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The Z2 and chiral spin liquids in an anisotropic kagome spin model

Gapped spin liquid is a topological phase of matter that hosts deconfined fractionalized quasiparticles. Quasiparticle statistics are an intrinsic property of spin liquid. However, it is difficult to identify explicitly in a microscopic model. Here we perform an exact diagonalization study of the Z2 quasiparticle statistics in the ‘easy-axis’ S = 1/2 Heisenberg J1 − J2 − J3 model on the kagome lattice, which hosts a robust Z2 spin liquid. Using the entanglement entropy measurement, we identify multiple independent minimal entangled states (MESs) on the torus geometry, which represents the eigenstates of the Wilson loop operators encircling the torus that measure the Z2 flux threading it. Based on the MESs, we obtain that: (1) The scaling of the finite size entropy leads to a topological entanglement entropy γ ≈ 0.637, indicating a Z2 topological spin liquid. (2) The modular transformation of MESs gives rise to the modular matrices S ?> and U ?>, which unambiguously fixes the Z2 quasiparitcle statistics including quasiparticle quantum dimensions, fusion rules and topological spins. (3) Through adiabatical threading a 2π flux in the hole of the torus, spinon state ψs (vacuum state ψ1) evolves into ψ1 (ψs), as predicted by Z2 gauge theory. Away from the ‘easy-axis’ limit with intermediate spin-z coupling, we identify some signature evidence of the chiral spin liquid based on the entanglement spectrum of the state using density matrix renormalization group simulation.