Duality Relations for the Classical Ground States of SoftMatter Systems
Abstract
Bounded interactions are particularly important in softmatter systems, such as colloids, microemulsions, and polymers. We derive new duality relations for a class of soft potentials, including threebody and higherorder functions, that can be applied to ordered and disordered classical ground states. These duality relations link the energy of configurations associated with a realspace potential to the corresponding energy of the dual (Fouriertransformed) potential. We apply the duality relations by demonstrating how information about the classical ground states of shortranged potentials can be used to draw new conclusions about the ground states of longranged potentials and vice versa. The duality relations also lead to bounds on the system energies in density intervals of phase coexistence. Additionally, we identify classes of “selfsimilar” potentials, for which one can relate low and highdensity groundstate energies. We analyze the ground state configurations and thermodynamic properties of a onedimensional system previously thought to exhibit an infinite number of structural phase transitions and comment on the known ground states of purely repulsive monotonic potentials in the context of our duality relations.
pacs:
05.20.y, 82.35.Jk,82.70.Dd 61.50.AhI Introduction
The determination of the classical ground states of interacting manyparticle systems (minimum energy configurations) is a subject of ongoing investigation in condensedmatter physics and materials science Ul68 ; Ra87 . While such results are readily produced by slowly freezing liquids in experiments and computer simulations, our theoretical understanding of classical ground states is far from complete. Much of the progress to rigorously identify ground states for given interactions has been for lattice models, primarily in one dimension Ra87 . The solutions in dimensional Euclidean space for are considerably more challenging. For example, the ground state(s) for the wellknown LennardJones potential in or are not known rigorously LJ . Recently, a “collectivecoordinate” approach has been employed to study and ascertain ground states in two and three dimensions for a certain class of interactions Uc04 ; Su05 . A surprising conclusion of Ref. Uc04 is that there exist nontrivial disordered classical ground states without any longrange order footnote , in addition to the expected periodic ones. Despite these advances, new theoretical tools are required to make further progress in our understanding of classical ground states.
In a recent Letter, we derived duality relations for a certain class of soft pair potentials that can be applied to classical ground states whether they are disordered or not To08 . Soft interactions are considered because, as we will see, they are easier to treat theoretically and possess great importance in softmatter systems, such as colloids, microemulsions, and polymers Ru89 ; La00 ; Ml06 ; Ka07 . These duality relations link the energy of configurations associated with a realspace pair potential to the energy associated with the dual (Fouriertransformed) potential. Duality relations are useful because they enable one to use information about the ground states of certain soft shortranged potentials to draw conclusions about the nature of the ground states of longranged potentials and vice versa. The duality relations also lead to bounds on the zerotemperature energies in density intervals of phase coexistence.
In the present paper, we amplify and extend the results of Ref. To08 . We also study in detail a onedimensional system that Torquato and Stillinger claimed to possess an infinite number of structural phase transitions from Bravais to nonBravais lattices at as the density is changed To08 . A general set of potential functions that are selfsimilar under Fourier transform are described and studied. We also derive the generalizations of the duality relations for threebody as well as higherorder interactions.
Ii Definitions and Preliminaries
A point process in is a distribution of an infinite number of points at number density (number of points per unit volume) with configuration ; see Ref. To06b for a precise mathematical definition. It is characterized by a countably infinite set of particle generic probability density functions , which are proportional to the probability densities of finding collections of particles in volume elements near the positions . For a general point process, it is convenient to introduce the particle correlation functions , which are defined by
(1) 
Since for a completely uncorrelated point process, it follows that deviations of from unity provide a measure of the correlations between points in a point process. Of particular interest is the pair correlation function, which for a translationally invariant point process of density can be written as
(2) 
Closely related to the pair correlation function is the total correlation function, denoted by ; it is derived from via the equation
(3) 
Since as () for translationally invariant systems without longrange order, it follows that in this limit, meaning that is generally an function, and its Fourier transform is welldefined.
It is common in statistical mechanics when passing to reciprocal space to consider the associated structure factor , which for a translationally invariant system is defined by
(4) 
where is the Fourier transform of the total correlation function, is the number density, and is the magnitude of the reciprocal variable to . The dimensional Fourier transform of any integrable radial function is
(5) 
and the inverse transform of is given by
(6) 
Here is the wavenumber (reciprocal variable) and is the Bessel function of order .
