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arXiv:cond-mat/0605724 · Physical Review B

Modified triplet-wave expansion method applied to the alternating Heisenberg chain

A. Collins, Chris J. Hamer, Zheng Weihong

Published Oct 19, 2006• 17 citations
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Abstract

An alternative triplet-wave expansion formalism for dimerized spin systems is presented, a modification of the ``bond operator'' formalism of Sachdev and Bhatt. Projection operators are used to confine the system to the physical subspace, rather than constraint equations. The method is illustrated for the case of the alternating Heisenberg chain, and comparisons are made with the results of dimer series expansions and exact diagonalization. Some discussion is included of the phenomenon of ``quasiparticle breakdown'' as it applies to the two-triplon bound states in this model.

DOI
10.1103/physrevb.74.144414

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An alternative triplet-wave expansion formalism for dimerized spin systems is presented, a modification of the bond operator'' formalism of Sachdev and Bhatt. Projection operators are used to confine the system to the physical subspace, rather than constraint equations. The method is illustrated for the case of the alternating Heisenberg chain, and comparisons are made with the results of dimer series expansions and exact diagonalization. Some discussion is included of the phenomenon of quasiparticle breakdown'' as it applies to the two-triplon bound states in this model.
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 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 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.
Motivated by experiments on nonmagnetic triangular-lattice Mott insulators, we study one candidate paramagnetic phase, namely the columnar dimer (or valence-bond) phase. We apply variants of the bond-operator theory to a dimerized and spatially anisotropic spin-1/2 Heisenberg model and determine its zero-temperature phase diagram and the spectrum of elementary triplet excitations (triplons). Depending on model parameters, we find that the minimum of the triplon energy is located at either a commensurate or an incommensurate wave vector. Condensation of triplons at this commensurate-incommensurate transition defines a quantum Lifshitz point, with effective dimensional reduction that possibly leads to nontrivial paramagnetic (e.g., spin-liquid) states near the closing of the triplet gap. We also discuss the two-particle decay of high-energy triplons, and we comment on the relevance of our results for the organic Mott insulator EtMe3{}_{3}3​P[Pd(dmit)2{}_{2}2​]2{}_{2}2​.
Using the Bond-Operator Technique (BOT), we have studied the low energy excitation spectrum of a frustrated dimerized antiferromagnetic Heisenberg chain. In particular, we have compared our analytical results with previous Exact Diagonalization (ED) data. Qualitatively, the BOT results are in good agreement with the ED data. And even a very good quantitative agreement is obtained in some parameter region. It is clearly shown that there is only one elementary excitation branch (lowest triplet branch) and that the two other well defined excitations which appear below the continuum, one singlet and one triplet, are bound states of two elementary triplets.