Fractional-exponent behavior of magnetization near $T_c$ in ${\rm Bi_2Sr_2CaCu_2O_8}$

Using high-resolution torque magnetometry, we have investigated in detail how long-range phase coherence develops as the critical temperature $T_c$ (88.7 K) is approached in optimally-doped $\rm Bi_2Sr_2CaCuO_{8+\delta}$ with field $\bf H||c$. Three distinct regimes are observed. Above $\sim$92 K, $|M|$ increases rapidly as $T\rightarrow T_c$ in step with the vortex Nernst signal. $M$ is strictly linear in $H$ in weak $H$, but shows strong curvature at large $H$ (5-14 T). The curvature provides a determination of the correlation length $\xi_{sc}$ which grows as a power law, viz. $\xi_{sc}\sim 1/t^\nu$. In the second regime, $86 < T < 92$ K, $M$ becomes nonlinear in $H$, viz. $M\sim H^{\alpha(T)}$, where the exponent $\alpha(T)$ decreases from 1 to 0. This interesting fractional-exponent behavior is highly unusual and fits poorly with conventional pictures of `fluctuating diamagnetism.' As previously known, $M$ is virtually $H$ independent below 2 Tesla at the ``crossing temperature'' $T_{cr} $ = 86 K. Below $T_{cr}$, $M$ is a function of $\log H$. We compare this behavior with predictions of the 3DXY and Kosterlitz-Thouless theory. Supported by funds from the U.S. National Science Foundation under grant DMR 0213706.