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arXiv:1305.5323 · Physical Review B

Realization of stripes and slabs in two and three dimensions

Alessandro Giuliani, Élliott H. Lieb, Robert Seiringer

Published Aug 1, 2013• 9 citations
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Abstract

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 $J$ is larger than a critical value ${J}_{c}$, then the ground state is homogeneous and ferromagnetic. As the critical value is approached from smaller values of $J$, it is believed that the ground state consists of a periodic array of stripes ($d=2$) or slabs ($d=3$), 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 $J$ 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.

DOI
10.1103/physrevb.88.064401

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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 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.
Meandering stripes in the frustrated J1J_1J1​-J2J_2J2​ Ising model on the honeycomb lattice
We study the frustrated J1J_1J1​-J2J_2J2​ Ising model on the honeycomb lattice with ferromagnetic nearest-neighbor couplings fixed at J1=1J_1=1J1​=1 and strong antiferromagnetic next-nearest-neighbor interactions, i.e., J2≤−1/4J_2 \leq -1/4J2​≤−1/4. Little is known for this range of J2J_2J2​, whereas for less negative values of J2J_2J2​ the system orders ferromagnetically at low temperatures and appears to remain in the Ising universality class. In previous work it was shown that the model has a largely degenerate ground state, and it was conjectured that there is some kind of phase transition. We introduce a complex-valued nematic order parameter, which can differentiate between the high-temperature paramagnetic phase and the observed partially-disordered stripe phase at lower temperatures. Configurations in this phase consist of stripes of spins parallel with respect to one lattice direction, which collectively meander along the remaining two, producing partially disordered ground states. The sharp peaks in the specific heat observed in earlier work only appear when using periodic boundary conditions and are absent for free boundaries. Additionally, we reveal a striking dependence of the behavior on the aspect ratio of the considered samples. Ultimately, even a careful finite-size scaling analysis for J2=−0.5J_2 = -0.5J2​=−0.5 and J2=−1J_2 = -1J2​=−1 is unable to clearly discern between a crossover without any singularities and some form of continuous transition, including the possibility of an infinite-order transition of the Berezinskii-Kosterlitz-Thouless (BKT) type. For J2=−1/4J_2=-1/4J2​=−1/4 we find that the system remains disordered at all temperatures and that it exhibits a finite ground-state entropy per site, for which our simulations provide the accurate asymptotic estimate S(T=0)/N=0.230 960 93(14)S(T=0)/N = 0.230\,960\,93(14)S(T=0)/N=0.23096093(14) in the thermodynamic limit N→∞N\rightarrow\inftyN→∞.
The critical values of the ratio of interaction energies at which the ordering of spins in the ground state is indeterminate (and probably Tc=0{T}_{c}=0Tc​=0) are calculated exactly for a variety of Ising lattices with antiferromagnetic next-nearest-neighbor coupling, and for anisotropic nearest-neighbor coupling. Using these results and the exact two-dimensional solutions, the general dependence of the critical point on interaction ratio is sketched. The complex case of the antiferromagnetic fcc is reconsidered.
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 $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{0}$.
The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ds=2,{d}_{s}=2,ds​=2, the scaling of the roughness of the domain walls, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}},$ is consistent with the theoretical prediction $\ensuremath{\zeta}=2/3.$ As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent $\ensuremath{\theta},$ which determines the violation of hyperscaling, the correlation length exponent $\ensuremath{\nu},$ and the magnetization exponent $\ensuremath{\beta}.$ The value $\ensuremath{\beta}=0.017\ifmmode\pm\else\textpm\fi{}0.005$ computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.