  Hybrid spin-mechanical systems have great potentials in sensing, macroscopic
quantum mechanics, and quantum information science. In order to induce strong
coupling between an electron spin and the center-of-mass motion of a mechanical
oscillator, a large magnetic gradient is usually required, which is difficult
to achieve. Here we show that strong coupling between the electron spin of a
nitrogen-vacancy (NV) center and the torsional vibration of an optically
levitated nanodiamond can be achieved in a uniform magnetic field. Thanks to
the uniform magnetic field, multiple spins can strongly couple to the torsional
vibration at the same time. We propose to utilize this new coupling mechanism
to realize the Lipkin-Meshkov-Glick (LMG) model by an ensemble of NV centers in
a levitated nanodiamond. The quantum phase transition in the LMG model and
finite number effects can be observed with this system. We also propose to
generate torsional superposition states and realize torsional matter-wave
interferometry with spin-torsional coupling.
