Variational Monte Carlo Study of J₁-J_d-J_χ model on the Kagome Lattice

Quantum spin models on the kagome lattice are prominent candidates for studying the quantum spin liquid (QSL) phase. For instance, the J1-(J2)-Jd model serves as a typical example of ν=1/2 fractional quantum Hall system, and the staggered scalar spin chiral term leads to a non-Fermi liquid behavior of spinon excitation (J1 is the first-, J2 the second-, and Jd the third-neighbor exchange). In this work, we study the competition between the Jd and staggered scalar spin chiral term (with strength Jχ) on the kagome lattice using a variational Monte Carlo method with irreducible representation. As a function of the parameters Jd/J1 and Jχ/J1, we found a phase diagram consisting of four different QSLs: U(1)-Dirac spin liquid (U(1)-DSL), Dirac chiral spin liquid (Dirac-CSL), gapped chiral spin liquid (Gapped-CSL), and CSL with line Fermi surfaces (Line-CSL). For the Line-CSL, we show that an anti-symmetry relation between the vertical mirror plane and the mean-field Hamiltonian protects the line Fermi surfaces. Furthermore, we found that the Jχ induces a tricritical point between U(1)-DSL and gapped-CSL near (Jd/J1, Jχ/J1)~(0.3,0.2). We construct the symmetry-allowed Landau-Ginzburg theory that gives rise to the tricritical point. We also show static spin structure factor, chiral-chiral correlation, and longitudinal thermal conductivity as distinctive physical observables of each phase.