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arXiv:cond-mat/0211346 · Physical Review Letters

Spectral Weight Contributions of Many-Particle Bound States and Continuum

Weihong Zheng, Chris J. Hamer, Rajiv R. P. Singh

Published Jul 17, 2003• 12 citations
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Abstract

Cluster expansion methods are developed for calculating the spectral weight contributions of multiparticle excitations---continuum and bound states---to high orders. A complete 13th order calculation is carried out for the alternating Heisenberg chain. For $\ensuremath{\lambda}=0.27$, relevant to the material $\mathrm{C}\mathrm{u}({\mathrm{N}\mathrm{O}}_{3}{)}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}2.5{\mathrm{D}}_{2}\mathrm{O}$, we present detailed spectral weights for the two-triplet continuum and all bound states. We also examine the variation of the relative weights of one- and two-particle states with bond alternation from the dimerized to the uniform chain limit.

DOI
10.1103/physrevlett.91.037206

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Cluster expansion methods are developed for calculating the spectral weight contributions of multiparticle excitations---continuum and bound states---to high orders. A complete 13th order calculation is carried out for the alternating Heisenberg chain. For \ensuremathλ=0.27\ensuremath{\lambda}=0.27\ensuremathλ=0.27, relevant to the material Cu(NO3)2\ifmmode⋅\else\textperiodcentered\fi2.5D2O\mathrm{C}\mathrm{u}({\mathrm{N}\mathrm{O}}_{3}{)}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}2.5{\mathrm{D}}_{2}\mathrm{O}Cu(NO3​)2​\ifmmode⋅\else\textperiodcentered\fi2.5D2​O, we present detailed spectral weights for the two-triplet continuum and all bound states. We also examine the variation of the relative weights of one- and two-particle states with bond alternation from the dimerized to the uniform chain limit.
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 have developed strong-coupling series expansion methods to study the two-particle spectra in quantum lattice models. The properties of bound states and multiparticle excitations can reveal important information about the dynamics of a given model. At the heart of this method lies the calculation of an effective Hamiltonian in the two-particle subspace. We use an orthogonal transformation to perform this block diagonalising, and find that maintaining orthogonality is crucial for cases where the ground state and the two-particle subspace have identical quantum numbers. The two-particle Schrödinger equation is solved by using a finite lattice approach in coordinate space or an integral equation in momentum space. These methods allow us to determine precisely the low-lying excitation spectra and dispersion relations for the two-particle bound states. The method has been tested for the (1+1) D transverse Ising model, and applied to the two-leg spin-1/2 Heisenberg ladder. We study the coherence lengths of the bound states, and how they merge with the two-particle continuum. Finally, these techniques are applied to the frustrated alternating Heisenberg chain, which has been of considerable recent interest due to its relevance to spin-Peierls systems such as CuGeO 3. Starting from a limit corresponding to weakly-coupled dimers, we develop high-order series expansions for the effective Hamiltonian in the two-particle subspace. In the regime of strong dimerisation, various properties of the singlet and triplet bound states, and the quintet antibound states, can be accurately calculated. We also study the behaviour as the external bond alternation vanishes, and the way in which the bound states of triplet dimer excitations make the transition to a soliton-antisoliton continuum.
We present a new linked cluster expansion for calculating properties of multiparticle excitation spectra to high orders. We use it to obtain the two-particle spectra for systems of coupled spin-half dimers. We find that even for weakly coupled dimers the spectrum is very rich, consisting of many bound states. The number of bound states depends on both geometry of coupling and frustration. Many of the bound states can only be seen by going to sufficiently high orders in the perturbation theory, showing the extended character of the pair attraction.
We use recently developed strong-coupling expansion methods to study the two-particle spectra for the frustrated alternating Heisenberg model, consisting of an alternating nearest neighbor antiferromagnetic exchange and a uniform second neighbor antiferromagnetic exchange. Starting from the limit of weakly coupled dimers, we develop high order series expansions for the effective Hamiltonian in the two-particle subspace. In the limit of a strong applied dimerization, we calculate accurately various properties of singlet and triplet bound states and quintet antibound states. We also develop series expansions for bound state energies in various sectors, which can be extrapolated using standard methods to cases where the external bond alternation goes to zero. We study the properties of singlet and triplet bound states in the latter limit and suggest a crucial role for the bound states in the unbinding of triplets and deconfinement of spin-half excitations.
We calculate the excitation spectrum and spectral weights of the alternating antiferromagnetic-ferromagnetic spin-half Heisenberg chain with exchange couplings JJJ and −∣λ∣J-|λ|J−∣λ∣J as a power series in λλλ. For small ∣λ∣|λ|∣λ∣, the gapped one-particle spectrum has a maximum at k=0k=0k=0 and there is a rich structure of bound (and anti-bound) states below (and above) the 2-particle continuum. As ∣λ∣|λ|∣λ∣ is increased past unity the spectrum crosses over to the Haldane regime, where the peak shifts away from k=0k=0k=0, the one particle states merge with the bottom of the continuum near k=0k=0k=0, and the spectral weights associated with the one-particle states become very small. Extrapolation of the spectrum to large ∣λ∣|λ|∣λ∣ confirms that the ground state energy and excitation gap map onto those of the spin-one chain.
We study the spectral properties of the alternating antiferromagnetic-ferromagnetic spin-half Heisenberg chain with exchange couplings JJJ and $\ensuremath{-}\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}J$. The crossover from the gapped dimerized spectra at small $\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}$ to the spin-one chain behavior at large $\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}$ is signaled by the merging of the one-particle state with the bottom of the continuum, and its vanishing spectral weight, near k=0k=0k=0. At large $\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}$, the model represents a class of spin-one chains with varying degrees of spatial and temporal compositeness and provides an excellent quantitative description of the neutron scattering spectra for such materials as (CD3)2ND2CuCl3{(\mathrm{C}{\mathrm{D}}_{3})}_{2}\mathrm{N}{\mathrm{D}}_{2}\mathrm{Cu}{\mathrm{Cl}}_{3}(CD3​)2​ND2​CuCl3​ and $\mathrm{IPA}\text{\ensuremath{-}}\mathrm{Cu}{\mathrm{Cl}}_{3}$.