A special point process of central interest in this paper is a lattice. A lattice in is a subgroup consisting of the integer linear combinations of vectors that constitute a basis for , i.e., the lattice vectors ; see Ref. To03 for details. In a lattice , the space can be geometrically divided into identical regions called fundamental cells, each of which corresponds to just one point as in Figure 1. In the physical sciences, a lattice is equivalent to a Bravais lattice. Unless otherwise stated, for this situation we will use the term lattice. Every lattice has a dual (or reciprocal) lattice in which the sites of that lattice are specified by the dual (reciprocal) lattice vectors , where . The dual fundamental cell has volume , where is the volume of the fundamental cell of the original lattice , implying that the respective densities and of the real and dual lattices are related by . A periodic point process, or nonBravais lattice, is a more general notion than a lattice because it is is obtained by placing a fixed configuration of points (where ) within one fundamental cell of a lattice , which is then periodically replicated (see Figure 1). Thus, the point process is still periodic under translations by , but the points can occur anywhere in the chosen fundamental cell. Although generally a nonBravais lattice does not have a dual, certain periodic point patterns are known to possess formally dual nonBravais lattices. Roughly speaking, two nonBravais lattices are formal duals of each other if their average pair sums (total energies per particle) obey the same relationship as Poisson summation for Bravais lattices for all admissible pair interactions; for further details, the reader is referred to CoKuSc09 .
Iii Duality Relations
iii.1 Pair Potentials
For a configuration of particles in a bounded volume with stable pairwise interactions, the manybody function
(7) 
is twice the total potential energy per particle [plus the “selfenergy” ], where is a radial pair potential function and . A pair interaction is stable provided that for all and all . A nonnegative Fourier transform implies stability, but this is a stronger condition than the former footnote1 . A classical groundstate configuration (structure) within is one that minimizes . Since we will allow for disordered ground states, then we consider the general ensemble setting that enables us to treat both disordered as well as ordered configurations. The ensemble average of for a statistically homogeneous and isotropic system in the thermodynamic limit is given by
(8) 
where is the number density and is the pair correlation function. In what follows, we consider those stable radial pair potentials that are bounded and absolutely integrable. We call such functions admissible pair potentials. Therefore, the corresponding Fourier transform exists, which we also take to be admissible, and
(9) 
Lemma. For any ergodic configuration in , the following duality relation holds:
(10) 
If such a configuration is a ground state, then the left and right sides of (10) are minimized. Proof: We assume ergodicity, i.e., the macroscopic properties of any single configuration in the thermodynamic limit with constant are equal to their ensembleaverage counterparts. The identity (10) follows from Plancherel’s theorem, assuming that exists. It follows from (9) and (10) that both sides of (10) are minimized for any groundstate structure, although the duality relation (10) applies to general (i.e., nongroundstate) structures.
Remarks:

The general duality relation (10) does not seem to have been noticed or exploited before, although it was used for a specific pair interaction in Ref. To03 . The reason for this perhaps is due to the fact that one is commonly interested in the total energy or, equivalently, the integral of (8) for which Plancherel’s theorem cannot be applied because the Fourier transform of does not exist.

It is important to recognize that whereas always characterizes a point process To06b , its Fourier transform is generally not the total correlation function of a point process in reciprocal space. It is when characterizes a Bravais lattice (a special point process) that is the total correlation function of a point process, namely the reciprocal Bravais lattice .

The ensembleaveraged structure factor is related to the collective density variable via the expression .

On account of the “uncertainty principle” for Fourier pairs, the duality relation (10) provides a computationally fast and efficient way of computing energies per particle of configurations for a nonlocalized (longranged) potential, say , by evaluating the equivalent integral in reciprocal space for the corresponding localized (compact) dual potential .
Theorem 1. If an admissible pair potential has a Bravais lattice groundstate structure at number density , then we have the following duality relation for the minimum of :
(11) 
where the prime on the sum denotes that the zero vector should be omitted, denotes the reciprocal Bravais lattice footnote3 , and is the dual pair potential, which automatically satisfies the stability condition, and therefore is admissible. Moreover, the minimum of for any groundstate structure of the dual potential , is bounded from above by the corresponding realspace minimized quantity or, equivalently, the right side of (11), i.e.,
(12) 
Whenever the reciprocal lattice at reciprocal lattice density is a ground state of , the inequality in (12) becomes an equality. On the other hand, if an admissible dual potential has a Bravais lattice at number density , then
(13) 
where equality is achieved when the realspace ground state is the lattice reciprocal to .
