetoresistance is very weak 3,55 ).At low T , ρ yx is drastically suppressed even further (Supplementary Fig. 1), and the ratio ρ yx /ρ xx becomes even greater.This observation is significant in itself as discussed in the main text, but it also presents certain experimental challenges.
In With the single-shot 3 He cryostat, the temperature control below 1.6 K is usually complicated by the evaporation of the 3 He liquid, which induces a slow T drift with time.
To minimize its impact, we measured the traces with opposite fields in back-to-back experiments and did not consider the data when the T drift was too large.The maximum T drift between the two traces is typically ∼ 10 − 20 mK for T < 1.6 K.
To ensure the accuracy of our results, we have also repeated Hall measurements on La 1.7 Eu 0.2 Sr 0.1 CuO 4 at 0.71 K more than 10 times, by recondensing the 3 He liquid and resetting the temperature for each positive and negative field sweep.This ensures that, even if T drifts due to 3 He evaporation, the amount of the drift would be the same in the two traces.We carefully compared the field dependence of the Cernox ® thermometer reading, T r , for each positive and negative field sweep, a typical example of which is shown in Supplementary Fig. 4a inset.We note that the Cernox ® sensor is not calibrated in the field, and thus the increase of T r only reflects the magnetoresistance of the sensor, while the sample temperature (controlled by the sorb) is unchanged.As shown in the Supplementary Fig. 4a inset, the temperature is the same (within 1 mK) during the entire positive and negative field sweeps.
To determine the uncertainty of the Hall coefficient measurement results, we divide the R yx (H) and R H (H) data into bins (typical size is 1 T; see Supplementary Figs. 2 and 3), and calculate the mean and the standard deviation (SD) within each bin.Therefore, the error bars in Figs. 2 and 3, and Supplementary Figs. 1, 2, 3, 4, and 6, all correspond to ±1 SD of the data points within each bin.To reduce the SD even further, we averaged over five sets of measurements at 0.71 K to reduce the experimental error bar (i.e. 1 SD) from ∆R H ∼ 0.2 mm 3 /C to ∆R H ∼ 0.05 mm 3 /C (Supplementary Fig. 4b).This is comparable to, if not better than, ∆R H in other studies of the Hall effect on cuprates 14 , including those in which zero Hall coefficient (induced by superconductivity, not in the normal state) was found 30,33,40,43 , as well as on other systems, such as iron-based superconductors 56 .
We emphasize again that the experimental error for ρ yx (and R H ) is dominated by the imperfect cancellation of the contribution from the T -dependent longitudinal resistivity ρ xx , which is inevitably much larger than the (nearly) zero transverse contribution.At T = 0.71 K, where we achieved almost perfect temperature control (to within 1 mK) and thus the maximum cancellation of the longitudinal resistivity contribution (Supplementary Fig. 4a), we also determined, using standard error analysis, the ∼ 95% confidence intervals for R H , e.g.(−0.008 ± 0.020) mm 3 /C at 17 T.This further confirms our conclusion that the Hall coefficient (and Hall resistivity, see Supplementary Fig. 1e inset) remains zero in the high-field normal state, i.e. above the upper critical field H peak .
In principle, the error bar in the ρ yx measurement can also be reduced by increasing the excitation current density or, equivalently, by reducing the sample thickness or width for a fixed current.However, the applied current density still needs to remain below the limit above which Joule heating is induced.The effects of excitation currents have been studied thoroughly 3 , so that here we have used the highest excitation current density possible without inducing Joule heating.Therefore, reducing the sample thickness, for example, would not help to decrease the error bar further, because a smaller excitation current would also need to be used.3), while error bars correspond to ±1 SD (standard deviation) of the data points within each bin.The error bars are typically larger at lower T resulting from the use of lower excitation currents I (see Methods) necessary to avoid heating and to ensure that the measurements are taken in the I → 0 limit, because of the strongly nonlinear (i.e.non-Ohmic) transport in the presence of vortices 3 .At higher T , the error bars are 3-4 times smaller, ∆R H ∼ 0.2 − 0.3 mm 3 /C (see also Supplementary Fig. 3).However, similar ∆R H , and even ∆R H ∼ 0.05 mm 3 /C, have been achieved also at low T , as described in Methods (see also Supplementary Fig. 4).At high   28 ).c and d, ρ xx and R H , respectively, for La 1.48 Nd 0.4 Sr 0.12 CuO 4 ; T PG ∼ 150 K (ref. 28).The transition from the low-temperature orthorhombic to a low-temperature tetragonal structure occurs at T d2 ∼ 125 K in La 1.7 Eu 0.2 Sr 0.1 CuO 4 and T d2 ∼ 70 K in La 1.48 Nd 0.4 Sr 0.12 CuO 4 (ref. 3).The data in a and c are from refs. 3,4 At the highest fields, ρ xx ∝ ln(1/T ), as discussed in more detail elsewhere 3 .In both materials, R H decreases upon cooling, and reaches zero at T = T 0 (H).For H < H c2 ∼ H peak , R H becomes negative at even lower T , then goes through a minimum, and eventually reaches zero again at T = T ′ 0 (H), as shown; R H remains zero down to 0.019 K (b and d insets).For H > H c2 , R H = 0 for all H and T < T 0 (H).Similar to those in Fig. 2, error bars correspond to ±1 SD of the data points within each bin.All dashed lines guide the eye.
