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 anti-ferromagnetic interaction of strength g between the sites i and j, decaying as | i − j | −α , with α > 1. For α smaller than a certain threshold α0, 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 α and the ratio J/g. Our Monte Carlo study shows that the onsite 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 α 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 ∆/T temperature dependence of the correlation length and the uniform ferromagnetic ground state, characteristic of the g = 0 discrete Ising symmetry, are recovered for α > α0.

Periodic Striped Ground States in Ising Models with Competing Interactions

“…The first sum in (1.1) ranges over nearest neighbor pairs in Z d , while the second over pairs of distinct sites in Z d . For different values of the exponent p, this model is used to describe the effects of frustration induced in magnetic films by the presence of dipolar inter-actions (p = 3) or in charged systems by the presence of an unscreened Coulomb interaction (p = 1), as well as many other frustrated systems [1,3,4,5,6,7,8,9,11,22,27,28,30,32,33,34,35,38,41].…”

“…The setting is the following: consider Ising models defined by the formal Hamiltonian [1,[3][4][5][6][7][8][9]11,23,28,29,31,[33][34][35][36]39,42]. In this paper, we choose the exponent p to satisfy the constraint p > 2d.…”

“…where 8) and the * on the last sum indicates the following constraint: if the bubbles in G i are all vertical (resp. horizontal) we only sum over pairs of bubbles that do not overlap after arbitrary translations in the vertical (resp.…”

Formation of Stripes and Slabs Near the Ferromagnetic Transition

“…where the first sum ranges over nearest neighbor pairs in Z d , d = 2, 3, the second over pairs of distinct sites in Z d , and the exponent p is chosen to satisfy p > 2d, for reasons that will become clear below. For more general values of p, this model is used to describe the effects of frustration induced in thin magnetic films by the presence of dipolar interactions (p = 3) or in two-dimensional charged systems by the presence of an unscreened Coulomb interaction (p = 1) [1,2,4,5,6,7,8,9,10,19,21,22,23,24,25,26,27,31,32], see also [3,12,13,14,15,16,17,18] for a more detailed introduction to the subject, as well as for previous rigorous results. The competition between short range ferromagnetic and long-range antiferromagnetic interaction is believed to be responsible for the emergence of non-trivial "mesoscopic patterns" in the ground and low-temperature states of the system.…”

Realization of stripes and slabs in two and three dimensions

“…Particularly interesting is the formation of wide stripes, by which we mean that stripes have a width much larger than the microscopic length scales. Stripes of this kind are expected to display a sort of universal phenomenology, which is in fact observed in a variety of different systems, ranging from magnetic films [1][2][3][4][5], to manganites [6], to high-temperature superconductors [7][8][9][10][11], MOS-FETs [12,13], polymer suspensions [14,15], twinned martensites [16,17], Coulomb glasses [18], and many others [19][20][21][22][23][24][25][26].…”