Anisotropic Heisenberg Chain Research Articles

Excitations in the one-dimensional anisotropic classical Heisenberg chain

The classical equations of motion for the Heisenberg chain with single-site anisotropy are shown to be equivalent to a set of nonlinear wave equations. For a zero-field $\mathrm{XY}$ chain the excitations are linear. This agrees with magnetic specific heat measurements on ${(\mathrm{C}{\mathrm{H}}_{3})}_{4}$NMn${\mathrm{Cl}}_{3}$ (TMMC).

Solitons and two-magnon bound states in the anisotropic Heisenberg chain: Dynamic form factor

The two-magnon bound states for the S = 1 1D-easy-plane ferromagnets in a sufficiently strong longitudinal magnetic field ( H ⩾ 62 kG for CsNiF 3) are considered. It is shown that the two-magnon complexes affect the dynamic form factor in the same manner as the semiclassically quantized solitons.

Comparison of ground state properties for odd half-integer and integer spin antiferromagnetic Heisenberg chains

Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S=1/2, 3/2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S=1/2, 3/2) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.

On the eigenstates of a Heisenberg chain with complex wavenumbers not forming strings

The nature of those Sz=0 excitations of an anisotropic Heisenberg chain that have been found to be typical in the low-energy excited states is considered. A simple estimation is used to take into account the effect of complex phase shifts on the long-distance decay properties of the wavefunction. It is established that pairs of complex wavenumbers that do not form two-strings correspond to pairs of free spins, while quartets described two independent pairs of bound spins.

Anisotropic antiferromagnetic Heisenberg chains

We study anisotropic Heisenberg chains with antiferromagnetic exchange. It is shown that, within field theory, spin-Peierls transition occurs below a certain temperature T 0. Equations for the order parameter and susceptibility are obtained. The effect of a magnetic field on the transition temperature is also discussed.

Spin-peierls transition in anisotropic Heisenberg chains

We study spin-Peierls transition for anisotropic Heisenberg chains within mean field theory. We find that dimerization occurs in the absence of a magnetic field and when the field is smaller than a critical field H c. The transition temperature is obtained as a function of anisotropy and magnetic field.

On solitons in semiclassical limit of the quantum spin1/2Heisenberg chain

Using the exact Goldhirsch (1980) boson expansion of spin 1/2 operators and coherent states representation the author obtains an exact semiclassical equation of motion for the anisotropic Heisenberg chain. Comparison is made with the classical case and with other approaches in the quantum case. The possible contribution to the quasi-elastic neutron scattering cross section is estimated.

Critical properties of diluted anisotropic magnets near percolation threshold

In this paper we study some theoretical aspects of the multicritical behavior of a diluted Heisenberg magnet with Ising-like anisotropy near the percolation threshold. We present three approaches. The first approach is based on the observation that near the percolation threshold the magnetic correlations propagate dominantly along one-dimensional paths. A consequence of this assumption is argued to be that the magnetic correlation function factorizes into two parts, one of which is a purely geometric factor and the other a magnetic correlation function on a chain which appears to be a constrained self-avoiding walk. This simple assumption then helps us understand some salient features of magnetic correlations in diluted systems. Our second approach is based on the scaling theory due to Lubensky and Birgeneau et al. Here we present an analytical, though approximate, calculation of the longitudinal correlation length of an anisotropic Heisenberg chain of classical spins. Our expression is valid in both low- and high-temperature limits and for both low and high anisotropy. This enables us to study the crossover from Heisenberg-like behavior to Ising-like behavior as the percolation point is approached. In the third approach we perform a renormalization-group calculation for a $S=\frac{1}{2}$-spin Heisenberg model. The fixed points and the flows of the recursion relations are studied. The critical surface of the dominant fixed point in the temperature-concentration-anisotropy space is numerically obtained. From this, we calculate the Curie temperature versus concentration curves as the anisotropy is varied. These are found to be similar to those obtained in the scaling approach.

Finite-temperature quantum renormalization-group theory for the one-dimensional anisotropic Heisenberg model

A real-space renormalization-group theory is proposed for the anisotropic Heisenberg chain which yields physically reasonable results for all values of the temperature and of the anisotropy. The method is based on a four-spin cluster and a transformation which preserves the sublattice structure of the antiferromagnet. At low temperatures the transformation maps the four lowest energy levels of the four-spin cluster onto the four levels of two spins, giving ground-state energies in excellent agreement with exact results. The specific-heat exponent for the isotropic antiferromagnet is given correctly, but the exponent for larger transverse coupling is somewhat too small. A scaling form is proposed for the free energy near the isotropic antiferromagnetic fixed point, and the crossover to Ising-like behavior is discussed.

A new method for calculation of the magnetic properties in one- and two-dimensional spin systems with anisotropic Heisenberg exchange

A new approximation method is proposed for the calculation of the magnetic susceptibility of one-dimensional assembly of spins and the critical temperature of two-dimensional one both with the anisotropic Heisenberg exchange. In a linear chain system, every spins are grouped into pairs of adjacent spins (pair-approximation) or clusters of adjacent three spins (( q+1)-approximation), and the partition function of the total spin system is approximated as a sum of products of the partitions functions for the pairs or the clusters. Then the partition function of the anisotropic Heisenberg spin system is shown to reduce into a form of the Ising spin system with modified coupling constants. The exact result for the Ising chain system enables us to obtain an analytical expression for the magnetic susceptibility of anisotropic Heisenberg chain system. The same approximations are also applied to two-dimensional lattices, and the critical temperatures of the square, triangular, and honeycomb lattices with anisotropic Heisenberg exchange are calculated as a function of anisotropy parameter. The results are compared with those of the existing theories and shown to be quite excellent.

