The nature of the phase transition in the cuprates as revealed by a magnetic field free stiffness meter

A new method to measure the superconducting stiffness tensor $\overline{\rho}_s$, without subjecting the sample to magnetic field, is applied to La$_{1.875}$Sr$_{0.125}$CuO$_4$ (LSCO). The method is based on the London equation $\bf{J}=-\overline{\rho}_s \bf{A}$, where $\bf{J}$ is the current density and $\bf{A}$ is the vector potential. Using rotor free $\bf{A}$ and measuring $\bf{J}$ via the magnetic moment of superconducting rings, we extract $\overline{\rho}_s$ at $T\rightarrow T_c$. The technique, named Stiffnessometer, is sensitive to very small stiffness, which translates to penetration depth on the order of a few millimeters. We apply this method to two different LSCO rings: one with the current running only in the CuO$_2$ planes, and another where the current must cross planes. We find different transition temperatures for the two rings, namely, there is a temperature range with two dimensional stiffness. The Stiffnessometer results are accompanied by Low Energy $\mu$SR measurements on the same sample to determine the stiffness anisotropy at $T < T_c$.