Orbital magnetic field effects in spin liquid with spinon Fermi sea:   Possible application to $\kappa$-(ET)$_2$Cu$_2$(CN)$_3$
  We consider orbital magnetic field effects in a spin liquid phase of a half-filled triangular lattice Hubbard system close to the Mott transition, continuing an earlier exploration of a state with spinon Fermi surface. Starting from the Hubbard model and focusing on the insulator side, we derive an effective spin Hamiltonian up to four-spin exchanges in the presence of magnetic field, and find that the magnetic field couples linearly to spin chirality on the triangles. The latter corresponds to a flux of an internal gauge field in a gauge theory description of the spin liquid, and therefore a static such internal flux is induced. A quantitative estimate of the effect is obtained using a spinon mean field analysis, where we find that this orbital field seen by the spinons is comparable to or even larger than the applied field. We further argue that because the stiffness of the emergent internal gauge field is very small, such a spinon-gauge system is strongly susceptible at low temperatures to an instability of the homogeneous state due to strong Landau level quantization for spinons. This instability is reminiscent of the so-called strong magnetic interaction regime in metals with the usual electromagnetic field, but we estimate that the corresponding temperature--magnetic field range is significantly broader in the spinon-gauge system. 
Olexei I. Motrunich