Novel phase line in the mixed state of the multi-band MgB$_2$ single crystal

High-quality MgB$_2$ single crystals of submillimeter size were synthesized successfully by the vapor transport method for carrying out magnetic torque measurements. The magnetic torque of MgB$_2$ has been analyzed by a single-band model as well as a multi-band model for an anisotropic superconductor. The anisotropy in the coherence length $\gamma_\xi = \xi_a/\xi_c$ and the anisotropy in the penetration depth $\gamma_\lambda = \lambda_c/\lambda_a$ with the multi-band Kogan theory for the magnetic torque were simultaneously determined by means of the least-squares fittings. A systematic variation of two-sorts of effective superconducting anisotropies, $\gamma_\xi$ and $\gamma_\lambda$, in the field-temperature plane was obtained as contours, and can be interpreted as a manifestation of the two-band nature of the MgB$_2$ superconductivity. It is not a gradual crossover but a drastic change in the electronic state that an MgB$_2$ superconductor transits from a $\pi-\sigma$ multiband superconductor in lower fields to a $\sigma$ single band superconductor in higher fields.