Proof: The radially averaged total correlation function for a Bravais lattice, which we now assume to be a groundstate structure, is given by
(14) 
where is the surface area of a dimensional sphere of radius , is the coordination number (number of points) at the radial distance , and is a radial Dirac delta function. Substitution of this expression and the corresponding one for into (10) yields
(15) 
where is the coordination number in the reciprocal lattice at the radial distance . Recognizing that [equal to twice the minimized energy per particle given by (7) at its minimum in the limit ] and yields the duality relation (11). The fact that is stable footnote1 means that the dual potential is stable since the left side of (15) is nothing more than the sum given in Ref. footnote1 in the limit , which must be nonnegative. However, the minimum is generally not equal to the corresponding minimum associated with the ground state of the dual potential , i.e., there may be periodic structures that have lower energy than the reciprocal lattice so that . To prove this point, notice that for any nonBravais lattice by definition obeys the inequality . However, because the corresponding Fourier transform of total correlation function of the nonBravais lattice in real space generally does not correspond to a point process in reciprocal space (see Remark 2 under Lemma 1), we cannot eliminate the possibilities that there are nonBravais lattices in reciprocal space with lower than . Therefore, the inequality of (12) holds in general with equality applying whenever the ground state structure for the dual potential is the Bravais lattice at density . Inequality (13) follows in the same manner as (12) when the ground state of the dual potential is known to be a Bravais lattice.
Remarks:

Whenever equality in relation (12) is achieved, then a ground state structure of the dual potential evaluated at the realspace variable is the Bravais lattice at density .

The zerovector contributions on both sides of the duality relation (11) are crucial in order to establish a relationship between the real and reciprocalspace “lattice” sums indicated therein. To emphasize this point, consider in the wellknown Yukawa (screenedColoumb) potential , which has the dual potential . At first glance, this potential would seem to be allowable because the realspace lattice sum, given on the left side of (11), is convergent. However, the reciprocalspace lattice sum on the right side does not converge. This nonconvergence arises because is unbounded. Equality of “infinities” is established, but of course this is of no practical value and is the reason why we demand that an admissible potential be bounded.
Theorem 2. Suppose that for admissible potentials there exists a range of densities over which the ground states are side by side coexistence of two distinct structures whose parentage are two different Bravais lattices, then the strict inequalities in (12) and (13) apply at any density in this densitycoexistence interval.
Proof: This follows immediately from the Maxwell doubletangent construction in the  plane, which ensures that the energy per particle in the coexistence region at density is lower than either of the two Bravais lattices.
As we will see, the duality relations of Theorem 1 will enable one to use information about ground states of shortranged potentials to draw new conclusions about the nature of the ground states of longranged potentials and vice versa. Moreover, inequalities (12) and (13) provide a computational tool to estimate groundstate energies or eliminate candidate groundstate structures as obtained by annealing in Monte Carlo and molecular dynamics simulations. In the ensuing discussion, we will examine the ground states of several classes of admissible functions, focusing under what conditions the equalities or strict inequalities of the duality relations (12) and (13) apply.
iii.2 ThreeBody and HigherOrder Interactions
The aforementioned analysis can be extended to establish duality relations for manyparticle systems interacting via threebody and higherorder interactions. For simplicity of exposition, we begin with a detailed construction of the threebody duality relations and then generalize to the higherorder case.
We consider a statistically homogeneous particle interaction with one, two, and threebody contributions , , and , respectively. With the convention that and , we may write
(16) 
where is symmetric, bounded, and shortranged. Taking the ensemble average of this function implies
(17) 
involving averages over single particles, pairs, and triads. Duality relations for the former two contributions have already been considered, and we therefore direct our attention to the last term in (17).
