Mean-Field Theory of DNLS

• Mean-Field Theory of DNLS is a framework that decouples a one-dimensional quantum lattice model into site-level problems, capturing energy and mass conservation.
• It employs a grand-canonical partition function and small-w expansions to provide precise predictions of phase transitions between homogeneous positive-temperature and localized negative-temperature regimes.
• Comparisons with Monte Carlo and transfer-operator simulations validate the theory's asymptotically exact predictions and its ability to describe metastable states near critical manifolds.

The mean-field (MF) theory of the Discrete Nonlinear Schrödinger (DNLS) equation provides a comprehensive equilibrium description of a one-dimensional quantum lattice model characterized by both energy and norm (mass) conservation. The DNLS exhibits an equilibrium transition between homogeneous states at positive absolute temperatures and localized, negative absolute temperature regimes, a phenomenon enabled by the model's dual conservation laws. MF theory furnishes explicit, semiquantitative predictions throughout the $(a, h)$ phase diagram—where $a$ and $h$ denote mass and energy densities, respectively—and attains asymptotic exactness near the critical manifold separating the two thermal regimes.

1. Microscopic Hamiltonian and Conserved Quantities

The DNLS model is defined on a one-dimensional lattice of $N$ sites, with each site $n$ assigned a nonnegative amplitude (“mass”) $c_n \ge 0$ and a phase $\phi_n \in [0,2\pi)$. The Hamiltonian is given by

$$H = \sum_{n=1}^N \left[ c_n^2 + 2J \sqrt{c_n c_{n+1}} \cos(\phi_n - \phi_{n+1}) \right] \equiv N h,$$

where $J$ is the hopping strength (one often sets $J=1$ by rescaling), and $h$ is the energy density. The total mass (or norm) is

$A = \sum_{n=1}^N c_n \equiv N a,$

with $a$ the mass density. Both $H$ and $A$ are conserved under DNLS dynamics. The ground state (zero temperature) features uniform amplitudes and alternating phases, with energy density

$h_{\rm GS}(a) = a^2 - 2J a.$

The critical line (“infinite temperature,” $\beta=0$) identifying the boundary between positive and negative absolute temperatures is given by

$h_c(a) = 2 a^2.$

Below $h_c(a)$, the equilibrium state is homogeneous with $T>0$; above, the state is localized with $T<0$.

2. Mean-Field Grand-Canonical Partition Function

The grand-canonical partition function in the DNLS context reads

$$Z(\beta, \mu) = \int_{0}^{\infty} \prod_n dc_n \int_{0}^{2\pi} \prod_n d\phi_n\, \exp[-\beta (H + \mu A)],$$

with inverse temperature $\beta=1/T$ and chemical potential $\mu$. The tight coupling between lattice sites precludes factorization of $Z$ in the full model.

Applying a mean-field decoupling

$$\sqrt{c_n c_{n+1}} \rightarrow q \sqrt{c_n}, \quad \text{where} \quad q = \langle \sqrt{c_n} \rangle,$$

yields a site-factorizable Hamiltonian: $$H_{\rm MF} = \sum_{n=1}^N \left[ c_n^2 + 2J q \sqrt{c_n} \cos(\phi_n - \phi_{n+1}) \right].$$ Without loss of generality, $J=1$ is adopted. The reduced single-site partition function becomes

$$z(\beta,\mu; q) = \int_{0}^{\infty} dc \int_{0}^{2\pi} d\varphi\, e^{-\beta [c^2 + 2q \sqrt{c} \cos\varphi] + \beta\mu c},$$

where $\varphi = \phi_{n+1} - \phi_n$. The angular integral produces a modified Bessel function $I_0$: $$z = 2\pi \int_0^\infty dc\, \exp(-\beta c^2 + m c) I_0(2\beta q \sqrt{c}), \qquad m \equiv \beta\mu.$$ The MF grand-canonical partition function becomes

$$Z_{\rm MF}(\beta, \mu) = z^N, \qquad F_{\rm MF}(T,\mu) = -\frac{1}{\beta} \ln z.$$

MF thus yields an explicit (but integral) formula for the grand-potential and all thermodynamic observables.

