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Journal of Physics A Mathematical and General

Néel order in the ground state of Heisenberg antiferromagnetic chains with long-range interactions

J. Rodrigo Parreira, Oscar Bolina, J Fernando Perez

Published Feb 21, 1997• 11 citations
Publisher

Abstract

We consider the ground state of one-dimensional antiferromagnets with long-range interactions with Hamiltonian given by where , with J>0. We prove Néel order for all if the spin s is sufficiently large. We also prove the absence of long-range order when for any spin value.
DOI
10.1088/0305-4470/30/4/012

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We investigate the existence of N\'eel long-range order in a spatially anisotropic antiferromagnetic Heisenberg model. As the anisotropical coupling ${J}_{\ensuremath{\perp}}$ increases from 000 to 1,1,1, the model interpolates between an infinite number of decoupled one-dimensional gapped spin chains and a two-dimensional gapless isotropic Heisenberg antiferromagnet. By applying a method based on reflection positivity [F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys. 18,18,18, 335 (1978)], we determine the upper bound of the critical values of ${J}_{\ensuremath{\perp}}^{c}$ such that when ${J}_{\ensuremath{\perp}}>~{J}_{\ensuremath{\perp}}^{c},$ the ground state of this model has an antiferromagnetic long-range order. For the case of spin S=1,S=1,S=1, our result confirms and strengthens previous results derived by numerical calculations on finite-size samples, while for other values of S, our results are new.
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We study antiferromagnetic spin chains with unfrustrated long-range interactions that decay as power laws with exponent $\ensuremath{\beta},$ using the spin-wave approximation. We find for sufficiently large spin S that the Neel order is stable at T=0T=0T=0 for $\ensuremath{\beta}<3,$ and survives up to a finite Neel temperature for $\ensuremath{\beta}<2,$ validating the spin-wave approach in these regimes. We estimate the critical values of S and T for the Neel order to be stable. The spin-wave spectra are found to be gapless but have nonlinear momentum dependence at long wavelength, which is responsible for the suppression of quantum and thermal fluctuations and stabilizing the Neel state. We also show that for $\ensuremath{\beta}<~1$ and for a large but finite-size system size L, the excitation gap of the system approaches zero slower than ${L}^{\ensuremath{-}1},$ a behavior that is in contrast to the Lieb-Schultz-Mattis theorem.
When quantum effects are small, Heisenberg antiferromagnets have a Néel ordered ground state, but strong zero-point fluctuations can lead to new ‘‘exotic’’ spin-liquid ground states without long-range Néel order. These effects are strongest in low dimensions and frustrated systems, and always occur in one dimension. The nature of the new ground state can depend on whether the microscopic spins have integer or half-integer S. As an example of such systems the rich phase diagram of the one-dimensional S=1 Heisenberg antiferromagnet with additional biquadratic exchange and uniaxial on-site anisotropy will be discussed in detail.
It is well known that the ground state of low-dimensional antiferromagnets deviates from Neel states due to strong quantum fluctuations. Even in the presence of long-range order, those fluctuations produce a substantial reduction of the magnetic moment from its saturation value. Numerical simulations in anisotropic antiferromagnetic chains suggest that quantum fluctuations over Neel order appear in the form of localized reversal of pairs of neighboring spins. In this paper, we propose a coherent state representation for the ground state to describe the above situation. In the one-dimensional case, our wave function corresponds to a two-mode Glauber state, when the Neel state is used as a reference, while the boson fields are associated to coherent flip of spin pairs. The coherence manifests itself through the antiferromagnetic long-range order that survives the action of quantum fluctuations. The present representation is different from the standard zero-point spin wave state, and is asymptotically exact in the limit of strong anisotropy. The fermionic version of the theory, obtained through the Jordan-Wigner transformation, is also investigated.