Two tiny magnets sit in a piece of metal. Around them is a sea of conduction electrons. Usually, electrons try to "shield" the magnetism of the impurities: they line up around them and quench it. This phenomenon is called the Kondo effect. But in this work, physicists took an unusual step: they allowed electrons to interact with the impurities differently depending on the direction. Such asymmetry is described by non-Hermitian quantum mechanics, where the familiar conservation laws can be violated.
The authors considered a pair of spin impurities (each like half of an elementary magnet) coupled to $n$ electron channels. The key parameter is the dimensionless non-Hermiticity $\alpha$, which shows how strongly the symmetry is broken. As $\alpha$ grows, the system goes through four phases. First, the overscreened Kondo regime: electrons try too hard to quench the impurities, and a complex collective state forms. Then, the zero-mode regime: special excitations with zero energy appear in the spectrum, like stationary strings. Next, the Yu–Shiba–Rusinov phase, where $\mathscr{PT}$-symmetry (a combination of spatial and temporal inversion) is spontaneously broken. Finally, the local-moment regime, where the impurities behave like free magnets, and the renormalization-group flow becomes cyclic.
Using an exact solution (Bethe ansatz), the physicists computed the impurity free energy and the Affleck–Ludwig $g$-function — a measure of the entropy associated with the impurity. In the Kondo phase, the entropy flows from $2\ln2$ to $2\ln[2\cos(\pi/(n+2))]$, consistent with predictions from conformal field theory. But in other phases, the entropy behaves non-monotonically. This calls into question the famous $g$-theorem — the statement about the irreversibility of renormalization-group flows in the presence of defects. It turns out that the actual spectrum and agreement with conformal theory do not yet guarantee irreversibility.
In the zero-mode regime, the spectrum of the system resembles a multi-story building: each excitation gives rise to a whole "tower" of states with the same energy.