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 $\mathrm{\Delta }$ 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 ($\mathrm{\Delta }\sim e^{-aL}$), a magnetic-incommensurate (MI) phase ($\mathrm{\Delta }\sim e^{-aL}{f}_{MI}(L)$), a kink (K) phase ($\mathrm{\Delta }\sim L^{-2}$), and a kink-incommensurate (KI) phase ($\mathrm{\Delta }\sim L^{-2}{f}_{KI}(L)$); $f_{MI}(L)$ and $f_{KI}(L)$ 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., $\mathrm{\Delta }\sim L^{-4}$ for large values of $L$.