Comments on nonlinear excitations in the anisotropic Heisenberg chain

It is suggested that pulse soliton solutions for the classical, continuum, isotropic, ferromagnetic Heisenberg chain in a magnetic field continuously deform to sine Gordon breather solutions as anisotropy is increased to an extremeXY limit (theXY plane containing the applied field). The relevance of this observation for applications to magnetic chains such as CsNiF3 is emphasized, and comparisons with the zero-field limit are noted.

Effect of Nonmagnetic Impurities on the Low-Temperature Susceptibility of the Linear Antiferromagnet (CH3)4NMnCl3 (TMMC)

The magnetic susceptibilities, χ //c and χ ⊥c , for the field parallel and perpendicular to the chain axis are calculated exactly of an anisotropic classical Heisenberg chain diluted randomly with nonmagnetic impurities. These impurities break up completely the host chain into independent segments (finite chains) with various lengths. It is shown that the combined effect of the finite chain and the dipolar anisotropy leads to the crossover in the relative values of χ //c and χ ⊥c from χ //c >χ ⊥c to χ ⊥c >χ //c at sufficiently low temperatures, which has recently been observed in (CH 3 ) 4 NMn 1- x Cd x Cl 3 (TMMC: Cd) and (CH 3 ) 4 NMn 1- x Cu x Cl 3 (TMMC: Cu). Numerical results for the temperature dependence of χ //c and χ ⊥c of these two systems are in good quantitative agreement with the experimental data.

Scaling theory for classical anisotropic Heisenberg chain

The classical anisotropic Heisenberg chain (with Jperpendicular to /J/sub ///= alpha ) is treated by a simple scaling theory valid in the low-temperature limit (K identical to J/sub ////KBT>>1) for any alpha between 0 (Ising) and 1 (isotropic case). The correlation lengths are given in terms of universal single-variable scaling functions Phi /sub //, perpendicular to /: 1/ xi /sub //, perpendicular to /=(cosh-1(1/ alpha )) Phi /sub //, perpendicular to /(K(1- alpha 2)12/+F( alpha )). The term F( alpha ), given in the text, is negligible except when alpha approximately 0. The asymptotic forms of the functions are: Phi /sub ///(x) approximately Phi perpendicular to (x) approximately A/x for x >1 (where A, B and C are constants), giving the characteristic 'Heisenberg-like' and 'Ising-like' behaviours occurring on either side of the crossover which takes place at x approximately 1. Exact critical and crossover exponents are deduced.

Classical equivalents of one-dimensional quantum-mechanical systems

Trotter's formula is used to construct two-dimensional classical systems equivalent to some one-dimensional quantum-mechanical systems of interest. The finite-temperature properties of the completely anisotropic Heisenberg chain are expressed in terms of an eight-vertex model in which the vertex weights depend on the size of the lattice. Knowledge of only the largest eigenvalue of the transfer matrix of the eight-vertex model is not sufficient to find the free energy of the chain except in the limit of zero temperature, when Baxter's result for the ground-state energy is recovered. We also examine two models with two species of variables each, and point out that by constructing the equivalent classical problem the trace over one set of variables can be performed.

One-Dimensional spin model with arbitrary high power terms

Basic spectral and thermodynamical properties of a spin chain (S=1/2) with a longitudinal — transverse anisotropy and nearest-neighbour interaction are considered, assuming the Hamiltonian to be a sum of arbitrary high power terms. After the renormalization of the spectrum, the rigorous solutions of the bilinear X-Y-model turn out to be stable with respect to the arbitrary high power terms. As to the low-excited and magnon-bound states in the spectrum of the system considered, they can be readily obtained from the known results for a “usual” bilinear anisotropic Heisenberg chain.

Thermodynamic properties of extended Hubbard chains

In this contribution we discuss the thermodynamic behaviour of Hubbard chains with an additional site-diagonal interaction. It is shown rigorously that the strong coupling limit is equivalent to a generalized anisotropic Heisenberg system, which in some recently discussed limiting cases degenerates to an Ising orX-Y system. The whole procedure also works for a collection of chains with interchain interaction, thus it is not restricted to strictly one-dimensional systems. The effect of a magnetic field is discussed, and the results are compared with a recent perturbation approach by using numerical results for the anisotropic Heisenberg chain.

Eight-vertex model in lattice statistics and one-dimensional anisotropic heisenberg chain. III. Eigenvectors of the transfer matrix and hamiltonian

We obtain the eigenvectors of the transfer matrix of the zero-field eight vertex model. These are also the eigenvectors of the Hamiltonian of the corresponding one-dimensional anistropic Heisenberg chain.

Eight-vertex model in lattice statistics and one-dimensional anisotropic heisenberg chain. I. Some fundamental eigenvectors

We obtain some simple eigenvectors of the transfer matrix of the zero-field eight-vertex model. These are also eigenvectors of the Hamiltonian of the one-dimensional anisotropic Heisenberg chain. We also obtain new equations for the matrix Q( v) introduced in earlier papers.

Eight-vertex model in lattice statistics and one-dimensional anisotropic heisenberg chain. II. Equivalence to a generalized ice-type lattice model

We establish an equivalence between the zero-field eight-vertex model and an Ising model (with four-spin interaction) in which each spin has L possible values, labeled 1, …, L, and two adjacent spins must differ by one (to modulus L). Such an Ising model can also be thought of as a generalized ice-type model and we will later show that the eigenvectors of the transfer matrix can be obtained by a Bethe-type ansatz.

One-dimensional anisotropic Heisenberg chain

The minimum eigenvalue of the Hamiltonian of an infinitely long one-dimensional anisotropic Heisenberg chain is evaluated.