Abstract
The three central phenomena of cuprate (copper oxide) superconductors are linked by a common doping level p*—at which the enigmatic pseudogap phase ends and the resistivity exhibits an anomalous linear dependence on temperature, and around which the superconducting phase forms a domeshaped area in the phase diagram^{1}. However, the fundamental nature of p* remains unclear, in particular regarding whether it marks a true quantum phase transition. Here we measure the specific heat C of the cuprates EuLSCO and NdLSCO at low temperature in magnetic fields large enough to suppress superconductivity, over a wide doping range^{2} that includes p*. As a function of doping, we find that C_{el}/T is strongly peaked at p* (where C_{el} is the electronic contribution to C) and exhibits a log(1/T) dependence as temperature T tends to zero. These are the classic thermodynamic signatures of a quantum critical point^{3,4,5}, as observed in heavyfermion^{6} and ironbased^{7} superconductors at the point where their antiferromagnetic phase comes to an end. We conclude that the pseudogap phase of cuprates ends at a quantum critical point, the associated fluctuations of which are probably involved in dwave pairing and the anomalous scattering of charge carriers.
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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 author upon reasonable request.
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Acknowledgements
We thank J. Chang, M. Horio, M.H. Julien, S. Kivelson, R. Markiewicz, C. Proust, B. Ramshaw, S. Sachdev, A. Sacuto, J. Tallon, A.M. Tremblay and C. Varma for discussions. C.M. and T.K. acknowledge support from the Laboratoire d’excellence LANEF (ANR10LABX5101) and the Laboratoire National des Champs Magnétiques Intenses (LNCMI) in Grenoble. J.K. was supported by the Slovak Research and Development Agency under grant number APVV160372. L.T. acknowledges support from the Canadian Institute for Advanced Research (CIFAR) and funding from the Natural Sciences and Engineering Research Council of Canada (NSERC; PIN: 123817), 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 under Cooperative Agreement Number DMR1720595 in the US. H.T. acknowledges MEXT Japan for a GrantinAid for Scientific Research.
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B.M., C.G., C.M. and T.K. performed the specific heat measurements. J.K. assisted in the development of the measurement technique. B.M., S.B. and N.D.L. characterized the samples via resistivity and magnetization measurements. J.S.Z. prepared the NdLSCO crystals. S.P., T.T. and H.T. prepared the EuLSCO crystals. Q.M., M.D., H.A.D. and B.D.G. prepared the NdLSCO polycrystalline samples and measured the phonon energies of NdLSCO with neutron scattering. S.V. calculated the specific heat of NdLSCO from its threedimensional band structure. B.M., C.M., L.T. and T.K. wrote the manuscript, in consultation with all authors. L.T. and T.K. designed the study and supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Temperature–doping phase diagram.
Temperaturedoping phase diagram of LSCO (black), NdLSCO (red) and EuLSCO (green), showing the boundary of the phase of longrange commensurate antiferromagnetic order (T_{N}, brown line), the pseudogap temperature T* (blue line) and the superconducting transition temperature T_{c} of LSCO (grey line) and NdLSCO (pink line). T* is detected in two transport properties: resistivity (T_{ρ}, circles) and the Nernst effect (T_{ν}, squares). The open triangles show T* detected by ARPES as the temperature below which the antinodal pseudogap opens, in LSCO (black) and NdLSCO (red). We see that T_{ν} ≈ T_{ρ} ≈ T*, within error bars. The pseudogap phase ends at a critical doping p* = 0.18 ± 0.01 in LSCO (black diamond) and p* = 0.23 ± 0.01 in NdLSCO (red diamond). Figure adapted from figure 10 in ref. ^{35} (American Physical Society) (error bars defined therein).
Extended Data Fig. 2 Characterization of our EuLSCO and NdLSCO samples.
a, T_{c} versus p for our EuLSCO samples. T_{c} is defined as the onset of the drop in the magnetization upon cooling. Error bars on T_{c} reflect the uncertainty in defining the onset of the drop in the magnetization. b, Same for our NdLSCO samples. c, ρ versus T in our EuLSCO samples with p = 0.21 (red) and p = 0.24 (blue), at H = 0 and H = 33 T (short section below 40 K). d, Same for our NdLSCO samples with p = 0.22 (red) and p = 0.24 (blue)^{2}. The approximately linear ρ(T) as T → 0 at p = 0.24 (blue) shows that 0.24 is close to the critical point p* ≈ 0.23 in both materials. The large upturn in ρ(T) as T → 0 at p = 0.21 and p = 0.22 (red) shows that the pseudogap has opened in both materials (at p < 0.23).
