Solving Shape Optimization Using da Costa/Gravity Duality

Gravity theory and holography have well-known applications in condensed matter physics, such as through the ADS/CFT correspondence to model electric conductivity in strongly coupled systems. In this work, we attempt to expand this paradigm by introducing the da Costa (Phys. Rev. A 23, 1982)/Gravity duality to solve the shape optimization problems in semiconductors, especially when electric, magnetic and optical properties of lower-dimensional nanomaterials become sensitive to their geometries. We start by exploring the geometrodynamics of a quantumly-confined particle on a 2-dimensional curved semiconductor surface. By developing 2-dimensional action with curvature-induced potentials, we can effectively capture and understand the dynamics and behavior of the quantumly-confined particle through the lens of 2D dilaton gravity theory. The action principle for this 2D dilaton gravity exhibits striking similarities to our formulated 2D action principle for the da Costa quantum-constrained particle. We will conclude with a theoretical framework that allows us to converge the shape deformation on the original 2D metric ansatz through an iterative process, aiming to uncover an optimized nano-geometry that leads to the system's lowest eigenenergy configuration and minimized external work for confinement.