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arXiv · 2503.14364

Localization and "classical entanglement'' in the Discrete Non-Linear Schr\"odinger Equation

Martina Giachello·Stefano Iubini·Roberto Livi·Giacomo Gradenigo
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Abstract

We perform a detailed numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the Discrete Non-Linear Schr\"odinger Equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle relation between the presence of the localized phase and a property of the system that we have called {\it ``classical entanglement''}. Our main finding is that a quantity defined for our classical system in perfect analogy with the entanglement entropy of quantum ones, and that we have therefore called $S_{\mathrm{ent}}$, grows with the system size $N$ in the localized phase as $S_{\mathrm{ent}}(N) \sim \log(N)$, therefore revealing the presence of subtle non-local correlations between any finite portion of the system and the rest of it. This manifestation of {\it ``classical entanglement''} beautifully captures the lack of system separability in the DNLSE localized phase, revealing how statistical correlations specific to the microcanonical ensemble and non-reproducible in the canonical one, may concur to determine a property totally analogous to the one produced by non-local quantum correlations.

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Martina Giachello, Stefano Iubini, Roberto Livi, Giacomo Gradenigo. 2025-03-18. Localization and "classical entanglement'' in the Discrete Non-Linear Schr\"odinger Equation. https://doi.org/10.1088/1742-5468%2Fae17ab

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