Extended Data Fig. 3 Specific heat jump at T_{c} in EuLSCO.
a, C/T versus T in our EuLSCO sample with p = 0.24, at H = 0 (red) and H = 8 T (blue). The inset shows the difference between the two curves in the main panel (red). This is the difference between the superconductingstate C/T and the normalstate C/T. It reveals the jump at T_{c}, whose peak value, ΔC/T_{c}, is defined as drawn. The black curve is the magnetization of that sample. At p = 0.24, the bulk T_{c} = 10.5 ± 0.5 K (dashed line). b, As in a, for our sample with p = 0.21, at H = 0 (red) and H = 18 T (blue). At p = 0.21, the bulk T_{c} = 14.5 ± 0.5 K. c, As in a, for our sample with p = 0.16, at H = 0 (red) and H = 18 T (blue). At p = 0.16, the bulk T_{c} = 11.5 ± 0.5 K. d, As in a, for our sample with p = 0.11, at H = 0 (red) and H = 8 T (blue). At p = 0.11, the bulk T_{c} = 5.0 ± 0.5 K. e, Plot of ΔC/T_{c} versus C_{el}/T at T = 0.5 K, the latter being obtained from Fig. 3b (red squares). The error bar on ΔC/T_{c} comes mostly from the uncertainty on defining the baseline above T_{c}. The dashed line is a linear fit through the first three data points. The tenfold increase in ΔC/T_{c} from p = 0.11 to p = 0.24 is independent evidence of a similar increase in C_{el}/T.
Extended Data Fig. 4 Specific heat as a function of magnetic field.
a, C/T versus H in our EuLSCO samples with p = 0.21 (orange) and p = 0.24 (red), at T = 2 K. The upper critical field above which there is no remaining superconductivity is H_{c2} = 15 T at p = 0.21 and H_{c2} = 9 T at p = 0.24. Note that for p = 0.24, C/T has reached 99% of its normal state value by 8 T. b, C/T versus H in our NdLSCO sample with p = 0.23, at T = 2 K, in a semilog plot. The dashed line shows the expected field dependence of the Schottky contribution associated with Nd ions (C_{mag}, dashed line). The data are independent of field above H ≈ 9 T. Dotted lines are horizontal.
Extended Data Fig. 5 Specific heat of EuLSCO and NdLSCO down to base temperature.
a, Specific heat of the four EuLSCO samples of Fig. 2a measured in a field H = 8 T, down to 0.4 K. The rapid rise below 1 K is a nuclear Schottky anomaly (C_{nuclear}). b, Difference between the measured C/T of a and a constant term γ, plotted for each doping as a function of temperature, on a log–log plot (γ = 2.8 and 4.2 mJ K^{−2} mol^{−1}, at p = 0.11 and 0.16, respectively). The line marked T^{2} shows that the data at p = 0.11 and p = 0.16 obey ΔC = βT^{3} in the range from 1.5 K to about 5 K. The line marked T^{–3} shows that the data at p = 0.11 and p = 0.16 obey ΔC ≈ T^{–2} below 1 K, as expected for the upper tail of a Schottky anomaly. The ΔC curve at p = 0.16, ΔC(p = 0.16; T), therefore constitutes the nonelectronic, and weakly dopingdependent, background for C(T) in EuLSCO, made of phonon and Schottky contributions. c, Specific heat of our NdLSCO crystal with p = 0.12, plotted as C/T versus T at three different fields, as indicated. At H = 0 (green), we see the large Schottky anomaly associated with Nd ions, varying as C_{mag} ≈ T^{–2} at low T. At H = 8 T (red), it is pushed up above 2 K; at H = 18 T (blue), above 5 K. The line is a fit of the 18 T data to γ + βT^{2} below 5 K. d, Specific heat of the four NdLSCO samples of Fig. 2c, plotted as C/T versus T. Below the vertical dashed line, we show lowtemperature data taken at H = 8 T on three of these same samples. (See Supplementary Fig. 1 for the complete set of dopings and Extended Data Fig. 6 for further analysis and discussion.).
