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arXiv:cond-mat/9805042 · International Journal of Modern Physics B

BOUND STATES IN DIMERIZED AND FRUSTRATED HEISENBERG CHAINS

G. Bouzerar, S. SIL

Published Aug 10, 2001• 3 citations
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Abstract

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.

DOI
10.1142/s0217979201006537

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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.
We have studied the low-energy excitation spectrum of a dimerized and frustrated antiferromagnetic Heisenberg chain. We use an analytic approach, based on a description of the excitations as triplets above a strong-coupling singlet ground state. The quasiparticle spectrum is calculated by treating the excitations as a dilute Bose gas with infinite on-site repulsion. Additional singlet (S=0)(S=0)(S=0) and triplet (S=1)(S=1)(S=1) modes are found as two-particle bound states of the elementary triplets. We have also calculated the contributions of the elementary and collective excitations into the spin structure factor. Our results are in excellent agreement with exact diagonalizations and dimer series expansions data as long as the dimerization parameter \ensuremath{\delta} is not too small $(\ensuremath{\delta}>0.1),$ i.e., while the elementary triplets can be treated as localized objects.
We present a detailed numerical analysis of the low-energy excitation spectrum of a frustrated and dimerized spin S=1/2S=1/2S=1/2 Heisenberg chain. In particular, we show that in the commensurate spin-Peierls phase the ratio of the singlet and triplet excitation gap is a universal function which depends on the frustration parameter only. We identify the conditions for which a second elementary triplet branch in the excitation spectrum splits from the continuum. We compare our results with predictions from the continuum limit field theory. We discuss the relevance of our data in connection with recent experiments on CuGeO3,{\mathrm{CuGeO}}_{3},CuGeO3​, NaV2O5,{\mathrm{NaV}}_{2}{\mathrm{O}}_{5},NaV2​O5​, and (VO)2P2O7(\mathrm{VO}{)}_{2}{\mathrm{P}}_{2}{\mathrm{O}}_{7}(VO)2​P2​O7​.
We study the finite-size behavior of the low-lying excitations of spin-12\frac{1}{2}21​ Heisenberg chains with dimerization and next-to-nearest neighbors interaction J2{J}_{2}J2​. The numerical analysis, performed using density-matrix renormalization group, confirms previous exact diagonalization results and shows that, for different values of the dimerization parameter $\ensuremath{\delta}$, the elementary triplet and singlet excitations present a clear scaling behavior in a wide range of $l=L∕\ensuremath{\xi}$ (where LLL is the length of the chain and $\ensuremath{\xi}$ is the correlation length). At J2=J2c{J}_{2}={J}_{2c}J2​=J2c​, where no logarithmic corrections are present, we compare the numerical results with finite-size predictions for the sine-Gordon model obtained using L\"uscher's theory. For small $\ensuremath{\delta}$ we find a very good agreement for $l\ensuremath{\gtrsim}4$ or 7 depending on the excitation considered.
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.
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We present a self-consistent quantum-theoretical analysis of the ground-state energy and low-energy excitation spectrum of a dimerized and frustrated Heisenberg chain by the bosonization representation in the continuum-limit approach. The frustration effects on the ground-state energy and energy gap are investigated for α>αc where α is the frustration parameter and αc is its critical value, above which a frustration-induced energy gap is opened. It is shown that as α increases, the ground-state energy decreases and the energy gap increases. We find that the dimerization dependence of the ground-state energy and energy gap for small dimerization obeys the Cross-Fisher power law only at α = αc, but departs significantly from it when α>αc. The present results are in good agreement with numerical analyses from the density-matrix renormalization-group and exact diagonalization methods. The relevance of our calculated results to recent experiments on the spin-Peierls compounds CuGeO3 is also discussed.