Since as , this function is generally not integrable, and we therefore introduce the associated threebody total correlation function . Application of a double Fourier transform and Plancherel’s theorem implies the following threebody analog of the Lemma (10):
(18) 
where
(19) 
One can verify directly that the following relationship defines the threebody correlation function for any statistically homogeneous particle point pattern:
(20) 
For a Bravais lattice, ergodicity should hold, and we can rewrite (20) as
(21) 
where the set in the summations includes all points of the lattice excluding the origin.
The dual Bravais lattice will possess a threeparticle correlation function of the form , where is the real space number density. Substituting (21) and the corresponding for the dual Bravais lattice into (18) gives the following duality relation for threeparticle interactions:
(22) 
where we have defined .
The extension of this analysis to higherorder interactions is straightforward. Specifically, we consider a particle bounded, symmetric, and shortranged potential with a statistically homogeneous point distribution and the associated Plancherel identity
(23) 
The particle correlation function of a Bravais lattice is
(24) 
where denotes all sets of distinct vectors in a Bravais lattice, excluding the origin, and indexes the lattice points. Using this relationship, we find the following general particle duality relation:
(25) 
where .
Iv Applications
iv.1 Admissible functions with compact support
Recently, the ground states have been studied corresponding to a certain class of oscillating realspace potentials as defined by the family of Fourier transforms with compact support such that is positive for and zero otherwise Uc04 ; Su05 . Clearly, is an admissible pair potential. Sütő Su05 showed that in three dimensions the corresponding realspace potential , which oscillates about zero, has the bodycentered cubic (bcc) lattice as its unique ground state at the realspace density (where we have taken ). Moreover, he demonstrated that for densities greater than , the ground states are degenerate such that the facecentered cubic (fcc), simple hexagonal (sh), and simple cubic (sc) lattices are ground states at and above the respective densities , , and .
The longrange behavior of the realspace oscillating potential might be regarded to be unrealistic by some. However, since all of the aforementioned ground states are Bravais lattices, the duality relation (11) can be applied here to infer the ground states of realspace potentials with compact support. Specifically, application of the duality theorem in and Sütő’s results enables us to conclude that for the realspace potential that is positive for and zero otherwise, the fcc lattice (dual of the bcc lattice) is the unique ground state at the density and the ground states are degenerate such that the bcc, sh and sc lattices are ground states at and below the respective densities , , and (taking ). Specific examples of such realspace potentials, for which the ground states are not rigorously known, include the “squaremound” potential Hi57 [ for and zero otherwise] and what we call here the “overlap” potential, which corresponds to the intersection volume of two dimensional spheres of diameter whose centers are separated by a distance , divided by the volume of a sphere. The latter potential, which has support in the interval , remarkably arises in the consideration of the variance in the number of points within a spherical “window” of diameter for point patterns in and its minimizer is an open problem in number theory To03 . The dimensional Fourier transforms of the square mound and overlap potentials are and , respectively, with . Figure 2 shows the realspace and dual potentials for these examples in three dimensions. The densities at which the aforementioned lattices are ground state structures are easily understood by appealing to either the squaremound or overlap potential. The fcc lattice is the unique ground state at the density because at this value (where the nearestneighbor distance is unity) and lower densities the lattice energy is zero. At a slightly higher density, each of the 12 nearest neighbors contributes an amount of to the lattice energy. At densities lower than , there is an uncountably infinite number of degenerate ground states. This includes the bcc, sh and sc lattices, which join in as minimumenergy configurations at and below the respective densities , , and because those are the threshold values at which these structures have lattice energies that change discontinuously from some positive value (determined by nearest neighbors only) to zero. Moreover, any structure, periodic or not, in which the nearestneighbor distance is greater than unity is a ground state.
However, at densities corresponding to nearestneighbor distances that are less than unity, rigorous prediction of the possible groundstate structures is considerably more difficult. For example, it has been argued in Ref. Ml06 (with good reason) that realspace potentials whose Fourier transforms oscillate about zero will exhibit polymorphic crystal phases in which the particles that comprise a cluster sit on top of each other. The squaremound potential is a special case of this class of potentials and the fact that it is a simple piecewise constant function allows for a rigorous analysis of the clustered ground states for densities in which the nearestneighbor distances are less than the distance at which the discontinuity in occurs Ml06 .
iv.2 Nonnegative admissible functions
Another interesting class of admissible functions are those in which both and are nonnegative (i.e., purely repulsive) for their entire domains. The “overlap” potential discussed above is an example.
