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arXiv:0810.0828 · Physical Review B

Finite-temperature dynamical structure factor of alternating Heisenberg chains

A. J. A. James, Fabian H. L. Eßler, Robert Konik

Published Sep 16, 2008• 39 citations
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Abstract

We develop a low-temperature expansion for the finite-temperature dynamical structure factor of the spin half Heisenberg chain with alternating nearest-neighbor exchange in the limit of strong alternation of the exchange constants. We determine both the broadening of the low-lying triplet lines and the contribution of the thermally activated intraband scattering.

DOI
10.1103/physrevb.78.094411

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We develop a low-temperature expansion for the finite-temperature dynamical structure factor of the spin half Heisenberg chain with alternating nearest-neighbor exchange in the limit of strong alternation of the exchange constants. We determine both the broadening of the low-lying triplet lines and the contribution of the thermally activated intraband scattering.
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We present results for the dynamic structure factor of the S=1∕2S=1∕2S=1∕2 bond alternating Heisenberg chain over a large range of frequencies and temperatures. Data are obtained from a numerical evaluation of thermal averages based on the calculation of all eigenvalues and eigenfunctions for chains of up to 20 spins. Interpretation is guided by the exact temperature dependence in the noninteracting dimer limit which remains qualitatively valid up to an interdimer exchange $\ensuremath{\lambda}\ensuremath{\approx}0.5$. The temperature induced central peak around zero frequency is clearly identified and aspects of the crossover to spin diffusion in its variation from low to high temperatures are discussed. The one-magnon peak acquires an asymmetric shape with increasing temperature. The two-magnon peak is dominated by the S=1S=1S=1 bound state which remains well defined up to temperatures of the order of JJJ. The variation with temperature and wave vector of the integrated intensity for one-magnon and two-magnon scattering and of the central peak are discussed.
Dynamical structure factors of the S =1 bond-alternating spin chains in the dimer state are calculated at finite temperature using the pair dynamical correlated-effective-field approximation. At T =0, the delta-function-type peak corresponding to the one-magnon mode appears. When temperature increases, such a sharp peak is broadened and the additional weak peak caused by the excitation from the triplet state to the quintet state emerges in the higher energy region. The results are discussed in comparison with those obtained by the exact diagonalization method.
We develop a linked cluster method to calculate the spectral weights of many-particle excitations at zero temperature. The dynamical structure factor is expressed as a sum of ``exclusive'' structure factors, each representing contributions from a given set of excited states. A linked cluster technique to obtain high-order series expansions for these quantities is discussed. We apply these methods to the alternating Heisenberg chain around the dimerized limit $(\ensuremath{\lambda}=0),$ where complete wave vector and frequency-dependent spectral weights for one- and two-particle excitations (continuum and bound states) are obtained. For small to moderate values of the interdimer coupling parameter $\ensuremath{\lambda},$ these lead to extremely accurate calculations of the dynamical structure factors. We also examine the variation of the relative spectral weights of one- and two-particle states with bond alternation all the way up to the limit of the uniform chain $(\ensuremath{\lambda}=1).$ In agreement with Schmidt and Uhrig, we find that the spectral weight is dominated by 2-triplet states even at $\ensuremath{\lambda}=1,$ which implies that a description in terms of triplet-pair excitations remains a good quantitative description of the system even for the uniform chain.
We determine the dynamical structure factor of the two-leg spin-12\frac{1}{2}21​ Heisenberg ladder at low temperatures in the regime of strong rung coupling. The dominant feature at zero temperature is the coherent triplon mode. We show that the line shape of this mode broadens in a nonsymmetric way at finite temperatures and that the degree of asymmetry increases with temperature. We also show that at low frequencies, a temperature-induced resonance akin to the Villain mode in the spin-12\frac{1}{2}21​ Heisenberg Ising chain emerges.