The emergent of the chiral spin liquid in kagome Heisenberg model

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The quantum spin liquid is an emergent new state of matter, which has attracted a lot of recent attention. In particular, the time reversal symmetry broken spin liquid (Kalmeyer et. al. and Wen et. al.), characterized by the chiral ordering and fractionalized quasi-particle as a realization of the fractional quantum Hall state had been proposed for more than 20 years, but never identified as the true ground state in any more generic (e.g. Heisenberg spin exchange) models with time reversal symmetry. Here I will report two concrete examples where chiral spin liquid (CSL) emerge for a range of parameter space for kagome J1-J2-J3 (three nearest neighbors) model based on accurate density matrix renormalization group (DMRG) simulations. We identify long-range chiral ordering, ground state degeneracy, characteristic entanglement spectra, and the fractionalized topological Chern number to establish the topological state in such a system as the long-sought CSL. We further explicitly extract the modular matrix, which captures all the information of the fractional statistics of the quasi-particles in the system. I will also discuss the close relation of our model to some frustrated kagome antiferromagnets, and make a conjecture that J1 kagome model is near an unconventional critical point. I will conclude the talk with some discussions of the open directions.