Latent Ewald Summation in Lattice Systems

• Latent Ewald summation is a unified analytical framework that extends standard Ewald methods to arbitrary inverse power-law interactions in lattice systems.
• It decomposes lattice sums into real- and reciprocal-space contributions using incomplete gamma and Bessel functions, enabling efficient evaluation.
• The method applies to fully periodic and quasi-2D systems, controlling IR divergences via electroneutrality and balancing convergence with a parameter.

Latent Ewald summation is a unified analytical framework for evaluating lattice sums with inverse power-law interactions, extending the canonical Ewald method beyond the Coulomb limit to general power exponents η and supporting both fully periodic (3D) and quasi–two-dimensional (quasi-2D) systems. This formulation provides deep insight into the mathematical structure hidden—“latent”—in standard summation techniques, introducing new analytical connections involving incomplete gamma and special functions that underlie the robust convergence and flexibility of Ewald-type decompositions (Mazars, 2010).

1. Generalized Ewald Decomposition for Inverse Power-Law Interactions

The latent Ewald method addresses lattice sums of the form

$$\phi_{(\eta)}(\mathbf{r}) = \sum_{\mathbf{n}} \frac{1}{|\mathbf{r} + \mathbf{L}_n|^\eta}$$

for arbitrary exponent $\eta > 0$, generalizing the standard Coulomb case ($\eta=1$). The method introduces an integral representation

$$\frac{1}{|\mathbf{r}|^\eta} = \frac{1}{\Gamma(\eta/2)} \int_0^\infty \mathrm{d}t\, t^{\eta/2-1} \exp(-r^2 t)$$

and uses a convergence parameter $\alpha$ to partition the sum into “real-space” and “reciprocal-space” contributions. The resulting decomposition in periodic 3D can be compactly written as

$$\phi_{(\eta)}(\mathbf{r}) = \sum_{\mathbf{L}_n} \Phi_R^{(3)}(\eta, \alpha; |\mathbf{r}+\mathbf{L}_n|) + \sum_{\mathbf{k} \ne 0} \Phi_k^{(3)}(\eta, \alpha; k) e^{i\mathbf{k}\cdot \mathbf{r}} + [k=0\ \mathrm{term}]$$

where the analytical forms of the contributions are

$$\Phi_R^{(3)}(\eta, \alpha; r) = \frac{\Gamma(\eta/2, \alpha^2 r^2)}{\Gamma(\eta/2)\, r^\eta},\qquad \Phi_k^{(3)}(\eta, \alpha; k) = \frac{\pi^{3/2} V}{\Gamma(\eta/2)} \left(\frac{4}{k^2}\right)^{\frac{3-\eta}{2}} \Gamma\left(\frac{3-\eta}{2}, \frac{k^2}{4\alpha^2}\right)$$

with $\Gamma(a,z)$ the complementary incomplete gamma function. These formulas recover the standard Ewald sum for $\eta=1$ and extend seamlessly to arbitrary $\eta$.

2. Quasi–Two–Dimensional Systems and Analytical Equivalences

For systems with periodicity in only two dimensions and an unbounded third dimension (quasi-2D), the latent formulation derives two equivalent representations:

• Parry’s Limit: Starting from the full 3D lattice sum and letting the period $L_z \rightarrow \infty$, the analysis decomposes vectors and reciprocal vectors accordingly, leading to a representation where the summation over $k_z$ becomes continuous. This limit controls the procedure for extracting quasi-2D physics.
• Poisson–Jacobi Identity (2D Approach): The sum over a 2D lattice is transformed using the Poisson–Jacobi (theta function) identity, leading directly to a mixed reciprocal-space formulation incorporating the nonperiodic direction $z$.

The equivalence of these two derivations yields new analytical Fourier transform expressions for the incomplete gamma function, notably

$$\frac{1}{2\pi} \int_{-\infty}^{+\infty} \frac{\Gamma\left(\frac{3-\eta}{2}, \frac{G^2 + k^2}{4\alpha^2}\right)}{(G^2 + k^2)^{(3-\eta)/2}}\, e^{ikz} \,dz = \frac{(2\alpha)^{\eta-2}}{2\sqrt{\pi}} K_{\eta/2 - 1}\left(\frac{G^2}{4\alpha^2},\, \alpha^2 z^2\right)$$

where $K_\nu(x,y)$ is a generalized incomplete Bessel function. These relationships expose the latent structure between real- and reciprocal-space summations for general $\eta$ and across dimensionality-reduced systems.

