Localization and "classical entanglement'' in the Discrete Non-Linear Schrödinger Equation

TL;DR

This work establishes that the high-energy localized phase of the Discrete Non-Linear Schrödinger Equation is a genuine microcanonical equilibrium with negative temperature, demanding a global (nonlocal) definition of temperature. It introduces a classical entanglement measure $S_{ent}$, showing it remains constant in the homogeneous phase but grows as $S_{ent}(N)\sim \log(N)$ in localization, signaling nonlocal correlations akin to quantum entanglement. The study combines microcanonical large-deviation thermodynamics with direct Hamiltonian dynamics and analytical marginal distributions $\mu_E(\varepsilon)$ to validate theoretical predictions for the participation ratio $Y_2$, temperature scaling $T_{micro}$, and the entanglement structure. This bridges classical and quantum perspectives by revealing non-additivity and ensemble inequivalence in a classical field theory, with implications reminiscent of Many-Body Localization phenomena.

Abstract

We perform a detailed numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the Discrete Non-Linear Schrödinger 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.

Figures (11)

• Figure 1: Microcanonical phase diagram of the DNLSE model. Solid blue line identifies the ground state $e=a^2-2a$ at $T=0$ (see RCKG00 for details) below which there are forbidden states. The red dashed line refers to $e_{\mathrm{th}}(a)=2a^2$ and is characterized by infinite temperature. Above it $T$ is negative: delocalized states are contained in the orange region $e_{\mathrm{th}}(a) < e < e_c(a,N)$, while localized ones for $e > e_c(a,N)$ are highlighted in green color. The solid orange curve here refers to $e_c(a,N)=e_{\mathrm{th}}(a) + 11.05\, a^2 N^{-1/3}$ with $N=100$, according to GILM21aGILM21b.
• Figure 2: $\mathrm{Upper panel:}$ Finite-size scaling of the participation ratio $Y_2(N)$ of the DNLSE for $a=1$ and different energy densities $e$. Statistical expectations are computed as time averages in the time interval $10^5$-$10^6$. Error bars are the standard deviation of fluctuations, sampled at intervals $\Delta t =10$, in the measurement window. $\mathrm{Lower panel:}$ Finite-size scaling of minus the inverse temperature, $-\beta$, vs. $N$ in the metastable and stable localized phases. The system has been initialized with an artificial breather added to an infinite temperature background, with $a = 1$. The running time of the simulations is $\Delta \tau = 1.5 \times 10^3$, and the microcanonical temperature has been measured globally across all the sites of the system. The uncertainties have been evaluated as the standard deviation of the temperature measurements taken at every $d\tau = 10$ as the simulations progress. The red line corresponds to the fit with the theoretical prediction.
• Figure 3: Marginal probability distribution of the local energy $\varepsilon_j=|z_j|^4$ for a single site of the lattice, averaged over all sites. The system is in the homogeneous positive temperature phase: $E/N = 1.5 < e_{\mathrm{th}}$.
• Figure 4: Marginal probability distribution of the local energy $\varepsilon_j=|z_j|^4$ for a single site of the lattice, averaged over all sites. The system is in the localized negative temperature phase: $E/N = 4.5 > e_c$. Secondary peaks represent the characteristic condensate bump.
• Figure 5: Behavior of the classical entanglement $S_{\mathrm{ent}}(N)$ measured numerically as a function of lattice size $N$. Orange squares: classical entanglement of the homogeneous phase at positive temperature, $(e = 1.5) < e_{th}$: Green circles: localized phase at negative temperature, $(e = 4.5) > e_c$. In the homogeneous phase $S_{\mathrm{ent}}(N)$ is constant while in the localized phase it grows as $S_{\mathrm{ent}}\sim \log(N)$, as emphasized by the straight line in the semi-log scale.