arXiv:cond-mat/9803180 · Physical Review Letters

Bound States of Magnons in the S=1/2 Quantum Spin Ladder

Oleg P. Sushkov, Valeri N. Kotov

Published Aug 31, 1998• 97 citations

Abstract

The excitation spectrum of the two-chain Heisenberg ladder with antiferromagnetic interactions is studied. Our approach is based on the 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 ( $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$) and triplet ( $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$) excitations are found as quasiparticle bound states. Thus it is demonstrated that the spectrum consists of one elementary triplet, one composite triplet, and one composite singlet.

DOI: 10.1103/physrevlett.81.1941

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arXiv:cond-mat/9803180 · Physical Review Letters

Bound States of Magnons in the S=1/2 Quantum Spin Ladder

Oleg P. Sushkov, Valeri N. Kotov

Published Aug 31, 1998• 97 citations

arXiv HTML PDF

Abstract

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

) and triplet (

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1

) excitations are found as quasiparticle bound states. Thus it is demonstrated that the spectrum consists of one elementary triplet, one composite triplet, and one composite singlet.

DOI: 10.1103/physrevlett.81.1941

Related papers

Spectrum of elementary and collective excitations in the dimerized S=1/2 Heisenberg chain with frustration

Pavel V. Shevchenko, Valeri N. Kotov, Oleg P. Sushkov

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 $(S=0)$ and triplet $(S=1)$ 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 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 $(S=0)$ and triplet $(S=1)$ 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.

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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.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.

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Heisenberg antiferromagnetic spin ``ladders'' (two coupled spin chains) are low-dimensional magnetic systems which for S=1/2 interpolate between half-integer-spin chains, when the chains are decoupled, and effective integer-spin one-dimensional chains in the strong-coupling limit. The spin-1/2 ladder may be realized in nature by vanadyl pyrophosphate, (VO ${)}_{2}$ ${\mathrm{P}}_{2}$ ${\mathrm{O}}_{7}$ . In this paper we apply strong-coupling perturbation theory, spin-wave theory, Lanczos techniques, and a Monte Carlo method to determine the ground-state energy and the low-lying excitation spectrum of the ladder. We find evidence of a nonzero spin gap for all interchain couplings ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$>0. A band of spin-triplet excitations above the gap is also analyzed. These excitations are unusual for an antiferromagnet, since their long-wavelength dispersion relation behaves as (k- ${\mathit{k}}_{0}$ ${)}^{2}$ (in the strong-coupling limit ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\gg}J, where J is the in-chain antiferromagnetic coupling). Their band is folded, with a minimum energy at ${\mathit{k}}_{0}$ =\ensuremath{\pi}, and a maximum between ${\mathit{k}}_{1}$ =\ensuremath{\pi}/2 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=0) and 0 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=\ensuremath{\infty}). We also give numerical results for the dynamical structure factor S(q,\ensuremath{\omega}), which can be determined in neutron scattering experiments. Finally, possible experimental techniques for studying the excitation spectrum are discussed.Heisenberg antiferromagnetic spin ``ladders'' (two coupled spin chains) are low-dimensional magnetic systems which for S=1/2 interpolate between half-integer-spin chains, when the chains are decoupled, and effective integer-spin one-dimensional chains in the strong-coupling limit. The spin-1/2 ladder may be realized in nature by vanadyl pyrophosphate, (VO${)}_{2}$${\mathrm{P}}_{2}$${\mathrm{O}}_{7}$. In this paper we apply strong-coupling perturbation theory, spin-wave theory, Lanczos techniques, and a Monte Carlo method to determine the ground-state energy and the low-lying excitation spectrum of the ladder. We find evidence of a nonzero spin gap for all interchain couplings ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$>0. A band of spin-triplet excitations above the gap is also analyzed. These excitations are unusual for an antiferromagnet, since their long-wavelength dispersion relation behaves as (k-${\mathit{k}}_{0}$${)}^{2}$ (in the strong-coupling limit ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\gg}J, where J is the in-chain antiferromagnetic coupling). Their band is folded, with a minimum energy at ${\mathit{k}}_{0}$=\ensuremath{\pi}, and a maximum between ${\mathit{k}}_{1}$=\ensuremath{\pi}/2 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=0) and 0 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=\ensuremath{\infty}). We also give numerical results for the dynamical structure factor S(q,\ensuremath{\omega}), which can be determined in neutron scattering experiments. Finally, possible experimental techniques for studying the excitation spectrum are discussed.

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Motivated by recent measurements on CuGe ${\mathrm{O}}_{3}$ , the spectrum of magnetic excitations of an antiferromagnetic $S=\frac{1}{2}$ chain with alternating coupling strength is investigated. Wave-vector-dependent magnons and a continuum with square-root behavior at the band edges are found. The spectral density of the continua is calculated. Spin-rotation symmetry fixes the gap of the continuum to be twice the elementary magnon gap. This is in excellent agreement with experimental results. The existence of (anti)bound states of two magnons is predicted: below the continuum a singlet and a triplet, above the continuum a quintuplet.Motivated by recent measurements on CuGe${\mathrm{O}}_{3}$, the spectrum of magnetic excitations of an antiferromagnetic $S=\frac{1}{2}$ chain with alternating coupling strength is investigated. Wave-vector-dependent magnons and a continuum with square-root behavior at the band edges are found. The spectral density of the continua is calculated. Spin-rotation symmetry fixes the gap of the continuum to be twice the elementary magnon gap. This is in excellent agreement with experimental results. The existence of (anti)bound states of two magnons is predicted: below the continuum a singlet and a triplet, above the continuum a quintuplet.

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S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

) and triplet (

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1

) excitations are found as quasiparticle bound states. Thus it is demonstrated that the spectrum consists of one elementary triplet, one composite triplet, and one composite singlet.

(S=0)

and triplet

(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.

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.

{)}_{2}

{\mathrm{P}}_{2}

{\mathrm{O}}_{7}

. In this paper we apply strong-coupling perturbation theory, spin-wave theory, Lanczos techniques, and a Monte Carlo method to determine the ground-state energy and the low-lying excitation spectrum of the ladder. We find evidence of a nonzero spin gap for all interchain couplings ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$>0. A band of spin-triplet excitations above the gap is also analyzed. These excitations are unusual for an antiferromagnet, since their long-wavelength dispersion relation behaves as (k-

{\mathit{k}}_{0}

{)}^{2}

(in the strong-coupling limit ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\gg}J, where J is the in-chain antiferromagnetic coupling). Their band is folded, with a minimum energy at

{\mathit{k}}_{0}

=\ensuremath{\pi}, and a maximum between

{\mathit{k}}_{1}

=\ensuremath{\pi}/2 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=0) and 0 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=\ensuremath{\infty}). We also give numerical results for the dynamical structure factor S(q,\ensuremath{\omega}), which can be determined in neutron scattering experiments. Finally, possible experimental techniques for studying the excitation spectrum are discussed.

Motivated by recent measurements on CuGe

{\mathrm{O}}_{3}

, the spectrum of magnetic excitations of an antiferromagnetic

S=\frac{1}{2}

chain with alternating coupling strength is investigated. Wave-vector-dependent magnons and a continuum with square-root behavior at the band edges are found. The spectral density of the continua is calculated. Spin-rotation symmetry fixes the gap of the continuum to be twice the elementary magnon gap. This is in excellent agreement with experimental results. The existence of (anti)bound states of two magnons is predicted: below the continuum a singlet and a triplet, above the continuum a quintuplet.