iv.2.1 OneDimensional Overlap Potential
Here we examine the onedimensional groundstate structures associated with the dual of the socalled overlap potential
(26) 
which is equal to the intersection volume, scaled by , of two rods of radius with centers separated by a distance . The dual potential is
(27) 
Figure 3 shows that both potentials are bounded and repulsive. However, while the overlap potential possesses the compact support , the dual potential is longranged with a countably infinite number of global minima determined by the zeros () of . Torquato and Stillinger have shown To03 that the unique ground state of the overlap potential is the integer lattice with density ; Theorem 1 therefore implies that the integer lattice at reciprocal density is the unique ground state of the dual potential (27). This result intuitively corresponds to placing each point in an energy minimum of the dual potential, thereby driving the total potential energy to zero. This argument immediately implies that the integer lattice at reciprocal density for all is also a ground state of the dual potential; however, the ground states at intermediate densities are generally nonBravais lattices and have heretofore been unexplored. Based on these observations, previous work has suggested that the dual interaction (27) undergoes an infinite number of structural phase transitions from Bravais or simple nonBravais lattices to complex nonBravais lattices over the entire density range lieb .
We have characterized the ground states of the dual overlap potential numerically using the MINOP algorithm DeMe79 , which applies a dogleg strategy using a gradient direction when one is far from the energy minimum, a quasiNewton direction when one is close, and a linear combination of the two when one is at intermediate distances from a solution. The MINOP algorithm has been shown to provide more reliable results than gradientbased algorithms for similar manybody energy minimization problems UcToSt06 . We fix the length of the simulation box and use a modified version of the dual potential
(28) 
where is the number of particles. Note that provides the unit of length for the problem, allowing us to control the density of the resulting configuration by varying .
For the case , we have numerically verified that the the integer lattice is the unique ground state (up to translation) of the dual potential; indeed, direct calculation shows that the integer lattice minimizes the potential energy (28) for all as expected from the arguments above. However, we have also identified degenerate ground states that are nonBravais lattices; these systems are shown in Figure 4. Our results suggest that for the ground states are complex superpositions of Bravais lattices with a minimum interparticle spacing determined by . Additionally, the conjectured infinite structural phase transitions are not observed in this density range owing to the high degeneracy of the ground state. We remark that although the integer lattice is a ground state for any , it is never observed in our numerical simulations because the energy landscape possesses a large number of global minima. Furthermore, although the ground states for integral and nonintegral values of are visually similar, we emphasize that the integer lattice is never a groundstate candidate for . For , the ground states are more difficult to resolve numerically because finitesize effects become more pronounced in this region; justification for this behavior is provided below. Nevertheless, we observe a “clustered” integer lattice structure in which several points occupy a single lattice site for .
Our numerical results suggest an exact approach to characterizing the ground states of the dual potential (27). For simplicity and without loss of generality, we will henceforth consider the scaled pair interaction
(29) 
corresponding to a normalized dual potential with . To facilitate the approach to the thermodynamic limit, we first examine a compact subset of subject to periodic boundary conditions. The entropy of this system for can be determined by relating the problem to the classic model of distributing balls into jars such that no more than one ball occupies each jar (FermiDirac statistics). Specifically, choosing the parameter in (28) is equivalent to choosing a density in the general problem (29). Therefore, for any , there are “jars” for the particles (“balls”). Assuming that the particles are indistinguishable, the number of distinct ways of distributing the particles into the potential energy minima is the binomial coefficient . For large (approaching the thermodynamic limit), Stirling’s formula implies that the entropy is
(30)  
(31) 
where we have chosen units with . Rearranging terms and substituting for the density, we find
(32) 
which is fixed in the thermodynamic limit and is plotted in Figure 5. Note that as , which is expected from the observation that the integer lattice is the unique ground state at unit density. This unusual residual entropy reflects the increasing degeneracy of the ground state with decreasing density and implies that in general the aforementioned infinite structural phase transitions from Bravais to nonBravais lattices are not thermodynamically observed. Instead, one finds an increasing number of countable coexisting groundstate structures as seen in our numerical energy minimizations.
