One-dimensional Ising ferromagnet frustrated by long-range interactions at finite temperatures

Abstract

We consider a one-dimensional lattice of Ising-type variables where the ferromagnetic exchange interaction $J$ between neighboring sites is frustrated by a long-ranged antiferromagnetic interaction of strength $g$ between the sites $i$ and $j$, decaying as ${|i\ensuremath{-}j|}^{\ensuremath{-}\ensuremath{\alpha}}$, with $\ensuremath{\alpha}>1$. For $\ensuremath{\alpha}$ smaller than a certain threshold ${\ensuremath{\alpha}}_{0}$, which is larger than 2 and depends on the ratio $J/g$, the ground state consists of an ordered sequence of segments with equal length and alternating magnetization. The width of the segments depends on both $\ensuremath{\alpha}$ and the ratio $J/g$. Our Monte Carlo study shows that the on-site magnetization vanishes at finite temperatures and finds no indication of any phase transition. Yet, the modulation present in the ground state is recovered at finite temperatures in the two-point correlation function, which oscillates in space with a characteristic spatial period. The latter depends on $\ensuremath{\alpha}$ and $J/g$ and decreases smoothly from the ground-state value as the temperature is increased. Such an oscillation of the correlation function is exponentially damped over a characteristic spatial scale, the correlation length, which asymptotically diverges roughly as the inverse of the temperature as $T=0$ is approached. This suggests that the long-range interaction causes the Ising chain to fall into a universality class consistent with an underlying continuous symmetry. The ${e}^{\ensuremath{\Delta}/T}$ temperature dependence of the correlation length and the uniform ferromagnetic ground state, characteristic of the $g=0$ discrete Ising symmetry, are recovered for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{0}$.