## 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 dome-shaped 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 Eu-LSCO and Nd-LSCO 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 heavy-fermion^{6} and iron-based^{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 *d*-wave 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.

## Additional information

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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 (ANR-10-LABX-51-01) 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 APVV-16-0372. 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 DMR-1720595 in the US. H.T. acknowledges MEXT Japan for a Grant-in-Aid for Scientific Research.

## Author information

### Affiliations

#### Institut Néel, Université Grenoble Alpes, Grenoble, France

- B. Michon
- , C. Girod
- & T. Klein

#### Institut quantique, Département de physique and RQMP, Université de Sherbrooke, Sherbrooke, Québec, Canada

- B. Michon
- , C. Girod
- , S. Badoux
- , S. Verret
- , N. Doiron-Leyraud
- & L. Taillefer

#### CNRS, Institut Néel, Grenoble, France

- B. Michon
- , C. Girod
- & T. Klein

#### Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia

- J. Kačmarčík

#### Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada

- Q. Ma
- & B. D. Gaulin

#### Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario, Canada

- M. Dragomir
- , H. A. Dabkowska
- & B. D. Gaulin

#### Canadian Institute for Advanced Research, Toronto, Ontario, Canada

- B. D. Gaulin
- & L. Taillefer

#### Materials Science and Engineering Program, Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas, USA

- J.-S. Zhou

#### Department of Advanced Materials Science, University of Tokyo, Kashiwa, Japan

- S. Pyon
- , T. Takayama
- & H. Takagi

#### Université Grenoble Alpes, CEA, INAC, PHELIQS, LATEQS, Grenoble, France

- C. Marcenat

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### Contributions

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 Nd-LSCO crystals. S.P., T.T. and H.T. prepared the Eu-LSCO crystals. Q.M., M.D., H.A.D. and B.D.G. prepared the Nd-LSCO polycrystalline samples and measured the phonon energies of Nd-LSCO with neutron scattering. S.V. calculated the specific heat of Nd-LSCO from its three-dimensional 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.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to L. Taillefer or T. Klein.

## Extended data figures and tables

### Extended Data Fig. 1 Temperature–doping phase diagram.

Temperature-doping phase diagram of LSCO (black), Nd-LSCO (red) and Eu-LSCO (green), showing the boundary of the phase of long-range 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 Nd-LSCO (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 anti-nodal pseudogap opens, in LSCO (black) and Nd-LSCO (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 Nd-LSCO (red diamond). Figure adapted from figure 10 in ref.^{35}(American Physical Society) (error bars defined therein).### Extended Data Fig. 2 Characterization of our Eu-LSCO and Nd-LSCO samples.

**a**,*T*_{c}versus*p*for our Eu-LSCO 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 Nd-LSCO samples.**c**,*ρ*versus*T*in our Eu-LSCO 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 Nd-LSCO 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 Eu-LSCO.**a**,*C*/*T*versus*T*in our Eu-LSCO 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 superconducting-state*C*/*T*and the normal-state*C*/*T*. It reveals the jump at*T*_{c}, whose peak value, Δ*C*/*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 Δ*C*/*T*_{c}versus*C*_{el}/*T*at*T*= 0.5 K, the latter being obtained from Fig. 3b (red squares). The error bar on Δ*C*/*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 Δ*C*/*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 Eu-LSCO 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 Nd-LSCO sample with*p*= 0.23, at*T*= 2 K, in a semi-log 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 Eu-LSCO and Nd-LSCO down to base temperature.

**a**, Specific heat of the four Eu-LSCO 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 Δ*C*=*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 Δ*C*≈*T*^{–2}below 1 K, as expected for the upper tail of a Schottky anomaly. The Δ*C*curve at*p*= 0.16, Δ*C*(*p*= 0.16;*T*), therefore constitutes the non-electronic, and weakly doping-dependent, background for*C*(*T*) in Eu-LSCO, made of phonon and Schottky contributions.**c**, Specific heat of our Nd-LSCO 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 Nd-LSCO samples of Fig. 2c, plotted as*C*/*T*versus*T*. Below the vertical dashed line, we show low-temperature 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 Eu-LSCO at*p*= 0.11 (blue), 0.16 (green) and 0.24 (red), plotted as*C*/*T*versus log*T*. 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 Eu-LSCO crystals (dark blue squares) and Nd-LSCO 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 Nd-LSCO 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 Nd-LSCO 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 Nd-LSCO*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 (dash-dotted), 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 Nd-LSCO 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 single-crystal data at*p*= 0.24 for Eu-LSCO (orange) and Nd-LSCO (red), taken at*H*= 18 T. (The two raw curves agree beautifully at*T*< 5 K, where*C*_{mag}in Nd-LSCO is negligible; Extended Data Fig. 5c.) The upper dotted line is a linear guide to the eye showing that the Eu-LSCO 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 Nd-LSCO band structure.

Comparison of the measured specific heat of Nd-LSCO (red; Fig. 3b, with error bars defined in the legend of Fig. 3b) and the specific heat calculated for the band structure of Nd-LSCO (blue; see Methods), with the van Hove point

*p*_{vHs}set to be at*p**. The calculations include the three-dimensional dispersion in the Fermi surface (along the*c*axis) and the disorder scattering, both consistent with the measured properties of our Nd-LSCO 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 cusp-like peak at*p*_{vHs}(top left panel) and a log(1/*T*) dependence of*C*/*T*at*p*_{vHs}in a perfectly two-dimensional system with no disorder (lower left panel), these features inevitably disappear when the considerable three-dimensional 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*c*-axis 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 non-superconducting LSCO.

**a**, Normal-state electronic specific heat*C*_{el}of Eu-LSCO (squares; from Fig. 2b) and Nd-LSCO (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 Nd-LSCO 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 non-superconducting 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 Nd-LSCO (Extended Data Fig. 1). All solid lines are a guide to the eye.**b**, Comparison of*C*_{el}/*T*versus*p*in our sam*p*les of Eu-LSCO and Nd-LSCO at*T*= 10 K (green squares and circles in**a**) with published data on non-superconducting 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 non-superconducting 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 Eu-LSCO and Nd-LSCO, 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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