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arXiv:cond-mat/0602387 · Physical Review B

Finite-temperature dynamics of the spin- 12 bond alternating Heisenberg antiferromagnetic chain

H.-J. Mikeska, C. Luckmann

Published May 19, 2006• 19 citations
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Abstract

We present results for the dynamic structure factor of the $S=1∕2$ 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 $S=1$ bound state which remains well defined up to temperatures of the order of $J$. 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.

DOI
10.1103/physrevb.73.184426

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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 \ensuremathλ\ensuremath≈0.5\ensuremath{\lambda}\ensuremath{\approx}0.5\ensuremathλ\ensuremath≈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 calculate the dynamical structural factor of the S=1S=1S=1 bond-alternating Heisenberg chain. In the Haldane phase, the lowest excited states form the lower edge of the multimagnon continuum in 0< q< qc0<~q<~{q}_{c}0< q< qc​ and the one-magnon mode in ${q}_{c}<~q<~\ensuremath{\pi}.$ As the system approaches the gapless point, qc{q}_{c}qc​ shifts towards $q=\ensuremath{\pi}$ and the largest integrated intensity of the one-magnon mode is decreased. In the singlet-dimer phase, the one-magnon mode appears in 0< q< qc.0<~q<~{q}_{c}.0< q< qc​. As the bond-alternation becomes strong, qc{q}_{c}qc​ shifts towards $q=\ensuremath{\pi}.$ In the antiferromagnetic-ferromagnetic bond-alternation region with a strong ferromagnetic coupling, the lowest excited states form the lower edge of the multimagnon continuum in $0<~q<~0.2\ensuremath{\pi}$ and $0.8\ensuremath{\pi}<~q<~\ensuremath{\pi},$ and the one-magnon mode appears in $0.2\ensuremath{\pi}<q<0.8\ensuremath{\pi}.$ The largest integrated intensity of the one-magnon mode is 9393%,93 which is slightly smaller than that in the S=1S=1S=1 Haldane-gap system. We further discuss the dynamical structural factor in connection with the inelastic neutron-scattering experiments.
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.
The thermal evolution of equal-time spin correlations in the antiferromagnetic isotropic spin-1/2 Heisenberg chain is examined by full exact diagonalization of a ten-site periodic cluster. The calculation retains the complete spectrum and evaluates canonical ensemble averages without decoupling approximations. Correlations through the maximum independent separation are reported over 0.10 ≤ kBT/J ≤ 5. The low-temperature state displays the expected alternating antiferromagnetic pattern and appreciable correlations across the finite ring. Heating suppresses the magnitude of every nonlocal correlator, with longer-range terms disappearing first. The nearest-neighbour correlation approaches the high-temperature limit continuously, while an effective finite-cluster correlation length falls rapidly around kBT/J of order unity. Low- and high-temperature consistency checks, SU (2) symmetry, and comparison with Bethe-ansatz expectations are used to validate the computation. The results provide a compact reproducible reference for interpreting thermal loss of short-range order, while clearly separating finite-size numerical evidence from thermodynamic-limit conclusions.
The large-scale dynamics of quantum integrable systems is often dominated by ballistic modes due to the existence of stable quasiparticles. We here consider as an archetypical example for such a system the spin-12\frac{1}{2}21​ XXX Heisenberg chain that features magnons and their bound states. An interesting question, which we here investigate numerically, arises with respect to the fate of ballistic modes at finite temperatures in the limit of zero magnetization m=0m=0m=0. At a finite magnetization density mmm, the spin-autocorrelation function $\mathrm{\ensuremath{\Pi}}(x,t)$ (at high temperatures) typically exhibits a trimodal behavior with left- and right-moving quasiparticle modes and a broad center peak with slower dynamics. The broadening of the fastest propagating modes exhibits a subdiffusive t1/3{t}^{1/3}t1/3 scaling at large magnetization densities $m\ensuremath{\rightarrow}\frac{1}{2}$, familiar from noninteracting models; it crosses over into a diffusive scaling t1/2{t}^{1/2}t1/2 upon decreasing the magnetization to smaller values. The behavior of the center peak appears to exhibit a crossover from transient superdiffusion to ballistic relaxation at long times. In the limit $m\ensuremath{\rightarrow}0$, the weight carried by the propagating peaks tends to zero; the residual dynamics is carried only by the central peak; it is sub-ballistic and characterized by a dynamical exponent zzz close to the value 32\frac{3}{2}23​ familiar from Kardar-Parisi-Zhang (KPZ) scaling. We confirm that, employing elaborate finite-time extrapolations, that the spatial scaling of the correlator $\mathrm{\ensuremath{\Pi}}$ is in excellent agreement with KPZ-type behavior and analyze the corresponding corrections.
The antiferromagnetic spin-one chain is considerably one of the most fundamental quantum many-body systems, with symmetry-protected topological order in the ground state. Here, we present results for its dynamical spin structure factor at finite temperatures, based on a combination of exact numerical diagonalization, matrix-product-state calculations, and quantum Monte Carlo simulations. Open finite chains exhibit a subgap band in the thermal spectral functions, indicative of localized edge states. Moreover, we observe the thermal activation of a distinct low-energy continuum contribution to the spin spectral function with an enhanced spectral weight at low momenta and its upper threshold. This emerging thermal spectral feature of the Haldane spin-one chain is shown to result from intraband magnon scattering due to the thermal population of the single-magnon branch, which features a large bandwidth-to-gap ratio. These findings are discussed with respect to possible future studies on spin-one chain compounds based on inelastic neutron scattering.