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arXiv:cond-mat/0006282 · Physical review. B, Condensed matter

Magnetization profiles and NMR spectra of doped Haldane chains at finite temperatures

Fabien Alet, Erik Schwartz Sørensen

Published Dec 1, 2000• 13 citations
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Abstract

Open segments of $S=1$ 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 $S=1/2$ excitations at the end of open $S=1$ chain segments. At temperatures below the gap, neglecting interchain couplings, we still find well defined peaks in the calculated NMR spectra corresponding to the $S=1/2$ chain end excitations. At low temperatures, interchain couplings could become important, resulting in a more complicated phase.

DOI
10.1103/physrevb.62.14116

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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 Y2BaNi1\ensuremath−xMgxO5.{\mathrm{Y}}_{2}{\mathrm{BaNi}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{O}}_{5}.Y2​BaNi1\ensuremath−x​Mgx​O5​. 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.
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.
The response of an open spin chain with isotropic antiferromagnetic interactions to a uniform magnetic field is studied by classical Monte Carlo simulations. It is observed how the induced on-site magnetization is nonuniform, due to the occurrence of edge staggered terms which decay exponentially over a distance equal to the zero-field correlation length of the infinite chain. The total magnetic moment associated to each staggered term is found to be about half of the original single-spin magnitude and to decrease as the inverse of temperature (i.e., to behave as a Curie-like moment). The numerical results are compared to recent NMR findings in spinless-doped Y2BaNiO5;{\mathrm{Y}}_{2}{\mathrm{BaNiO}}_{5};Y2​BaNiO5​; the remarkable agreement found shows that, for temperatures above the Haldane gap, the classical approach gives a correct picture of the boundary effects observed in the Heisenberg S=1S=1S=1 chain.
The effect of spin-vacancy doping on the ground state of the quasi-one-dimensional S=1S=1S=1 quantum antiferromagnet PbNi2V2O8{\mathrm{PbNi}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}PbNi2​V2​O8​ is studied by X band as well as high-field ESR. A very broad ESR line is observed at low temperatures in powdered ${\mathrm{PbNi}}_{2\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$ for 0< x< 0.24.0<~x<~0.24.0< x< 0.24. The frequency dependence of the observed ESR line rules out the possibility that the low-temperature ESR signal comes from the isolated S=12S=\frac{1}{2}S=21​ spins. Instead, it is proposed that a magnetic coupling (most likely of second order) between the chain-end spins on both sides of the Mg2+{\mathrm{Mg}}^{2+}Mg2+ impurity plays an important role in this system. Impurity-induced spin states, due to the strong interchain coupling, also induce a staggered magnetization on the neighboring chains with a characteristic length \ensuremath{\xi}. \ensuremath{\xi} increases with decreasing temperature, as suggested by the additional broadening and the shift of the ESR signal at low temperatures. On the other hand the temperature dependence of the low temperature susceptibility suggests a much more complicated picture in which impurity-induced spins are exposed to coupling to thermal excitations on each chain as well as to three-dimensional antiferromagnetic correlations.
Experimental bulk susceptibility $\ensuremath{\chi}(T)$ and magnetization M(H,T)M(H,T)M(H,T) of the S=1S=1S=1 Haldane chain system doped with nonmagnetic impurities, ${\mathrm{Y}}_{2}{\mathrm{BaNi}}_{1\ensuremath{-}x}{\mathrm{Zn}}_{x}{\mathrm{O}}_{5}$ (x< 0.08),(x<~0.08),(x< 0.08), are analyzed. A numerical calculation for the low-energy spectrum of noninteracting open segments describes well experimental data above 4 K. Below 4 K, we observe power-law behaviors $\ensuremath{\chi}(T)\ensuremath{\sim}{T}^{\ensuremath{-}\ensuremath{\alpha}}$ and $M(H,T)\ensuremath{\sim}{T}^{1\ensuremath{-}\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}(H/T),$ with $\ensuremath{\alpha}(<1)$ depending on the doping concentration x. This observation suggests the appearance of a gapless quantum phase due to a broad distribution of effective couplings between the dilution-induced moments.
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.
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=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 Heisenberg antiferromagnetic) chain ${\mathrm{Y}}_{2}{\mathrm{BaNi}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{O}}_{5}$ 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 $\ensuremath{\xi}$ 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.