Unconventional thermodynamics arising from nonuniform level scaling

Global manipulations of energy levels in finite systems, such as through external potential fields or volume changes, typically lead to uniform scaling or shifts in the energy spectrum of a generic Hamiltonian. However, a recent geometric technique called the size-invariant shape transformation, changing the shape of a domain while preserving the Lebesgue measure, has been shown to cause nonuniform scaling in energy spectra [1,2]. This technique enables precise tuning of the potential energy landscape of quantum systems. Unlike global transformations of Hamiltonian parameters, which uniformly affect all energy levels, local-parameter transformations selectively modulate specific regions of the spectrum, allowing controlled adjustments. We show that nonuniform level scaling induces unconventional thermodynamic behaviors. For example, thermodynamic spontaneity—characterized by a decrease in free energy—may be driven solely by entropy or internal energy, even when the other quantity works in opposition. We construct a rich thermodynamic process diagram, where energy-driven and entropy-driven spontaneous processes represent additional types of thermodynamic spontaneity. We show that even in a simple two-level system, geometric level couplings can induce opposite responses in the ground and excited states. We further analyze contributions from different types of confinements, showing that geometric level couplings provide a novel method for controlling the spectral gap, which could have significant applications for robustly separating computational and leakage states in quantum computing devices.

[1] A. Aydin, “Spectral properties of size-invariant shape transformation”, Phys. Rev. E, 107, 054108, (2023).

[2] A. Aydin and A. Sisman, “Origin of the quantum shape effect”, Phys. Rev. E, 108, 024105, (2023).