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

TL;DR

This study investigates a one-dimensional Ising model with competing short-range ferromagnetic and long-range anti-ferromagnetic interactions, revealing complex ground states and correlation behaviors without phase transitions at finite temperatures.

Contribution

It introduces a detailed analysis of the effects of long-range interactions on the Ising chain, including the emergence of modulated ground states and the temperature dependence of correlations.

Findings

Ground state consists of segments with alternating magnetization for certain alpha.

No phase transition occurs at finite temperatures, magnetization vanishes.

Correlation functions show oscillations with temperature-dependent periods.

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 anti-ferromagnetic interaction of strength g between the sites i and j, decaying as |i-j|^-alpha, with alpha>1. For alpha smaller than a certain threshold 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 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 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^(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 alpha > alpha_0.