Online Supplementary Material: Low temperature vortex liquid in $\rm La_{2-x}Sr_xCuO_4$

We report here the supplementary material for Ref. [1]. Using torque magnetometry, we measured the magnetization anisotropy of 7 crystals of $\rm La_{2-x}Sr_xCuO_4$ (LSCO). In optimally-doped cuprates, the bulk susceptibility is dominated by the paramagnetic van-Vleck orbital term $\chi_{orb}$ which has a significant anisotropy ($\chi^{orb}_c>\chi^{orb}_{ab}$) that changes weakly with $T$ (subscripts $c$ and ${ab}$ identify quantities measured with $\bf H||c$ and $\bf H\perp c$, respectively). Moreover, in the lightly-doped regime, the paramagnetic spin susceptibilities $\chi^s_c$ and $\chi^s_{ab}$ become significantly large below the interval 40-60 K. However, the spin susceptibility is very nearly isotropic (except below 10 K where its anisotropy becomes measurable). Against the large orbital and spin terms, the weak diamagnetic signal is very difficult to resolve using standard bulk magnetometry in lightly-doped cuprates. By contrast, torque magnetometry selectively detects the orbital diamagnetism generated by supercurrents confined to the CuO$_2$ layers while ignoring the large spin contribution when it is isotropic. The orbital van-Vleck contribution is also detected, but as a ``background'' that is $H$-linear to intense fields and only mildly $T$ dependent.