For , determination of the ground states of the dual potential (29) is nontrivial since it is no longer possible to distribute all of the particles into potential energy wells. Therefore, FermiDirac statistics are no longer applicable for the manyparticle system. Nevertheless, we can make some quantitative observations concerning the ground states in this density regime. We first consider the scenario of adding one particle to a local region, subject to periodic boundary conditions, of the integer lattice of unit spacing. Since the potential energy minima of the pair interaction (29) occur on the sites of the integer lattice, the total potential energy cannot be driven to its global minimum. Symmetry of the lattice implies that, without loss of generality, we can limit the location of the particle to the interval . Since the energy of the underlying integer lattice is zero and the particle, by construction, will not interact with periodic images of itself, the total potential energy of the system after addition of the particle is exactly
(33)  
(34) 
where is the trigamma function AbSt72 . From the reflection property of the trigamma function AbSt72 , the latter expression is exactly equal to unity [] for any value of the parameter .
Determination of the ground state then depends on “relaxing” the system by making a small perturbation in the underlying integer lattice (see Figure 6). The energy of this perturbed system is then parametrized by the displacements and as in Figure 6 and is given by
(35)  
(36) 
where we have utilized reflection and recurrence relations for the polygamma function AbSt72 with given by (29). Figure 6 illustrates that possesses a unique minimum value at . Because the pair interaction (29) is longranged, it is unclear if can be further decreased by additional local deformation of the structure. Nevertheless, our analysis suggests that the ground state structures at densities slightly above unity are perturbed integer lattices with “defects” in the crystal structure.
Upon reaching , it is possible to “stack” two integer lattices with unit spacing for an energy per particle ; interestingly, the longrange nature of the pair potential (29) implies that these lattices can be mechanically decoupled from each other without increasing the energy of the system. Furthermore, the symmetry of this configuration implies that no local perturbation of the lattice structure can decrease the energy per particle, meaning that this “stacked” integer lattice and its translates within are at least local minima of the pair interaction (29); a similar argument will hold for any . The energy of this stacked configuration is
(37) 
Remarks:

If the stacked integer lattices are global minima of the pair interaction (29) for any number density , then the ground states are unique (up to translation of layers) at these densities, and the residual entropy will therefore vanish. However, for there is a combinatorial degeneracy associated with local deformations of the underlying integer lattice, implying that the residual entropy is nonanalytic over the full density range. This behavior in combination with the thermodynamic relation
(38) where is the thermal expansion coefficient and is the isothermal compressibility, suggests that there exist densities where the ground state exhibits negative thermal expansion as .

One special case of the aforementioned “stacked” integer lattice configurations occurs when multiple particles occupy the same lattice sites (i.e., with no translation between layers). For these “clustered” integer lattices, pair interactions are localized to include only those particles on the same lattice site, meaning that there are no longrange interactions for these systems. However, we have seen that the inclusion of longrange pair interactions, such as with the integer lattice, does not affect the total energy of the system. Since relative displacements between layers are uniformly distributed on , the average displacement of indeed corresponds to the integer lattice.

Our results imply that numerical methods are in general not appropriate for identifying the ground states for since truncation of the summation (33) (e.g., with the minimal image convention) breaks the translational degeneracy of the system.

The ground states of the overlap potential also exhibit rich behavior for . Since the interactions are localized to nearest neighbors, one can verify that addition of a particle to the unit density integer lattice increases the energy of the system by one unit, regardless of the position of the particle. However, unlike the dual potential (29), no local perturbation of the integer lattice can drive the system to lower energy, resulting in a large number of degenerate structures.

The twodimensional ground states of the generalized dual overlap potential
(39) have also been numerically investigated BaStTo09 ; the topology of the plane significantly increases the difficulty in analytically characterizing the groundstate configurations.
