Commensurate-incommensurate transitions and the Lifshitz point in the quantum asymmetric clock model

The asymmetric three-state clock model is studied in the context of a one-dimensional quantum Hamiltonian. Series expansions are used to investigate the commensurate-incommensurate transition and the incommensurate-liquid Kosterlitz-Thouless transition. Evidence is presented for a Lifshitz point at which the critical indices $\alpha$, $\beta$, and $\gamma$ are Potts-type, while the mass-gap index appears to take its Ising value, $\nu =1\mathrm{}$.