Quantum phase transitions and stable edge magnetism in effective spin models for graphene zigzag nanoribbons

We derive and examine effective quantum spin models that allow to analyse the interaction-induced magnetic correlations among the edge states of graphene nanoribbons beyond previous mean-field theory approaches, fully accounting for quantum fluctuations. We find that the extended nature of the intra-edge couplings in the effective spin model for zigzag nanoribbons leads to a quantum phase transition at a large, finite value of the inter-edge coupling. This quantum critical point separates a quantum disordered region from a gapless phase of stable edge magnetism at weak intra-edge coupling, which includes the ground states of spin-ladder models for sufficiently wide zigzag nanoribbons. We find that narrow zigzag ribbons of only a few zigzag lines instead reside within the quantum disordered regime. To study the quantum critical behavior, the effective spin model can be related to a model of two antiferromagnetically coupled Haldane-Shastry spin-half chains with long-ranged ferromagnetic intra-chain couplings. The results for the critical exponents compare well to recent renormalization group calculations for related long-ranged interacting quantum systems.