iv.2.2 GaussianCore Potential
Another interesting example of nonnegative admissible functions is the Gaussian core potential St76 , which has been used to model interactions in polymers Fl50 ; La00 . The corresponding dual potentials are selfsimilar Gaussian functions for any . The potential function pairs for the case with and are and . It is known from simulations St76 that at sufficiently low densities in , the fcc lattices are the ground state structures for . It is also known that for the range , fcc is favored over bcc St79 . If equality in (12) is achieved for this density range, the duality theorem would imply that the bcc lattices in the range (i.e., high densities) are the ground state structures for the dual potential. Latticesum calculations and the aforementioned simulations for the Gaussian core potential have verified that this is indeed the case, except in a narrow density interval of fccbcc coexistence around . In the coexistence interval, however, the corollary states the strict inequalities in (12) and (13) must apply. Importantly, the ground states here are not only nonBravais lattices, they are not even periodic. The ground states are sidebyside coexistence of two macroscopic regions, but their shapes and relative orientations are expected to be rather complicated functions of density, because they depend on the surface energies of grain boundaries between the contacting crystal domains. Proposition 9.6 of Ref. Co07 enables us to conclude that the integer lattices are the ground states of the Gaussian core potential for all densities in one dimension. Note that in , the triangular lattices apparently are the ground states for the Gaussian core potential at all densities (even if there is no proof of such a conclusion), and therefore would not exhibit a phase transition. Similar behavior has also been observed in four and eight dimensions, where the selfdual and lattices are the apparent ground states ZaStTo08 ; CoKuSc09 . Cohn, Kumar, and Schürmann have recently identified nonBravais lattices in five and seven dimensions with lower groundstate energies than the densest known Bravais lattices and their duals in these dimensions CoKuSc09 . Interestingly, these nonBravais lattices, which are deformations of the and packings, possess the unusual property of formal selfduality, meaning that their average pair sums (total energies per particle) obey the same relation as Poisson summation for Bravais lattices for all admissible pair interactions. It is indeed an open problem to explain why formallydual ground states exist for this pair potential.
It is also instructive to apply our higherorder duality relations (25) to the simple example of a threebody generalization of the aforementioned Gaussiancore potential. Specifically, we consider a threebody potential of the form
(40) 
Applying a double Fourier transform to this function shows that the dual potential, given by
(41) 
is selfsimilar to (40). As with the twobody version of the Gaussiancore potential, this selfsimilarity implies that if a Bravais lattice is the ground state of the threebody Gaussiancore interaction at low density, then its dual lattice will be the ground state at high density with the exception of a narrow interval of coexistence around the selfdual density . However, we have been unable to find either numerical or analytical studies of the ground states of this higherorder interaction in the literature, and determining whether it shares ground states with its twobody counterpart is an open problem.
iv.3 Completely monotonic admissible functions
A radial function is completely monotonic if it possesses derivatives for all and if . A radial function is completely monotonic if and only if it is the Laplace transform of a finite nonnegative Borel measure on , i.e., Wi41 . Not all completely monotonic functions are admissible (e.g., the pure powerlaw potential in is inadmissible). Examples of completely monotonic admissible functions in include for and for , . Importantly, the Fourier transform of a completely monotonic radial function is completely monotonic in Sc42 .
Remarkably, the ground states of the pure exponential potential have not been investigated. Here we apply the duality relations to the realspace potential in and its corresponding dual potential [where ], which has a slow powerlaw decay of for large . Note that the dual potential is a completely monotonic admissible function in , and both and also fall within the class of nonnegative admissible functions. We have performed latticesum calculations for the exponential potential for a variety of Bravais and nonBravais lattices in and . In , we found that the triangular lattices are favored at all densities (as is true for the Gaussian core potential). If equality in (12) is achieved, then the triangular lattices are also the ground states for the slowly decaying dual potential at all densities. In , we found that the fcc lattices are favored at low densities () and bcc lattices are favored at high densities (). The Maxwell doubletangent construction reveals that there is a very narrow density interval of fccbcc coexistence. We see that qualitatively the exponential potential appears to behave like the Gaussian core potential. If equality in (12) applies outside the coexistence interval, then the duality theorem would predict that the ground states of the slowlydecaying dual potential are the fcc lattices for and the bcc lattice for . Note that in one dimension, it also follows from the work of Cohn and Kumar Co07 that since the integer lattices are the ground states of the Gaussian potential, then these unique Bravais lattices are the ground states of both the exponential potential and its dual evaluated at (i.e., ).
