Fundamental Work Scaling and Non-Extensivity in Critical Quantum Stirling Engines

Abstract

We present a general analytical framework for quasi-static quantum Stirling engines operating across ground-state level crossings (GLC). In the low-temperature regime, we derive the Primarch Formula, an exact universal expression linking extracted work and efficiency directly to macroscopic ground-state degeneracies. We analytically prove that these engines achieve Carnot efficiency without a classical regenerator, and that thermal excitations strictly degrade this performance. Validated against exact numerical simulations of generalized \textit{N}-th spin-1/2 Heisenberg models with nontrivial interactions, the framework is applied to the one-dimensional antiferromagnetic Ising model, revealing a profound connection to number theory. Governed by Fibonacci, Lucas, and parity-dependent critical degeneracies, the engine exhibits distinct operational regimes that permanently violate classical thermodynamic extensivity while operating at the absolute Carnot limit, regardless of macroscopic system size.