3. Infrared Divergences and Electroneutrality Constraints

For $\eta \leq d$ (where $d$ is the number of periodic dimensions), the direct lattice sums are only conditionally convergent and exhibit infrared (IR) divergences. The latent approach systematically groups and cancels the diverging terms—often in the $k=0$ (or $G=0$) component of the reciprocal sum—by enforcing global electroneutrality ($\sum_i Q_i = 0$ for point charges). For example, the divergence in the 3D energy expression

$$\left[ \int_0^{\alpha^2} \frac{dt}{t^{(5-\eta)/2}} \right] \left(\sum_i Q_i\right)^2$$

vanishes under electroneutrality. This grouping is essential for physically meaningful energies and stable numerical implementation.

4. Analytical and Numerical Results for Generalized Models

The derived latent Ewald sums are applied to:

• Generalized Restricted Primitive Model (η-RPM): A system of equal numbers of oppositely charged hard spheres interacting via $1/r^\eta$.
• Generalized One-Component Plasma (η-OCP): Point charges of the same sign with neutralizing background.

In monolayer (strictly 2D) systems, the energies (e.g., $E_{\eta,\mathrm{RPM}}^{2D}$, $E_{\eta,\mathrm{OCP}}^{2D}$) have a reciprocal part that factors into one-particle sums for charged particles, while the neutralizing background introduces a $z$-dependent correction in quasi-2D settings.

Monte Carlo simulations indicate:

• For $\eta=1.0$ (Coulomb), the analytical and numerical results match established results for charged monolayers.
• For $\eta<1$, energies are less negative (softer interactions); for $\eta>1$, the energy is more negative and the pair correlation functions show increased short-range order.
• Structural transitions (e.g., sharper $g_{++}(r)$ peaks) as $\eta$ increases confirm the method quantitatively predicts the influence of the potential’s exponent on spatial correlations.

5. Mathematical Insights: Latent Fourier Relations and Special Functions

A critical insight of the latent Ewald summation formalism is the emergence of new relations for the Fourier transforms of incomplete gamma functions, connecting them to incomplete Bessel functions: $$\frac{1}{2\pi} \int_{-\infty}^{+\infty} \frac{\Gamma(\nu, k^2/4)}{k^{2\nu}} e^{ikz} \,dk = \frac{1}{|z|\sqrt{\pi} (z/2)^{2\nu}}\, \gamma(\nu - 1/2, z^2)$$ where $\gamma(\mu, y)$ is the lower incomplete gamma function. These relations bridge the analytic structure of the Ewald split for generic power-law potentials, highlighting the “latent” special function content.

6. Practical Implementation and Applicability

The latent formalism yields explicit, convergent sums for arbitrary power-law ($1/r^\eta$) potentials and arbitrary periodic boundary conditions (3D, 2D, or reduced symmetry). Implementations involve:

• Selecting the convergence parameter $\alpha$ to balance real- and reciprocal-space convergence.
• Using incomplete gamma and Bessel function routines for evaluating terms, for which efficient numerical libraries are available.
• Enforcing electroneutrality to guarantee energy convergence.

The unification across $\eta$ circumvents case-by-case derivation and applies to simulation of electrolytes, plasmas, polytropic or generalized soft-matter models, and transitions between 3D and quasi-2D physical regimes.

7. Significance and Theoretical Impact

The latent Ewald summation approach is notable for:

• Providing a comprehensive, unified framework for arbitrary inverse power-law potentials in periodic and quasi-periodic systems.
• Revealing latent analytical connections between incomplete gamma functions, Bessel-type functions, and their Fourier transforms, with direct practical implications for efficiently evaluating periodic sums.
• Enabling a deeper understanding of the dependence of structural and thermodynamic properties on the potential exponent $\eta$ in classical charged and soft-matter systems.

These results make the latent Ewald summation an essential analytical and computational tool for the theoretical and numerical paper of lattice sums, interaction energies, and correlation functions in broad classes of classical and quantum many-body systems (Mazars, 2010).