in La 1.48 Nd 0.4 Sr 0.12 CuO 4 for T 5 K at 9 T is due to the onset of freezing of the vortex motion.Recently, R H = 0 was reported 40 also in La 2−x Ba x CuO 4 with x = 1/8, in the regime of nonlinear (i.e.non-Ohmic) transport analogous to the VL in Fig.1, in which the negative R H arising from the vortex motion decreases towards zero as the doping approaches x = 1/8 (Figs.3b,d).The vanishing Hall response in La 1.875 Ba 0.125 CuO 4 was indeed attributed[40][41][42] to the presence of SC phase fluctuations and Cooper pairs that survive within the charge stripes after the interstripe SC phase coherence has been destroyed by H. Likewise, in YBa 2 Cu 3 O y thin films near a disorder-tuned superconductor-insulator transition, R H = 0 was found43 below the onset T (∼ 80 K) for SC fluctuations, at low fields up to 9 T and in the regime of strong positive MR consistent with the suppression of superconductivity.Both refs.40,43

1 . 7
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, and the results did not depend on the position of the contacts.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 La Eu 0.2 Sr 0.1 CuO 4 (0.5 Ω for La 1.48 Nd 0.4 Sr 0.12 CuO 4 ) at room temperature.Meanwhile, we found no change in the superconducting properties of the samples before and after the annealing.MeasurementsThe standard ac lock-in techniques (∼ 13 Hz) were used for measurements of R xx and R yx with the magnetic field parallel and anti-parallel to the c axis.The Hall resistance was determined from the transverse voltage by extracting the component antisymmetric in the magnetic field.The Hall coefficient R H = R yx d/H = ρ yx /H, where d is the sample thickness.The ρ xx data measured simultaneously with ρ yx agree well with the previously reported results of magnetoresistance measurements3,4 .The resistance per square per CuO 2 layer R /layer = ρ xx /l, where l = 6.6 Å is the thickness of each layer.

4 Figure 1 :
Figure 1: In-plane Hall coefficient R H across the T -H phase diagram of striped cuprates.a and b, Regions of T and H with different signs of R H for La 1.7 Eu 0.2 Sr 0.1 CuO 4 and La 1.48 Nd 0.4 Sr 0.12 CuO 4 , respectively.c and d, Comparison of the results for R H to the other transport data 3, 4 for La 1.7 Eu 0.2 Sr 0.1 CuO 4 and La 1.48 Nd 0.4 Sr 0.12 CuO 4 , respectively.T c (H) (black squares): boundary of the vortex solid in which ρ xx (T < T c ) = 0 and R H = 0, as expected for a superconductor.The upper critical field H c2 (T ) ∼ H peak (T ); H peak (T ) (dark green dots) are the fields above which the magnetoresistance changes from positive to negative 3, 4 .The low-T , viscous vortex liquid (VL) regime (light violet) is bounded by T c (H) and, approximately, by T peak (H) (positions of the peak in ρ xx (T ); open blue diamonds), H * (T ) (crossover between non-Ohmic and Ohmic behavior 3 ; open royal squares), or H peak (T ); here the behavior is metallic (dρ xx /dT > 0) with ρ xx (T → 0) = 0 and R H = 0.The field-revealed normal state (blue) exhibits anomalous behavior: ρ xx (T ) has an insulating, ln(1/T ) dependence 3, 4 , but R H = 0 despite the absence of superconductivity.At high T (yellow), R H > 0 and drops to zero at T = T 0 (H) (magenta triangles).In the high-T VL regime (H < H peak ; dark beige), R H becomes negative before vanishing at lower T = T ′ 0 (H) (magenta squares), as the vortices become less mobile.The h/4e 2 symbols (open brown diamonds) show the (T, H) values where the sheet resistance changes from R /layer < R Q = h/4e 2 at higher T , to R /layer > R Q at lower T .Zero-field values of T SO and T CO are also shown; T PG ∼175 K and ∼150 K for La 1.7 Eu 0.2 Sr 0.1 CuO 4 and La 1.48 Nd 0.4 Sr 0.12 CuO 4 , respectively 28 .All dashed lines guide the eye.In all panels, grey horizontal marks indicate measurement temperatures in different runs, the resolution of which defines vertical error bars for T 0 and T ′ 0 ; horizontal error bars reflect the uncertainty in defining T ′ 0 within our experimental resolution (see Supplementary Fig. 3 for the raw R H (H) data).

