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Journal of Applied Physics

Excitation spectrum of ferromagnetic Heisenberg chains

T. Schneider, E. Stoll

Published Mar 1, 1982• 16 citations
Publisher

Abstract

We present a review of the developments leading to our present understanding of the excitation spectrum of planar and uniaxial Heisenberg chains (xxz) as probed by dynamic form factors. Exact zero-temperature results, finite-chain calculations for s = 1/2, and molecular-dynamics results, realistic for s≫1 and T≳0, are discussed, and the role of envelope solitons clarified.

DOI
10.1063/1.330608

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We present a review of the developments leading to our present understanding of the excitation spectrum of planar and uniaxial Heisenberg chains (xxz) as probed by dynamic form factors. Exact zero-temperature results, finite-chain calculations for s = 1/2, and molecular-dynamics results, realistic for s≫1 and T≳0, are discussed, and the role of envelope solitons clarified.
Results are presented and interpreted for the zero- and finite-temperature excitation spectrum of ferromagnetic Ising-Heisenberg chains, as probed by dynamic form factors. Finite-chain calculations demonstrate the occurrence of a thermally induced resonance centered around zero frequency, and a broadening of the magnon and two-magnon bound-state resonances at low TTT. The occurrence of bound states in the associated transitions demonstrates the failure of finite-order magnon perturbation theory.
The excitation spectrum of the ferromagnetic s=12s=\frac{1}{2}s=21​ Ising-Heisenberg chain as probed by dynamic form factors is studied. Exact results are presented for zero temperature, and the modifications introduced by thermally induced transitions are elucidated by finite-chain calculations. A particularly interesting result is the occurrence of a thermally induced resonance around zero frequency for an Ising-like model. Its origin is traced back to transitions involving magnon bound states. The resonance structure of the dynamic form factors associated with a Heisenberg-like system turns out to be less spectacular, because the contributions of different induced transitions nearly overlap on the frequency scale; so that rather broad resonance structures result. Our findings have an important bearing on related models with s>12s>\frac{1}{2}s>21​, such as CsNiF3{\mathrm{F}}_{3}F3​, and on the soliton aspect of the classically treated systems.
We present analytic weak-coupling and numerical finite-chain results for the dynamic form factor associated with the out-of-plane fluctuations at zero temperature and zero field. The dynamic form factor associated with the in-plane fluctuations is treated numerically only. In the ferromagnetic regime, the dominant features are traced back to particle-hole pair continua and the associated bound states. With the approach toward the isotropic ferromagnet, the resonance associated with the bound state of the particle-hole pairs gradually exhausts the spectrum, corresponding in this limit to the magnon peak. In the antiferromagnetic regime, the spectrum is dominated by the particle-hole pair continuum. The analytic results for the dynamic form factor associated with the out-of-plane fluctuations reproduce and extend the well-known properties of the continuum model to leading order in the coupling constant. As expected from the relation between the six-vertex model and the xxz\mathrm{xxz}xxz chain, these properties include singular behavior of dynamic form factors depending continuously on the coupling constant.
The anisotropic XXZXXZXXZ spin-12\frac{1}{2}21​ Heisenberg chain is studied using renormalization-group theory. The specific heats and nearest-neighbor spin-spin correlations are calculated throughout the entire temperature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Ising-like ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found: The in-plane interaction sixsjx+siysjy{s}_{i}^{x}{s}_{j}^{x}+{s}_{i}^{y}{s}_{j}^{y}six​sjx​+siy​sjy​ induces an antiferromagnetic correlation in the out-of-plane siz{s}_{i}^{z}siz​ component, at higher temperatures in the antiferromagnetic XXZXXZXXZ chain, dominantly at low temperatures in the ferromagnetic XXZXXZXXZ chain, and, in-between, at all temperatures in the XYXYXY chain. We find that the converse effect also occurs in the antiferromagnetic XXZXXZXXZ chain: an antiferromagnetic sizsjz{s}_{i}^{z}{s}_{j}^{z}siz​sjz​ interaction induces a correlation in the sixy{s}_{i}^{xy}sixy​ component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or XYXYXY) of anisotropy; and (ii) in complete contrast, in the ferromagnet, the value and temperature of the specific heat peak increase with either type of anisotropy.
The natural explanation of the excitation spectrum of the spin-1 antiferromagnetic Heisenberg chain is given from the viewpoint of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain. The energy spectrum of the latter is calculated with fixed momentum kkk by numerical diagonalization of finite size systems. It consists of a branch of propagating triplet pair (triplet wave) and the continuum of multiple triplet waves for weak ferromagnetic coupling. As the ferromagnetic coupling increases, the triplet wave branch is absorbed in the continuum for small kkk, reproducing the characteristics of the spin-1 antiferromagnetic Heisenberg chain.