## Massive electrons signify correlations

Thirty years on, and the mechanism of superconductivity in copper-oxide superconductors remains a mystery. Knowledge of their normal nonsuperconducting state is also incomplete; however, we do know that the more robust the superconductivity, the higher the magnetic fields required to suppress it. Ramshaw *et al.* studied samples of three different compositions of the copper-oxide YBa_{2}Cu_{3}O_{6+δ} in magnetic fields exceeding 90 T. They found that as the oxygen content increased toward the point of the maximum transition temperature, the conducting electrons became heavier and heavier. This mass enhancement reflected an increase in electronic correlations, which in turn may be a signature of a quantum critical point.

*Science*, this issue p. 317

## Abstract

In the quest for superconductors with higher transition temperatures (*T*_{c}), one emerging motif is that electronic interactions favorable for superconductivity can be enhanced by fluctuations of a broken-symmetry phase. Recent experiments have suggested the existence of the requisite broken-symmetry phase in the high-*T*_{c} cuprates, but the impact of such a phase on the ground-state electronic interactions has remained unclear. We used magnetic fields exceeding 90 tesla to access the underlying metallic state of the cuprate YBa_{2}Cu_{3}O_{6+δ} over a wide range of doping, and observed magnetic quantum oscillations that reveal a strong enhancement of the quasiparticle effective mass toward optimal doping. This mass enhancement results from increasing electronic interactions approaching optimal doping, and suggests a quantum critical point at a hole doping of *p*_{crit} ≈ 0.18.

In several classes of unconventional superconductors, superconductivity has been linked to a quantum critical point (QCP). At a QCP, the system undergoes a phase transition and a change in symmetry at zero temperature; the associated quantum fluctuations enhance interactions, which can give rise to (or enhance) superconductivity (*1*, *2*). As the QCP is approached, these fluctuations produce increasingly stronger electronic correlations, resulting in an experimentally observable enhancement of the electron effective mass (*1*, *3*–*5*). It is widely believed that spin fluctuations in the vicinity of an antiferromagnetic QCP are important for superconductivity in many heavy-fermion, organic, and pnictide superconductors (*2*, *6*), leading to the ubiquitous phenomenon of a superconducting dome surrounding a QCP. The role of quantum criticality in cuprate high-temperature superconductors is more controversial (*7*): Do the collapsing experimental energy scales (*8*), enhanced superconducting properties (see Fig. 1), and evidence for a change in ground-state symmetry near optimal doping (*9*–*16*) support the existence of strong fluctuations that are relevant to superconductivity (*2*, *17*–*19*)? Alternative explanations for the phenomenology of the cuprate phase diagram focus on the physics of a lightly doped Mott insulator (*7*, *20*) rather than that of a metal with competing broken-symmetry phases. Several investigations, both theoretical and experimental, suggest that competing order is present in the cuprates and is associated with the charge (rather than spin) degree of freedom [such as charge density wave (CDW) order, orbital current order, or nematicity; see Fig. 1] (*12*, *15*–*18*, *21*–*28*). What has been missing is direct, low-temperature evidence that the disappearance of competing order near optimal doping, and the associated change in ground-state symmetry, are accompanied by enhanced electronic interactions in the ground state.

A powerful technique for measuring low-temperature Fermi surface properties is the magnetic quantum oscillation phenomenon, which directly accesses quasiparticle interactions through the effective mass (*29*). Such measurements have been successful in identifying mass enhancements near QCPs in lower-*T*_{c} materials [e.g., CeRhIn_{5} and Ba(FeAs* _{x}*P

_{1–}

*)*

_{x}_{2}(

*3*,

*5*)], but the robustness of superconductivity near optimal doping in the cuprates has impeded access to the metallic ground state. The Fermi surface in underdoped cuprates is known to be relatively small and electron-like (

*30*–

*34*), in contrast to overdoped cuprates, in which a much larger hole-like surface is observed (

*35*). This suggests the existence of broken translational symmetry in the underdoped cuprates that “reconstructs” the large hole-like surface into the smaller electron-like surface, and this translational symmetry breaking is likely related to the charge order observed in the same doping range as the small Fermi pockets (

*15*,

*27*). Thus, it is desirable to perform a systematic study of the doping dependence of these small pockets as optimal doping is approached within a single cuprate family. We used high magnetic fields, extending to >90 T, to suppress superconductivity and access quantum oscillations of the underlying Fermi surface over nearly twice the range of for a range of underdoped YBa

_{2}Cu

_{3}O

_{6+δ}compositions with

*T*

_{c}s of up to 91 K. (Fig. 1).

