Variational Monte Carlo study of J₁-J_d-J_χ Hamiltonian on the kagome lattice

A quantum spin model on the kagome lattice has been extensively studied as a prominent candidate for quantum spin liquids (QSL). In this work, we study the extended Heisenberg model with staggered scalar spin chiral term on the kagome lattice using a variational Monte Carlo method. Varying third-nearest-neighbor Heisenberg exchange strength J_d and staggered scalar spin chiral strength J_χ, we find out four different spin liquid phases: U(1)-Dirac spin liquid, gapped chiral sipn liquid, and two gapless chiarl spin liquids. One of the gapless chiral spin liquids has three line Fermi surfaces, and each of them is protected by anti-commutation relation between mean-field Hamiltonian and three mirror planes. As J_d increased, chiral flux is induced and staggered one is suppresed. This breaks anti-commutation relation, and three line Fermi surfaces merges into one large Fermi surface. We suggest static and dynamical spin structure factors, and DC thermal condcutance behavior as an experimental signature of these four phases.