Dynamical Correlation Functions for Linear Spin Chains

Dynamical spin correlation functions are calculated numerically for cyclic linear Heisenberg chains contain- ing up to 10 spins with S = 12 and S = 1 . We consider ferro- and antiferromagnets including single-site and exchange anisotropies. The results agree well with the neutron scattering cross sections on quasi one- dimensional systems.


Gerhard Müller and Hans Beck
Institut für Physik, Universität Basel, CH-4056 Basel, Switzerland Dynamical spin correlation functions are calculated numerically for cyclic linear Heisenberg chains containing up to 10 spins with S = 1 2 and S = 1. We consider ferro-and antiferromagnets including single-site and exchange anisotropies. The results agree well with the neutron scattering cross sections on quasi onedimensional systems.
The properties of quasi one-dimensional magnetic materials have recently been reviewed [1]. Some prominent examples are: TMMC (S = 5 2 Heisenberg antiferromagnet (HB AF)), CPC (S = 1 2 HB AF), CsNiF 3 (S = 1 planar HB Ferromagnet (FM)). The dynamics of such weakly coupled spin chains is investigated by neutron scattering. The experimental results show rather well defined spinwave peaks at low temperatures. Unfortunately, a rigorous theoretical treatment of the dynamics of HB chains is impossible. Thus besides various analytical approaches (see [1]), some authors have evaluated the dynamical spin correlation functions numerically by diagonalizing the Hamiltonian of finite chains. Richards and Carboni [2] demonstrated the existence of spin-wave peaks at low T for isotropic HB AF S = 1 2 chains. The purpose of this work is to extend these calculations to various anisotropic systems and to S > 1 2 . We treat the Hamiltonian for a chain of N sites with periodic boundary conditions. The eigenfunctions of (1) can be classified by S T z (z-component of total spin) and a k-vector (k = n2π/N, n = 0, . . . , N − 1). Using the eigenvalues E λ and eigenvectors |λ we evaluate For finite systems these functions are best represented, for fixed q, as histograms in frequency space. In the following we describe our main results for various cases: (i) Isotropic HB AF. In agreement with [2] we obtain Gaussian line shapes (spin diffusion) for T → ∞ and spin-wave peaks for low T . These peaks are predominantly produced by matrix elements between the ground state, which has K 0 = 0 or K 0 = π depending on N , and the lowest eigenstates with wave vector q + k 0 . The latter were determined exactly by Des Cloiseaux and Pearson (DP), see [1], for infinite chains. However, even at T = 0, states with higher energies also contribute in agreement with theoretical considerations by Hohenberg and Brinkman [3].
(ii) Isotropic HB FM. Here, at T = 0, the spin-wave peaks are sharp. All nonzero matrix elements, i.e. those between each of the degenerate ground states and the corresponding spin-wave states, contribute to G αα at the same frequency. For finite, but low, T additional contributions arise from spin-wave bound states, which, at least for small q, again contribute at frequencies close to the T = 0 spin-wave frequency. Therefore, for low T , the peak is narrower for a FM than for an AF chain.
(iii) HB FM with anisotropic exchange (α < β, γ = 0). For α = β the lowering of the symmetry partially lifts the degeneracies of the isotropic HB chain: the energies depend on |S T z |, and G xx and G zz are no more identical. Due to selection rules, only states with the same S T z are connected for G zz . However, these states are all affected in a similar way by the anisotropy. The matrix elements for G xx are those with ∆S T z = ±1, i.e. between states that are shifted differently by anisotropy. Thus the peak of G zz is narrower than the one of G xx for α < β. In the extreme case α = 0 (XY -chain) G zz has one sharp peak at T = 0 and the smallest q (= 2π/N ), whereas for larger wave-vectors several peaks appear. G xx shows a broad 'background' accompanying the main peak, which is due to the one-fermion states in the treatment of Lieb, Schultz and Mattis (LSM), see [1].

Figure 1.
In-plane (Gxx) and out-of-plane (Gzz) correlation function at q = π/3 for the planar HB FM S = 1 chain of 6 particles. The value γ = 0.212J for the anisotropy is appropriate for CsNiF3, [4] and q is close to qz = 0.35π used in neutron scattering [4]. The three temperatures correspond to those of ref. (iv) Planar HB FM (α = β, γ > 0. This model is appropriate for CsNiF 3 [I, 4]. Histograms of G xx and G zz are shown in fig. 1 for q = π/3 and various T . Our results are in good qualitative agreement with neutron scattering data. The main peak of G zz is narrow and decreases rather rapidly with rising T , without shifting appreciably in energy. In contrast G xx shows a broader shape. Its width and intensity both increase with growing T . The energies of the lowest states connected with the ground state by S x (q) and S z (q) follow closely the dispersion relation given by Villain [1,4]. The local anisotropy (γ > 0) splits the degenerate eigenvalues of the isotropic system in a way similar to the case α = β described before. Thus the rather distinct behaviour of G zz and G xx is again due to the shifts produced by the (single-site) anisotropy and the S T z selection rules. More details will be published elsewhere. We have used a modified cmpj.sty style file.