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Journal of Applied Physics

Solitons in the antiferromagnetic X Y chain

M. Staudinger, R. W. Gerling, Challa S. S. Murty, David Paul Landau

Published Apr 15, 1985• 17 citations
Publisher

Abstract

An important sampling Monte Carlo method is used to study the classical, antiferromagnetic, XY chain in a symmetry breaking uniform magnetic field. From equilibrium configurations we extract values for the bulk properties as well as explicit pictures showing individual spin directions. An analysis of the angles which the spin makes with the field shows clearly the presence of both ‘‘π-like’’ and ‘‘2π-like’’ solitons. We also see two distinct types of ‘‘π-like’’ solitons which differ in their behavior as a function of field. The bulk properties are discussed in terms of both spin waves and soliton excitations.

DOI
10.1063/1.335087

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A variational estimate of the free energy of a quantum spin chain based on the stochastic behavior of the quantization axes is proposed. It allows a description of the nonlinear excitations and quantum spin waves within the same framework. For an easy-plane ferromagnet with an external field in the plane it reveals that the nonlinear excitations (solitons) are predominantly described by the classical degrees of freedom. Considerable effort has been devoted to the theoretical description1 and experimental verification of the existence of nonlinear excitations. In this field magnetic chain systems have played an important part.2 Chain systems with an easy-plane anisotropy, such as (C6 H11 NH3 )CuBr3 (CHAB) and CsNiF3 , which are subjected to a symmetry breaking field, were found to be soliton bearing.3 Theoretically the description of the nonlinear effect is based on the mapping of the spin dynamics on a sine-Gordon model. Further research4 indicated that the interpretation of the thermodynamical properties of real systems in terms of the sine-Gordon model required however quantum and anisotropy corrections. Since the dipolar interactions between the spins in the chain are considered to be negligible, the direction of the spin chain in configuration space with respect to any preferred direction in spin space does not enter explicitly in the Hamiltonian. This makes it interesting to define a site-dependent z direction in spin space which makes an angle (Θn ,φn ) with the z direction in configuration space. The model Hamiltonian can then be written in terms of the spin operators defined with respect to the site-dependent quantization axis. The representation with equal quantization axis for all the sites is related to the representation with different quantization axes at each site by a rotation in spin space. The unitary transformation that describes the rotation contains the angles (Θn ,φn ), which we have treated as random variables. Based on this observation we have proposed a new variational approach for the quantum spin chain. We will outline the main results here. A full account is published elsewhere.5 In our approach the free energy of the system can be separated in a quantum part and a classical part. The free energy F of the chain is then approximated by an upper bound F<Fcl +Fqm , where βFcl =−ln[E(exp−βVcl )] and βFqm =−ln Tr(exp−β〈Hqm 〉). The first part is a configurational classical free energy, depending on the choice of the probability distribution for the quantization axes via the expectation E. The second contribution is the free energy of a modified quantum chain, whose parameters are classical averages and can be obtained from the configurational classical free energy. For easy-plane ferromagnetic chains, the nonlinear excitations are comprised in the classical part, whereas the quantum part contains predominantly linear excitations. For those thermodynamical quantities that are insensitive to the contribution of the quantum excitations, the chain is thermodynamically equivalent with a classical sine-Gordon system. The out-of-plane corrections are incorporated in the quantum part of the free energy. This behavior is corroborated experimentally by measurements of the excess heat capacity of CHAB for different magnetic fields with the same in-plane component.6 The validity of our approach is, however, not restricted to easy-plane ferromagnetic chains. Also for more complex systems with, for example, an orthorhombic or helical symmetry, the introduction of stochastic quantization axes can shed new light on some of the thermodynamical properties of these quantum systems in terms of their classical equivalent.

