Abstract
The nature of the pseudogap phase of the copper oxides (‘cuprates’) remains a puzzle. Although there are indications that this phase breaks various symmetries, there is no consensus on its fundamental nature1. Fermi-surface, transport and thermodynamic signatures of the pseudogap phase are reminiscent of a transition into a phase with antiferromagnetic order, but evidence for an associated long-range magnetic order is still lacking2. Here we report measurements of the thermal Hall conductivity (in the x–y plane, κxy) in the normal state of four different cuprates—La1.6−xNd0.4SrxCuO4, La1.8−xEu0.2SrxCuO4, La2−xSrxCuO4 and Bi2Sr2−xLaxCuO6+δ. We show that a large negative κxy signal is a property of the pseudogap phase, appearing at its critical hole doping, p*. It is also a property of the Mott insulator at p ≈ 0, where κxy has the largest reported magnitude of any insulator so far3. Because this negative κxy signal grows as the system becomes increasingly insulating electrically, it cannot be attributed to conventional mobile charge carriers. Nor is it due to magnons, because it exists in the absence of magnetic order. Our observation is reminiscent of the thermal Hall conductivity of insulators with spin-liquid states4,5,6, pointing to neutral excitations with spin chirality7 in the pseudogap phase of cuprates.
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Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank L. Balents, K. Behnia, S. Chatterjee, B. D. Gaulin, H. J. Han, S. M. Hayden, C. Hess, S. A. Kivelson, H. Y. Kee, P. A. Lee, Y. S. Lee, A. Rosch, S. Sachdev, M. Scheurer, T. Senthil, A.-M. S. Tremblay, C. M. Varma and S. Verret for helpful and stimulating discussions. L.T. acknowledges support from the Canadian Institute for Advanced Research (CIFAR) as a CIFAR Fellow and funding from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Québec–Nature et Technologies (FRQNT), the Canada Foundation for Innovation (CFI), and a Canada Research Chair. This research was undertaken thanks in part to funding from the Canada First Research Excellence Fund. Part of this work was funded by the Gordon and Betty Moore Foundation’s EPiQS Initiative (grant GBMF5306 to L.T.). J.-S.Z was supported by NSF MRSEC DMR-1720595 in the US.
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G.G., A.L., S.B., E.L., V.Z., M.L., F.L., A.G. and N.D.-L. performed the thermal Hall conductivity measurements. G.G., A.L., S.B., E.L., V.Z., M.L. and N.D.-L. performed the electrical Hall conductivity measurements. J.-S.Z. grew the Nd-LSCO single crystals. S.P., T.T. and H.T. grew the Eu-LSCO and LSCO single crystals. S.O. grew the Bi2201 single crystal. G.G., N.D.-L. and L.T. wrote the manuscript, in consultation with all authors. L.T. supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Magnetic field dependence of κxx.
a–f, Field dependence of κxx in Eu-LSCO p = 0.08 (top panels) and LSCO p = 0.06 (bottom panels), displayed in three ways. a, d, Plot of κxx/T versus T at H = 1 T (blue) and H = 15 T (red) (data points). The difference between the two curves is very small, not visible by eye. b, e, Plot of the change in κxx with field measured relative to its value at H = 1 T, that is, [κxx(H) − κxx(1 T)] versus H, for various temperatures as indicated (data points). c, f, Change in κxx between 15 T and 1 T, plotted as [κxx(H) − κxx(1 T)]/T versus T (blue, right axis), compared to κxy(15 T)/T versus T (red, left axis) (data points). Markers represent data and the line is a guide to the eye. Note how at low T the transverse response grows to be as large, if not larger, than the longitudinal response.
Extended Data Fig. 2 Comparison of cuprates to other oxides.
a, Thermal conductivity of two isostructural oxides, plotted as κxx/T versus T at H = 0, namely Y2Ti2O7 (red) and Tb2Ti2O7 (blue) (data points39). The presence of disordered magnetic moments in Tb2Ti2O7 produces a strong scattering of phonons, seen as a massive suppression of κxx (15-fold at T = 15 K). b, Field dependence of κxx, plotted as Δκxx(H)/κxx(0) versus H, with Δκxx = κxx(H) − κxx(0), at T = 15 K (blue data points12). The strong effect of field (30% in 8 T) is a direct signature of the strong coupling between phonons and spins in Tb2Ti2O7. Also shown is the transverse response in Tb2Ti2O7 at T = 15 K, plotted as κxy/T versus H (red data points12). Note that in Y2Ti2O7, κxy = 0 (ref. 12). c, Thermal conductivity of two Nd-LSCO samples, on either side of p* (red, p = 0.24; blue, p = 0.21), plotted as κxx/T versus T at H = 18 T (data points). We see that contrary to Tb2Ti2O7 (a), the appearance of the negative κxy signal in Nd-LSCO below p* is not accompanied by a large suppression of κxx (see Extended Data Fig. 3). d, Same as b but for LSCO p = 0.06, with the same x-axis and y-axis scales and data taken at (nearly) the same temperature (data points). We see that the situation in LSCO is very different to that found in Tb2Ti2O7 (b): instead of having a small κxy and a large Δκxx (b), we now have a large κxy and a small Δκxx. Quantitatively, κxy/Δκxx ≈ 1 in LSCO and approximately 0.01 in Tb2Ti2O7, at T = 15 K and H = 8 T (Table 1).