Extended Data Fig. 6 Analysis of specific heat data and doping dependence of β.
a, Raw data for EuLSCO at p = 0.11 (blue), 0.16 (green) and 0.24 (red), plotted as C/T versus logT. The width of the pale band tracking each curve is the uncertainty on the absolute measurement of C (±4%). The solid green line is a fit to C_{nuclear} ≈ T^{–2} for the p = 0.16 data. b, Same three curves as in a, from which the same Schottky anomaly, C_{nuclear}/T, the green line in a, has been subtracted. The straight dotted lines show that (C – C_{nuclear})/T is flat as T → 0 for p = 0.11 and 0.16, while it rises as log(1/T) for p = 0.24. The solid green line is a fit of the green curve at p = 0.16 to (C – C_{nuclear})/T = γ + C_{ph}/T up to 10 K, where C_{ph}/T = βT^{2} + δT^{4} is the phonon contribution. c, Same three curves as in b, from which the same phonon contribution C_{ph}/T, the green line in b, has been subtracted. We see that within error bars the resulting C_{el}/T is constant up to 10 K for p = 0.11 (blue) and 0.16 (green), while it varies as log(1/T) up to 10 K for p = 0.24 (red). The dotted lines are a linear fit to the data. d, Doping dependence of the phonon specific heat parameter β, in C_{ph}/T = βT^{2} + δT^{4}, obtained from a fit to (C – C_{nuclear})/T = γ + C_{ph}/T up to 10 K, for EuLSCO crystals (dark blue squares) and NdLSCO crystals (red dots). For crystals, the error bars are the sum of two uncertainties: on the magnitude of the raw data, defined in the legend of Fig. 3, plus on the fitting procedure to extract β, described in a, b and c. For NdLSCO powders (orange dots), the values are obtained from Extended Data Figs. 7, 8; the error bars are the sum of two uncertainties: on the magnitude of the raw data, defined in the legend of Fig. 3, plus on the fitting procedure to extract β, described in the legend of Extended Data Figs. 7, 8. The black diamonds are 1/E_{ph}^{3} (right axis), where E_{ph} is the phonon energy (top of the acoustic branch) measured by neutron scattering on three NdLSCO single crystals: E_{ph} = 14.6, 14.7 and 14.8 meV at p = 0.12, 0.19 and 0.24, respectively (measurements performed by Q.M., M.D., H.A.D. and B.D.G.). We see that E_{ph} varies very little with doping, and β ≈ 1/E_{ph}^{3}, justifying our assumption that C_{ph}(T) does not change appreciably between p = 0.11 and p = 0.25.
Extended Data Fig. 7 Specific heat of our polycrystalline samples at p = 0.07 and 0.12.
a, C/T versus T for NdLSCO p = 0.12, comparing raw data on crystal and powder, as indicated. The solid red line is a fit to the crystal data, consisting of the sum of three contributions, plotted below: electrons (dashdotted), phonons (dashed) and Schottky (C_{mag} ≈ T^{–2}, dotted). b, Specific heat data for our powders with p = 0.07 and 0.12, at H = 0, from which the Schottky term (C_{mag}) has been subtracted, plotted as (C(H = 0) – C_{mag})/T versus T^{2}. The dashed lines are linear fits to the data (γ + βT^{2}). For p = 0.12, the fit yields γ ≈ 4.0 mJ K^{−2} mol^{−1}, in reasonable agreement with the value obtained by applying 18 T to suppress the Schottky anomaly in our NdLSCO crystal with p = 0.12 (Fig. 2c), namely γ = 3.6 ± 0.5 mJ K^{−2} mol^{−1} (Fig. 3b). For the two powder samples, the γ values are plotted in Fig. 3b (purple dots) and the β values are plotted in Extended Data Fig. 6d (orange dots). The value of (C(H = 0) – C_{mag})/T extrapolated at T = 2 K is plotted in Fig. 3a (open red circles).
