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Physical Review Letters

Doping Induced Quantum Bound States within the Haldane Gap

Zhong‐Yi Lu, Zhao-Bin Su, Lu Yu

Published May 22, 1995• 16 citations
Publisher

Abstract

Using the approach proposed by Zittartz and M\"uller-Hartmann [Z. Phys. 232, 11 (1970)], we show explicitly that quantum bound states at the centre of the Haldane energy gap can be induced by doping in a spin-1 linear chain Heisenberg antiferromagnet. We calculate the dynamic structure factor $S(q,\ensuremath{\omega})$ due to these bound states and use it to interpret the recent neutron scattering experiments on doped ${\mathrm{Y}}_{2}{\mathrm{BaNiO}}_{5}$ [J. F. DiTusa et al., Phys. Rev. Lett. 73, 1857 (1994)].

DOI
10.1103/physrevlett.74.4297

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Using the approach proposed by Zittartz and M"uller-Hartmann [Z. Phys. 232, 11 (1970)], we show explicitly that quantum bound states at the centre of the Haldane energy gap can be induced by doping in a spin-1 linear chain Heisenberg antiferromagnet. We calculate the dynamic structure factor S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) due to these bound states and use it to interpret the recent neutron scattering experiments on doped Y2BaNiO5{\mathrm{Y}}_{2}{\mathrm{BaNiO}}_{5}Y2​BaNiO5​ [J. F. DiTusa et al., Phys. Rev. Lett. 73, 1857 (1994)].
Using an impurity density-matrix renormalization-group scheme, we establish a reliable picture of how the low-lying energy levels of an S=1S=1S=1 Heisenberg antiferromagnetic chain change quantitatively upon bond doping. An impurity state gradually occurs in the Haldane gap as ${J}^{\ensuremath{'}}<J$, while it appears only if $\frac{{J}^{\ensuremath{'}}}{J}>{\ensuremath{\gamma}}_{c}$ with $\frac{1}{{\ensuremath{\gamma}}_{c}}=0.708$ as ${J}^{\ensuremath{'}}>J$. The spin polarization in the impurity state shows a localization-delocalization transition at ${\ensuremath{\gamma}}_{c}$. The system is nonperturbative as $1<~\frac{{J}^{\ensuremath{'}}}{J}<~{\ensuremath{\gamma}}_{c}$. It can explain the appearance of a new state in the Haldane gap recently observed experimentally on ${\mathrm{Y}}_{2\ensuremath{-}x}{\mathrm{Ca}}_{x}\mathrm{BaNi}{\mathrm{O}}_{5}$ [J.F. DiTusa et al., Phys. Rev. Lett. 73, 1857 (1994)].
Using the valence-bond-solid (VBS) approach and the Schwinger-boson mean-field approximation, we study the dependence of the Haldane gap of a spin-1 linear-chain Heisenberg antiferromagnet on impurity doping with different spins. The impurity spins affect the singlet pairing order parameter \ensuremath{\Delta} and the constraint factor \ensuremath{\lambda}. As a result, the Haldane gap is reduced by a factor \ensuremath{\sim}ni2/3{\mathit{n}}_{\mathit{i}}^{2/3}ni2/3​, with ni{\mathit{n}}_{\mathit{i}}ni​ as the impurity concentration, and eventually collapses at ni{\mathit{n}}_{\mathit{i}}ni​\ensuremath{\sim}1/\ensuremath{\xi} with \ensuremath{\xi} as the VBS correlation length. This theoretical prediction can be verified by neutron-scattering experiments.
The s=1 Heisenberg antiferromagnet is studied in the presence of two kinds of local impurities. First, a perturbed antiferromagnetic bond with J'\ensuremath{\ne}J at the center of an even-length open chain is considered. Using the density-matrix renormalization-group method we find that, for sufficiently strong or weak J', a bound state is localized at the impurity site, giving rise to an energy level in the Haldane gap. The energy of the bound state is in agreement with perturbative results, based on s=1/2 chain-end excitations, both in the weak- and strong-coupling limit. In a region around the uniform limit, J'=J, no states are found with energy below the Haldane gap. Second, a s=1/2 impurity at the center of an otherwise even-length open chain is considered. The coupling to the s=1/2 impurity is varied. Bound states in the Haldane gap are found only for sufficiently weak (antiferromagnetic) coupling. For a s=1/2 impurity coupled with a strong (antiferromagnetic) bond, no states are found in the Haldane gap. Our results are in good qualitative agreement with recent experiments on doped NENP and Y2{\mathrm{Y}}_{2}Y2​BaNiO5{\mathrm{BaNiO}}_{5}BaNiO5​.
Using the density matrix renormalization-group technique, we calculate the impurity energy levels for two different effective models of off-chain doping for quasi-one-dimensional Heisenberg chain compound Y2BaNiO5:{\mathrm{Y}}_{2}{\mathrm{BaNiO}}_{5}:Y2​BaNiO5​: ferromagnetic bond doping and antiferromagnetic site spin-1/2 doping. Thresholds of the impurity strength for the appearance of localized states are found for both models. However, the ground-state and low-energy excitations for weak impurity strength are different for these two models and the difference can be detected by experiments.
Inelastic neutron scattering was used to probe edge states in a quantum spin liquid. The experiment was performed on finite length antiferromagnetic spin-1 chains in ${\mathrm{Y}}_{2}{\mathrm{B}\mathrm{a}\mathrm{N}\mathrm{i}}_{1\ensuremath{-}x}{\mathrm{M}\mathrm{g}}_{x}{\mathrm{O}}_{5}$. At finite fields, there is a Zeeman resonance below the Haldane gap. The wave-vector dependence of its intensity provides direct evidence for staggered magnetization at chain ends, which decays exponentially towards the bulk [$\ensuremath{\xi}=8(1)$ at T=0.1 KT=0.1\text{ }\mathrm{K}T=0.1 K]. Continuum contributions to the chain-end spectrum indicate interchain segment interactions. We also observe a finite size blueshift of the Haldane gap.
We study the evolution of the dynamical spin structure factor in a spin-1 antiferromagnetic Heisenberg chain which is randomly doped with spin-1/2 moments. Using stochastic series expansion quantum Monte Carlo simulations combined with the maximum entropy method, we monitor the crossover from the spin-1 chain, dominated by a single resonance at the Haldane energy, to the spin-1/2 chain with a continuum of spinon states which diverges at zero frequency. Upon increasing the doping level, spectral weight is rapidly transferred from the Haldane peak to lower energies. If the exchange couplings between the spin-1/2 substituents and the spin-1 sites of the host are sufficiently small, finite-frequency bound states are observed below the spin gap for small doping concentrations.