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Physical Review B

Absence of long-range order in an antiferromagnetic chain with long-range interactions: Green’s function approach

Ren-Gui Zhu, An Min Wang

Published Jul 19, 2006• 6 citations
Publisher

Abstract

The antiferromagnetic spin-$1∕2$ chain with power-law decaying interactions $\ensuremath{\mid}{J}_{n}\ensuremath{\mid}\ensuremath{\propto}1∕{n}^{p}$ is studied by a Green's function method. The dispersion relation ${\ensuremath{\omega}}_{k}$ and an expression for the critical temperature ${T}_{N}$ as a function of $p$ are presented. By analyzing the small-$k$ behavior of ${\ensuremath{\omega}}_{k}$, it is shown that there will be no long-range order at finite temperature when $p\ensuremath{\geqslant}2$, which is in good agreement with Bruno's rigorous proof. Our work refurbishes a previous work also by Green's function method, and validates the Tyablikov decoupling approximation to long-range interactions with $1<p<2$.

DOI
10.1103/physrevb.74.012406

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The antiferromagnetic spin-1∕21∕21∕2 chain with power-law decaying interactions \ensuremath∣Jn\ensuremath∣\ensuremath∝1∕np\ensuremath{\mid}{J}_{n}\ensuremath{\mid}\ensuremath{\propto}1∕{n}^{p}\ensuremath∣Jn​\ensuremath∣\ensuremath∝1∕np is studied by a Green's function method. The dispersion relation \ensuremathωk{\ensuremath{\omega}}_{k}\ensuremathωk​ and an expression for the critical temperature TN{T}_{N}TN​ as a function of ppp are presented. By analyzing the small-kkk behavior of \ensuremathωk{\ensuremath{\omega}}_{k}\ensuremathωk​, it is shown that there will be no long-range order at finite temperature when p\ensuremath⩾2p\ensuremath{\geqslant}2p\ensuremath⩾2, which is in good agreement with Bruno's rigorous proof. Our work refurbishes a previous work also by Green's function method, and validates the Tyablikov decoupling approximation to long-range interactions with 1<p<21<p<21<p<2.
The thermodynamic properties of a classical spin-SSS Heisenberg ferromagnetic chain with long-range interactions in the presence of an external magnetic field are investigated by means of the spectral density method within the framework of the two-time Green's functions in classical statistical mechanics. For exchange interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}},$ it is shown that long-range order exists at finite temperature for $1<\ensuremath{\alpha}<2$ and that no transition to a ferromagnetic phase at any finite temperature occurs for $\ensuremath{\alpha}>~2,$ consistently with a recent extension of the Mermin-Wagner theorem to spin models with long-range interactions. Besides, by means of a suitable modification of the lowest-order moment equations for the spectral density, the critical temperature and the main critical exponents for $1<\ensuremath{\alpha}<2$ and the low-temperature behavior of the paramagnetic susceptibility for $\ensuremath{\alpha}>~2$ are also obtained. Our results appear in a reasonable agreement with the available exact ones and Monte Carlo calculations.
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.
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Antiferromagnetic Hamiltonians with short range, non-frustrating interactions are well known to exhibit long range magnetic order in dimensions d ≥2 but exhibit only quasi-long-range order, with power-law decay of correlations, in d = 1 (for half-integer spin). On the other hand, non-frustrating long range interactions can induce long range order in d = 1. We study Hamiltonians in which the long range interactions have an adjustable amplitude λ, as well as an adjustable power law 1/| x | α , using a combination of quantum Monte Carlo and analytic methods: spin-wave, large N non-linear σ model, and renormalization group methods. We map out the phase diagram in the λ–α plane and study the nature of the critical line separating the phases with long range and quasi-long-range order. We find that this corresponds to a novel line of critical points with continuously varying critical exponents and a dynamical exponent, z <1.
We investigate the magnetic susceptibility $\ensuremath{\chi}(T)$ of quantum spin chains of N=1280N=1280N=1280 spins with power-law long-range antiferromagnetic couplings as a function of their spatial decay exponent $\ensuremath{\alpha}$ and cutoff length $\ensuremath{\xi}$. The calculations are based on the strong disorder renormalization method which is used to obtain the temperature dependence of $\ensuremath{\chi}(T)$ and distribution functions of couplings at each renormalization step. For the case with only algebraic decay $(\ensuremath{\xi}=\ensuremath{\infty})$ we find a crossover at ${\ensuremath{\alpha}}^{*}=1.066$ between a phase with a divergent low-temperature susceptibility $\ensuremath{\chi}(T\ensuremath{\rightarrow}0)$ for $\ensuremath{\alpha}>{\ensuremath{\alpha}}^{*}$ to a phase with a vanishing $\ensuremath{\chi}(T\ensuremath{\rightarrow}0)$ for $\ensuremath{\alpha}<{\ensuremath{\alpha}}^{*}$. For finite cutoff lengths $\ensuremath{\xi}$, this crossover occurs at a smaller ${\ensuremath{\alpha}}^{*}(\ensuremath{\xi})$. Additionally, we study the localization of spin excitations for $\ensuremath{\xi}=\ensuremath{\infty}$ by evaluating the distribution function of excitation energies, and we find a delocalization transition that coincides with the opening of the pseudogap at ${\ensuremath{\alpha}}_{c}={\ensuremath{\alpha}}^{*}$.