The derivation by Wysin and Bishop [Phys. Rev. B 42, 810 (1990)] of a general result for the dynamic structure function, S(q,t), of classical isotropic Heisenberg systems is shown to be incorrect. A correct quantum-mechanical proof of a general theorem, which is valid for all translationally invariant systems and which produces the result of Wysin and Bishop as a special classical case, is provided. It is also pointed out that another condition on S(q,t) for rotationally invariant systems such as the isotropic Heisenberg model does not appear to be satisfied by the results of Wysin and Bishop.
Brown [Phys. Rev. B 45, 10 809 (1991)] points out an error made in an article on the dynamics of a Heisenberg model by Wysin and Bishop [Phys. Rev. B 42, 810 (1990)]. After correcting the error, he arrives at the conclusion that the dynamic structure in this model cannot be in the form of Lorentzians. We show that this conclusion represents an orthogonality condition of certain basis vectors that span Hilbert space. Hence, it is applicable to all Hermitian many-body models. Also, a simple model is provided to illustrate both the orthogonality and transformation ideas. They can be very useful in the dynamical studies of Hermitian many-particle systems.
A theoretical approach to the one-dimensional isotropic Heisenberg model at low temperatures is presented which uses rotationally invariant combinations of spin operators only and is thus adapted to treat correctly effects resulting from the absence of long-range order in these systems. Analytic expressions for space- and time-dependent spin correlation functions are given for small values of
JS2T and
S1. The results include an expression for the spin correlation function in the antiferromagnetic ground state and a discussion of changes in line shape, due to quantum effects, for short-wavelength magnons as observed in neutron scattering experiments.
The usual spin-wave theory, when applied to the calculation of rotationally invariant averages in the Heisenberg model, permits exact calculation of static and dynamical correlation functions in one dimension. The results are also believed to be exact in two dimensions, although this has not been show rigorously here. The one-dimensional ferromagnet in the hydrodynamic regime and the two-dimensional ferromagnet at all wave vectors do not satisfy the dynamical scaling hypothesis. There is no characteristic exponent in these cases. The present results contradict the prediction of the mode-coupling theory.
We revisit our initial derivations of the exact 2-spinon
S2 and 4-spinon
S4 dynamical structure factors (DSF). First we show that the latter derivations had normalization factors that are twice and quadriple the correct ones respectively. This means that
S2 contributes not 72% as was previously thought but 36% to the total DSF. We also calculated the contribution of
S4 to be between 18% and 20% and not 27% as was calculated by Caux and Hagemans. In fact we show that the latter reference had also the normalization factor twice the correct value and had it done the numeric integrations correctly it would have obtained a contribution between 36% and 40% for
S4. Furthermore, we prove that its claim that our initial derivation of
S4 was also incorrect in its dependency on the spectral parameters is incorrect because fixing the momentun transfer
k up to
2π as the latter reference did to justify its claim only amounts to mutltiplying the overall factor by 2 because as we will prove in this paper
S4 is periodic in
k with period
2π. Also in this paper we derive
Sn for general
n by following a different approach compared to our initial derivation of
S4. Although for
S4 both the new derivation and the initial one lead to equivalent formulas that are expressed differently, the new form presented in this paper is much more elegant and compact and also reveals new hidden and nontrivial symmetries which substantially simplify the numeric evaluation of
S4 and its sum rules. Moreover based on the results of this paper we propose a simple approximation to the total DSF of the Heisenberg model. Finally we comment on how our work might resolve the discrepancy between the exact theoretic results and experiemntal data as reported by Zaliznyak et al.
We consider a “generalized Heisenberg model”, with a Hamiltonian given by ℋ︁(D) = –JΣij S(D)j, where the quantities Si are isotropically-interacting D-dimensional classical spins and the summation is restricted to nearest-neighbor pairs of sites <i,j>. Thus when D = 1, 2, and 3, this model reduces to the S = 1/2 Ising, plane rotator, and classical Heisenberg models which are useful for describing phase transitions in, respectively, liquid-gas systems, Bose fluids, and magnetic materials. Unfortunately this Hamiltonian is exactly soluble only in the limit D → ∞, in which case the solution corresponds to the spherical model of Berlin and Kac. For finite D, we must consider approximation procedures in order to obtain any useful information. In this paper we develop a diagrammatic representation, valid for arbitrary D, for successive terms in the high-temperature expansions of the free energy and the spin-spin correlation function—from which we can obtain series expansions for all the thermodynamic properties of the system.