Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0007192 · Physical Review Letters

Strong-Coupling Expansions for Multiparticle Excitations: Continuum and Bound States

Simon Trebst, H. Monien, Chris J. Hamer, Zheng Weihong, Rajiv R. P. Singh

Published Nov 13, 2000• 89 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.85.4373

Related papers

LINKED CLUSTER SERIES EXPANSIONS FOR TWO-PARTICLE STATES IN QUANTUM LATTICE MODELS

Weihong Zheng, Chris J. Hamer, Rajiv R. P. Singh +2 more

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.

International Journal of Modern Physics B•Nov 10, 2003•0 citations

Deconfinement transition and bound states in frustrated Heisenberg chains: Regimes of forced and spontaneous dimerization

Weihong Zheng, Chris J. Hamer, Rajiv R. P. Singh +2 more

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.

arXiv:cond-mat/0010243•Physical review. B, Condensed matter•Mar 19, 2001•21 citations

Spectral Weight Contributions of Many-Particle Bound States and Continuum

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

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.

arXiv:cond-mat/0211346•Physical Review Letters•Jul 17, 2003•12 citations

Linked cluster series expansions for two-particle bound states

Weihong Zheng, Chris J. Hamer, Rajiv R. P. Singh +2 more

We develop strong-coupling series expansion methods to study two-particle spectra of quantum lattice models. At the heart of the method lies the calculation of an effective Hamiltonian in the two-particle subspace. We explicitly consider an orthogonality transformation to generate this block diagonalization, and find that maintaining orthogonality is crucial for systems where the ground state and the two-particle subspace are characterized by identical quantum numbers. We discuss the solution of the two-particle Schr\"odinger equation by using a finite lattice approach in coordinate space or by an integral equation in momentum space. These methods allow us to precisely determine the low-lying excitation spectra of the models at hand, including all two-particle bound/antibound states. Further, we discuss how to generate series expansions for the dispersions of the bound/antibound states. These allow us to employ series extrapolation techniques, whereby binding energies can be determined even when the expansion parameters are not small. We apply the method to the (1+1)-dimensional transverse Ising model and the two-leg $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ Heisenberg ladder. For the latter model, we also calculate the coherence lengths and determine the critical properties where bound states merge with the two-particle continuum.

arXiv:cond-mat/0010354•Physical review. B, Condensed matter•Mar 19, 2001•69 citations

Dynamical structure factor for the alternating Heisenberg chain: A linked cluster calculation

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

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.

arXiv:cond-mat/0307517•Physical review. B, Condensed matter•Dec 8, 2003•27 citations

Excitation spectrum of the S=1/2 quantum spin ladder with frustration: Elementary quasiparticles and many-particle bound states

Valeri N. Kotov, Oleg P. Sushkov, R. Eder

The excitation spectrum of the two-chain $S=1/2$ Heisenberg spin ladder with additional second neighbor frustrating interactions is studied by a variety of techniques. A description, based on a mapping of the model onto a Bose gas of hard-core triplets is used to determine the one- and two-particle excitation spectra. We find that low-lying singlet and triplet bound states are present and their binding energy increases with increasing frustration. In addition, many-particle bound states are found by exact diagonalization and variational methods. We prove that the larger the number of bound particles the larger the binding energy. Thus the excitation spectrum has a complex structure and consists of elementary triplets and composite many-particle singlet and triplet bound states. The composite excitations mix strongly with the elementary ones in the coupling regime where quantum fluctuations are strong. The quantum phase transition, known to take place in this model at critical frustration, is interpreted as a condensation process of (infinitely) large many-particle bound states.

arXiv:cond-mat/9808169•Physical review. B, Condensed matter•Mar 1, 1999•65 citations
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 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 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.
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.
We develop strong-coupling series expansion methods to study two-particle spectra of quantum lattice models. At the heart of the method lies the calculation of an effective Hamiltonian in the two-particle subspace. We explicitly consider an orthogonality transformation to generate this block diagonalization, and find that maintaining orthogonality is crucial for systems where the ground state and the two-particle subspace are characterized by identical quantum numbers. We discuss the solution of the two-particle Schr\"odinger equation by using a finite lattice approach in coordinate space or by an integral equation in momentum space. These methods allow us to precisely determine the low-lying excitation spectra of the models at hand, including all two-particle bound/antibound states. Further, we discuss how to generate series expansions for the dispersions of the bound/antibound states. These allow us to employ series extrapolation techniques, whereby binding energies can be determined even when the expansion parameters are not small. We apply the method to the (1+1)-dimensional transverse Ising model and the two-leg $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ Heisenberg ladder. For the latter model, we also calculate the coherence lengths and determine the critical properties where bound states merge with the two-particle 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.
The excitation spectrum of the two-chain S=1/2S=1/2S=1/2 Heisenberg spin ladder with additional second neighbor frustrating interactions is studied by a variety of techniques. A description, based on a mapping of the model onto a Bose gas of hard-core triplets is used to determine the one- and two-particle excitation spectra. We find that low-lying singlet and triplet bound states are present and their binding energy increases with increasing frustration. In addition, many-particle bound states are found by exact diagonalization and variational methods. We prove that the larger the number of bound particles the larger the binding energy. Thus the excitation spectrum has a complex structure and consists of elementary triplets and composite many-particle singlet and triplet bound states. The composite excitations mix strongly with the elementary ones in the coupling regime where quantum fluctuations are strong. The quantum phase transition, known to take place in this model at critical frustration, is interpreted as a condensation process of (infinitely) large many-particle bound states.