We investigate two exotic states in condensed matter: mesoscopic magnetism of integrable systems and Cooper pairing mediated by multiple-spin exchange.
To study mesoscopic magnetism of integrable systems, we investigate, analytically and numerically, the orbital magnetism of free electron gas in a rectangular box, which is, classically, a model integrable system. We find that both the average orbital susceptibility and its fluctuations are determined by the two-level van Vleck susceptibility that involves the last occupied (Fermi) level and the first unoccupied level. This is in line with previous results for disordered (classically chaotic) systems. The mesoscopic fluctuations, however, are much larger in integrable systems and there is a pronounced non-self-averaging effect for phyical quantities in integrable systems.
To study multiple-spin interactions introduced by ring exchange process, we investigate the Cooper pairing state of a modified t-J model on a square lattice. In this work, we developed the renormalized mean-field theory appropriate to the modified t-J model, where the J term is the 4-spin interaction introduced by the ring exchange, instead of the 2-spin interaction. Our result shows that a mixing state of spin singlet and triplet pairing matches the short-range correlations and optimizes the energy. At half filling, the pairing state is unphysical, due to the fact that there is no double occupancy. Upon doping or with intrinsic vacancies, the paring state becomes physical and may give rise to d+p or s+p wave superfluidity. Such a mechanism may introduce supersolidity in bulk solid Helium 3 and solid Helium 3 absorbed on a substrate at very low temperature.