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Journal of Physics C Solid State Physics

Quantum versus classical domain walls in the Ising chain with a transverse field

Peter Prelovšek, Igor Sega

Published Aug 10, 1981• 3 citations
Publisher

Abstract

Within the classical and quantum S= 1 / 2 versions of the one-dimensional Ising model in a transverse field non-linear domain-wall-like excitations are studied. For the quantum case a domain wall (DW) wavefunction is constructed from the first (improper) excitation band. Various one-DW expectation values are then calculated. In particular, it is shown that the proposed DW wavefunction indeed yields the expected stationary-wall-like profile for the (S l z ) expectation value. Results are compared with the ones obtained for the classical model and used to analyse the quantum-classical correspondence of domain walls in both models. The extension to non-stationary quantum DW is briefly discussed.

DOI
10.1088/0022-3719/14/22/017

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Within the classical and quantum S= 1 / 2 versions of the one-dimensional Ising model in a transverse field non-linear domain-wall-like excitations are studied. For the quantum case a domain wall (DW) wavefunction is constructed from the first (improper) excitation band. Various one-DW expectation values are then calculated. In particular, it is shown that the proposed DW wavefunction indeed yields the expected stationary-wall-like profile for the (S l z ) expectation value. Results are compared with the ones obtained for the classical model and used to analyse the quantum-classical correspondence of domain walls in both models. The extension to non-stationary quantum DW is briefly discussed.
We discuss several problems concerning domain walls in the spin-S Ising model at zero temeprature in a magnetic field, H/(2S), applied in the x direction. Some results are also given for the planar (y-z) model in a transverse field. We treat the quantum problem in one dimension by perturbation theory at small H and numerically over a large range of H. We obtain the spin-density profile by fixing the spins at opposite ends of the chain to have opposite signs of Sz{\mathit{S}}_{\mathit{z}}Sz​. One dimensional is special in that there the quantum width of the wall is proportional to the size L of the system. We also study the quantitative features of the ``particle'' band which extends up to energies of order H above the ground state. Except for the planar limit, this particle band is well separated from excitations having energy J/S involving creation of more walls. At large S this particle band develops energy gaps and the lowest subband has tunnel splittings of order H${2}^{1\mathrm{\ensuremath{-}}2\mathit{S}}$. This scale of of energy gives rise to anomalous scaling with respect to (a) finite size, (b) temperature, or (c) random potentials. The intrinsic width of the domain wall and the pinning energy are also defined and calculated in certain limiting cases. The general conclusion is that quantum effects prevent the wall from being sharp and in higher dimension would prevent sudden excursions in the configuration of the wall.
We investigate analytically and numerically the dynamics of domain walls in a spin chain with ferromagnetic Ising interaction and subject to an external magnetic field perpendicular to the easy magnetization axis (transverse field Ising model). The analytical results obtained within the continuum approximation and numerical simulations performed for the discrete classical model are used to analyze the quantum properties of domain walls using the semiclassical approximation. We show that the domain wall spectrum shows a band structure consisting of 2S2S2S nonintersecting zones.
A variational calculation yields an eigenstate representing a domain wall-type solution for various quantum-mechanical spin models. Heisenberg Hamiltonians with local and exchange anisotropy, and an Ising model in a transverse field (including its critical regime) are discussed in detail. The domain walls are viewed as localized solutions of non-linear equations. The extent to which the results for the transverse field Ising model may be mapped on to a currently interesting “double well” anhurmonic lattice problem are analysed.
We study effects of a transverse magnetic field on the properties of the one-dimensional spin-1/2 Ising-like antiferromagnet. In order to emphasize the relevant role of domain walls between opposite antiferromagnetic domains, the spin Hamiltonian is exactly mapped to the interacting fermion Hamiltonian, via the dual transformation as well as the Jordan-Wigner transformation. Within the generalized Hartree-Fock approximation, we find that the excitation spectrum exhibits characteristic features, which are explained as a result of the different roles of the transverse field and of the transverse components of the exchange interactions in the propagation of the domain wall. At a critical field where the excitation gap vanishes, a ground-state phase transition is found to take place in the same manner as that in the transverse Ising model. In addition, we discuss the relevance of our magnetization curve with regard to experimental data in CsCoCl 3 .
The possibility of the creation of the bound state of a domain wall and surrounding lattice distortion was investigated in the framework of the one-dimensional Ising model with transverse field. It was found that the existence of such an entity is very sensitive to the mutual ratio of the basic physical parameters of the system. The authors obtain that, in the weak-coupling regime, such a 'dressed' excitation can arise in the so-called unadiabatic limit when the maximal photon energy greatly exceeds the nearest-neighbour exchange energy. In that case, the 'dressing' effect can significantly modify domain-wall properties, causing the increase of its effective mass and lowering of the ground-state energy, while the domain-wall velocity cannot exceed the speed of sound.
Dynamics and the approximate thermodynamical properties of the classical Ising model in a transverse field (CIMTF) are studied. This is done by elucidating a close correspondence of the CIMTF to a nonlinear lattice field in which each atom, while coupled harmonically with its neighbours, sits on on-site anharmonic potential. It is shown that when the ratio of the Ising interaction to a transverse field-strength exceeds a certain critical value, the on-site potential is of double-well type with saturable nonlinearity. Domain wall modes and magnon modes in continuous one-dimensional CIMTF are studied for both systems to substantiate the correspondence. By exploiting this, a brief study is made on the approximate properties of the free energy and of the dynamical structure factor S ( k ,ω) of longitudinal spin fluctuations of the CIMTF. These quantities are shown to be composed of two contributions, one from domain-wall modes and the other from magnon modes, which manifest themselves as a central peak and side peaks in S ( k ,ω), respectively.