Cohn and Kumar Co07 have rigorously proved that certain configurations of points interacting with completely monotonic potentials on the surface of the unit sphere in arbitrary dimension were energyminimizing. They also studied ways to possibly generalize their results for compact spaces to Euclidean spaces and conjectured that the densest Bravais lattices in for the special cases , 8 and 24 are the unique energyminimizing configurations for completely monotonic functions. These particular lattices are selfdual and therefore phase transitions between different lattices is not possible. Note that if the ground states for completely monotonic functions of squared distance in (the Gaussian function being a special case) can be proved for any , it immediately follows from Ref. Co07 that the completely monotonic functions of distance share the same ground states. Thus, proofs for the Gaussian core potential automatically apply to the exponential potential as well as its dual (i.e., ) because the latter is also completely monotonic in .
Based upon the work of Cohn and Kumar Co07 , it was conjectured that the Gaussian core potential, exponential potential, the dual of the exponential potential, and any other admissible potential function that is completely monotonic in distance or squared distance share the same groundstate structures in for and , albeit not at the same densities To08 . Moreover, it was also conjectured for any such potential function, the ground states are the Bravais lattices corresponding to the densest known sphere packings packing for and the corresponding reciprocal Bravais lattices for , where and are the density limits of phase coexistence of the low and highdensity phases, respectively. In instances in which the Bravais and reciprocal lattices are selfdual ( and ) , otherwise (which occurs for and 7). The second conjecture was recently shown by Cohn and Kumar to be violated for and .
V SelfDual Families of Pair Potentials
Our discussion of the Gaussian core model above suggests that one can exactly map the energy of a lattice at density to that of its dual lattice at reciprocal density for pair potentials that are selfsimilar (defined below) under Fourier transform. Here we provide additional examples of selfsimilar pair potentials, including radial functions that are eigenfunctions of the Fourier transform. Only some of these results are known in the mathematics literature CoEl03 , and this material has not previously been examined in the context of duality relations for classical ground states.
v.1 Eigenfunctions of the Fourier transform
Pair potentials that are eigenfunctions of the Fourier transform are unique in the context of the duality relations above since they preserve length scales for all densities; i.e., with no scaling factor in the argument. We therefore briefly review these eigenfunctions and the associated eigenvalues for radial Fourier transforms. In order to simply the discussion, we will adopt a unitary convention for the Fourier transform in this section
(42) 
which differs from our previous usage only by a scaling factor. The slight change in notation ( instead of ) is intended to clarify which convention is being used.
The eigenfunctions of the Fourier transform for can be derived from the generating function for the Hermite polynomials, which, when scaled by a Gaussian, is given by
(43) 
Taking the Fourier transform of both sides, one obtains
(44) 
implying
(45) 
By collecting powers of in (45), we immediately conclude
(46) 
thereby identifying both the eigenfunctions and eigenvalues of the Fourier transform. Note that the eigenvalues are real when is even.
We now seek eigenfunctions of the radiallysymmetric Fourier transform, defined here as
(47) 
for an isotropic function . Direct substitution shows that is an eigenfunction for all with eigenvalue . Other eigenfunctions of the Fourier transform can be identified by noting that they are also eigenfunctions of the dimensional Schrödinger equation for the radial harmonic oscillator
(48) 
where we have used the relation
(49) 
for radiallyisotropic functions in dimensions. The eigenvalues of the Schrödinger equation are for some . The general solutions to (48) are then given by
(50) 
where is the associated Laguerre polynomial AbSt72 and is a dimensiondependent constant. Note that for
(51) 
and we recover the even eigenfunctions of the harmonic oscillator.
To determine the eigenvalues of the radial Fourier transform, we note that if is an eigenfunction, then it must be true that
(52) 
for some eigenvalue . However, it is also true that
(53) 
Equation (53) implies that either or ; for a nontrivial solution we conclude that the eigenvalues of the radiallysymmetric Fourier transform are , which is in contrast to the general case on . This result is exactly consistent with the constraint that the index of an eigenstate of the radial Schrödinger equation (48) be even. Note that when , the Fourier transform changes the nature of the interaction (i.e., repulsive to attractive and viceversa).
v.2 PolyGaussian potential
The results above can be extended to include linear combinations of eigenfunctions of the Fourier transform; furthermore, we can generalize these functions to be simply selfsimilar under Fourier transform, meaning that length scales are not preserved by the transformation. Specifically, our interest is in functions for which:
(54) 
where and are constants.
As an example, we consider the Gaussian pair potential of the Gaussian core model
(55) 
The corresponding Fourier transform is given by