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arXiv:cond-mat/9508132 · Physical review. B, Condensed matter

Impurity state in the Haldane gap for an S=1 Heisenberg antiferromagnetic chain with bond doping

Xiaoqun Wang, Steffen Mallwitz

Published Jan 1, 1996• 37 citations
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Abstract

Using an impurity density-matrix renormalization-group scheme, we establish a reliable picture of how the low-lying energy levels of an $S=1$ 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)].

DOI
10.1103/physrevb.53.r492

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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{J}^{\ensuremath{'}}<JJ\ensuremath′<J, while it appears only if J\ensuremath′J>\ensuremathγc\frac{{J}^{\ensuremath{'}}}{J}>{\ensuremath{\gamma}}_{c}JJ\ensuremath′​>\ensuremathγc​ with 1\ensuremathγc=0.708\frac{1}{{\ensuremath{\gamma}}_{c}}=0.708\ensuremathγc​1​=0.708 as J\ensuremath′>J{J}^{\ensuremath{'}}>JJ\ensuremath′>J. The spin polarization in the impurity state shows a localization-delocalization transition at \ensuremathγc{\ensuremath{\gamma}}_{c}\ensuremathγc​. The system is nonperturbative as 1< J\ensuremath′J< \ensuremathγc1<~\frac{{J}^{\ensuremath{'}}}{J}<~{\ensuremath{\gamma}}_{c}1< JJ\ensuremath′​< \ensuremathγc​. It can explain the appearance of a new state in the Haldane gap recently observed experimentally on Y2\ensuremath−xCaxBaNiO5{\mathrm{Y}}_{2\ensuremath{-}x}{\mathrm{Ca}}_{x}\mathrm{BaNi}{\mathrm{O}}_{5}Y2\ensuremath−x​Cax​BaNiO5​ [J.F. DiTusa et al., Phys. Rev. Lett. 73, 1857 (1994)].
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​.
An impurity bond ${\mathit{J}}^{\ensuremath{'}}$ in a periodic one-dimensional antiferromagnetic spin-1 chain with exchange J is considered. Using the numerical density matrix renormalization-group method, we find an impurity energy level in the Haldane gap, corresponding to a bound state near the impurity bond. When ${\mathit{J}}^{\ensuremath{'}}$J the level changes gradually from the edge of the Haldane gap to the ground-state energy as the deviation dev=(J-${\mathit{J}}^{\ensuremath{'}}$)/J changes from 0 to 1. It seems that there is no threshold. Yet, there is a threshold when ${\mathit{J}}^{\ensuremath{'}}$\ensuremath{\gtrsim}J. The impurity level appears only when the deviation dev=(${\mathit{J}}^{\ensuremath{'}}$-J)/${\mathit{J}}^{\ensuremath{'}}$ is greater than Bc{\mathit{B}}_{\mathit{c}}Bc​, which is close to 0.3 in our calculation. \textcopyright{} 1996 The American Physical Society.
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.
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)].
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.
Low-lying excited states of the spin-1 antiferromagnetic Heisenberg chain with an impurity bond are investigated in terms of domain-wall excitations by means of analytical methods as well as a method of numerical diagonalization.It is shown that 1) the impurity bond brings about a massive triplet mode in the Haldane gap, 2) the triplet state comprises three of the four ground states of an open chain, 3) the excited states with the domain wall trapped at the impurity bond, including the massive triplet mode, exhibit a Néel-type configuration of the magnetic moment around the impurity bond.On the basis of the present results, we propose a new energy versus magnetic-field diagram.