Cubic Modified Energy in Quantum Systems

• Cubic modified energy is defined as the additional energy required to deform a quantum many-body system from a spherical ground state to a cubic configuration.
• The variational framework uses trial wavefunctions with a cubicity parameter that interpolates between Gaussian (spherical) and hypercuboid (cubic) forms, yielding measurable deformation energies.
• Implications include denser atomic packing, reduced shear modulus, and softened equations of state in systems like compressed solids and neutron stars.

Cubic modified energy refers to the energetic cost and physical implications associated with deforming a quantum many-body system—specifically atoms, nucleons, or bound composite objects—from their usual spherically symmetric ground-state wave functions to configurations with cubic (octahedral) symmetry. This concept has manifold relevance in theoretical atomic physics, nuclear structure, statistical mechanics of crystalline matter, and astrophysical contexts such as neutron stars, where extreme pressures can favor symmetry breaking away from the sphere. A "cubic modified energy" quantifies how much additional energy must be injected into the system to enforce cube-like deformation, as compared to the native spherical state. The framework is both variational and geometric, linking microscopic deformation energies to macroscopic properties—such as crystal packing, shear modulus, or the equation of state of dense matter.

1. Variational Framework for Cubic Deformation

Cubic modified energy is typically analyzed via variational trial wavefunctions that interpolate between perfect spherical and perfect cubic symmetry. The deformation is controlled by a "^^^^1^^^^" parameter—denoted $N$ or $\eta$—entering the norm or exponential argument of the wavefunction. For a two-electron atomic system (Helium), the trial function in momentum space is

$$\psi_\rho(\vec{\pi}_\rho) = N_\rho \exp \left[ -\left( |\pi_{\rho, x}|^\eta + |\pi_{\rho, y}|^\eta + |\pi_{\rho, z}|^\eta \right)^{1/\eta} \right],$$

and similarly for $\psi_\lambda$. When $\eta=2$, the wavefunction is a spherically symmetric Gaussian; as $\eta \to \infty$, it becomes a hypercuboid (perfect cube). For baryons modeled in QCD (Cornell Hamiltonian in Coulomb gauge), a similar ansatz employs the generalized "Minkowski-norm,"

$p_N = \left( p_x^N + p_y^N + p_z^N \right)^{1/N},$

with $N=2$ for spherical and $N \to \infty$ for cubic cases (Portela et al., 2015, Llanes-Estrada et al., 2011). The variational energy is minimized for each value of the symmetry-breaking parameter, yielding $E_{\text{spherical}}$ and $E_{\text{cubic}}$. The cubic modified energy is then defined as

$$\Delta E = E_{\text{cubic}} - E_{\text{spherical}}.$$

For He, $\Delta E$ rises smoothly with $\eta$, reaching $\sim9.5$ eV for $\eta \to \infty$; for the neutron in the Cornell model, $\Delta E \approx 150$ MeV between $N=2$ and $N=\infty$ (Llanes-Estrada et al., 2011, Portela et al., 2015).

2. Volume Gain, Pressure Thresholds, and Mechanical Equilibrium

Deforming each constituent from a sphere to a cube enables denser packing in a crystal or lattice, reducing the interstitial (void) volume per particle. The reduction is quantified geometrically; e.g., in a close-packed atomic array, the packing fraction increases from $\sim0.74$ (spheres) to $1$ (cubes). The corresponding volume gain is

$$\Delta V = V_{\text{sph}} \left[ \frac{1}{\eta_{\text{sph}}} - 1 \right].$$

Energetic favorability requires that the mechanical work gained by shrinking the void compensates the deformation cost: $\Delta E(\eta) = P\,\Delta V(\eta),$ defining a threshold pressure $P_\text{t}$: $P_\text{t} = \frac{\Delta E}{\Delta V}.$ For Helium, significant cubic distortion (up to $25\%$) sets in at $P\sim 500$ GPa; complete cubicity requires $P$ of order $1$ TPa (Portela et al., 2015). For neutrons in dense matter, $P_t \sim 200$ MeV/fm$^3$ (i.e., several times nuclear saturation density, $\rho_0$) (Llanes-Estrada et al., 2011).