Extended Data Fig. 8 Specific heat of our polycrystalline samples at p = 0.27, 0.36 and 0.40.
a, Raw specific heat data for our powders with p = 0.27, 0.36 and 0.40, at H = 0. b, The data in a from which the magnetic Schottky term (C_{mag} = A/T^{2}) has been subtracted, plotted as (C – C_{mag})/T versus T^{2}. The dotted lines are linear fits to the data (γ + βT^{2}). For the three powder samples, the γ values are plotted in Fig. 3b (purple circles) and the β values are plotted in Extended Data Fig. 6d (orange circles). The value of (C(H = 0) – C_{mag})/T extrapolated at T = 2 K is plotted in Fig. 3a (open red circles). c, The data in b from which the phonon term (C_{ph} = βT^{3}) has been subtracted, plotted as (C – C_{mag} – C_{ph})/T versus T^{2}. All three curves are seen to be constant. d, The data in b, compared to raw singlecrystal data at p = 0.24 for EuLSCO (orange) and NdLSCO (red), taken at H = 18 T. (The two raw curves agree beautifully at T < 5 K, where C_{mag} in NdLSCO is negligible; Extended Data Fig. 5c.) The upper dotted line is a linear guide to the eye showing that the EuLSCO curve deviates from linearity at low temperature, unlike the curves at p = 0.27, 0.36 and 0.40.
Extended Data Fig. 9 Calculated specific heat from the NdLSCO band structure.
Comparison of the measured specific heat of NdLSCO (red; Fig. 3b, with error bars defined in the legend of Fig. 3b) and the specific heat calculated for the band structure of NdLSCO (blue; see Methods), with the van Hove point p_{vHs} set to be at p*. The calculations include the threedimensional dispersion in the Fermi surface (along the c axis) and the disorder scattering, both consistent with the measured properties of our NdLSCO samples, namely their resistivity anisotropy (ρ_{c}/ρ_{a}) and their residual resistivity (see Methods). We see that while the van Hove singularity can give rise to a cusplike peak at p_{vHs} (top left panel) and a log(1/T) dependence of C/T at p_{vHs} in a perfectly twodimensional system with no disorder (lower left panel), these features inevitably disappear when the considerable threedimensional dispersion of the real material and the high disorder of the real samples are included (right panels). The calculations only quantify what is naturally expected: the rise in specific heat due to the van Hove singularity is cut off when k_{B}T < ħΓ, where Γ is the scattering rate, or when k_{B}T < t_{z}, where t_{z} is the caxis hopping parameter. The fact that we see C/T continuing to increase down to 0.5 K (lower right panel) excludes the van Hove singularity as the underlying mechanism.
Extended Data Fig. 10 Comparing with data on nonsuperconducting LSCO.
a, Normalstate electronic specific heat C_{el} of EuLSCO (squares; from Fig. 2b) and NdLSCO (circles; from Fig. 2d), at T = 0.5 K (red), 2 K (blue) and 10 K (green), plotted as C_{el} /T versus p. (At p = 0.08, 0.11 and 0.16, the red and green squares are split apart slightly so they can both be seen.) Open symbols are extrapolated values (dashed lines in Fig. 2b, d). Data on NdLSCO at p = 0.07, 0.12, 0.27, 0.36 and 0.40 (purple) are γ values obtained on polycrystalline samples, as described in Extended Data Figs. 7, 8. The black, purple and red data points are taken from Fig. 3b, with error bars defined in the legend of Fig. 3b. Error bars on the blue and green data points are defined in the same way as for the red data points (see legend of Fig. 3b). We also include γ for nonsuperconducting LSCO from published work (diamonds), obtained by extrapolating C/T = γ + βT^{2} to T = 0 from data below 10 K (p < 0.06, ref. ^{32}; p = 0.33, ref. ^{13}). The vertical dashed line marks the pseudogap critical point p* in NdLSCO (Extended Data Fig. 1). All solid lines are a guide to the eye. b, Comparison of C_{el}/T versus p in our samples of EuLSCO and NdLSCO at T = 10 K (green squares and circles in a) with published data on nonsuperconducting LSCO (diamonds). Open diamonds are γ measured in single crystals of LSCO at dopings where there is no superconductivity (p = 0.33, ref. ^{13}; p < 0.06, ref. ^{32}; remainder^{17}). Solid diamonds are data from powders made nonsuperconducting by Zn substitution^{17}; γ values are obtained from fits to C/T = γ + βT^{2} between about 4 K and about 8 K. We see that these early data on LSCO are quantitatively consistent with our data on EuLSCO and NdLSCO, apart from a downward shift in the position of the peak, consistent with a lower p* in LSCO (Extended Data Fig. 1). Lines are a guide to the eye.
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Michon, B., Girod, C., Badoux, S. et al. Thermodynamic signatures of quantum criticality in cuprate superconductors. Nature 567, 218–222 (2019). https://doi.org/10.1038/s415860190932x
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