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arXiv:cond-mat/9801224 · Physical review. B, Condensed matter

S=12 alternating chain using multiprecision methods

Ted Barnes, José A. Riera, D. M. Tennant

Published May 1, 1999• 105 citations
arXivHTMLPDF

Abstract

In this paper we present results for the ground state and low-lying excitations of the $S=1/2$ alternating Heisenberg antiferromagnetic chain. Our more conventional techniques include perturbation theory about the dimer limit and numerical diagonalization of systems of up to 28 spins. An application of multiple precision numerical diagonalization allows us to determine analytical perturbation series to high order; the results found using this approach include ninth-order perturbation series for the ground state energy and one magnon gap. We also determine the fifth-order dispersion relation and third-order exclusive neutron scattering structure factor for one-magnon modes and numerical and analytical binding energies of $S=0$ and $S=1$ two-magnon bound states.

DOI
10.1103/physrevb.59.11384

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In this paper we present results for the ground state and low-lying excitations of the S=1/2S=1/2S=1/2 alternating Heisenberg antiferromagnetic chain. Our more conventional techniques include perturbation theory about the dimer limit and numerical diagonalization of systems of up to 28 spins. An application of multiple precision numerical diagonalization allows us to determine analytical perturbation series to high order; the results found using this approach include ninth-order perturbation series for the ground state energy and one magnon gap. We also determine the fifth-order dispersion relation and third-order exclusive neutron scattering structure factor for one-magnon modes and numerical and analytical binding energies of S=0S=0S=0 and S=1S=1S=1 two-magnon bound states.
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0{\mathit{e}}_{0}e0​\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
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.
We numerically investigate elementary excitations of the Heisenberg alternating-spin chains with two kinds of spins, 1 and 1/2,1/2,1/2, antiferromagnetically coupled to each other. Employing a recently developed efficient Monte Carlo technique as well as an exact-diagonalization method, we verify the spin-wave argument that the model exhibits two distinct excitations from the ground state which are gapless and gapped. The gapless branch shows a quadratic dispersion in the small-momentum region, which is of the ferromagnetic type. With the intention of elucidating the physical mechanism of both excitations, we make a perturbation approach from the decoupled-dimer limit. The gapless branch is directly related to spin 1's, while the gapped branch originates from cooperation of the two kinds of spins.
Abstract Using quantum Monte Carlo, exact diagonalization, and perturbation theory, we study the spectrum of the S = 1/2 antiferromagnetic Heisenberg trimer chain by varying the ratio g = J2/J1 of the intertrimer and intratrimer coupling strengths. The doublet ground states of trimers form effective interacting S = 1/2 degrees of freedom described by a Heisenberg chain. Therefore, the conventional two-spinon continuum of width ∝ J1 when g = 1 evolves into to a similar continuum of width ∝ J2 when g → 0. The intermediate-energy and high-energy modes are termed doublons and quartons which fractionalize with increasing g to form the conventional spinon continuum. In particular, at g ≈ 0.716, the gap between the low-energy spinon branch and the high-energy band with mixed doublons, quartons, and spinons closes. These features should be observable in inelastic neutron scattering experiments if a quasi-one-dimensional quantum magnet with the linear trimer structure and J2 < J1 can be identified. Our results may open a window for exploring the high-energy fractional excitations.
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.
A zero-temperature renormalization group (RG) approach is applied to the one-dimensional, spin-1/2 antiferromagnetic Heisenberg dimerized (alternating) chain. Specifically, the ground state energy and lowest-lying spectral excitations are examined. The calculation indicates the existence of a gap in the spectrum of the dimerized chain which vanishes only in the limit of a uniform spin chain, in contrast to a recent Green’s function approach. The RG results are in reasonable agreement with numerical extrapolations on the exact eigenvalue spectrum of finite chains of up to 12 spins. Both methods are compared with several other approximate treatments of the Heisenberg system, and tested by comparison with exact results for the spin-1/2 XY dimerized chain.