3. Thermodynamic and Structural Consequences

When the pressure exceeds $P_\text{t}$, cubic deformation is energetically favored, altering macroscopic properties:

• Equation of State Softening (Neutron Matter): Once cubic neutrons arise, the extra $\Delta E$ per baryon and reduced void volume modifies the free energy, leading to a softer equation of state $P(\rho)$. This affects neutron-star mass–radius relations and may delay the onset of other exotic phases (hyperons, deconfinement) (Llanes-Estrada et al., 2011).
• Shear Modulus Reduction (Atomic Crystals): For compressed atomic solids, an increase in cubicity reduces resistance to tangential stress—the shear modulus drops, saturating when cubicity is high. At sufficiently large pressure the scaling of $G$ with $P$ flattens, implying a finite limit (Portela et al., 2015).
• Form Factor Anisotropy: The atomic form factor $F(q)$ develops pronounced angular dependence, with up to 10% azimuthal variation observed for substantial cubicity.

4. Energetics in Lattice, Finite-Volume, and Statistical Mechanics

The concept of cubic modified energy extends to the analysis of optimal lattice packings and energy functionals:

• Cubic Lattice Energies: In Bravais lattices, modified energies are considered via parametrized sums over lattice sites for various interaction potentials (e.g., theta function, Lennard–Jones). Simple cubic, face-centered cubic, and body-centered cubic configurations are critical points of such functionals. The local optimality (minimizer, saddle, or maximizer) is determined by the second derivatives (Hessian) of the energy (Bétermin, 2016).
• Finite-Volume Quantum Systems: For three-body bound states in a cubic box, the leading finite-size correction to the binding energy, $\Delta E$, has a cubic symmetry and shows exponential suppression with box length $L$. This "cubic modified energy" provides a direct finite-volume artifact for Lattice QCD extrapolations (Rusetsky, 2015).

5. Modified Energies in Field Theory and Nonlinear Dynamics

Cubic modifications to energy functionals appear in several modern field theories:

• Teleparallel $f(T)$ Gravity: The energy of spherically symmetric black holes in $f(T)=T+\epsilon[\alpha T^2+\beta T^3]$ gravity exhibits an $\mathcal{O}(\epsilon)$ correction proportional to $\beta$, which can raise or lower the gravitational mass depending on the sign of the cubic coefficient. The leading correction is constant at large radius, reflecting the nontrivial effect of cubic torsion invariants (Nashed, 2021).
• Nonlinear Schrödinger Equation (NLS): In the context of the defocusing cubic NLS, the "modified energy" functional $E_{\text{mod}}(t)$ is constructed to accommodate long-range scattering data and enables robust global-in-time estimates. This functional incorporates time-dependent, cubic-weighted quadratic forms and is central to recent advances in the theory of modified wave operators (Kawamoto et al., 2 Jun 2025).

6. Astrophysical and High-Pressure Physics Implications

Cubic modified energy is directly relevant in astrophysical and high-density matter scenarios:

• Neutron Stars: The transition to cubic neutrons in the dense core leads to macroscopic observables—mass–radius relation, glitch amplitudes, and moment of inertia variations—that are potentially testable against pulsar timing and gravitational wave measurements. The associated softening of the equation of state may delay or alter the thresholds for the appearance of hyperons and deconfined quark matter (Llanes-Estrada et al., 2011, Portela et al., 2015).
• Compressed Atomic Solids: For elemental solids like Helium under TPa pressures, cubic deformation of atomic wavefunctions implies salient changes to the elastic and scattering properties—potentially measurable if such conditions are realized experimentally or in planetary interiors (Portela et al., 2015).

7. Methodological Significance and Context

The determination and application of cubic modified energy relies on high-dimensional variational optimization (Monte Carlo integration for wavefunctions), geometric analysis of packing, and symmetry-breaking parametrization. These methods provide quantitative predictions for threshold pressures, deformation energies, and associated macroscopic effects in regimes unattainable with linear or perturbative approximations. In the context of lattice energetics, the precise parametrization of cubic deformations provides insight into structural transitions and energetically preferred symmetries under varying density and interaction regimes (Bétermin, 2016, Rusetsky, 2015).

The framework unifies quantum variability at the microscopic level with macroscopic observability, establishing cubic modified energy as a foundational tool in the analysis of dense, deformed, or strongly interacting matter.