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arXiv:cond-mat/9907431 · Physical review. B, Condensed matter

Three-magnon contribution to the spin correlation function in integer-spin antiferromagnetic chains

Matthew D. P. Horton, Ian K. Affleck

Published Nov 1, 1999• 23 citations
arXivHTMLPDF

Abstract

The exact form factor for the O(3) nonlinear $\ensuremath{\sigma}$ model is used to predict the three-magnon contribution to the spin correlation function, $S(q,\ensuremath{\omega}),$ near wave vector $q=\ensuremath{\pi}$ in an integer spin, one-dimensional antiferromagnet. The three-magnon contribution is extremely broad and extremely weak; the integrated intensity is $<2%$ of the single-magnon contribution.

DOI
10.1103/physrevb.60.11891

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The exact form factor for the O(3) nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model is used to predict the three-magnon contribution to the spin correlation function, S(q,\ensuremathω),S(q,\ensuremath{\omega}),S(q,\ensuremathω), near wave vector q=\ensuremathπq=\ensuremath{\pi}q=\ensuremathπ in an integer spin, one-dimensional antiferromagnet. The three-magnon contribution is extremely broad and extremely weak; the integrated intensity is <2<2%<2 of the single-magnon contribution.
Two quantum spin chains problems are discussed. Firstly, using the exact form factors of the 0(3) nonlinear a model the three-magnon contribution to the spin correlation function of a spin-1 chain is calculated. A very broad peak with a small amplitude is predicted. The integrated intensity of this contribution relative to the single magnon peak is of the order 2%. Experimentally, such a contribution would be difficult, if not impossible, to observe. In the second problem we consider the problem of doping spin-1 magnetic materials with nonmagnetic impurities. In the case where interchain coupling is weak, the nonmagnetic impurities effectively cut decoupled spin chains into finite lengths. We impose free boundary conditions at the chain ends and apply bosonization, mapping the problem onto a torus. Using this formalism we are able to numerically calculate the alternating susceptibility for any reasonable length chain at any temperature. The Neel temperature is then calculated for a three dimensional square lattice as a function of average chain length.
We consider spin-1 Haldane chains with single-ion anisotropy, which exists in known Haldane chain materials. We develop a perturbation theory in terms of anisotropy, where the magnon-magnon interaction is important even in the low-temperature limit. The exact two-particle form factor in the O(3)O(3)O(3) nonlinear sigma model leads to quantitative predictions on several dynamical properties, including the dynamical structure factor and electron-spin-resonance frequency shift. These agree very well with numerical results, and with experimental data on the Haldane chain material Ni(C5{}_{5}5​H14{}_{14}14​N2{}_{2}2​)2{}_{2}2​N3{}_{3}3​(PF6{}_{6}6​).
The z-component dynamical spin correlation function S zz (q,w) of the s= 1 / 2 anisotropic XY chain is calculated exactly in the fermion representation. S zz consists of two parts: one (S m ) represents two fermion excitations and the other (S s ) describes scattering from thermally excited fermions. In the continuum limit, S s has the structure of the solitonic part of a classical sine-Gordon correlation function, but the soliton form factor is modified by quantum effects. S m has similarities with the spin-wave continuum form of S zz an isotropic spin chains. The continuum results are compared with those for a discrete chain.
We study weakly coupled antiferromagnetic spin chains in their ordered phase by combining an exact solution of the single-chain problem with a random-phase-approximation analysis of the interchain interaction. A single chain is described by a quantum Sine-Gordon model and dynamical staggered susceptibilities are determined by employing the form-factor approach to quantum correlation functions. We consider both antiferromagnetic order encountered in quasi-one-dimensional materials such as KCuF3{\mathrm{KCuF}}_{3}KCuF3​ and spin-Peierls order as found in CuGeO3.{\mathrm{CuGeO}}_{3}.CuGeO3​.
We analyze the three-particle scattering continuum in quasi-one-dimensional integer spin Heisenberg antiferromagnets within a low-energy effective field theory framework. We exactly determine the zero-temperature dynamical structure factor in the O(3) nonlinear \ensuremath{\sigma} model and in Tsvelik's Majorana fermion theory. We study the effects of interchain coupling in a random phase approximation. We discuss the application of our results to recent neutron-scattering experiments on the spin-1 Haldane-gap material CsNiCl3.{\mathrm{CsNiCl}}_{3}.CsNiCl3​.