Ground states of pair potentials with nonnegative compactly supported Fourier transform: the regimes below the threshold density, above close packing, and the bcc–fcc window

For a pair potential $\varphi$ whose Fourier transform is nonnegative and supported in a ball of radius $K_0$, Sütő (Phys. Rev. B 74, 104117 (2006)) classified the infinite-volume ground states at densities $\rho\ge\rho_d$ as exactly the unions of Bravais lattices whose reciprocal vectors all have length at least $K_0$, and proved that adding a nonnegative finite-range repulsion $\psi$ of range $r_0\le\sqrt6\pi/K_0$ makes the ground state unique at the two ends of a degenerate interval: the bcc lattice at $\rho_3$ and the fcc lattice at the close-packing density $\rho_3'=\sqrt2/r_0^3$. The classification stops at three boundaries — densities below $\rho_d$, densities above $\rho_d'$, and, at the maximal range $r_0K_0=\sqrt6\pi$, a density window between the bcc and fcc stability points in which no Bravais lattice is a ground state — and the necessity of its hypotheses is unknown. This problem asks for the ground-state classification in these unresolved regimes (including the conjectured persistence of bcc below $\rho_d$, the conjectured fcc selection above $\rho_d'$, and the nature of the conjectured 'liquid-like' ground state between bcc and fcc).

Background

Consider identical classical particles in $d$-dimensional continuous space $\mathbb{R}^d$ interacting through a translation-invariant pair potential. A ground state configuration (GSC), in the sense fixed by the source paper, is an infinite, locally uniformly finite configuration of particle positions — a locally finite counting measure, i.e. a multiset of positions, so that several particles may sit at the same point (every bounded region contains finitely many particles, counted with multiplicity, with the number in any translate of a given bounded region uniformly bounded) — that is stable against every local perturbation: replacing the particles in any bounded region by any other configuration with the same number of particles, in the presence of the frozen exterior, never decreases the total interaction energy (a chemical-potential variant fixes the grand-canonical form). The energy density and the particle density are limits over growing boxes. The central question of the rigorous crystallization program is which configurations are ground states at a given density $\rho$.

Sütő identified a class of pair potentials for which this question can be answered exactly on a whole density interval. Let $\hat\varphi\in L^1(\mathbb{R}^d)$ be real, even, continuous at the origin, nonnegative, and zero outside the ball $|\mathbf k|\le K_0$, and let $\varphi$ be its inverse Fourier transform. Such a $\varphi$ is bounded, oscillating and long-ranged — in three dimensions its decay can be as slow as $r^{-2}$, and the class contains potentials of asymptotic form $\cos(K_0 r)/r^3$, the same decay as the RKKY (Ruderman–Kittel–Kasuya–Yosida) interaction, the oscillating effective interaction that conduction electrons mediate between ions or localized moments in a metal. The tool is the Poisson summation formula, which for a Bravais lattice $B$ (a discrete lattice generated by $d$ linearly independent vectors) with reciprocal lattice $B^*$ (the lattice of wave vectors whose scalar products with all lattice vectors are integer multiples of $2\pi$) and particle density $\rho(B)$ converts the real-space lattice sum of $\varphi$ into a reciprocal-space sum of $\hat\varphi$:

$$\sum_{\mathbf R\in B}\varphi(\mathbf r+\mathbf R)=\rho(B)\sum_{\mathbf K\in B^*}\hat\varphi(\mathbf K)\,e^{i\mathbf K\cdot\mathbf r}.$$

In the source, the possibly only conditionally convergent lattice sum on the left is defined by a Gaussian-tempered limit, $\lim_{\varepsilon\downarrow0}\sum_{\mathbf R\in B}e^{-\varepsilon|\mathbf r+\mathbf R|^2}\varphi(\mathbf r+\mathbf R)$, and the identity holds whenever $\hat\varphi$ is continuous at every point of $B^*$.

