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arXiv:cond-mat/9902084 · Physical Review Letters

Y89 NMR Imaging of the Staggered Magnetization in the Doped Haldane Chain Y2BaNi1−xMgxO5

Fabio Tedoldi, Raoul Santachiara, M. Horvatić

Published Jul 12, 1999• 68 citations
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Abstract

We present a static, model-independent, experimental determination of the spin-spin correlation length $\ensuremath{\xi}$ in a quantum spin system. This is achieved in the doped Haldane (i.e., $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ Heisenberg antiferromagnetic) chain ${\mathrm{Y}}_{2}{\mathrm{BaNi}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{O}}_{5}$ by ${}^{89}\mathrm{Y}$ NMR imaging of the staggered magnetization induced around the Mg impurities (i.e., chain boundaries) by a magnetic field. The magnitude of this magnetization is found to decay exponentially, with $\ensuremath{\xi}$ equal to the theoretical prediction for an infinite $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ chain and the staggered magnetic moment at the edge site showing the Curie behavior of an effective $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ spin.

DOI
10.1103/physrevlett.83.412

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We present a static, model-independent, experimental determination of the spin-spin correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ in a quantum spin system. This is achieved in the doped Haldane (i.e., S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 Heisenberg antiferromagnetic) chain Y2BaNi1\ensuremath−xMgxO5{\mathrm{Y}}_{2}{\mathrm{BaNi}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{O}}_{5}Y2​BaNi1\ensuremath−x​Mgx​O5​ by 89Y{}^{89}\mathrm{Y}89Y NMR imaging of the staggered magnetization induced around the Mg impurities (i.e., chain boundaries) by a magnetic field. The magnitude of this magnetization is found to decay exponentially, with \ensuremathξ\ensuremath{\xi}\ensuremathξ equal to the theoretical prediction for an infinite S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 chain and the staggered magnetic moment at the edge site showing the Curie behavior of an effective S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 spin.
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We present the effect of Zn (S=0)(S=0)(S=0) and Cu (S=1/2)(S=1/2)(S=1/2) substitution at the Ni site of S=1S=1S=1 Haldane chain compound Y2BaNiO5.{\mathrm{Y}}_{2}{\mathrm{BaNiO}}_{5}.Y2​BaNiO5​. 89Y{}^{89}\mathrm{Y}89Y nuclear-magnetic resonance (NMR) allows us to measure the local magnetic susceptibility at different distances from the defects. The 89Y{}^{89}\mathrm{Y}89Y NMR spectrum consists of one central peak and several less intense satellite peaks. The central peak represents the chain sites far from the defect. Its shift measures the uniform susceptibility, which displays a Haldane gap $\ensuremath{\Delta}\ensuremath{\approx}100\mathrm{K}$ and it corresponds to an antiferromagnetic (AF) coupling $J\ensuremath{\approx}260\mathrm{K}$ between the nearest neighbor Ni spins. Zn or Cu substitution does not affect the Haldane gap. The satellites, which are evenly distributed on the two sides of the central peak, probe the antiferromagnetic staggered magnetization near the substituted site. The spatial variation of the induced magnetization is found to decay exponentially from the impurity for both Zn and Cu for T>50K.T>50\mathrm{K}.T>50K. Its extension is found identical for both impurities and corresponds accurately to the correlation length $\ensuremath{\xi}(T)$ determined by Monte Carlo simulations for the pure compound. In the case of nonmagnetic Zn, the temperature dependence of the induced magnetization is consistent with a Curie law with an ``effective'' spin S=0.4S=0.4S=0.4 on each side of Zn. This staggered effect is quantitatively well accounted for in all the explored range by quantum Monte Carlo (QMC) computations of the spinless-defect-induced magnetism. In the case of magnetic Cu, the similarity of the induced magnetism to the Zn case implies a weak coupling of the Cu spin to the nearest-neighbor Ni spins. The slight reduction of about $20\ensuremath{-}30%$ of the induced polarization with respect to Zn is reproduced by QMC computations by considering an antiferromagnetic coupling of strength ${J}^{\ensuremath{'}}=0.1J\ensuremath{-}0.2J$ between the S=1/2S=1/2S=1/2 Cu spin and nearest-neighbor Ni spin. Macroscopic susceptibility measurements confirm these results as they display a clear Curie contribution due to the impurities nearly proportional to their concentration. This contribution is indeed close to that of two spin half for Zn substitution. The Curie contribution is smaller in the Cu case, which confirms that the coupling between Cu and near-neighbor Ni is antiferromagnetic.
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.
Open segments of S=1S=1S=1 antiferromagnetic spin chains are studied at finite temperatures and fields using continuous time quantum Monte Carlo techniques. By calculating the resulting magnetization profiles for a large range of chain lengths with fixed field and temperature we reconstruct the experimentally measured NMR spectrum of impurity doped ${\mathrm{Y}}_{2}{\mathrm{BaNi}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{O}}_{5}.$ For temperatures above the gap the calculated NMR spectra are in excellent agreement with the experimental results, confirming the existence of S=1/2S=1/2S=1/2 excitations at the end of open S=1S=1S=1 chain segments. At temperatures below the gap, neglecting interchain couplings, we still find well defined peaks in the calculated NMR spectra corresponding to the S=1/2S=1/2S=1/2 chain end excitations. At low temperatures, interchain couplings could become important, resulting in a more complicated phase.
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The phenomenological expression $\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins S=1S=1S=1, whose size nnn is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For nnn large, the coefficients of the expression converge towards C1=0.125{C}_{1}=0.125C1​=0.125, W1=0.451J{W}_{1}=0.451JW1​=0.451J, C2=0.564{C}_{2}=0.564C2​=0.564, W2=1.793J{W}_{2}=1.793JW2​=1.793J (JJJ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at T=0T=0T=0 and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of Y2BaNiO5{\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}Y2​BaNiO5​ and of Ni(C2H8N2)2NO2ClO4\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}Ni(C2​H8​N2​)2​NO2​ClO4​ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.