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arXiv:cond-mat/0007280 · Physical review. B, Condensed matter

On-site magnetization in open antiferromagnetic chains: A classical analysis versus NMR experiments in a spin-1 compound

Silvana Botti, Alberto Rosso, Raoul Santachiara, Fabio Tedoldi

Published Dec 12, 2000• 7 citations
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Abstract

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 ${\mathrm{Y}}_{2}{\mathrm{BaNiO}}_{5};$ 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 $S=1$ chain.

DOI
10.1103/physrevb.63.012409

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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.
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.
Effects of an uniaxial single-ion anisotropy tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​Dz{\mathit{D}}_{\mathit{z}}Dz​(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2 on the magnetic properties of an S=1 antiferromagnetic Heisenberg chain with open boundaries are studied by a quantum Monte Carlo method. In the ground stae, how the anisotropy affects the staggered magnetization 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the spin correlation functions 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 is investigated. Amplitudes of 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 decrease with with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​. The correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{z}}$ of 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 monotonously decreases with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​, while the correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{x}}$ of 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 has the opposite but weak Dz{\mathit{D}}_{\mathit{z}}Dz​ dependence. The fourfold degeneracy of the ground state also holds with nonzero Dz{\mathit{D}}_{\mathit{z}}Dz​'s in the thermodynamic limit although it is lifted up in finite chains. At finite temperatures, the magnetic susceptibilities along the z axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{z}}$) and the x axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{x}}$) are calculated. Temperature dependences of the bulk susceptibilites, ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{x}}$, are consistent with the previous experimental result on Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​ (NENP). Finite-size corrections, ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{x}}$, and the Curie susceptibilities resulting from chain ends, ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{x}}$, are also investigated in detail. Both ${\mathrm{\ensuremath{\chi}}}_{0}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ diverge as temperature goes to zero and they are combined through a relation ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$=${\mathrm{\ensuremath{\chi}}}_{0}$+${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}$. Temperature dependences of ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ are dominated by the edge effect at low temperatures, while at high temperatures they are attributed to an S=1 free spin.
Properties of the ground state of open chains of the S=1 Heisenberg antiferromagnet are studied by a quantum Monte Carlo method. Size (length) dependences of the ground-state energy, the energy gap, the staggered magnetization 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the spin-correlation function 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 are investigated under free, fixed, and periodic boundary conditions. The size dependences are significant in chains with less than 25 spins, where the magnitude of the staggered magnetization is rather large. However, the fourfold degeneracy of the ground state and the exponential decay of correlations that are inherent to the Haldane system are recovered when the chain length becomes long enough. Both the staggered magnetization and the spin-correlation function decay exponentially. The correlation length is about six in lattice spacing, which agrees with the value obtained in the bulk.
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.
A recent study [J. Lou, et al., Phys. Rev. Lett. 94, 217207 (2005)] reveals that topological edge effects induce bound midgap states in antiferromagnetic Heisenberg chains with a staggered magnetic field. We report density matrix renormalization group calculations that examine the effects of high magnetic field on the gap and midgap states and the range of validity of the field theory prediction on their scaling behavior. We also investigate the effects of the site parity (i.e., even or odd number of sites in the chain), the dilution of the spin chain by dopant atoms, and the uniform magnetic field on the emergence and evolution of the midgap states. The results provide useful insights into the behavior of the edge excitations in spin chain materials, which should facilitate the comparison between theory and experiment and stimulate further investigations.
Magnetic-field tuning of equal-time longitudinal and transverse correlations is investigated for the antiferromagnetic isotropic spin-1/2 Heisenberg chain. Full exact diagonalization of a ten-site periodic ring is used at kBT/J=0.15 over 0≤h/J≤2.6, where h=gμBB. The field breaks spin-rotational symmetry, increases uniform magnetization, redistributes longitudinal and transverse correlation weight, and drives the system toward the fully polarized state above the saturation field hs=2J. Below saturation, longitudinal correlations acquire a uniform m² background and field-dependent oscillatory structure, whereas transverse antiferromagnetic correlations weaken as the density of reversed spins decreases. The finite-temperature magnetization rounds the zero-temperature level crossings and saturation threshold. The numerical results are interpreted using the Tomonaga-Luttinger-liquid description and checked against exact limiting values. The analysis provides a reproducible finite-chain account of field-controlled spin correlations and identifies the distinctions required when connecting static calculations to scattering experiments.