In metals, orbital motions of conduction electrons on the Fermi surface are quantized in magnetic fields, which is manifested by quantum oscillations in electrical resistivity. This Landau quantization is generally absent in insulators. Here, we report a notable exception in an insulator-ytterbium dodecaboride (YbB). The resistivity of YbB, which is of a much larger magnitude than the resistivity in metals, exhibits distinct quantum oscillations. These unconventional oscillations arise from the insulating bulk, even though the temperature dependence of the oscillation amplitude follows the conventional Fermi liquid theory of metals with a large effective mass. Quantum oscillations in the magnetic torque are also observed, albeit with a lighter effective mass.

In copper oxide superconductors, the lightly doped (small dopant concentration, x) region is of major interest 1,2 because superconductivity, antiferromagnetism and the pseudogap state come together near a critical doping value, x c . But the way in which superconductivity is destroyed as x is decreased at very low temperatures, T , is not clear [3][4][5][6][7] . Does the pair condensate vanish abruptly at a critical value, x c ? Or is phase coherence of the condensate destroyed by spontaneous vortices-as is the case at elevated T (refs 8-10)? So far, magnetization data at low T are very sparse in this region of the phase diagram. Here, we report torque magnetometry measurements on La 2−x Sr x CuO 4 , which show that, in zero magnetic field, quantum phase fluctuations destroy superconductivity at x c ≈ 0.055. The phase-disordered condensate survives to x = 0.03. In finite field H, the vortex solidto-liquid transition occurs at H lower than the depairing field, H c2 . The resulting phase diagram reveals the large fraction of the x-H plane occupied by the quantum vortex liquid.In underdoped La 2−x Sr x CuO 4 (LSCO), the magnetic susceptibility is dominated by the Curie-like spin susceptibility and the van Vleck orbital susceptibility [11][12][13] . These strongly T-dependent terms render weak diamagnetic signals difficult to detect using standard magnetometry. However, because the spin susceptibility is nearly isotropic 13 whereas the incipient diamagnetism is anisotropic, torque magnetometry has proved to be effective in resolving the diamagnetic signal [14][15][16][17] (the resistivity anisotropy is 6,000-8,000 below 40 K, so the supercurrent is predominantly in-plane; see Supplementary Information). With H tilted at a slight angle, φ, to the crystal c axis, the torque, τ, may be expressed as an effective magnetization, M obs ≡ τ/μ 0 H x V , where V is the sample volume, μ 0 is the permeability and H x = H sin φ (we take z c). In cuprates, M obs comprises three terms 14,15M s is the anisotropy of the spin local moments and χ orb is the anisotropy of the van Vleck orbital susceptibility (see the Supplementary Information).We label the seven samples studied as 03 (with = 0.030), 04 (0.040), 05 (0.050), 055 (0.055), 06 (0.060), 07 (0.070) and 09 (0.090). To start, we confirmed that, above ∼25 K, M obs derived from the torque experiment in sample 03 is in good, quantitative agreement with the anisotropy inferred from previous bulk susceptibility measurements 13 on a large crystal of LSCO (x = 0.03) (see the Supplementary Information). In Fig. 1b, the evolution is similar, except that the larger diamagnetism forces M obs to negative values at low H. As mentioned, the spin contribution, M s , is unresolved above ∼40 K in both panels. To magnify the diamagnetic signal, we subtract the orbital term, χ orb H (the three terms are also apparent in the total observed susceptibility χ obs = M obs /H; see the Supplementary Information).The resulting curves, M obs (T , H ) ≡ M obs − χ orb H, are shown for sample 05 in Fig. 1c...

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