We study finite-size scaling for the magnetic observables of an impurity residing at the end point of an open quantum Ising chain with transverse magnetic field, realized by locally rescaling the field by a factor $\ensuremath{\mu}\ensuremath{\ne}1$. In the homogeneous chain limit at $\ensuremath{\mu}=1$, we find the expected finite-size scaling for the longitudinal impurity magnetization, with no specific scaling for the transverse magnetization. At variance, in the classical impurity limit $\ensuremath{\mu}=0$, we recover finite scaling for the longitudinal magnetization, while the transverse one basically does not scale. We provide both analytic approximate expressions for the magnetization and the susceptibility as well as numerical evidences for the scaling behavior. At intermediate values of $\ensuremath{\mu}$, finite-size scaling is violated, and we provide a possible explanation of this result in terms of the appearance of a second, impurity-related length scale. Finally, by going along the standard quantum-to-classical mapping between statistical models, we derive the classical counterpart of the quantum Ising chain with an end-point impurity as a classical Ising model on a square lattice wrapped on a half-infinite cylinder, with the links along the first circle modified as a function of $\ensuremath{\mu}$.
The quantum wetting transition in the one-dimensional transverse-field Ising model with random bonds is studied. Opposite boundary fields are applied at the two ends of the Ising chain. The interface between two domains with oppositely oriented magnetization is localized near the boundary or in the middle of the chain. The wetting transition refers to the phase transition of localization and delocalization of the interface. The tuning parameter
hL is the boundary field at one end, e.g., the left end. At the transition point
hL=hw, the wetting transition occurs. First, we study the wetting transition with one defect bond. It is a first-order transition, and the interface jumps from the left end of the chain to the defect bond at the transition point. Second, we study the wetting transition with two defect bonds and find that the weaker defect bond dominates the phase transition. The wetting transition is still first order, and the interface is localized at the left end of the chain or at the weaker defect bond. Lastly, the random bond case is studied. The random bonds have a rectangular distribution. The wetting transition is still first order. The finite-size effects are studied. The statistics of the transition points and the energy gaps are obtained. We study lattices with sizes
N=100, 150, 200, 250, 300, and 350. The deviation of the phase transition points
hw does not decrease as the lattice size increases. It is argued that the transition point is sample dependent, i.e., the variation in the transition point
hw does not approach zero, even at the thermodynamic limit.
Based on a relationship with continuous-time random walks discovered by Igl\'oi, Turban, and Rieger [Phys. Rev. E 59, 1465 (1999)], we derive exact lower and upper bounds on the lowest energy gap of open transverse-field Ising chains, which are explicit in the parameters and are generally valid for arbitrary sets of possibly random couplings and fields. In the homogeneous chain and in the random chain with uncorrelated parameters, both the lower and upper bounds are found to show the same finite-size scaling in the ferromagnetic phase and at the critical point, demonstrating the ability of these bounds to infer the correct finite-size scaling of the critical gap. Applying the bounds to random transverse-field Ising chains with coupling-field correlations, a model which is relevant for adiabatic quantum computing, the finite-size scaling of the gap is shown to be related to that of sums of independent random variables. We determine the critical dynamical exponent of the model and reveal the existence of logarithmic corrections at special points.