Interplay between multi-spin and chiral spin interactions on triangular lattice
Abstract
We investigate the spin-12 nearest-neighber Heisenberg model with the four-site ring-exchange J4 and chiral interaction J on the triangular lattice by using the variational Monte Carlo method. The J4 term induces the quadratic band touching (QBT) quantum spin liquid (QSL) with only a d+id spinon pairing (without hopping term), the nodal d-wave QSL and U(1) QSL with a finite spinon Fermi surface progressively. The effect of the chiral interaction J can enrich the phase diagram with two interesting chiral QSLs (topological orders) with the same quantized Chern number C = 12 and ground-state degeneracy GSD = 2, namely the U(1) chiral spin liquid (CSL) and Z2 d+id-wave QSL. The nodal d-wave QSL is fragile and will turn to the Z2 d+id QSL with any finite J within our numerical calculation. However, in the process from QBT to the Z2 d+id QSL with the increase of J, an exotic crossover region is found. In this region, the previous QBT state acquires a small hopping term so that it opens a small gap at the otherwise band touching points, and leads to an energy minimum which is energetically more favorable compared to another competitive local minimum from the Z2 d+id QSL. We dub this state as the proximate QBT QSL and it gives way to the Z2 d+id QSL eventually. Therefore, the cooperation of the J4 and J terms favors mostly the Z2 d+id-wave QSL, so that this phase occupies the largest region in the phase diagram.