Interplay between multi-spin and chiral spin interactions on triangular lattice

TL;DR

This study explores how multi-spin and chiral interactions influence quantum spin liquids on a triangular lattice, revealing new phases and a crossover region that enhances understanding of topological quantum states.

Contribution

It introduces a detailed phase diagram showing the effects of four-site ring-exchange and chiral interactions on various quantum spin liquids, including the discovery of a proximate QBT state.

Findings

Identification of multiple quantum spin liquid phases including QBT, U(1), Z2 d+id, and chiral spin liquids.

Discovery of a crossover region where the QBT state acquires a small gap and becomes energetically favorable.

The Z2 d+id wave QSL dominates the phase diagram due to cooperative effects of J4 and Jχ.

Abstract

We investigate the spin-$\frac{1}{2}$ nearest-neighber Heisenberg model with the four-site ring-exchange $J_4$ and chiral interaction $J_\chi$ on the triangular lattice by using the variational Monte Carlo method. The $J_4$ 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 $J_\chi$ can enrich the phase diagram with two interesting chiral QSLs (topological orders) with the same quantized Chern number $\mathcal{C} = \frac{1}{2}$ and ground-state degeneracy GSD = 2, namely the U(1) chiral spin liquid (CSL) and Z$_2$ $d+id$-wave QSL. The nodal $d$-wave QSL is fragile and will turn to the Z$_2$ $d+id$ QSL with any finite $J_\chi$ within our numerical calculation. However, in the process from QBT to the Z$_2$ $d+id$ QSL with the increase of $J_\chi$, 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 Z$_2$ $d+id$ QSL. We dub this state as the proximate QBT QSL and it gives way to the Z$_2$ $d+id$ QSL eventually. Therefore, the cooperation of the $J_4$ and $J_\chi$ terms favors mostly the Z$_2$ $d+id$-wave QSL, so that this phase occupies the largest region in the phase diagram.