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arXiv:1106.0922 · Physical Review B

Checkerboards, stripes, and corner energies in spin models with competing interactions

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

Published Aug 30, 2011• 39 citations
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Abstract

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 $J$. 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 $J$ is close to ${J}_{c}$, 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 ${r}^{\ensuremath{-}p}$, $p>2$, 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 $p>3$, and for $J$ 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.

DOI
10.1103/physrevb.84.064205

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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{r}^{\ensuremath{-}p}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≲JcJ\ensuremath{\lesssim}{J}_{c}J\ensuremath≲Jc​ if p>3p>3p>3, and for JJJ sufficiently large if 2<p\ensuremath⩽32<p\ensuremath{\leqslant}32<p\ensuremath⩽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 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 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 ${r}^{\ensuremath{-}p}$, 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 $d<p\ensuremath{\leqslant}d+1$ 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 $0<p\ensuremath{\leqslant}1$ 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 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 have studied the phase diagrams of the one-dimensional spin-1 Blume-Capel model with anisotropy constant D, in which equivalent-neighbor ferromagnetic interactions of strength -J are superimposed on nearest-neighbor antiferromagnetic interactions of strength K. A rich critical behavior is found due to the competing interactions. At zero temperature two ordered phases exist in the D/J-K/J plane, namely the ferromagnetic (F) and the antiferromagnetic one (AF). For lower values of D/J(D/J<0.25) these two ordered phases are separated by the point K_{c}=0.25J. For 0.25<D/J≤0.50, the paramagnetic phase P emerges in a region separated between the lines determined by D/J=0.5-K/J and D/J=K/J. For D/J>0.5, only phases AF and F exist and are separated by a line given by D/J=K/J. At finite temperatures, we found that the ferromagnetic region of the phase diagram in the k_{B}T/J-D/J plane is enriched by another ferromagnetic phase F^{^{'}} above a first-order line for 0.195<K/J<0.250. This first-order line, which separates phases F and F^{^{'}}, begins at a coexistence point, where phases F,F^{^{'}}, and P coexist, and ends at an ordered critical point. Similarly, we found that the phase F^{^{'}} is present in the phase diagram in the k_{B}T/J-K/J plane for 0.228<D/J<0.286.
Due to intrinsic frustrations of interaction, the 2d Ising model with competing ferromagnetic short-range nearest-neighbour and antiferromagnetic long-range dipole interactions possesses a rich phase diagram. The order of the phase transition from the striped h = 1 phase to the tetragonal phase that is observed in this model has been a subject of recent controversy. We address this question by using the partition function density analysis in the complex temperature plane. Our results support the second-order phase transition scenario. To measure the strength of the phase transition, we calculate the values of specific heat critical exponent a. Along with the space dimension, it appears to depend on the ratio of strengths of the short-range and long-range interactions.
The d=1 Ising ferromagnet and spin glass with long-range power-law interactions J r^-a is studied for all interaction range exponents a by a renormalization-group transformation that simultaneously projects local ferromagnetism and antiferromagnetism. In the ferromagnetic case, J>0, a finite-temperature ferromagnetic phase occurs for interaction range 0.742, the phase transition temperature discontinuously drops to zero and for a>2 there is no ordered phase above zero temperature, also as predicted by rigorous results. At the other end, on approaching a=0.74 from above, namely increasing the range of the interaction, the phase transition temperature diverges to infinity, meaning that, at all non-infinite temperatures, the system is ferromagnetically ordered. Thus, the equivalent-neighbor interactions regime is entered before (a > 0) the neighbors become equivalent, namely before the interactions become equal for all separations. The critical exponents alpha,beta, gamma,delta,eta,nu are calculated, from a large recursion matrix, varying as function of a. For antiferromagnetic J<0, all triplets of spins at all ranges have competing interactions and this highly frustrated system does not have an ordered phase. In the spin-glass system, where all couplings for all separations are randomly ferromagnetic or antiferromagnetic (with probability p), a finite-temperatures spin-glass phase is obtained in the absence of antiferromagnetic phase. In the spin-glass phase, the signature chaotic behavior under scale change occurs in a richer version than previously: In the long-range interaction of this system, the interactions at every separation become chaotic, yielding a piecewise chaotic interaction function.
We study an Ising model in one dimension with short-range ferromagnetic and long-range (power law) antiferromagnetic interactions. We show that the zero temperature phase diagram in a (longitudinal) field HHH involves a sequence of up and down domains whose size varies continuously with HHH, between $\ensuremath{-}{H}_{c}$ and Hc{H}_{c}Hc​, which represent the edges of the ferromagnetic up and down phases. The implications of long-range interaction in many body systems are discussed.