Geometry-induced asymmetric level coupling

Tailoring energy levels in quantum systems via Hamiltonian control parameters is essential for designing quantum thermodynamic devices and materials. However, conventional methods for manipulating finite-size systems, such as tuning external fields or system size, typically lead to uniform spectral shifts, limiting precise control. A recently introduced technique, called the size-invariant shape transformation, overcomes this by introducing a new control parameter that deforms the potential landscape without altering system size, enabling nonuniform level scaling. This shape parameter gives rise to quantum shape effects in confined systems, conceptually distinct from quantum size effects. We explore the limits of this phenomenon by asking: what is the minimal system in which such spectral behavior can emerge? We show that even a two-level system can exhibit thermodynamic consequences of quantum shape effects, including spontaneous transitions into lower-entropy states, a feature absent in classical thermodynamics for non-interacting systems. We identify the origin as geometry-induced asymmetric level coupling, where the ground-state energy and level spacing respond oppositely to shape changes. This extends to many-level systems, where the thermally averaged level spacing and ground-state energy evolve in opposite directions. We construct spontaneity maps revealing energy- and entropy-driven spontaneous processes. These behaviors emerge under quasistatic, isothermal deformations and show how geometry alone can induce thermodynamic effects typically exclusive to interacting or open systems. Our results offer a broadly applicable route to spectral gap control in quantum technologies.