First- and second-order quantum phase transitions in the one-dimensional transverse-field Ising model with boundary fields

Abstract

The one-dimensional transverse field Ising model with boundary fields is studied analytically and numerically. The phase diagram in the ordered state is obtained. We find that there exist two types of phase transitions. The longitudinal boundary fields applied to the left and right ends of the Ising chain are ${h}_{L}$ and ${h}_{R}$, respectively. For $|{h}_{L}|,|{h}_{R}|<\sqrt{1\ensuremath{-}g}$, where $g$ is the transverse field and ${h}_{L}{h}_{R}<0$, a first-order phase transition occurs with the changing ${h}_{L}$ or ${h}_{R}$. The energy gap and boundary magnetization are solved exactly. The analytical expressions of the finite-size scaling for the first-order phase transition are obtained. For $|{h}_{R}|>\sqrt{1\ensuremath{-}g}$ and ${h}_{L}{h}_{R}<0$, there exists a continuous phase transition with changing ${h}_{L}$ and vice versa. This transition is identified as a quantum wetting transition. The singularity of the boundary magnetization in this phase transition is explicitly shown. A simple computational procedure with high accuracy and efficiency is proposed to calculate the magnetization. The magnetization profile, correlation functions, and wetting layer thickness are studied numerically.