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 approaches to manipulating finite-size quantum systems, such as tuning external fields or system size, typically lead to uniform shifts across the spectrum, limiting the scope of spectral engineering. A recently introduced technique, known as the size-invariant shape transformation, overcomes this limitation by introducing a new control parameter that deforms the potential landscape without altering the system's size parameters, thereby enabling nonuniform scaling of energy levels. This new degree of freedom-referred to as the shape parameter-gives rise to quantum shape effects in the thermodynamics of confined systems, which are conceptually distinct from quantum size effects. Here, we explore the fundamental limits of nonuniform level scaling in the spectra by asking: what is the minimal quantum system in which such behavior can arise? We demonstrate that even a two-level system can exhibit the thermodynamic consequences of quantum shape effects, including spontaneous transitions into lower-entropy states, a phenomenon absent in classical thermodynamics for noninteracting systems. We identify the spectral origin of these unconventional thermodynamic behaviors as geometry-induced asymmetric level coupling, in which the ground-state energy and energy gap respond in opposite ways to changes in a shape parameter. This asymmetry naturally extends to many-level systems, where the thermally averaged energy spacing and ground-state energy evolve in opposite directions. To characterize unconventional thermodynamic behaviors, we construct thermodynamic spontaneity maps, identifying regions of energy-driven and entropy-driven spontaneous processes in ground-state energy versus energy gap space. These effects emerge under quasistatic, isothermal changes of a shape degree of freedom and illustrate how the confinement geometry alone can enable unconventional thermodynamic behaviors that are otherwise exclusive to interacting or open systems. We argue that any scaling-invariant local parameter transformation that induces asymmetric-level coupling can be used to engineer similar responses, making this a broadly applicable framework. Our results deepen the theoretical foundations of the quantum shape effect and introduce a route to spectral gap control, with potential applications in isolating computational subspaces within quantum information platforms.