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Phase Transitions in the Transverse-field Ising Ladder

A.M. Rigas

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

In this thesis I have successfully recreated the results that describe the behavior of a transverse-field Ising chain. There is a phase transition when the parameter that controls the influence of the transverse field is one (that is: the factor in front of the transverse-field term and the interaction term are equally big). However, I could not determine exactly what the order parameter should be.\nThen for the transverse-field Ising ladder I have used perturbation theory in the parameter that controls the interaction between the chains. I have found that the phase transition doesn't change position, but that the ground state is a different state compared to the transverse-field Ising chain, when the transverse-field term is smaller than the interaction.

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In this thesis I have successfully recreated the results that describe the behavior of a transverse-field Ising chain. There is a phase transition when the parameter that controls the influence of the transverse field is one (that is: the factor in front of the transverse-field term and the interaction term are equally big). However, I could not determine exactly what the order parameter should be.\nThen for the transverse-field Ising ladder I have used perturbation theory in the parameter that controls the interaction between the chains. I have found that the phase transition doesn't change position, but that the ground state is a different state compared to the transverse-field Ising chain, when the transverse-field term is smaller than the interaction.
Phase transitions in Ising models are a broadly studied subject within physics.In this thesis we will study the quantum mechanical transverse-field Ising model in order to determine analytical critical points for quantum phase transitions.This will be done by introducing the concept of fermionic operators and the Jordan-Wigner transformation.Hereafter it is possible to Fourier transform the transverse Ising Hamiltonian and derive an exact eigenenergy spectrum.With these eigenenergies it is possible to derive the condition(s) for which a phase transition can occur.This is done by looking when the eigenenergy solutions are equal to zero.This causes the system to become unstable and thus indicates a point of phase transition.Further in this thesis we will cover two different variations of the transverse-field Ising model which are the di-atomic transverse Ising chain and the frustrated Ising chain.The frustrated Ising chain will be a modification of the mono-atomic Ising chain where there is an extra interaction term introduced which will require an assumption of lowest order approximation of perturbation theory to solve analytically.I would like to thank Lars Fritz for the great learning experience as a physicist within my bachelor thesis.
We examine the ground state properties of the s=12s=\frac{1}{2}s=21​ transverse Ising chain with regularly alternating bonds and fields, using exact analytical results and exact numerical data for long (up to N=900)N=900)N=900) and short (N=20)(N=20)(N=20) chains. For a given period of alternation the system may exhibit a series of quantum phase transitions, where the number of transitions depends on the concrete set of the parameters of the Hamiltonian. The critical behavior for the nonuniform and uniform chains is the same.
This chapter is devoted to a discussion of quantum phase transitions in regularly alternating spin-1/2 Ising chain in a transverse field. After recalling some generally-known topics of the classical (temperature-driven) phase transition theory and some basic concepts of the quantum phase transition theory I pass to the statistical mechanics calculations for a one-dimensional spin-1/2 Ising model in a transverse field, which is the simplest possible system exhibiting the continuous quantum phase transition. The essential tool for these calculations is the Jordan-Wigner fermionization. The latter technique being completed by the continued fraction approach permits to obtain analytically the thermodynamic quantities for a `slightly complicated' model in which the intersite exchange interactions and on-site fields vary regularly along a chain. Rigorous analytical results for the ground-state and thermodynamic quantities, as well as exact numerical data for the spin correlations computed for long chains (up to a few thousand sites) demonstrate how the regularly alternating bonds/fields effect the quantum phase transition. I discuss in detail the case of period 2, swiftly sketch the case of period 3 and finally summarize emphasizing the effects of periodically modulated Hamiltonian parameters on quantum phase transitions in the transverse Ising chain and in some related models.
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 hL{h}_{L}hL​ and hR{h}_{R}hR​, respectively. For $|{h}_{L}|,|{h}_{R}|<\sqrt{1\ensuremath{-}g}$, where ggg is the transverse field and hLhR<0{h}_{L}{h}_{R}<0hL​hR​<0, a first-order phase transition occurs with the changing hL{h}_{L}hL​ or hR{h}_{R}hR​. 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 hLhR<0{h}_{L}{h}_{R}<0hL​hR​<0, there exists a continuous phase transition with changing hL{h}_{L}hL​ 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.
Phase transitions- both in the classical and in the quantum version- are the perfect playground for appreciating universality at work. Indeed, the fine details become unimportant and a classification in very few universality classes is possible. Very recently, a striking deviation from this picture has been discovered: some antiferromagnetic spin chains with competing interactions show a different set of phase transitions depending on the parity of number of spins in the chain. The aim of this article is to demonstrate that the same behavior also characterizes the most simple quantum spin chain: the Ising model in a transverse field. By means of an exact solution based on a Wigner-Jordan transformation, we show that a first order quantum phase transition appears at zero applied field in the odd spin case, while it is not present in the even case. A hint of a possible physical interpretation is given by the combination of two fact: at the point of the phase transition, the degeneracy of the ground state in the even and the odd case substantially differ, being respectively 2 and 2N, with N the number of spins; the spin of the most favorable kink states changes at that point.
We investigate phase transitions in the Ising model and the ANNNI model in transverse field using the interface approach. The exact result of the Ising chain in a transverse field is reproduced. We find that apart from the interfacial energy, there are two other response functions which show simple scaling behavior. For the ANNNI model in a transverse field, the phase diagram can be fully studied in the region where a ferromagnetic to paramagnetic phase transition occurs. The other region can be studied partially; the boundary where the antiphase vanishes can be estimated.