Write $q_{B^*}$ for the length of the shortest nonzero vector of $B^*$ and $r_B$ for the nearest-neighbor distance of $B$. Because $\hat\varphi\ge0$, energies are minimized by configurations whose spectrum avoids the support of $\hat\varphi$: a union of translated Bravais lattices $\cup_j(B_j+\mathbf y_j)$ is a GSC of $\varphi$ whenever every nonzero vector of $\cup_j B_j^*$ has length at least $K_0$ (equality admissible only where $\hat\varphi$ is continuous at $|\mathbf k|=K_0$), and its energy density is $\epsilon(\rho)=\tfrac12\rho[\rho\hat\varphi(\mathbf 0)-\varphi(\mathbf 0)]$, independent of the lattices — a continuously degenerate ground-state set containing periodic and aperiodic configurations. This is possible only for $\rho\ge\rho_d$, the threshold density at and above which the condition $q_{B^*}\ge K_0$ can be met by some Bravais lattice; in the band $\rho_d\le\rho<2\rho_d$ only single Bravais lattices qualify. In three dimensions $\rho_d=\rho_3=K_0^3/(8\sqrt2\,\pi^3)$ and the unique GSC at $\rho_3$ is the body-centered cubic (bcc) lattice, which maximizes $q_{B^*}$ at fixed density.

Now add a nonnegative finite-range repulsion $\psi(\mathbf r)\in[0,\infty]$, zero for $r\ge r_0$, with $r_0\le\gamma_d/K_0$ ($\gamma_1=2\pi$, $\gamma_2=4\pi/\sqrt3$, $\gamma_3=\sqrt6\pi$). For $\varphi+\psi$ the GSCs in the density interval $[\rho_d,\rho_d']$ — $\rho_d'$ being the close-packing density of hard balls of diameter $r_0$, in three dimensions $\rho_3'=\sqrt2/r_0^3$ — are exactly the Bravais lattices satisfying both $r_B\ge r_0$ and $q_{B^*}\ge K_0$. The degeneracy is reduced but stays continuous inside the interval, while at its ends the ground state is unique: bcc at $\rho_3$ and the face-centered cubic (fcc) lattice at $\rho_3'$ (fcc maximizes $r_B$; it is the close packing). The bcc stability interval is $I_{\rm bcc}=[K_0^3/(8\sqrt2\pi^3),\,3\sqrt3/(4r_0^3)]$ and the fcc one is $I_{\rm fcc}=[K_0^3/(6\sqrt3\pi^3),\,\sqrt2/r_0^3]$; they are disjoint when $3\pi/2^{1/3}<r_0K_0\le\sqrt6\pi$, and at the maximal admissible range $r_0K_0=\sqrt6\pi$ both shrink to single points, leaving a window $(\rho_3,\rho_{\rm fcc})$, $\rho_{\rm fcc}=(4\sqrt2/3\sqrt3)\rho_3$, in which no Bravais lattice qualifies at all — the regime the source calls the 'liquid-like' ground state between the fcc and bcc phases.

The class is the rigorous archetype of an active physics literature: configurations whose structure factor $S(\mathbf k)$ (the scaled intensity of the Fourier transform of the point set) vanishes for $0<|\mathbf k|\le K_0$ are called stealthy hyperuniform, and the degenerate ground-state manifold of precisely these potentials has been studied by simulation and ensemble theory for a decade. The classification above nevertheless stops at three boundaries — below $\rho_d$, above $\rho_d'$, and inside the bcc–fcc window — and the necessity of its hypotheses is unknown. Determining the ground states there is the present problem.