We observed quantum oscillations in YBa_{2}Cu_{3}O_{6+δ} at δ = 0.75, 0.80, and 0.86 (corresponding to hole doping *p* = 0.135, 0.140, and 0.152) (Fig. 2A). Three regimes are clearly seen in the data: zero resistance in the vortex solid state; finite resistance that increases strongly with field in the crossover to the normal state; and magnetoresistance accompanied by quantum oscillations in the normal state. Subtracting a smooth and monotonic background from the magnetoresistance yields the oscillatory component (*36*). One can make two immediate observations: (i) At higher doping, the oscillation amplitude grows faster with decreasing temperature; and (ii) the oscillation frequency changes very little between *p* = 0.135 and *p* = 0.152. The first observation directly indicates an increasing effective mass; the second observation constrains the doping where the reconstruction from a large to a small Fermi surface takes place. We quantify these observations below.

The evolution of the cyclotron effective mass with doping, and how it relates to the temperature dependence of the quantum oscillations, can be understood quantitatively within the Lifshitz-Kosevich formalism, which has been used successfully to analyze oscillations in cuprates at lower hole doping (*23*, *32*–*34*, *37*). The effective mass extracted from the quantum oscillation amplitude (*36*) is plotted as a function of doping in Fig. 3C, which reveals an increase in the mass by almost a factor of 3 from *p* = 0.116 to *p* = 0.152. Note that electron-phonon coupling is generally observed to decrease with increasing hole doping in the cuprates, ruling it out as the mechanism of mass enhancement (*38*) and suggesting instead that the mass enhancement comes from increased electron-electron interactions. The enhancement of the effective mass toward *p* ≈ 0.18 is consistent with the doping-dependent maxima observed in the upper critical field *H*_{c2} (*39*) and in the jump in specific heat (Δγ) at *T*_{c} (*40*, *41*), both of which are expected to be enhanced by the effective mass through the density of states (see Fig. 1). Maxima in thermodynamic quantities are typical of quantum critical systems at their QCPs, having been observed in many heavy-fermion systems (*42*) and in an iron pnictide superconductor (*5*). Some physical quantities, such as superfluid density, do not show an enhancement toward optimal doping in YBa_{2}Cu_{3}O_{6+δ} (*43*); this is possibly because different physical properties experience different renormalizations from interactions (*44*) or because they are measured in the superconducting state, where the gap may serve as a cutoff.

The quantum oscillation frequency *F* gives the Fermi surface area through the Onsager relation (*29*), *A*_{k} = (2π*e/ħ*)*F*, where *A*_{k} is the Fermi surface area in momentum space perpendicular to the magnetic field and *ħ* is Planck’s constant divided by 2π. In contrast to the effective mass, which is enhanced by almost a factor of 3, the Fermi surface area only evolves weakly toward optimal doping: Fig. 2B shows *F* increasing by roughly 20% from *p* ≈ 0.09 to *p* ≈ 0.152. The observation of the small Fermi surface pocket up to *p* ≈ 0.152 requires that the reconstruction of the Fermi surface also persists up to this doping, which strongly suggests that the reconstruction is related to the incommensurate charge order also observed in this doping range (*15*, *25*–*27*). The large increase in effective mass with no accompanying large change in Fermi surface area is reminiscent of what is seen on approach to QCPs in CeRh_{2}Si_{2}, CeRhIn_{5}, and BaFe_{2}(As_{1–}* _{x}*P

*)*

_{x}_{2}(

*5*,

*42*).

The connection between the mass enhancement we observe in quantum oscillations and high-*T*_{c} superconductivity is evident in Fig. 4, which shows successive *T*_{c} curves in increasing magnetic field. By 30 T—the third-highest curve in Fig. 4—superconductivity persists only in two small domes centered around *p* ≈ 0.08 and *p* ≈ 0.18; by 50 T, only the region around *p* = 0.18 remains. This phase diagram of YBa_{2}Cu_{3}O_{6+δ} in high field, with *T*_{c} first suppressed to zero around *p* ≈ 0.125, closely resembles that of La_{2–}* _{x}*Ba

*CuO*

_{x}_{4}in zero field, where static charge and stripe order are observed (

*45*). To emphasize the enhancement of the effective mass, we plot 1

*/m** on this phase diagram [including previous

*m** measurements at lower doping (

*46*)]. This shows a trend toward maximum mass enhancement at

*p*≈ 0.08 and

*p*≈ 0.18—the same dopings at which superconductivity is the most robust to applied magnetic fields. One possible scenario for YBa

_{2}Cu

_{3}O

_{6+δ}is that critical fluctuations surrounding

*p*

_{crit}≈ 0.08 and

*p*

_{crit}≈ 0.18 provide two independent pairing mechanisms, analogous to the two superconducting domes in CeCu

_{2}Si

_{2}that originate at antiferromagnetic and valence-transition QCP (

*47*). A second scenario is a single underlying pairing mechanism whose strength varies smoothly with doping (

*7*,

*48*), but where

*T*

_{c}is enhanced at

*p*

_{crit}≈ 0.08 and

*p*

_{crit}≈ 0.18 by an increased density of states and/or by quantum critical dynamics.