Extended Data Fig. 3 Change in phonon κxx across p* in Nd-LSCO.
a, Thermal conductivity of Nd-LSCO at four different dopings, above p* (p = 0.24) and below p* (p = 0.20, 0.21, 0.22), plotted as κxx/T versus T, at H = 18 T (data points). We see that κxx increases below p*. b, Same as a but for Nd-LSCO p = 0.21 (blue; H = 18 T) and LSCO p = 0.06 (green, H = 16 T). We see that κxx continues to increase as we lower p further. This shows that phonons conduct better at lower p. A natural explanation is that they are less scattered by charge carriers as the material becomes less metallic. c, Same data as in a for Nd-LSCO p = 0.21 (blue data points) and p = 0.24 (red data points), compared to the electrical conductivity of those same samples, plotted as L0/ρ versus T (lines; measured at H = 33 T (ref. 17)). The latter curves are a reasonable estimate of the electronic thermal conductivity \({\kappa }_{xx}^{{\rm{el}}}\), exact at T → 0 (since the Wiedemann–Franz law is satisfied40), as seen in Fig. 2a. d, Estimate of the phonon conductivity, defined as \({\kappa }_{xx}^{{\rm{ph}}}={\kappa }_{xx}-{L}_{0}T/\rho \), plotted as \({\kappa }_{xx}^{{\rm{ph}}}/T\) versus T (using data from c) (data points). We see that \({\kappa }_{xx}^{{\rm{ph}}}(T)\) increases upon crossing below p*, most probably because electron–phonon scattering is weakened by the loss of carrier density. There is no evidence that the phonons suddenly suffer from the onset of strong spin scattering below p* (which would cause \({\kappa }_{xx}^{{\rm{ph}}}(T)\) to drop below p*), such as would be required to explain the appearance of the large negative κxy signal below p* (Fig. 3) as being due to phonon transport.
Extended Data Fig. 4 Magnetic field dependence of κxy in LSCO.
a, Field dependence of the thermal Hall conductivity of LSCO at p = 0.06, plotted as κxy versus H at various temperatures, as indicated (data points). The dependence of κxy on H is linear at high T and it becomes sublinear at lower T. b, Deviation from linearity displayed by plotting κxy/(TH) versus T at four different fields H, as indicated (data points).
Extended Data Fig. 5 Magnon thermal conductivity in La2CuO4.
Thermal conductivity of magnons in La2CuO4, plotted as κmag/T versus T (blue data points, right axis; ref. 37). The solid blue line is a fit to the data using the standard calculation for two magnon branches in 2D, with gaps as measured by neutron inelastic scattering36, namely Δ1 = 26 K and Δ2 = 58 K. Below T ≈ 5 K, thermally excited magnons are exponentially rare and κmag/T ≈ 0. In sharp contrast, the thermal Hall conductivity of La2CuO4, |κxy/T| (red data points, left axis; the red line is a guide to the eye; Fig. 1b), is largest at T ≈ 5 K. This comparison shows that the κxy signal in La2CuO4 cannot come from magnon transport.
Extended Data Fig. 6 Electrical resistivity in La2CuO4.
Electrical resistivity, ρxx, of two of our samples—La2CuO4 (LCO, red) and LSCO at p = 0.06 (blue)—compared with published data for La2CuO4 (yellow34) and LSCO at p = 0.01 (green35). This shows that our LCO sample is very close to the Mott insulator La2CuO4, being more insulating than LSCO with p = 0.01 and much more insulating than our LSCO sample with p = 0.06.
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Grissonnanche, G., Legros, A., Badoux, S. et al. Giant thermal Hall conductivity in the pseudogap phase of cuprate superconductors. Nature 571, 376–380 (2019). https://doi.org/10.1038/s41586-019-1375-0
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DOI: https://doi.org/10.1038/s41586-019-1375-0
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