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arXiv:cond-mat/9301035 · Physical review. B, Condensed matter

Haldane energy gap of a doped linear-chain Heisenberg antiferromagnet

Zhong‐Yi Lu, Zhao-Bin Su, Lu Yu

Published May 1, 1993• 2 citations
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Abstract

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}${\mathit{n}}_{\mathit{i}}^{2/3}$, with ${\mathit{n}}_{\mathit{i}}$ as the impurity concentration, and eventually collapses at ${\mathit{n}}_{\mathit{i}}$\ensuremath{\sim}1/\ensuremath{\xi} with \ensuremath{\xi} as the VBS correlation length. This theoretical prediction can be verified by neutron-scattering experiments.

DOI
10.1103/physrevb.47.12276

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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 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 Y2BaNiO5{\mathrm{Y}}_{2}{\mathrm{BaNiO}}_{5}Y2​BaNiO5​ [J. F. DiTusa et al., Phys. Rev. Lett. 73, 1857 (1994)].
We have investigated Haldane's conjecture for the S = 2 antiferromagnetic quantum spin chain with nearest-neighbour exchange J . Using a density matrix renormalization group algorithm for chains up to L = 350 spins, we show that the ground state has hidden topological order: the physics of the S = 2 chain can be captured by a valence bond solid description. We also observe effective free spin-1 states at the ends of an open S = 2 chain. We find in the thermodynamic limit a gap Δ = 0.085(5) J and a spin correlation length ξ = 49(1) lattice spacings in the isotropic case. We give the phase diagram in the presence of single-ion and exchange anisotropy, showing how the Haldane phase is squeezed out when the spin S increases.
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)].
The spin correlations of the S=1 Heisenberg antiferromagnetic chains in AgVP2{\mathrm{AgVP}}_{2}AgVP2​S6{\mathrm{S}}_{6}S6​ have been studied by neutron scattering at temperatures of the order of T\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ J/kB{\mathit{k}}_{\mathit{B}}kB​. On the polycrystalline samples, with a careful choice of experimental conditions, we have observed a Haldane gap of Eg{\mathit{E}}_{\mathit{g}}Eg​=26 meV at the antiferromagnetic point at q=\ensuremath{\pi}. The excitation spectra are consistent with a mode that has a spin-wave velocity C=150 meV and a correlation length \ensuremath{\xi}\ensuremath{\approxeq}5.5, close to C/Eg{\mathit{E}}_{\mathit{g}}Eg​. These results are in quantitative agreement with the predictions and numerical results on the Haldane state.
A Monte Carlo simulation study based on the heat-bath algorithm on a checkerboard arrangement is applied to the spin-1 antiferromagnetic chain model with both bilinear and biquadratic exchange interactions. Changing the parameter \ensuremath{\lambda} which denotes strength of the biquadratic interaction against the bilinear one from -1.0 to 2.0, we obtain the following results according to the measurements of energy and two-point correlation function. At \ensuremath{\lambda}=1 and -(1/3 where some physical quantities are derived rigorously, our Monte Carlo simulation results agree well with the rigorous ones. For ${\ensuremath{\lambda}}_{c}$\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}${\ensuremath{\lambda}}_{1}$ with taking 0.25\ensuremath{\le}${\ensuremath{\lambda}}_{c}$\ensuremath{\le}0.5 and ${\ensuremath{\lambda}}_{1}$\ensuremath{\ge}2.0, its model has the gapless ground state accompanied by a power-law decaying correlation function. This implies the existence of a critical line. For -1.0${\ensuremath{\lambda}}_{c}$, on the other hand, its model has the ground state with energy gap accompanied with exponentially decaying correlation function. This indicates that the valence-bond-solid (VBS) state asserted by Affleck et al. is stabilized there. As the point of \ensuremath{\lambda}=0 comes to the standard Heisenberg antiferromagnetic model and is in the region of -1.0${\ensuremath{\lambda}}_{c}$, the Haldane's conjecture is supported that this ground state is described by the VBS state for the case of integer spin, e.g., S=1.