The Spectrum and the Quantum Hall Effect on the Square Lattice with Next-Nearest-Neighbor Hopping: Statistics of Holons and Spinons in the t-J Model

Abstract

We investigate the energy spectrum and the Hall effect of electrons on the square lattice with next-nearest-neighbor (NNN) hopping as well as nearest-neighbor hopping. General rational values of magnetic flux per unit cell φ = p/q are considered. In the absence of NNN hopping, the two bands at the center touch for q even, thus the Hall conductance is not well defined at half filling. An energy gap opens there by introducing NNN hoping. When φ =1/2, the NNN model coincides with the mean field Hamiltonian for the chiral spin state proposed by Wen, Wilczek and Zee (WWZ). The Hall conductance is calculated from the Diophantine equation and the E-φ diagram. We find that gaps close for other fillings at certain values of NNN hopping strength. The quantized value of the Hall conductance changes once this phenomena occurs. In a mean field treatment of the t-J model, the effective Hamiltonian is the same as our NNN model. From this point of view, the statistics of the quasi-particles is not always semion and depends on the filling and the strength of the mean field.