We investigate the quantum corrections to the energy of the static soliton in the
XY-like ferromagnetic spin chain in a symmetry-breaking external field. We apply a semiclassical continuum approximation which allows us to treat the leading-order deviations from idealized planar behavior. The equivalent quantum-field theory, an extension of the quantum sine-Gordon theory, is renormalized and its soliton energy is shown to depend strongly on this deviation. The quantum-field-theoretical renormalization, however, does not affect the energy of the magnetic soliton, which is calculated to leading order in
S1, where
S is the spin length. The sine-Gordon limit is found applicable only for very large anisotropies; for realistic anisotropies, the reduction of the soliton energy owing to quantum effects is enhanced and depends only on
S. A quantitative calculation for the
S=1 system CsNi
F3 is likely to require higher-order terms in
S1.
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.
We investigate the relation between magnon, magnon bound states, and the classical soliton solutions in the isotropic, the anisotropic-exchange, and the easy-axis ferromagnetic Heisenberg chains. The Dyson-Maleev boson representation is used to study the interaction between magnons, and bound states are investigated in terms of two-spin Green's functions. For easy-axis and anisotropic-exchange magnets, a mapping to the Bose gas with attractive $\ensuremath{\delta}$-function interactions is established, and it yields the eigenvalues of the magnon and the
mth bound state in the weak-coupling and continuum limits. The classical limit of the two-spin Green's function reveals that, to leading order in temperature, the bound-state resonance and the associated effects on the two-magnon continuum survives. An important result is a
T2 dependence of the "binding" energy of the bound-state resonance. As a consequence, in a classical description the bound states enter only in order
T2. Finally, we quantize the soliton solutions according to the Bohr-Sommerfeld and de Broglie rules. This approach is found to be exact for the
s=21 isotropic Heisenberg chain and for sufficiently small wave numbers for all
s values. In the anisotropic-exchange and easy-axis models, it agrees with the results obtained from the mapping on the Bose gas, for large quantum numbers and in the easy-axis case for $s\ensuremath{\gg}\frac{1}{2}$ in addition. On this basis, we conclude that the envelope solitons considered here lead to bound-state resonances and associated effects in a classical treatment. Moreover, their semiclassical quantization gives remarkably accurate energy levels.
We study the quantum-statistical mechanics, at low temperatures, of a one-dimensional antiferromagnetic Heisenberg model with two anisotropies. In the weak-coupling limit we determine the temperature dependences of the soliton energy and the soliton density. We have found that the leading correction to the sine-Gordon (SG) expression for the soliton density and the quantum soliton energy comes from the out-of-plane magnon mode, not present in the pure SG model. We also show that when an external magnetic field is applied, the chain supports a new type of kink, where the sublattices rotate in opposite directions.
We suggest a new variational approach for quantum spin chains, based on the stochastic behavior of the quantization axes, which describes a chain in terms of a classical and a quantum contribution to the free energy. For an easy-plane ferromagnetic chain with the external field in the plane it reveals that the nonlinear excitations are predominantly described by the classical degrees of freedom. Hence it offers a fundamental basis for the description of the elementary excitations of such systems in terms of classical solitons and quantum spin waves, as suggested by various experiments.
We consider the determination of the ground state energy and the lowest lying excitations for ferromagnetic Heisenberg spin chains with uniaxial anisotropy of the “easy plane” type in an external magnetic field directly along the anisotropy axis. The ratio of the anisotropy and exchange constants (α) is assumed small, but the size of the site spin S is arbitrary. To first order in α the exact results are shown to differ from the quasiclassical results only by the substitution α→α˜=α[1−(1/2)S]. The asymptotic behavior of the transverse spin correlation function is discussed, and a comparison is made between the theoretical results and recent experiment data.