Lévy distribution in many-particle quantum systems

The Lévy distribution, previously used to describe complex behavior of classical systems, is shown to characterize that of quantum many-body systems. Using two complimentary approaches, the canonical and grand-canonical formalisms, we discovered that the momentum profile of a Tonks-Girardeau gas—a one-dimensional gas of $N$ impenetrable (hard-core) bosons–harmonically confined on a lattice at finite temperatures obeys a Lévy distribution. Finally, we extend our analysis to different confinement setups and demonstrate that the tunable Lévy distribution properly reproduces momentum profiles in experimentally accessible regions. Our finding allows for calibration of complex many-body quantum states by using a unique scaling exponent.