Mean-field theory of the DNLS equation at positive and negative absolute temperatures

TL;DR

The paper addresses the equilibrium thermodynamics of the discrete nonlinear Schrödinger (DNLS) model, which features two conserved quantities and a negative-temperature phase with breather formation. It develops a mean-field (MF) theory that replaces the inter-site coupling by a site-averaged term, so the grand canonical partition function factorizes and explicit expressions for the mass density $a$, energy density $h$, and the nonlinear and interaction energy densities $h_{nl}$ and $h_{int}$ are obtained; this framework applies to both positive and negative temperatures, with a cutoff needed for $\beta<0$. The MF theory is exact on the transition line between the positive-$T$ homogeneous phase and the negative-$T$ localized regime and remains semi-quantitatively accurate across the phase diagram, including the $T=0$ limit and metastable negative-$T$ states. A key finding is the near-linear dependence of $h_{nl}$ and $h_{int}$ on the total energy and a predictive ratio $R=(h_c-h_{nl})/(h_c-h)$ in the small-$w$ regime, with a consistent physical picture of negative-temperature homogeneous states regulated by a cutoff and characterized by metastability. Overall, the MF approach provides a tractable analytic description bridging DNLS equilibrium thermodynamics and negative-temperature phenomena, with potential for extension to non-equilibrium dynamics.

Abstract

The Discrete Non Linear Schrödinger (DNLS) model, due to the existence of two conserved quantities, displays an equilibrium transition between a homogeneous phase at positive absolute temperature and a localized phase at negative absolute temperature. Here, we provide a mean-field theory of DNLS and show that this approximation is semi-quantitatively correct in the whole phase diagram, becoming exact in proximity of the transition. Our mean-field theory shows that the passage from stable positive-temperature to metastable negative-temperature states is smooth.

Figures (10)

• Figure 1: Value of $h_{nl}$ along three isothermal curves ($\beta=0.01,0.1,10$ from top to bottom) comparing the results of equilibrium numerical simulations (see \ref{['app.tech']}) of the DNLS equation (symbols) with the prediction of the MF theory (dashed lines) and with the energy of the C2C model (dotted lines) as derived in Szavits2014_PRLgotti22.
• Figure 2: Relative difference between the MF partition function $z$ computed by numerically integrating the exact expression (\ref{['eq_z1']}) and its analytical approximation $z_k$ obtained as a series expansion at order $k$ in $w$. The various symbols correspond to different $w$ values as from the legend. The value of $\beta$ is always equal to $\pm 0.4$ except for the lowermost curve, where $\beta=-0.01$. Notice that the sign of $\beta$ coincides with the sign of $w$. For negative $\beta$, the numerical integration is performed by imposing the same cutoff $c^*$ as in the C2C model.
• Figure 3: Comparison between MF theory (curves) and exact results obtained via grandcanonical simulations (symbols). Each curve corresponds to a different $T=1/\beta$, increasing from bottom to top according to the legenda. The size of the system is $N=100$ and details about simulations are given in \ref{['app.tech']}.
• Figure 4: Same as in Fig. \ref{['fig:ahpb']}, but we now compare MF and exact numerical results in the planes $(\mu,a)$ and $(\mu,h)$ rather than in $(a,h)$. In this way a horizontal shift at low temperature is evident, due to a wrong determination of the chemical potential in the MF theory. Symbols: exact results from grandcanonical simulations. Full lines: MF approximation. Dashed lines: MF approximation corrected for the chemical potential: $\mu\to\mu -1$.
• Figure 5: The contributions $h_{int}$ and $h_{nl}$ are presented versus the total energy $h$ suitably rescaled (see the main text). Symbols are the exact results from microcanonical simulations, as described in Appendix A, but with the removal of the interaction with the heat bath and starting from an initial condition with the chosen values of $a$ and $h$.