Figure 2 :
Figure 2: Field dependence of the Hall coefficient R H at various temperatures.Higher-and lower-T data for La 1.7 Eu 0.2 Sr 0.1 CuO 4 are shown in a and b, respectively, i.e. in c and d for La 1.48 Nd 0.4 Sr 0.12 CuO 4 .Different symbols, corresponding to the data taken in different magnet systems, show good agreement between the runs.The data points represent R H values averaged over 1 T bins (Supplementary Fig.3), while error bars correspond to ±1 SD (standard deviation) of the data points within each bin.The error bars are typically larger at lower T resulting from the use of lower excitation currents I (see Methods) necessary to avoid heating and to ensure that the measurements are taken in the I → 0 limit, because of the strongly nonlinear (i.e.non-Ohmic) transport in the presence of vortices 3 .At higher T , the error bars are 3-4 times smaller, ∆R H ∼ 0.2 − 0.3 mm 3 /C (see also Supplementary Fig.3).However, similar ∆R H , and even ∆R H ∼ 0.05 mm 3 /C, have been achieved also at low T , as described in Methods (see also Supplementary Fig.4).At high T , R H is independent of H, but it decreases to zero at T = T 0 (H) upon cooling.As T is reduced further, R H becomes negative for lower H, within the VL regime [H < H c2 (T )].In the normal state [H > H c2 (T )], however, R H ≈ 0 down to the lowest T ; ∆R H ∼ 0.05 mm 3 /C.

Figure 3 :
Figure 3: Temperature dependence of the in-plane longitudinal resistivity ρ xx and the Hall coefficient R H for various perpendicular H. a and b, ρ xx and R H , respectively, for La 1.7 Eu 0.2 Sr 0.1 CuO 4 ; the pseudogap temperature T PG ∼ 175 K (ref.28 ).c and d, ρ xx and R H , respectively, for La 1.48 Nd 0.4 Sr 0.12 CuO 4 ; T PG ∼ 150 K (ref.28 ).The transition from the low-temperature orthorhombic to a low-temperature tetragonal structure occurs at T d2 ∼ 125 K in La 1.7 Eu 0.2 Sr 0.1 CuO 4 and T d2 ∼ 70 K in La 1.48 Nd 0.4 Sr 0.12 CuO 4 (ref.3 ).The data in a and c are from refs.3,4 .At the highest fields, ρ xx ∝ ln(1/T ), as discussed in more detail elsewhere3 .In both materials, R H decreases upon cooling, and reaches zero at T = T 0 (H).For H < H c2 ∼ H peak , R H becomes negative at even lower T , then goes through a minimum, and eventually reaches zero again at T = T ′ 0 (H), as shown; R H remains zero down to 0.019 K (b and d insets).For H > H c2 , R H = 0 for all H and T < T 0 (H).Similar to those in Fig.2, error bars correspond to ±1 SD of the data points within each bin.All dashed lines guide the eye.

Supplementary Fig. 3 :Supplementary Fig. 6 :
Hall coefficient R H vs H up to 31 T. a-f, La 1.7 Eu 0.2 Sr 0.1 CuO 4 ; g-l, La 1.48 Nd 0.4 Sr 0.12 CuO 4 .In all panels, black and grey traces represent the raw data obtained in two different runs with fields up to 18 T and 31 T, respectively.(The corresponding R yx (H) data are shown in Supplementary Fig.2.)The same R H data, averaged over 1 T bins, are shown by red and magenta symbols that correspond to the black and grey traces, respectively.Error bars correspond to 1 SD of the data points within each bin.At low T , the signals appear relatively noisy because extremely small excitation currents I are used to avoid heating and to ensure that the measurements are taken in the I → 0 limit, since prior work has demonstrated12 strongly nonlinear (i.e.non-Ohmic) transport in the presence of vortices; at higher T , the error bars are 3-4 times smaller, ∆R H ∼ 0.2 − 0.3 mm 3 /C, at the highest fields.Dashed lines mark R H = 0. Electrical transport in the vortex liquid regime.a and b, Hall resistance R yx (left axis) and longitudinal resistance R xx (right axis) of La 1.7 Eu 0.2 Sr 0.1 CuO 4 and La 1.48 Nd 0.4 Sr 0.12 CuO 4 , respectively, vs magnetic field at several temperatures within the vortex liquid regime.c and d, Scaling of the longitudinal and Hall resistance, ρ xx 2 /ρ yx ∝ H, for La 1.7 Eu 0.2 Sr 0.1 CuO 4 and La 1.48 Nd 0.4 Sr 0.12 CuO 4 , respectively, at the same temperatures shown in a and b.R xx and R yx were averaged over 0.5 T bins before ρ xx 2 /ρ yx was calculated.Error bars correspond to 1 SD of the data points within each bin.Solid lines are linear fits going through the origin.The observed scaling indicates a state of dissipating vortex motion 35 .
