A pseudo-Hermitian hybrid model at finite temperature: The role of exceptional points

Abstract

We study a hybrid system formed by an ensemble of nitrogen-vacancy (NV) centres in diamond interacting with a superconducting flux qubit at finite temperature. The presence of impurities is modelled through a pseudo-Hermitian Hamiltonian by introducing an asymmetry parameter in the interaction between the flux qubit and the NV ensemble. We construct the exact grand partition function of the system and derive the associated thermodynamic quantities, including the entropy, internal energy, and Helmholtz free energy. In the symmetry-broken phase, the partition function may develop zeros associated with the emergence of complex-conjugate eigenvalue pairs whose real parts lie in the low-energy sector of the spectrum. Within the Yang–Lee framework, these zeros signal first-order phase transitions. We determine the critical temperature at which the first zero of the partition function appears, as a function of the asymmetry parameter and the coupling strength between the NV ensemble and the superconducting flux qubit. Metastable regions in parameter space are characterized using Maxwell constructions and spinodal-decomposition analysis. To illustrate the physical implications of the thermodynamic structure, we briefly comment on thermodynamic cycles in the stability region where the partition function remains strictly positive. Finally, we outline a rescaling scheme that allows the extension of the model to larger Hilbert-space dimensions.