Inverse design of disordered stealthy hyperuniform spin chains

Positioned between crystalline solids and liquids, disordered many-particle systems which are stealthy and hyperuniform represent new states of matter that are endowed with novel physical and thermodynamic properties. Such stealthy and hyperuniform states are unique in that they are transparent to radiation for a range of wave numbers around the origin. In this work, we employ recently developed inverse statistical-mechanical methods, which seek to obtain the optimal set of interactions that will spontaneously produce a targeted structure or configuration as a unique ground state, to investigate the spin-spin interaction potentials required to stabilize disordered stealthy hyperuniform one-dimensional (1D) Ising-type spin chains. By performing an exhaustive search over the spin configurations that can be enumerated on periodic 1D integer lattices containing $N=2,3,...,36$ sites, we were able to identify and structurally characterize all stealthy hyperuniform spin chains in this range of system sizes. Within this pool of stealthy hyperuniform spin configurations, we then utilized such inverse optimization techniques to demonstrate that stealthy hyperuniform spin chains can be realized as either unique or degenerate disordered ground states of radial long-ranged (relative to the spin-chain length) spin-spin interactions. Such exotic ground states appear to be distinctly different from spin glasses in both their inherent structural properties and the nature of the spin-spin interactions required to stabilize them. As such, the implications and significance of the existence of these disordered stealthy hyperuniform ground-state spin systems warrants further study, including whether their bulk physical properties and excited states, like their many-particle system counterparts, are singularly remarkable, and can be experimentally realized.

Figure 1

(a) A visualization of how $\rho \left(k\right)$ cancels at the smallest positive wave number $k=2\pi /N$ for (b) the depicted stealthy hyperuniform spin configuration of size $N=12$ on the 1D integer lattice. The unit vectors represent the positive-spin (${\sigma }_j=+1$) exponential $e^{i2\pi j/N}$ terms from the collective density variables $\rho (k=2\pi /N)={\sum }_{j=1}^N{\sigma }_j{e}^{i2\pi j/N}$. The negative-spin terms also cancel in a similar fashion. The plot contains a gray regular polygon including all possible unit vectors. The unit vectors that are summed for the positive spins in this particular configuration are colored according to how they cancel with other vectors. In this case, the scattering cancellation can be decomposed into two doublets (green and blue) and one triplet (red), which correspond to the indicated spins in (b).

Figure 2

Four periodic spin configurations with $N=30$ sites and $m=0$. The order metric $\tau$ of each configuration is displayed, followed by the spin configuration, and the structure factor. (a)–(c) The top three spin configurations are stealthy hyperuniform with a decreasing degree of order from top to bottom. (d) The bottom configuration is a randomly generated Poisson spin pattern, which is hyperuniform but not stealthy.

Figure 3

(a) The number of stealthy hyperuniform spin configurations for spin configurations of periodic unit-cell sizes between $N=2$ and 36. In gray, for comparison, is the line $2^N$. In black are all of the stealthy hyperuniform configurations. In red are the irreducible stealthy configurations, which cannot be expressed in unit cells smaller than $N$. (b) Probability distributions of the magnetization per spin $m$ for irreducible spin configurations. For comparison, the symmetric binomial distributions about $m=0$ are shown as dashed lines and correspond to the magnetization distributions of all spin configurations enumerated on chains of length $N=18$ and $N=36$.

Figure 4

Histograms representing the probability density of the order metric $\tau$ for stealthy hyperuniform spin chains of size (a) $N=30$ and (b) $N=36$ with $m=0$. Displayed in dashed red lines are histograms of the $\tau$ distribution among all stealthy patterns of the specified sizes, including the reducible configurations which can be expressed in terms of smaller unit cells. Displayed in solid black lines are histograms of the $\tau$ distribution for irreducible stealthy configurations. A few extremely ordered configurations with $\tau >500$ are not shown. Note that there are a total of 315 size $N=30$ stealthy configurations (297 irreducible configurations) and 2306 size $N=36$ stealthy configurations (872 irreducible configurations). Evidently, for these system sizes, there appears to be a minimum value of $\tau$ of about 30, below which there are no stealthy configurations. An example of a highly disordered spin configuration with $\tau \approx 30$ of size 30 is displayed in Fig. 2.

Figure 5

The distribution of $M\left(K\right)$ [number of wave vectors for which $S\left(k\right)=0$ for $k\le K$] for all enumerated stealthy configurations of unit-cell size $N=36$.

Figure 6

(a) The number of irreducible stealthy 1D spin configurations in classes I, II, and III for sizes $N=12,...,36$ plotted on a log-scale. The first size with class II configurations is $N=30$. $N=12$ has one class III configuration. Configurations equivalent by a spin inversion are not counted. (b) The relative amounts of these configurations in classes I, II, and III. Most of the sizes studied have no irreducible stealthy configurations and so are not depicted. Only 10 out of 1068 irreducible stealthy configurations with period $N=30$ and 52 out of 5362 irreducible stealthy configurations with period $N=36$ are class II.

Figure 7

The probability distributions of relative potential cutoffs $R_C/N$, where $R_C$ is an integer between 1 and $N$, for $N=18$ and 36 periodic 1D irreducible stealthy spins systems on the integer lattice. Larger systems tend to have a greater spread in relative potential cutoff centered around lower values. For $N=18$ and 24 (not shown), all potential cutoffs were at least half of the period length $R_C\ge N/2$. While $N=30$ (not shown) and $N=36$ contained many configurations which could be represented as ground states of potentials of length $R_C<N/2$.

Figure 8

Three representative stealthy class I spin configurations of size 30 with $m=0$. Depicted are their unique spin-spin correlations $S_2\left(r\right)$ in blue and their stabilizing potentials $J\left(r\right)$ in red with potential cutoffs (a) $R_C=9$, (b) $R_C=12$, and (c) $R_C=14$. Note that the $J\left(r\right)$ are not unique and that other stabilizing interactions could exist, though not with cutoffs less than the given $R_C$.

Figure 9

All class II stealthy spin configurations of size 30. Five pairs of $S_2\left(r\right)$-degenerate configurations are grouped horizontally.