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Physica Scripta

On the size of boundary effects in the Ising chain

Oskar Amadeusz Prośniak

Published Mar 29, 2019• 5 citations
Publisher

Abstract

Abstract We discuss the length at which boundaries affect the magnetization of the quantum Ising chain with free ends. For this purpose we introduce two characteristic length scales describing the size of the boundary-modified region in respectively, longitudinal and transverse magnetization. We show that their finite-size scaling behaviour encodes the position of the critical point and the universal scaling exponent ν . The behaviour of proposed lengths in small systems is also discussed. These results should be useful in the context of both numerical and experimental studies.

DOI
10.1088/1402-4896/ab1492

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Abstract We discuss the length at which boundaries affect the magnetization of the quantum Ising chain with free ends. For this purpose we introduce two characteristic length scales describing the size of the boundary-modified region in respectively, longitudinal and transverse magnetization. We show that their finite-size scaling behaviour encodes the position of the critical point and the universal scaling exponent ν . The behaviour of proposed lengths in small systems is also discussed. These results should be useful in the context of both numerical and experimental studies.
We propose two easy-to-study observables in the quantum Ising chain with open boundary conditions. They measure the length at which boundaries affect the longitudinal or transverse magnetization. We show that their finite-size scaling behaviour encodes the position of the quantum critical point and the universal scaling exponent ννν. The applicability of proposed observables in small systems is also discussed. We expect that our results will be useful in quantum simulation of spin systems.
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 $\ensuremath{\mu}$.
Using free-fermionic techniques we study the entanglement entropy of a block of contiguous spins in a large finite quantum Ising chain in a transverse field, with couplings of different types: homogeneous, periodically modulated and random. We carry out a systematic study of finite-size effects at the quantum critical point, and evaluate subleading corrections both for open and for periodic boundary conditions. For a block corresponding to a half of a finite chain, the position of the maximum of the entropy as a function of the control parameter (e.g. the transverse field) can define the effective critical point in the finite sample. On the basis of homogeneous chains, we demonstrate that the scaling behavior of the entropy near the quantum phase transition is in agreement with the universality hypothesis, and calculate the shift of the effective critical point, which has different scaling behaviors for open and for periodic boundary conditions.
We present a detailed study of the finite one-dimensional quantum Ising chain in a transverse field in the presence of boundary magnetic fields coupled with the order-parameter spin operator. We consider two magnetic fields located at the boundaries of the chain that have the same strength and that are aligned in the same or in the opposite direction. We derive analytic expressions for the gap in all phases for large values of the chain length L , as a function of the boundary field strength. We also investigate the behaviour of the chain in the quantum ferromagnetic phase for oppositely aligned fields, focusing on the magnet-to-kink transition that occurs at a finite value of the magnetic field strength. At this transition we compute analytically the finite-size crossover functions for the gap, the magnetisation profile, the two-point correlation function, and the density of fermionic modes. As the magnet-to-kink transition is equivalent to the wetting transition in two-dimensional classical Ising models, our results provide new analytic predictions for the finite-size behaviour of Ising systems in a strip geometry at this transition.
We consider the behaviour of a critical system in the presence of a gradient perturbation of the couplings. In the direction of the gradient an interface region separates the ordered phase from the disordered one. We develop a scaling theory for the density profiles induced by the gradient perturbation which involves a characteristic length given by the width of the interface region. The scaling predictions are tested in the framework of the mean-field Ginzburg–Landau theory. Then we consider the Ising quantum chain in a linearly varying transverse field which corresponds to the extreme anisotropic limit of a classical two-dimensional Ising model. The quantum Hamiltonian can be diagonalized exactly in the scaling limit where the eigenvalue problem is the same as for the quantum harmonic oscillator. The energy density, the magnetization profile and the two-point correlation function are studied either analytically or by exact numerical calculations. Their scaling behaviour is in agreement with the predictions of the scaling theory.