The perfectly linear temperature dependence of the electrical resistivity observed as T → 0 in a variety of metals close to a quantum critical point 1,2,3,4 is a major puzzle of condensed matter physics 5 . Here we show that T-linear resistivity as T → 0 is a generic property of cuprates, associated with a universal scattering rate. We measured the low-temperature resistivity of the bi-layer cuprate Bi2Sr2CaCu2O8+δ and found that it exhibits a T-linear dependence with the same slope as in the single-layer cuprates Bi2Sr2CuO6+δ (ref. 6), La1.6-xNd0.4SrxCuO4 (ref. 7) and La2-xSrxCuO4 (ref. 8), despite their very different Fermi surfaces and structural, superconducting and magnetic properties.We then show that the T-linear coefficient (per CuO2 plane), A1 ☐ , is given by the universal relation A1 ☐ TF = h / 2e 2 , where e is the electron charge, h is the Planck constant and TF is the Fermi temperature. This relation, obtained by assuming that the scattering rate 1 / τ of charge carriers reaches the Planckian limit 9,10 , whereby ħ / τ = kB T, works not only for hole-doped cuprates 6,7,8,11,12 but also for electrondoped cuprates 13,14 , despite the different nature of their quantum critical point and strength of their electron correlations.

The nature of the pseudogap phase of cuprates remains a major puzzle. Although there are indications that this phase breaks various symmetries, there is no consensus on its fundamental nature 1 . Although Fermi-surface 2 , transport 3 and thermodynamic 4 signatures of the pseudogap phase are reminiscent of a transition into a phase with antiferromagnetic order 5,6 , there is no evidence for an associated long-range magnetic order. Here we report measurements of the thermal Hall conductivity κ xy in the normal state of four different cuprates (La 1.6-x Nd 0.4 Sr x CuO 4 , La 1.8-x Eu 0.2 Sr x CuO 4 , La 2-x Sr x CuO 4 , and Bi 2 Sr 2-x La x CuO 6+δ ) and show that a large negative κ xy signal is a property of the pseudogap phase, appearing with the onset of that phase at the critical doping p*. Since it is not due to charge carriersas it persists when the material becomes an insulator, at low doping -or magnons -as it exists in the absence of magnetic order -or phonons -since skew scattering

The specific heat C of the cuprate superconductors La2−xSrxCuO4 and Bi2+ySr2−x−yLaxCuO6+ was measured at low temperatures (down to 0.5 K) for dopings p close to p, the critical doping for the onset of the pseudogap phase. A magnetic field up to 35 T was applied to suppress superconductivity, giving direct access to the normal state at low temperatures, and enabling a determination of Ce, the electronic contribution to the normal-state specific heat at T→0. In La2−xSrxCuO4 at x=p=0.22, 0.24 and 0.25, Ce/T=15to16mJmol−1K−2 at T=2K, values that are twice as large as those measured at higher doping (p>0.3) and lower doping (p<0.15). This confirms the presence of a broad peak in the doping dependence of Ce at p0.19 as previously reported for samples in which superconductivity was destroyed by Zn impurities. Moreover, at those three dopings, we find a logarithmic growth as T→0 such that Ce/TBln(T0/T). The peak versus p and the logarithmic dependence versus T are the two typical thermodynamic signatures of quantum criticality. In the very different cuprate Bi2+ySr2−x−yLaxCuO6+, we again find that Ce/TBln(T0/T) at pp, strong evidence that this ln(1/T) dependence of the electronic specific heat-first discovered in the cuprates La1.8−xEu0.2SrxCuO4 and La1.6−xNd0.4SrxCuO4-is a universal property of the pseudogap critical point.

Five transport coefficients of the cuprate superconductor Bi 2 Sr 2−x La x CuO 6+δ were measured in the normal state down to low temperature, reached by applying a magnetic field (up to 66 T) large enough to suppress superconductivity. The electrical resistivity, Hall coefficient, thermal conductivity, Seebeck coefficient, and thermal Hall conductivity were measured in two overdoped single crystals, with La concentration x = 0.2 (T c = 18 K) and x = 0.0 (T c = 10 K). The samples have dopings p very close to the critical doping p where the pseudogap phase ends. The resistivity displays a linear dependence on temperature whose slope is consistent with Planckian dissipation. The Hall number n H decreases with reduced p, consistent with a drop in carrier density from n = 1 + p above p to n = p below p . This drop in n H is concomitant with a sharp drop in the density of states inferred from prior NMR Knight shift measurements. The thermal conductivity satisfies the Wiedemann-Franz law, showing that the pseudogap phase at T = 0 is a metal whose fermionic excitations carry heat and charge as do conventional electrons. The Seebeck coefficient diverges logarithmically at low temperature, a signature of quantum criticality. The thermal Hall conductivity becomes negative at low temperature, showing that phonons are chiral in the pseudogap phase. Given the observation of these same properties in other, very different cuprates, our study provides strong evidence for the universality of these five signatures of the pseudogap phase and its critical point.

The Seebeck coefficient (thermopower) S of the cuprate superconductor La 1.6−x Nd 0.4 Sr x CuO 4 was measured across its doping phase diagram (from p = 0.12 to p = 0.25), at various temperatures down to T 2 K, in the normal state accessed by suppressing superconductivity with a magnetic field up to H = 37.5 T. The magnitude of S/T in the T = 0 limit is found to suddenly increase, by a factor 5, when the doping is reduced below p = 0.23, the critical doping for the onset of the pseudogap phase. This confirms that the pseudogap phase causes a large reduction of the carrier density n, consistent with a drop from n = 1 + p above p to n = p below p , as previously inferred from measurements of the Hall coefficient, resistivity, and thermal conductivity. When the doping is reduced below p = 0.19, a qualitative change is observed whereby S/T decreases as T → 0, eventually to reach negative values at T = 0. In prior work on other cuprates, negative values of S/T at T → 0 were shown to result from a reconstruction of the Fermi surface caused by charge density wave (CDW) order. We therefore identify p CDW 0.19 as the critical doping beyond which there is no CDW-induced Fermi surface reconstruction. The fact that p CDW is well separated from p reveals that there is a doping range below p where the transport signatures of the pseudogap phase are unaffected by CDW correlations, as previously found in YBa 2 Cu 3 O y and La 2−x Sr x CuO 4 .