Monotonic Impurity Entropy beyond Unitarity: the $\mathscr{PT}-$Symmetric Quantum Impurity Model

Quantum impurity models provide a paradigmatic setting for studying Kondo screening, boundary criticality, and impurity entropies. While these phenomena are well understood in unitary systems, their fate in non-Hermitian many-body settings remains largely unexplored. We study a $\mathscr{PT}$-symmetric quantum impurity model consisting of a unitary $SU(2)_1$ Wess--Zumino--Witten bulk coupled to two impurity spins through complex-conjugate boundary Kondo interactions. Using an integrable lattice realization with $\mathscr{PT}$-symmetric boundary impurities, solved by the Bethe Ansatz and benchmarked against finite-temperature matrix-product-state calculations, we determine the impurity contribution to the free energy and entropy. In the Kondo-screened regime, where the spectrum remains entirely real and the impurities are screened by many-body Kondo clouds, we find that the impurity entropy decreases monotonically from $\ln 4$ in the ultraviolet to $0$ in the infrared. This monotonic flow persists despite the nonunitary nature of the boundary interaction, which places the system beyond the standard assumptions of the $g$-theorem.