Universality of Striped Morphologies

We present a method for predicting the low-temperature behavior of spherical and Ising spin models with isotropic potentials. For the spherical model the characteristic length scales of the ground states are exactly determined but the morphology is shown to be degenerate with checkerboard patterns, stripes and more complex morphologies having identical energy. For the Ising models we show that the discretization breaks the degeneracy causing striped morphologies to be energetically favored and therefore they arise universally as ground states to potentials whose Hankel transforms have nontrivial minima.

Figure 1

Reflection symmetry of triangles on a lattice causes the interaction matrices from any two radial potentials to commute.

Figure 2

Examples of eigenmodes of the two dimensional spherical model. (a)–(c) Any phase shift of a ground state is a new ground state, so checkerboards, stripes and everything between can be produced by the same model. Linear combinations of $f_{(2,2)}$ with different phase shifts are shown, all having the same energy. (d) Exchanging the elements of $\overrightarrow{k}$ gives a new ground state and linear combinations of them give rise to complex morphologies. Shown is $\frac{1}{2}{f}_{(3,4)}+\frac{1}{2}{f}_{(4,3)}$.

Figure 3

Predicting ground states in different two dimensional Ising models. (Top) Potentials (inset) and their energy spectra in $|\overrightarrow{k}|$-space from the transform (4). (a) A purely repulsive potential [18], (b) two competing interactions [12] [see Eq. (1)], and (c) an RKKY-like interaction [29]. For small $k$ (long wavelengths) the spectra only depend on the magnitude of $\overrightarrow{k}$, but for large $k$ (short wavelengths) lattice effects breaks the independence on the direction of $\overrightarrow{k}$ and $E(|\overrightarrow{k}|)$ becomes multivalued. To illustrate this, we link series with constant $k_y$ with lines and in (a) examples of such states are shown. (Below) Local minima as arrived by through Monte Carlo annealing as well as ground states of Ising spin-$1/2$ models with corresponding potentials.

Figure 4

Designing interactions to imitate observed striped patterns. (a) Stripes in a ferrofluid confined between two glass plates in a magnetic field [from [1, 6], reprinted with permission from AAAS and Elsevier] (b), the (negative) radial power spectrum (blue) of previous picture together with a Gaussian (red), (inset) a pair potential $V(r)$ constructed as the (inverse) transform (4) of said Gaussian and (c), metastable state of an Ising spin-$1/2$ model with potential $V(r)$. Note that the chosen energy spectrum is not unique. Many potentials having a spectrum minimized at the same wavelength will show similar low-temperature behavior.