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In underdoped La2−xSrxCuO4 (LSCO), the magnetic susceptibility is dominated by the Curie-like spin susceptibility and the van Vleck orbital susceptibility11,12,13. These strongly T-dependent terms render weak diamagnetic signals difficult to detect using standard magnetometry. However, because the spin susceptibility is nearly isotropic13 whereas the incipient diamagnetism is anisotropic, torque magnetometry has proved to be effective in resolving the diamagnetic signal14,15,16,17 (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, Mobs≡τ/μ0HxV, where V is the sample volume, μ0 is the permeability and Hx=Hsinφ (we take z || c). In cuprates, Mobs comprises three terms14,15

where Md(T,Hz) is the diamagnetic magnetization of interest, ΔMs is the anisotropy of the spin local moments and Δχorb 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, Mobs derived from the torque experiment in sample 03 is in good, quantitative agreement with the anisotropy inferred from previous bulk susceptibility measurements13 on a large crystal of LSCO (x=0.03) (see the Supplementary Information).

Figure 1 shows the magnetization, Mobs, in samples 055 and 06. The pattern of Mobs results from the sum of the three terms in equation (1). Figure 1a shows how it evolves in sample 055. At high T (60–200 K), the curves of Mobs versus H are fan-like, reflecting the weak T dependence14 of the orbital term Δχorb(T)H. At the onset temperature for diamagnetism, Tonset (55 K, bold curve), the diamagnetic term, Md, appears as a new contribution. The strong H dependence of Md causes Mobs to deviate from the H-linear behaviour. In Fig. 1b, the evolution is similar, except that the larger diamagnetism forces Mobs to negative values at low H. As mentioned, the spin contribution, ΔMs, is unresolved above ∼40 K in both panels. To magnify the diamagnetic signal, we subtract the orbital term, ΔχorbH (the three terms are also apparent in the total observed susceptibility χobs=Mobs/H; see the Supplementary Information).

Figure 1: Magnetization curves in lightly doped LSCO.
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a, Curves in sample 055 (Tc∼0.5 K) at T=4–200 K. b, Curves in 06 (Tc∼5 K). The crystal is glued to the tip of the cantilever with its c axis at a small angle, φ∼15∘, to the field H (we write Hz as H). Above Tonset (bold curves at 55 and 70 K in a and b, respectively), the fan-like pattern is due entirely to the paramagnetic term, Δχorb(T)H (see the Supplementary Information for a plot of Δχorb(T)). Below Tonset, the diamagnetic term, Md, becomes evident. c, The profiles of Mobs′=Mobs−ΔχorbH in sample 05. Note the oscillation in weak H at T=0.5 and 0.75 K and the approach towards saturation in high fields. These curves are separated into ΔMs and Md in Fig. 2b.

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The resulting curves, Mobs′(T,H)≡Mobs−ΔχorbH, are shown for sample 05 in Fig. 1c. At low fields, Mobs′, shows an interesting oscillatory behaviour (curves at 0.5 and 0.75 K), but at high fields it tends towards saturation. By examining how Mobs′ behaves in the two limits of weak and intense fields (see the Supplementary Information), we have found that Mobs′ comprises a diamagnetic term, Md(T,H), that resembles the ‘tilted hill’ profile of diamagnetism in the vortex-liquid state above the critical temperature, Tc, reported previously10,14, and a spin-anisotropy term, ΔMs.

Modelling the latter as free spin-1/2 local moments with anisotropic g factors measured with H∥c (gc) and (ga b), we have (see the Supplementary Information)

where μB is the Bohr magneton, β=1/kBT and (kB is Boltzmann’s constant). With gφ∼gc fixed at 2.1, the sole adjustable parameter at each T is the prefactor P(T). Equation (2) accounts very well for the curves in Fig. 1c, especially the oscillatory behaviour; at T=0.5 and 0.75 K, ΔMs∼1/kBT dominates Md in weak H, but for H>kBT/gcμB, the saturation of ΔMs implies that Mobs′(H) adopts the profile of Md(H) apart from a vertical shift. Lightly doped LSCO enters a spin- or cluster-glass state11,12 below the spin-glass temperature, Tsg, which is sensitive to sample purity (in 03 and 04, Tsg∼2.5 and 1 K, respectively).

Subtracting ΔMs from Mobs′, we isolate the purely diamagnetic term, Md(T,H). Figure 2 shows the curves of Md and ΔMs at selected T in samples 04, 05, 055 and 06. Samples 03, 04 and 05 do not show any Meissner effect. The strict reversibility of the Md–H curves confirms that we are in the vortex-liquid state in 03, 04 and 05. When x exceeds xc, the samples exhibit broad Meissner transitions (Tc∼0.5 and 5 K in 055 and 06, respectively). Hysteretic behaviour appears below a strongly T-dependent irreversibility field, Hirr(T), discussed below. The T dependences in the four panels reveal an important pattern. In the vortex liquid, the overall magnitude of Md grows rapidly as we cool from 35 to 5 K, but it stops changing below a crossover temperature, TQ (∼4 K in samples 05, 055 and 06, and ∼2 K in 04). Even in sample 06, where Hirr∼9 T at 0.35 K, Md recovers the T-independent profile when H>Hirr (diverging branches at 9 T in Fig. 2d). The insensitivity to T suggests that the excitations, which degrade the diamagnetic response in the liquid state, are governed by quantum statistics below TQ.

Figure 2: Separation of magnetization into the spin term, ΔMs(T,H), and the diamagnetic term, Md(T,H).
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a–d, Curves of ΔMs (positive) and Md (negative) versus H in samples 04 (a), 05 (b), 055 (c) and 06 (d). In each sample, the diamagnetic minimum (at 5 T) deepens rapidly between 30 and 5 K, but does not change below TQ. The depairing field, Hc2, is estimated to be 25, 35, 43 and 48 T in a–d, respectively. In d, the branching curves (with arrows) indicate the high-field limit of the vortex solid at 0.35 K. Above Hirr(T), the low-T curves merge with the vortex-liquid curve at 4 K.

