The Russian Doll Model — a generalization of Richardson's superconductivity model — with a time-reversal-breaking parameter theta exhibits reentrance transitions. As theta increases monotonically, the system passes from localized to ergodic to multifractal to localized again. The path from order to disorder is not monotone. The system visits the same phase more than once.
Reentrance means the phase diagram has loops. Increasing a control parameter doesn't simply push the system through a sequence of phases. It can return to a phase it already left, entering from a different direction with different internal structure. The localized phase encountered at large theta is not the same localized phase encountered at small theta — the wavefunctions have different spatial statistics — but the macroscopic classification is identical.
The cyclic renormalization group of the Russian Doll Model produces an extended non-ergodic multifractal phase that maps, through Bethe Ansatz, to ground states of vortex strings in supersymmetric gauge theory. The authors conjecture that the Hamiltonian describes mixing in a BPS sector — states protected by supersymmetry from quantum corrections. If correct, multifractality (the hallmark of criticality in disordered systems) and BPS protection (the hallmark of exactness in gauge theory) are two descriptions of the same phenomenon.
The connection, if it holds, would mean that the irregular, scale-dependent structure of wavefunctions at a disordered critical point has a dual life as an exactly protected quantity in a supersymmetric theory. Disorder and exactness would not be opposites but partners — the same mathematical structure viewed from different theoretical frameworks. The messiest corner of condensed matter physics would be the cleanest corner of high-energy physics, seen from a different angle.