Inverse design of disordered stealthy hyperuniform spin chains

Abstract

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 wavenumbers 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-like 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 longranged (relative to the spin chain length) spin-spin interactions. Such exotic ground states are 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 such 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.

Hyperuniform states of matter

“…Importantly, conditions under which equilibrium systems are hyperuniform can be derived from this fluctuationcompressibility theorem (78). For example, any ground state (T = 0) in which the isothermal compressibility κ T is bounded and positive must be hyperuniform because the structure factor S (k = 0) must be zero according to relation (78). This includes crystal ground states as well as exotic disordered ground states, such as stealthy ones [62,63,65] discussed in Sec.…”

“…A system at a thermal critical point, such as a liquid-vapor or magnetic critical point, has a fractal structure [11,12], which is characterized by hyperfluctuations, i.e., density fluctuations become unbounded, and in this sense is anti-hyperuniform. For general hyperfluctuating systems, we see from the fluctuation-compressibility theorem (78) thath(k = 0) or, equivalently, the volume integral of h(r) over all space is unbounded. Thus, h(r) is a long-ranged function characterized by a power-law tail that decays to zero slower than |r| −d [8,[10][11][12]171].…”

“…We note in passing that the hyperuniformity concept has recently been generalized to spin systems, including a capability to construct disordered stealthy hyperuniform spin configurations as classical ground states [78]. The discovered exotic disordered spin ground states, which are distinctly different from spin glasses [343], are the spin analogs of disordered stealthy hyperuniform many-particle ground states [62,63,65] that have been shown to be endowed with novel bandgap and wave characteristics [82-85, 87, 93, 95-97, 99].…”

Rational design of stealthy hyperuniform two-phase media with tunable order

“…For concreteness, we focus in this paper on twodimensional (2D) patterns discretized by (square) binary pixels that are arranged on a square (Z 2 ) lattice (subject to periodic boundary conditions along the x-and yaxes). Such patterns can be represented mathematically by σ(m, n), a function which takes two integers as input (m and n, the indices specifying the pixel location in the lattice) and yields a binary output (either 0 or 1) [38][39][40]. Accordingly, we can use this formalism to describe any two-state system, such as up/down spins (i.e., the Ising model [41,42] for ferromagnetism in statistical mechanics), occupancy/vacancy (i.e., the lattice gas model), or phase A/B in the case of digitized two-phase media.…”

Engineering Topological Models with a General-Purpose Symmetry-to-Hamiltonian Approach

“…Inverse methods are widely used throughout machine learning, such as in deep learning [2]. In physics, they have been used in classical systems to design interaction potentials that stabilize crystalline and magnetic order [3][4][5][6] and in quantum systems to design or reconstruct Hamiltonians from eigenstates or density matrices [7][8][9][10][11] as well as to build single-body Hamiltonians compatible with a given symmetry group [12]. The SHC algorithm extends ideas developed in the slow operator method [13] and is quite general.…”