Our observed mass increase establishes the enhancement of electronic interactions approaching *p*_{crit} ≈ 0.18. It is natural to ask whether this enhancement is caused by a QCP at *p*_{crit} ≈ 0.18 and, subsequently, what is the associated broken-symmetry phase. The hole doping *p*_{crit} ≈ 0.18 represents the juncture of several doping-dependent phenomena associated with underdoped cuprates. First, *p* ≈ 0.19 represents the collapse to zero of energy scales associated with the formation of the pseudogap and its onset temperature *T**. Second, the onset of an anomalous polar Kerr rotation and neutron spin flip scattering both terminate at *p* ≈ 0.18 (*12*, *13*), representing an unidentified form of broken symmetry (which persists inside the superconducting phase for the Kerr experiment). Third, in high magnetic fields, the sign change of the Hall coefficient in YBa_{2}Cu_{3}O_{6+δ} from positive to negative, and the anomaly in the Hall coefficient in Bi_{2}Sr_{0.51}La_{0.49}CuO_{6+δ}, occur near *p* ≈ 0.18 (*11*, *49*), which suggests that Fermi surface reconstruction from electron-like to hole-like occurs at this doping. Finally, *p* ≈ 0.18 represents the maximum extent of incommensurate CDW order reported in several different experiments (*15*, *26*, *27*). Although the Fermi surface reconstruction is likely related to this CDW order, its short correlation length and the weak doping dependence of its onset temperature appear to be at odds with the standard picture of long-range order collapsing to *T* = 0 at a QCP (*50*). Two scenarios immediately present themselves. In the first scenario, the suppression of superconductivity by an applied magnetic field allows the CDW to transition to long-range order, as suggested by x-ray, nuclear magnetic resonance, and pulsed-echo ultrasound experiments (*25*, *26*, *51*). In this first scenario, we would be observing a field-revealed QCP. In the second scenario, CDW order is coexistent with another form of order that also terminates near *p*_{crit} ≈ 0.18. Such a coexistence is suggested by multiple experimental results, including but not limited to Nernst anisotropy (*22*), polarized neutron scattering (*12*), and the anomalous polar Kerr effect (*13*). In this second scenario, the CDW reconstructs the Fermi surface and the other hidden form of order drives quantum criticality. Regardless of the specific mechanism, and regardless of whether *p*_{crit} = 0.18 is a QCP in the traditional sense, the observation of an enhanced effective mass coincident with the region of most robust superconductivity establishes the importance of competing broken symmetry for high-*T*_{c} superconductivity.

## Supplementary Materials

www.sciencemag.org/content/348/6232/317/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S15

## References and Notes

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**Acknowledgments:**This work was performed at the National High Magnetic Field Laboratory and was supported by the U.S. Department of Energy Office of Basic Energy Sciences “Science at 100 T” program, NSF grant DMR-1157490, the State of Florida, the Natural Science and Engineering Research Council of Canada, and the Canadian Institute for Advanced Research. S.E.S. acknowledges support from the Royal Society and the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement 337425. We thank S. Chakravarty, S. Kivelson, M. Le Tacon, K. A. Modic, C. Proust, A. Shekhter, and L. Taillefer for discussions; J. Baglo for sharing his results on the effect of quenched oxygen disorder on the microwave scattering rate in YBa_{2}Cu_{3}O_{6+δ}, without which oscillations would not have been observed; and the entire 100 T operations team at the pulsed-field facility for their support during the experiment. Full resistivity curves are available in the supplementary materials. B.J.R., S.E.S., R.D.M., B.T., Z.Z., J.B.B., and N.H. performed the high-field resistivity measurements at the National High Magnetic Field Laboratory Pulsed Field Facility. B.J.R., J.D., R.L., D.A.B., and W.N.H. grew and prepared the samples at the University of British Columbia. B.J.R. analyzed the data and wrote the manuscript, with contributions from S.E.S., R.D.M., N.H., J.D., D.A.B., and W.N.H.