Levy distribution in many-particle quantum systems

Levy 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 Levy distribution. Finally, we extend our analysis to different confinement setups and demonstrate that the tunable Levy 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.

💡 Research Summary

The paper investigates the momentum distribution of a one‑dimensional Tonks‑Girardeau (TG) gas—an ensemble of impenetrable bosons confined in a harmonic lattice—at finite temperature, and demonstrates that it follows a Lévy distribution. Two complementary statistical frameworks are employed: the canonical ensemble, where particle number N and temperature T are fixed, and the grand‑canonical ensemble, where chemical potential μ and temperature are fixed. In the canonical approach the authors map hard‑core bosons onto non‑interacting fermions via the Jordan‑Wigner transformation, compute the one‑body density matrix, and obtain the momentum profile n(k) by Fourier transformation. In the grand‑canonical approach they construct the finite‑temperature Green’s function, solve the associated Dyson equation, and recover the same n(k). Numerical results from both methods coincide, confirming that the findings are independent of the chosen ensemble.

The central discovery is that n(k) is not Gaussian or a simple power‑law, but is accurately described by a symmetric Lévy stable distribution

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