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In intense fields, the field suppression of Md provides an estimate of the depairing field, Hc2 (∼20, 25, 35, 43 and 48 T in samples 03, 04, 05, 055 and 06, respectively). We find that Hc2 is nominally T independent14. The remarkably large values of Hc2 at such low hole densities imply that the pair-binding energy is anomalously large down to x=0.03. In high fields, a sizeable diamagnetic signal is seen, but long-range phase rigidity (in zero H) is absent down to T=0.35 K unless x exceeds xc. At low T, this transition is quite abrupt. Next, we discuss the vortex solid and the phase diagram.

Hysteresis in Md versus H below Hirr(T) provides a sensitive indication of the presence of the vortex solid. The strong vortex pinning in LSCO leads to large hystereses as soon as the vortex system exhibits shear rigidity. The hysteretic loops, which appear in sample 055, expand very rapidly as x exceeds 0.055. By examining the hysteretic loops (Fig. 3a shows curves for 06), we can determine Hirr(T) quite accurately. Vortex avalanches—signatures of the vortex solid—are observed (for H<Hirr) unless the field-sweep rate is very slow (see theSupplementary Information).

Figure 3: The hysteresis curves and irreversibility field Hirr(T) in the vortex-solid phase.
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a, The hysteretic curves of Md versus H in sample 06 at T from 0.35 to 2 K. Although the hysteretic segments for H<Hirr(T) are very strongly T dependent, the reversible segments above Hirr(T) are not. The latter match the T-constant profile shown in Fig. 2d. b, The T dependences of Hirr(T) in samples 055, 06, 07 and 09 in semilog scale. At low T, the data approach equation (3). The steep decrease of the characteristic temperature T0 as x→xc implies a softening of the vortex solid (T0∼1 K in sample 06). The gradual convergence of the two branches of the irreversibility curves (see curves in a) leads to uncertainties in fixing Hirr(T) (shown as the error bars in b).

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The temperature dependence of Hirr(T) is plotted in Fig. 3b for samples x>xc. At low T, the dependence approaches the exponential form

The parameters H0 and T0 decrease steeply as x→xc. The field parameter, H0, provides an upper bound for the zero-Kelvin melting field, Hm(0). Equation (3), reminiscent of the Debye–Waller factor, strongly suggests that the excitations responsible for the melting transition follow classical statistics at temperatures down to 0.35 K. The classical nature of these excitations contrasts with the quantum nature of the excitations in the vortex liquid below TQ described above.

The inferred values of H0 (2, 13, 25 and 40 T in 055, 06, 07 and 09, respectively) are much smaller than Hc2(0). Hence, after the vortex solid melts, there exists a broad field range in which the vortices remain in the liquid state at low T. The existence of the liquid at T<TQ implies very large zero-point motion associated with a small vortex mass, mv, which favours a quantum-mechanical description.

Finally, we construct the low-T phase diagram in the x–H plane. Figure 4 shows that the x dependence of Hc2(0), the depairing field scale, is qualitatively distinct from that of H0, the boundary of the vortex solid. The former varies roughly linearly with x between 0.03 and 0.07 with no discernible break-in-slope at xc, whereas H0 falls steeply towards zero at xc with large negative curvature. This sharp decrease—also reflected in the 1,000-fold shrinkage of the hysteresis amplitude between x=0.07 and 0.055 (see the Supplementary Information)—is strong evidence that the collapse of the vortex solid is a quantum critical transition. This is shown by examining the variation of Hirr versus x at several fixed T (dashed lines). At 4 K, Hirr approaches zero gently with positive curvature, but at lower T, the trajectories tend towards negative curvature. In the limit T=0, H0 approaches zero at xc with a nearly vertical slope. The focusing of the trajectories to the point (xc, 0) is characteristic of a sharp transition at xc, and is strikingly different from the smooth decay suggested by viewing lightly doped LSCO as a system of superconducting islands with a broad distribution of Tc values.

Figure 4: The phase diagram of LSCO in the x–H plane at low temperatures.
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The field H0=limT→0Hirr (the boundary between vortex-solid and vortex-liquid states) falls steeply to zero as x→xc (solid curve). The dashed lines indicate the variation of Hirr(T) versus x at fixed T, as indicated. In contrast, the depairing field, Hc2(0) (closed circles), is nominally linear in x. Below xc, the vortex liquid is stable and coexists with a growing magnetic background (graded shading). The error bars on Hc2(0) and H0 (vertical lines) are estimated uncertainties in field values. When the uncertainty in H0 is smaller than ±1 T (nominal size of the open circles), the error bars are not shown.

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In Fig. 4, the high-field vortex liquid is seen to extend continuously to x<xc where it coexists with the cluster/spin-glass state11,12 (samples 03, 04 and 05). As shown in Fig. 2 (see the Supplementary Information), the robustness of Md to intense fields attests to unusually large pairing energy even at x=0.03, but the system stays as a vortex liquid down to 0.35 K.

In the limit H→0, the vortex liquid (x<xc) has equal populations of vortices and antivortices. This implies that, as x falls below xc at low T and in zero field, superconductivity is destroyed by the spontaneous appearance of free (anti) vortices engendered by increased charge localization and strong phase fluctuation3,4,6. In zero field, superconductivity first transforms to a vortex-liquid state with strong phase disordering. The rapid growth of the spin-/cluster-glass state in LSCO suggests that incipient magnetism also plays a role in destroying superconductivity.