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arXiv:cond-mat/0604668 · Physical Review B

Ising models with long-range antiferromagnetic and short-range ferromagnetic interactions

Alessandro Giuliani, Joel L. Lebowitz, Élliott H. Lieb

Published Aug 25, 2006• 70 citations
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Abstract

We study the ground state of a $d$-dimensional Ising model with both long-range (dipole-like) and nearest-neighbor ferromagnetic (FM) interactions. The long-range interaction is equal to ${r}^{\ensuremath{-}p}$, $p>d$, while the FM interaction has strength $J$. If $p>d+1$ and $J$ is large enough the ground state is FM, while if $d<p\ensuremath{\leqslant}d+1$ the FM state is not the ground state for any choice of $J$. In $d=1$ we show that for any $p>1$ the ground state has a series of transitions from an antiferromagnetic state of period 2 to $2h$-periodic states of blocks of sizes $h$ with alternating sign, the size $h$ growing when the FM interaction strength $J$ is increased (a generalization of this result to the case $0<p\ensuremath{\leqslant}1$ is also discussed). In $d\ensuremath{\geqslant}2$ we prove, for $d<p\ensuremath{\leqslant}d+1$, that the dominant asymptotic behavior of the ground-state energy agrees for large $J$ with that obtained from a periodic striped state conjectured to be the true ground state. The geometry of contours in the ground state is discussed.

DOI
10.1103/physrevb.74.064420

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We study the ground state of a ddd-dimensional Ising model with both long-range (dipole-like) and nearest-neighbor ferromagnetic (FM) interactions. The long-range interaction is equal to r\ensuremath−p{r}^{\ensuremath{-}p}r\ensuremath−p, p>dp>dp>d, while the FM interaction has strength JJJ. If p>d+1p>d+1p>d+1 and JJJ is large enough the ground state is FM, while if d<p\ensuremath⩽d+1d<p\ensuremath{\leqslant}d+1d<p\ensuremath⩽d+1 the FM state is not the ground state for any choice of JJJ. In d=1d=1d=1 we show that for any p>1p>1p>1 the ground state has a series of transitions from an antiferromagnetic state of period 2 to 2h2h2h-periodic states of blocks of sizes hhh with alternating sign, the size hhh growing when the FM interaction strength JJJ is increased (a generalization of this result to the case 0<p\ensuremath⩽10<p\ensuremath{\leqslant}10<p\ensuremath⩽1 is also discussed). In d\ensuremath⩾2d\ensuremath{\geqslant}2d\ensuremath⩾2 we prove, for d<p\ensuremath⩽d+1d<p\ensuremath{\leqslant}d+1d<p\ensuremath⩽d+1, that the dominant asymptotic behavior of the ground-state energy agrees for large JJJ with that obtained from a periodic striped state conjectured to be the true ground state. The geometry of contours in the ground state is discussed.
We consider Ising models in two and three dimensions with nearest neighbor ferromagnetic interactions and long-range, power law decaying, antiferromagnetic interactions. If the strength of the ferromagnetic coupling JJJ is larger than a critical value Jc{J}_{c}Jc​, then the ground state is homogeneous and ferromagnetic. As the critical value is approached from smaller values of JJJ, it is believed that the ground state consists of a periodic array of stripes (d=2d=2d=2) or slabs (d=3d=3d=3), all of the same size and alternating magnetization. Here we prove rigorously that the ground state energy per site converges to that of the optimal periodic striped or slabbed state, in the limit that JJJ tends to the ferromagnetic transition point. While this theorem does not prove rigorously that the ground state is precisely striped or slabbed, it does prove that in any suitably large box the ground state is striped or slabbed with high probability.
We study the zero-temperature phase diagram of Ising spin systems in two dimensions in the presence of competing interactions: long-range antiferromagnetic and nearest-neighbor ferromagnetic of strength JJJ. We first introduce the notion of a ``corner energy,'' which shows, when the antiferromagnetic interaction decays faster than the fourth power of the distance, that a striped state is favored with respect to a checkerboard state when JJJ is close to Jc{J}_{c}Jc​, the transition to the ferromagnetic state, i.e., when the length scales of the uniformly magnetized domains become large. Next, we perform detailed analytic computations on the energies of the striped and checkerboard states in the cases of antiferromagnetic interactions with exponential decay and with power-law decay ${r}^{\ensuremath{-}p}$, p>2p>2p>2, which depend on the Manhattan distance instead of the Euclidean distance. We prove that the striped phase is always favored compared to the checkerboard phase when the scale of the ground-state structure is very large. This happens for $J\ensuremath{\lesssim}{J}_{c}$ if p>3p>3p>3, and for JJJ sufficiently large if $2<p\ensuremath{\leqslant}3$. Many of our considerations involving rigorous bounds carry over to dimensions greater than two and to more general short-range ferromagnetic interactions.
We consider a one-dimensional lattice of Ising-type variables where the ferromagnetic exchange interaction JJJ between neighboring sites is frustrated by a long-ranged antiferromagnetic interaction of strength ggg between the sites iii and jjj, 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/gJ/gJ/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/gJ/gJ/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/gJ/gJ/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=0T=0T=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=0g=0g=0 discrete Ising symmetry, are recovered for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{0}$.
It is known that, after a quench to zero temperature (T=0), two-dimensional (d=2) Ising ferromagnets with short-range interactions do not always relax to the ordered state. They can also fall in infinitely long-lived striped metastable states with a finite probability. In this paper, we study how the abundance of striped states is affected by long-range interactions. We investigate the relaxation of d=2 Ising ferromagnets with power-law interactions by means of Monte Carlo simulations at both T=0 and T≠0. For T=0 and the finite system size, the striped metastable states are suppressed by long-range interactions. In the thermodynamic limit, their occurrence probabilities are consistent with the short-range case. For T≠0, the final state is always ordered. Further, the equilibration occurs at earlier times with an increase in the strength of the interactions.
The question is posed of how the ground state of the Heisenberg Hamiltonian $H=\ensuremath{-}\ensuremath{\Sigma}{F}_{\mathrm{ij}}{\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{j}$ depends on the magnitude sss of NNN interacting spins, particularly in the case of long-ranged oscillatory interactions Fij{F}_{\mathrm{ij}}Fij​. It is discussed whether fixing the geometry and the bond strengths Fij{F}_{\mathrm{ij}}Fij​ suffices to determine the nature of the spin correlations in the ground state, and a review is given of known instances when this is the case; those are special situations in ferromagnetism and antiferromagnetism when qualitative ground-state properties such as "lack of nodes" can be proved to be independent of sss. These are valuable examples for the application of semiclassical methods, which are strictly valid only for $s\ensuremath{\rightarrow}\ensuremath{\infty}$ and depend on the convergence of a series in powers of ${s}^{\ensuremath{-}1}$. But these examples are, after all, only special cases, and it is argued that, in general, the nature of the ground state can depend sensitively on sss. The following situation is considered in some detail: An oscillatory interaction which leads to a ferromagnetic ground state in the correspondence limit $s\ensuremath{\gg}1$, but for which the ferromagnetic state of saturation magnetization may be unstable for small quantum mechanical spins, e.g., s=12or1s=\frac{1}{2} or 1s=21​or1. Two distinct types of interaction are considered which lead to this result, and it is seen that the ferromagnetic instability is a consequence, not so much of the long range of the interaction as of the presence of some relatively strong antiferromagnetic (negative) bonds. However, the variational approach which is used casts no light on the nature of the true ground state or of the thermal properties, problems which are increasingly interesting in these instances when semiclassical procedures are seen to fail.