Problem Statement

Fix the setting of Sütő (Phys. Rev. B 74, 104117 (2006)). Let $\hat\varphi\in L^1(\mathbb{R}^d)$ be real, even, continuous at the origin, nonnegative, and zero for $|\mathbf k|>K_0$, with $\varphi$ its inverse Fourier transform; let $\psi(\mathbf r)\in[0,\infty]$ vanish for $r\ge r_0$ with $r_0\le\gamma_d/K_0$ ($\gamma_1=2\pi$, $\gamma_2=4\pi/\sqrt3$, $\gamma_3=\sqrt6\pi$). For a Bravais lattice $B$ write $\rho(B)$ for its particle density, $r_B$ for its nearest-neighbor distance, and $q_{B^*}$ for the length of the shortest nonzero vector of its reciprocal lattice $B^*$ (the lattice of wave vectors whose scalar products with all lattice vectors are integer multiples of $2\pi$); write $\rho_d$ for the threshold density — the minimal density at which a Bravais lattice can satisfy $q_{B^*}\ge K_0$ (in three dimensions $\rho_d=\rho_3=K_0^3/(8\sqrt2\pi^3)$, attained by the bcc lattice). Ground state configuration (GSC) means the source's definition: an infinite, locally uniformly finite configuration (every bounded region contains finitely many particles, with the number in any translate of a given bounded region uniformly bounded), stable against every replacement of a finite subset by a finite configuration of the same cardinality, with pair sums understood through the source's summation prescription — where a Bravais-lattice sum is not absolutely convergent it is defined by the Gaussian-tempered limit $\lim_{\varepsilon\downarrow0}\sum_{\mathbf R\in B}e^{-\varepsilon|\mathbf r+\mathbf R|^2}\varphi(\mathbf r+\mathbf R)$, and energies and densities are limits over growing boxes.

Known (proved in the source and its predecessor Letter, and not in question): for $\rho\ge\rho_d$ the GSCs of $\varphi$ are the unions of Bravais lattices whose reciprocal vectors all have length at least $K_0$ (in the source, the exclusion of all other configurations is proved over the full class of locally uniformly finite configurations when $\varphi$ is strongly tempered, and otherwise among unions of periodic configurations); for $\rho\in[\rho_d,\rho_d']$ the GSCs of $\varphi+\psi$ are the Bravais lattices with $r_B\ge r_0$ and $q_{B^*}\ge K_0$, unique bcc at $\rho_3=K_0^3/(8\sqrt2\pi^3)$ and unique fcc at $\rho_3'=\sqrt2/r_0^3$ in three dimensions.

Determine the ground states in the regimes where this classification does not reach:

1. Below the threshold: characterize the GSCs of $\varphi$ at densities $\rho<\rho_d$. In particular, decide whether the bcc lattice — the unique ground state at $\rho_3$ — persists (up to Euclidean motions) to lower densities for some subclass of admissible $\hat\varphi$, or whether some other structure, periodic or aperiodic, with or without superposed particles (several particles sitting at the same point), has strictly lower energy. The characterization is demanded for every admissible $\hat\varphi$; the clause 'for some subclass of admissible $\hat\varphi$' is an existential sub-question, settled by exhibiting a single admissible $\hat\varphi$ for which either alternative holds, and answering it does not by itself yield the characterization.

2. Above close packing: characterize the GSCs of $\varphi+\psi$ at densities $\rho>\rho_d'$, where $\rho_d'$ is the close-packing density of hard balls of diameter $r_0$. In particular, decide whether the continuously degenerate set below $\rho_d'$ gives way to a unique ground state, and whether for a subclass of positive finite-range $\psi$ that unique ground state is the fcc lattice. The characterization is demanded for every admissible $\hat\varphi$ and every finite-valued (hard-core-free) $\psi$; a hard-core $\psi$, taking the value $\infty$, gives infinite energy to every configuration of density above $\rho_d'$, so its ground-state set is empty there. The sub-questions are separate claims, the clause 'for a subclass of positive finite-range $\psi$' being existential and settled by a single admissible finite-valued $\psi$ that selects the fcc lattice, and answering the sub-questions does not by itself yield the characterization.

3. The liquid-like window: in three dimensions at the maximal admissible range $r_0K_0=\sqrt6\pi$, no Bravais lattice satisfies both conditions for $\rho_3<\rho<\rho_{\rm fcc}=(4\sqrt2/3\sqrt3)\rho_3$. Determine the ground state in this window: whether it exists at all within the locally uniformly finite configurations, whether it is unique, and whether it is a non-Bravais periodic structure (for example a multi-lattice, a finite union of translated Bravais lattices), an aperiodic configuration, or a 'liquid-like' state that changes continuously from bcc to fcc as the density increases.

The common objective of parts 1–3 is to complete the zero-temperature phase diagram of this interaction class: a classification of ground-state configurations across all densities. Answers must hold at this generality; in particular, a classification restricted to Bravais or periodic configurations is acceptable only if the restriction is itself proved for the regime in question.

References

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