The quantum XY chain with boundary fields: finite-size gap and phase behavior

We present a detailed study of the finite-size one-dimensional quantum XY chain in a transverse field in the presence of boundary fields coupled with the order-parameter spin operator. We consider fields located at the chain boundaries that have the same strength and that are oppositely aligned. We derive exact expressions for the gap Δ as a function of the model parameters for large values of the chain length L. These results allow us to characterize the nature of the ordered phases of the model. We find a magnetic (M) phase (Δ∼e−aL), a magnetic-incommensurate (MI) phase (Δ∼e−aLfMI(L) ), a kink (K) phase (Δ∼L−2) and a kink-incommensurate (KI) phase (Δ∼L−2fKI(L)); fMI(L) and fKI(L) are bounded oscillating functions of L. We also analyze the behavior along the phase boundaries. In particular, we characterize the universal crossover behavior across the K-KI phase boundary. On this boundary, the dynamic critical exponent is z = 4, i.e. Δ ∼ L−4 for large values of L.