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arXiv:cond-mat/0010246 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum phase transitions in the Ising model in a spatially modulated field

Parongama Sen

Published Dec 21, 2000• 27 citations
arXivHTMLPDF

Abstract

The phase transitions in the transverse field Ising model in a competing spatially modulated (periodic and oscillatory) longitudinal field are studied numerically. There is a multiphase point in absence of the transverse field where the degeneracy for a longitudinal field of wavelength lambda is [(1+sqrt[5])/2](2N/lambda) for a system with N spins, an exact result obtained from the known result for lambda=2. The phase transitions in the Gamma (transverse field) versus h(0) (amplitude of the longitudinal field) phase diagram are obtained from the vanishing of the mass gap Delta. We find that for all the phase transition points obtained in this way, Delta shows finite size scaling behavior signifying a continuous phase transition everywhere. The values of the critical exponents show that the model belongs to the universality class of the two dimensional Ising model. The longitudinal field is found to have the same scaling behavior as that of the transverse field, which seems to be a unique feature for the competing field. The phase boundaries for two different wavelengths of the modulated field are obtained. Close to the multiphase point at h(c), the phase boundary behaves as (h(c)-h(0))(b), where b is also lambda dependent.

DOI
10.1103/physreve.63.016112

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The phase transitions in the transverse field Ising model in a competing spatially modulated (periodic and oscillatory) longitudinal field are studied numerically. There is a multiphase point in absence of the transverse field where the degeneracy for a longitudinal field of wavelength lambda is (1+sqrt[5])/2 for a system with N spins, an exact result obtained from the known result for lambda=2. The phase transitions in the Gamma (transverse field) versus h(0) (amplitude of the longitudinal field) phase diagram are obtained from the vanishing of the mass gap Delta. We find that for all the phase transition points obtained in this way, Delta shows finite size scaling behavior signifying a continuous phase transition everywhere. The values of the critical exponents show that the model belongs to the universality class of the two dimensional Ising model. The longitudinal field is found to have the same scaling behavior as that of the transverse field, which seems to be a unique feature for the competing field. The phase boundaries for two different wavelengths of the modulated field are obtained. Close to the multiphase point at h(c), the phase boundary behaves as (h(c)-h(0))(b), where b is also lambda dependent.
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.
I investigate the quantum phase transition of the transverse-field quantum Ising model in which nearest neighbors are defined according to the connectivity of scale-free networks. Using a continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, I identify the quantum critical point and study its scaling characteristics. For the degree exponent λ=6, I obtain results that are consistent with the mean-field theory. For λ=4.5 and 4, however, the results suggest that the quantum critical point belongs to a non-mean-field universality class. Further simulations indicate that the quantum critical point remains mean-field-like if λ>5, but it continuously deviates from the mean-field theory as λ becomes smaller.
We have studied the antiferromagnetic Ising chain in a transverse magnetic field hx{h}_{x}hx​ and uniform longitudinal field hz.{h}_{z}.hz​. Using the density-matrix renormalization group calculation combined with a finite-size scaling the ground-state phase diagram in (hx,hz){(h}_{x}{,h}_{z})(hx​,hz​) plane is determined. It is shown that there is an order-disordered transition line in this plane and the critical properties belong to the universality class of the two-dimensional Ising model. Based on the perturbation theory in hz{h}_{z}hz​ the scaling behavior of the mass gap in the vicinity of the critical point (hx=1/2,hz=0){(h}_{x}{=1/2,h}_{z}=0)(hx​=1/2,hz​=0) is established. It is found that the form of the transition line near the classical multicritical point (hx=0,hz=1){(h}_{x}{=0,h}_{z}=1)(hx​=0,hz​=1) is linear. The connection of the considered quantum model with the quasi-one-dimensional classical Ising model in the magnetic field is discussed.
Abstract The interplay of correlated spatial modulation and symmetry breaking leads to quantum critical phenomena intermediate between those of the clean and randomly disordered cases. By performing a detailed analytic and numerical case study of the quasi-periodically (QP) modulated transverse field Ising chain, we provide evidence for the conjectures of reference (Crowley et al 2018 Phys. Rev. Lett. 120 175702) regarding the QP-Ising universality class. In the generic case, we confirm that the logarithmic wandering coefficient w governs both the macroscopic critical exponents and the energy-dependent localisation length of the critical excitations. However, for special values of the phase difference Δ between the exchange and transverse field couplings, the QP-Ising transition has different properties. For Δ = 0, a generalised Aubry–André duality prevents the finite energy excitations from localising despite the presence of logarithmic wandering. For Δ such that the fields and couplings are related by a lattice shift, the wandering coefficient w vanishes. Nonetheless, the presence of small couplings leads to non-trivial exponents and localised excitations. Our results add to the rich menagerie of quantum Ising transitions in the presence of spatial modulation.
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.
A transverse Ising spin system, in the presence of time-dependent longitudinal field, is studied by the effective-field theory (EFT). The effective-field equations of motion of the average magnetization are given for the simple cubic lattice (Z = 6) and the honeycomb lattice (Z = 3). The Liapunov exponent λ is calculated for discussing the stability of the magnetization and it is used to determine the phase boundary. The dynamic phase transition diagrams in h 0 /ZJ — Γ/ZJ plane and in h 0 /ZJ — T/ZJ plane have been drawn, and there is no dynamical tricritical point on the dynamic phase transition boundary. The effect of the thermal fluctuations upon the dynamic phase boundary has been discussed.