in striped La 1.905 Ba 0.095 CuO 4 , but in contrast to the strong H-dependence of R H in YBa 2 Cu 3 O 6+x and YBa 2 Cu 4 O 8 (YBCO materials; ref. 14 ), i.e. in the absence of spin order, or in La 2−x Sr x CuO 4 (ref.
3, the value of H c2 ≈ H peak is also consistent with the spectroscopic data for the closing of the SC gap in other cuprates.Although other experiments might be needed to definitively rule out the presence of any preformed pairs at H > H peak , it appears far more likely that pairs cannot be responsible for R H = 0 in the field-revealed normal state of La 1.7 Eu 0.2 Sr 0.1 CuO 4 68x Sr x CuO 4 (ref.6).While weak localization does not produce a correction to the classical R H value, electronelectron interactions in weakly disordered 2D metals give rise48to logarithmic corrections to ρ xx and R H , which are related such that δR H /R H = 2(δρ xx /ρ xx ).However, just like in La 2−x Sr x CuO 4 (ref.6),this is not consistent with our observation 3 of a large ln(1/T ) term in ρ xx , and it does not describe the vanishing R H . Hence, standard localization mechanisms cannot explain R H = 0 observed over a wide range of T < T 0 and H > H c2 .On the other hand, a confinement of carriers within 1D charge stripes, associated with the suppression of the cyclotron motion with increasing H, was proposed to understand the drop of the positive R H (T ) towards zero observed 31 in La 2−x−y Nd y Sr x CuO 4 near 4u 0.2 Sr 0.1 CuO 4 sample "B" changed, which was attributed to a small change (increase) in the effective doping, but its phases remained qualitatively the same4.We repeated the Hall measurements after the sample had changed, and obtained the same 3,4−x Eu 0.2 Sr x CuO 4 with a nominal x = 0.10 and La 1.6−x Nd 0.4 Sr x CuO 4 with a nominal x = 0.12 were grown by the traveling-solvent floatingzone technique 53 .The high quality of the crystals was confirmed by several techniques, as discussed in detail elsewhere 3, 4 .The samples were shaped as rectangular bars suitable for direct measurements of the longitudinal and transverse (Hall) resistance, R xx and R yx , respectively.Detailed measurements of R xx and R yx were performed on La 1.7 Eu 0.2 Sr 0.1 CuO 4sample "B" with dimensions 3.06×0.53×0.37 mm 3 (a×b×c, i.e. length×width×thickness) and a La 1.48 Nd 0.4 Sr 0.12 CuO 4 crystal with dimensions 3.82 × 1.19 × 0.49 mm 3 .The same two samples were also studied previously3,4.After ∼3 years, the low-T properties of the La 1.7 a standard Hall measurement, any contribution of ρ xx , which results from a slight misalignment of voltage contacts, is removed and ρ yx is isolated by antisymmetrization of the transverse voltage drops measured with a field both parallel and antiparallel to the c axis.However, a perfect cancellation of the ρ xx contribution can only be achieved if the two measurements in opposite field directions are conducted at exactly the same T (and other experimental conditions).Otherwise, ρ xx can contaminate the Hall resistivity even after the conventional antisymmetrization procedure, especially if ρ xx is much larger than ρ yx and it has a strong temperature dependence as in La 1.7 Eu 0.2 Sr 0.1 CuO 4 and La 1.48 Nd 0.4 Sr 0.12 CuO 4 .Therefore, careful temperature control during the experiment and meticulous data analysis afterwards are key to our Hall measurements on these two systems.
T , R H is independent of H, but it decreases to zero at T = T 0 (H) upon cooling.As T is reduced further, R H becomes negative for lower H, within the VL regime [H < H c2 (T )].In the normal state [H > H c2 (T )], however, R H ≈ 0 down to the lowest T ; ∆R H ∼ 0.05 mm 3 /C.