Journal of Applied Physics•May 1, 1990•0 citations
An important sampling Monte Carlo method is used to study the classical, antiferromagnetic, XY chain in a symmetry breaking uniform magnetic field. From equilibrium configurations we extract values for the bulk properties as well as explicit pictures showing individual spin directions. An analysis of the angles which the spin makes with the field shows clearly the presence of both ‘‘π-like’’ and ‘‘2π-like’’ solitons. We also see two distinct types of ‘‘π-like’’ solitons which differ in their behavior as a function of field. The bulk properties are discussed in terms of both spin waves and soliton excitations.
The time-dependent behavior of the classical, ferromagnetic xy chain in a symmetry-breaking magnetic field was studied using a very fast, vectorized spin-dynamics method. The equations of motion were integrated using starting configurations determined by Monte Carlo simulations. By calculating spin-spin correlation functions and taking space-time Fourier transforms, we determined S(q,\ensuremath{\omega}) for a wide range of fields and temperatures. We identify and measure the characteristics of single spin-wave peaks and two spin-wave sum peaks, as well as central peaks due to both soliton and two spin-wave difference processes. These results are discussed in light of existing theory and experiments.
We have studied the time-dependent behavior of the classical XY chain in a symmetry breaking magnetic field using an ultrafast spin-dynamics method in which we integrate the equations of motion using starting configurations determined by Monte Carlo simulations. We calculated the three polarizations of S(q,ω) separately for temperatures from 0.1 to 1.0 J/kB and fields up to 1.5 J. The soliton contribution, the single-magnon peaks, and both the sum and difference two-magnon contributions are separated; we discuss them in detail and compare them with theoretical predictions.
We have investigated the time-dependent behavior of the classical XY chain in a symmetry-breaking anisotropy field using an ultrafast spin-dynamics method in which we integrate the equation of motion with equilibrium spin configurations determined by Monte Carlo simulations. All three polarizations of the scattering function S(q,\ensuremath{\omega}) were determined for temperature kB{\mathit{k}}_{\mathit{B}}kB​T=0.2J and fields varying from 0.1J to 0.6J. Substantial differences are found as compared with the isotropic ferromagnetic chain in a transverse magnetic field. The soliton contribution, single-spin-wave peaks, and both the sum and difference two-spin-wave contributions are separated. Half-widths and integrated intensities are extracted and compared with theoretical predictions based on mapping onto the sine-Gordon model.
The spin configurations of a one-dimensional, weakly anisotropic Heisenberg antiferromagnet were examined by means of an importance sampling Monte Carlo simulation. The spin interactions display weak anisotropies of the same form in each of the orthogonal XY and XZ planes. This allows for two soliton modes, with one in each of the planes. The soliton density within the XZ plane has been determined as a function of the relative strengths of the two anisotropies as well as of temperature. These results are compared with theoretical analysis of similar systems.
We have studied two models for anisotropic quantum spin chains. (i) XY-chain with a field in the plane: The magnetization of the ferromagnet behaves as h1/3 for small fields, in agreement with scaling laws. The antiferromagnet shows a critical field at which the ground state is a simple Néel state and which separates power law from exponential decay of spatial correlations. (ii) Anisotropic XY-chain: The dynamic z-component spin correlation function can be decomposed into a spin wave and a soliton contribution. The nature of quantum soliton excitations is studied and their form compared to soliton solutions of classical equations of motion.
A variational estimate of the free energy of a quantum spin chain based on the stochastic behavior of the quantization axes is proposed. It allows a description of the nonlinear excitations and quantum spin waves within the same framework. For an easy-plane ferromagnet with an external field in the plane it reveals that the nonlinear excitations (solitons) are predominantly described by the classical degrees of freedom. Considerable effort has been devoted to the theoretical description1 and experimental verification of the existence of nonlinear excitations. In this field magnetic chain systems have played an important part.2 Chain systems with an easy-plane anisotropy, such as (C6 H11 NH3 )CuBr3 (CHAB) and CsNiF3 , which are subjected to a symmetry breaking field, were found to be soliton bearing.3 Theoretically the description of the nonlinear effect is based on the mapping of the spin dynamics on a sine-Gordon model. Further research4 indicated that the interpretation of the thermodynamical properties of real systems in terms of the sine-Gordon model required however quantum and anisotropy corrections. Since the dipolar interactions between the spins in the chain are considered to be negligible, the direction of the spin chain in configuration space with respect to any preferred direction in spin space does not enter explicitly in the Hamiltonian. This makes it interesting to define a site-dependent z direction in spin space which makes an angle (Θn ,φn ) with the z direction in configuration space. The model Hamiltonian can then be written in terms of the spin operators defined with respect to the site-dependent quantization axis. The representation with equal quantization axis for all the sites is related to the representation with different quantization axes at each site by a rotation in spin space. The unitary transformation that describes the rotation contains the angles (Θn ,φn ), which we have treated as random variables. Based on this observation we have proposed a new variational approach for the quantum spin chain. We will outline the main results here. A full account is published elsewhere.5 In our approach the free energy of the system can be separated in a quantum part and a classical part. The free energy F of the chain is then approximated by an upper bound F<Fcl +Fqm , where βFcl =−ln[E(exp−βVcl )] and βFqm =−ln Tr(exp−β〈Hqm 〉). The first part is a configurational classical free energy, depending on the choice of the probability distribution for the quantization axes via the expectation E. The second contribution is the free energy of a modified quantum chain, whose parameters are classical averages and can be obtained from the configurational classical free energy. For easy-plane ferromagnetic chains, the nonlinear excitations are comprised in the classical part, whereas the quantum part contains predominantly linear excitations. For those thermodynamical quantities that are insensitive to the contribution of the quantum excitations, the chain is thermodynamically equivalent with a classical sine-Gordon system. The out-of-plane corrections are incorporated in the quantum part of the free energy. This behavior is corroborated experimentally by measurements of the excess heat capacity of CHAB for different magnetic fields with the same in-plane component.6 The validity of our approach is, however, not restricted to easy-plane ferromagnetic chains. Also for more complex systems with, for example, an orthorhombic or helical symmetry, the introduction of stochastic quantization axes can shed new light on some of the thermodynamical properties of these quantum systems in terms of their classical equivalent.