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arXiv:cond-mat/0312042 · Journal of Physics Condensed Matter

Dynamical structure factors for dimerized spin systems

Matthias A. Müller, H.-J. Mikeska, N. Cavadini

Published Nov 25, 2003• 9 citations
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Abstract

We discuss the transition strength between the disordered ground state and the basic low-lying triplet excitation for interacting dimer materials by presenting theoretical calculations and series expansions as well as inelastic neutron scattering results for the material KCuCl 3 . We describe in detail the features resulting from the presence of two differently oriented dimers per unit cell and show how energies and spectral weights of the resulting two modes are related to each other. We present results from the perturbation expansion in the interdimer interaction strength and thus demonstrate that the wavevector dependence of the simple dimer approximation is modified in higher orders. Explicit results are given in tenth order for dimers coupled in 1D, and in second order for dimers coupled in 3D with application to KCuCl 3 and TlCuCl 3 .

DOI
10.1088/0953-8984/15/49/025

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We discuss the transition strength between the disordered ground state and the basic low-lying triplet excitation for interacting dimer materials by presenting theoretical calculations and series expansions as well as inelastic neutron scattering results for the material KCuCl 3 . We describe in detail the features resulting from the presence of two differently oriented dimers per unit cell and show how energies and spectral weights of the resulting two modes are related to each other. We present results from the perturbation expansion in the interdimer interaction strength and thus demonstrate that the wavevector dependence of the simple dimer approximation is modified in higher orders. Explicit results are given in tenth order for dimers coupled in 1D, and in second order for dimers coupled in 3D with application to KCuCl 3 and TlCuCl 3 .
The subject of the present work are dimerized spin 1/2 systems in one, two and three dimensions.The models are investigated perturbatively by means of cluster expansion.This method makes it possible to study systems even in two and three dimension up to high orders.Starting with a generic HAMILTON-Operator which includes both bilinear and biquadratic interaction terms, the groundstate energies and the low lying excitations of the alternating spin chain und the spin ladder are calculated.Well-known results from BETHE ansatz and bosonization are used to verify the series.Based on this model, two speci£c one-dimensional problems are investigated in more detail: the £rst one is the incommensurate regime of the generalized spin ladder, which was previously analyzed by other approximation techniques.This regime is characterized by a minimum of the dispersion between 0 and π.The second problem deals with a quantum phase transition in the spin ladder with additional cyclic four-spin interaction (ring exchange).The excitation gap which is always present in the regular spin ladder can be reduced by taking the ring exchange into account.Within the region which is accessible by perturbation theory a transition line is presented on which the gap vanishes.For further characterization of the phase transition critical exponents are calculated which give a connection between exchange strength and gap size.Within the range of the higher-dimensional spin dimer systems the compounds KCuCl 3 , TlCuCl 3 (3d) and (C 4 H 12 N 2 )Cu 2 Cl 6 (2d) are studied.To clarify the microscopic interactions, dispersion relations which take all reasonable (i.e.evident from the chemical structure) interactions into account are computed.In the case of KCuCl 3 and TlCuCl 3 there are nine and in the case of (C 4 H 12 N 2 )Cu 2 Cl 6 there are ten spin spin interactions considered.The strength of the exchange parameters is determined by comparison with inelastic neutron scattering data.For the above-mentioned substances the interaction constants are evaluated to match the dispersion relation with the experimental data to a high accuracy.Finally, the dynamical structure factor which is limited to the singlet-triplet transition is calculated perturbatively for the compounds KCuCl 3 and TlCuCl 3 .Since it is proportional to the differential cross section, one is able to establish a relationship to inelastic neutron scattering experiments.
We investigate the elementary excitations of quasi-one-dimensional S = ½ systems built up from zigzag chains with general isotropic exchange constants, using exact (Lanczos) diagonalization for 24 spins and series expansions starting from the decoupled dimer limit. For the ideal one-dimensional zigzag chain we discuss the systematic variation of the basic (magnon) triplet excitation with general exchange parameters and in particular the presence of practically flat dispersions in certain regions of phase space. We extend the dimer expansion in order to include the effects of three-dimensional interactions on the spectra of weakly interacting zigzag chains. In an application to KCuCl 3 we show that this approach allows us to determine the exchange interactions between individual pairs of spins from the spectra as determined in recent neutron scattering experiments.
We study the spin-1/2 Heisenberg chain with alternating nearest neighbor interactions ${J}_{1}(1+\ensuremath{\delta})$ and ${J}_{1}(1\ensuremath{-}\ensuremath{\delta})$ and a uniform second neighbor interaction ${J}_{2}=y(1\ensuremath{-}\ensuremath{\delta})$ by series expansions around the limit of decoupled dimers $(\ensuremath{\delta}=1).$ By extrapolating to $\ensuremath{\delta}=0$ and tuning y, we study the critical point separating the power-law and spontaneously dimerized phases of the spin-1/2 antiferromagnet. We then focus on the disorder line y=0.5,y=0.5,y=0.5, $0<~\ensuremath{\delta}<~1,$ where the ground states are known exactly. We calculate the triplet excitation spectrum, their spectral weights, and wave vector dependent static susceptibility along this line. It is well known that as $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}0,$ the spin gap is still nonzero but the triplets are replaced by spinons as the elementary excitations. We study this dynamical transition by analyzing the series for the spectral weight and the static susceptibility. In particular, we show that the spectral weight for the triplets vanishes and the static spin susceptibility changes from a simple pole at imaginary wave vectors to a branch cut at the transition.
By using the perturbation expansion up to the fifth order, we study the two-triplet-dimer excitation spectra in the Shastry-Sutherland model, where the localized nature of a triplet-dimer, the propagation of a triplet-dimer pair by the correlated hopping and the long-range interactions between triplet-dimers play an essential role. It is found that the dispersion relations for first-neighbor triplet-dimer pair excitations with S=1 and p-type symmetry qualitatively explain the second-lowest branch observed in the neutron inelastic scattering experiment. It is also predicted that the second-lowest branch consists of two components, p_x- and p_y-states, with slightly different excitation energies. The origin of the singlet mode at 3.7meV observed in the Raman scattering experiment is also discussed.
The compound TlCuCl(3) represents a model system of dimerized quantum spins with strong interdimer interactions. We investigate the triplet dispersion as a function of temperature by inelastic neutron scattering experiments on single crystals. By comparison with a number of theoretical approaches we demonstrate that the description of Troyer, Tsunetsugu, and Würtz [Phys. Rev. B 50, 13 515 (1994)10.1103/Phys. Rev. B 50, 13515] provides an appropriate quantum statistical model for dimer spin systems at finite temperatures, where many-body correlations become particularly important.