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We study the quantum phases and phase transitions of the Kane-Mele Hubbard(KMH) model on a zigzag ribbon of honeycomb lattice at a finite size via theweak-coupling renormalization group (RG) approach. In the non-interactinglimit, the KM model is known to support topological edge states where electronsshow helical property with orientations of the spin and momentum being locked.The effective inter-edge hopping terms are generated due to finite-size effect.In the presence of an on-site Coulomb repulsive interaction and the inter-edgehoppings, special focus is put on the stability of the topological edge states(TI phase) in the KMH model against (i) the charge and spin gaped (II) phase,(ii) the charge gaped but spin gapless (IC) phase and (iii) the spin gaped butcharge gapless (CI) phase depending on the number (even/odd) of the zigzagribbons, doping level (electron filling factor) and the ratio of the Coulombinteraction to the inter-edge tunneling. We discuss different phase diagramsfor even and odd numbers of zigzag ribbons. We find the TI-CI, II-IC, and II-CIquantum phase transitions are of the Kosterlitz-Thouless (KT) type. Bycomputing various correlation functions, we further analyze the nature andleading instabilities of these phases.

Kane-Mele Hubbard model on a zigzag ribbon: stability of the topological edge states and quantum phase transitions