3. Free Energy and Thermodynamic Observables

The per-site MF free energy is

$F(T, \mu) = -\frac{1}{\beta} \ln z(\beta, \mu; q),$

with $z$ as above. Systematic expansions are feasible in the small parameter

$w \equiv \frac{1}{\beta\mu^2} = \frac{\beta}{m^2},$

valid for $|w| \ll 1$. Explicitly,

$$z = \frac{2\pi}{|m|} - \frac{4\pi}{|m|}w + \frac{\pi (24 + 2|m|^3 q^2)}{|m|} w^2 + O(w^3),$$

$$F = -\frac{1}{\beta} \left[ \ln \frac{2\pi}{|m|} - \frac{4w}{2\pi/|m|} + O(w^2) \right].$$

Thermodynamic observables—including $a = \langle c \rangle$, $h_{nl} = \langle c^2 \rangle$, $h_{int} = \langle 2q\sqrt{c}\cos\varphi \rangle$—are computed by differentiating $F$ or evaluating moments: $$\langle c^\alpha \rangle = \frac{1}{z} \int dc\, d\varphi\, c^\alpha e^{-\beta(c^2 + 2q \sqrt{c} \cos\varphi) + m c}.$$ This framework allows one to recover all critical manifolds, $T=0$ and $T = \pm\infty$ lines, and to approximate the limit of metastability on the negative-$T$ side.

4. Self-Consistency and Leading-Order Expansions

The MF parameter $q$ is determined by the self-consistency equation

$$q = \langle \sqrt{c} \rangle = \frac{1}{z} \int_0^\infty dc \int_0^{2\pi} d\varphi\, \sqrt{c}\, e^{-\beta (c^2 + 2q\sqrt{c}\cos\varphi) + m c},$$

which in closed form becomes

$$q = \frac{2\pi}{z} \int_0^\infty dc\, \sqrt{c}\, \exp(-\beta c^2 + m c) I_0(2\beta q \sqrt{c}).$$

Other quantities are similarly expressed: $$a = \langle c \rangle, \quad h_{nl} = \langle c^2 \rangle, \quad h_{int} = -\frac{4\pi q}{z} \int_0^\infty dc\, \sqrt{c}\, e^{-\beta c^2 + m c} I_1(2\beta q \sqrt{c}).$$ Solving these equations order-by-order in $w$, the leading-order expansions (with $m<0$ along the critical line) are: $$\begin{aligned} q &= \frac{\sqrt{\pi}}{2\sqrt{|m|}} - \frac{7\sqrt{\pi}}{8\sqrt{|m|}}w + O(w^2), \ a &= \frac{1}{|m|} - \frac{4 w}{|m|} + O(w^2), \ h_{nl} &= \frac{2}{m^2} - \frac{20 w}{m^2} + O(w^2), \ h_{int} &= -\frac{\pi}{2} w + O(w^2), \ h &= h_{nl} + h_{int} = \frac{2}{m^2} - \left( \frac{\pi}{2} + \frac{20}{m^2} \right)w + O(w^2). \end{aligned}$$ These formulae are valid for both signs of $\beta$, applying on both sides of the infinite-temperature line.

5. Critical Manifolds and Phase Structure

The $(a,h)$ thermodynamic plane forms the natural backdrop for the DNLS phase diagram. The critical separation between positive- and negative-$T$ states occurs at

$h_c(a) = 2 a^2, \qquad \beta \to 0, \; m= -1/a.$

For $h < h_c(a)$, the system is homogeneous with $T > 0$; for $h > h_c(a)$, the system enters the $T < 0$ (formally negative temperature) regime, which is localized. The $T = 0$ ground-state line is $h = a^2 - 2a$.

Within the MF formalism, the region of metastability on the negative-$T$ side (homogeneous states persisting above the infinite temperature line) is identified by examining the MF potential for local minima at $c=0$: $|\beta| \lesssim a^{-4/3}.$ Translated to the $(a, h)$ plane,

$$h - h_c < \left( \frac{\pi}{2} + 20a^2 \right) a^{2/3}.$$

This demarcates the regime of long-lived homogeneous negative-$T$ states.

A schematic phase diagram is:

Region	$(a, h)$ constraint	Description
hom. $T>0$	$a^2 - 2a \leq h < 2a^2$	Homogeneous phase
localized $T<0$	$h > 2a^2$	Localized (negative-$T$) phase
ground state	$h = a^2 - 2a$	Zero-temperature boundary

6. Smooth Transition Across the Critical Line

A defining feature of the MF theory for DNLS is the smooth crossover from thermodynamically stable positive-$T$ to metastable negative-$T$ states at $\beta = 0$:

• All macroscopic observables, including $a, h, q, h_{nl}, h_{int}$, remain continuous at the critical line.
• The small-$w$ expansions for key quantities are identical for $\beta \to 0^+$ and $\beta \to 0^-$, with only exponentially small cutoff corrections for $\beta < 0$.
• Near the critical manifold, spatial correlations vanish, $h_{int} \to 0$, and MF theory becomes asymptotically exact.

A plausible implication is that MF theory fully captures the leading approach to the infinite-temperature transition despite the underlying microcanonical requirements for true negative-$T$ equilibrium (i.e., breather formation).

7. Comparison with Exact and Numerical Results

Extensive heat-bath and Monte Carlo simulations of the full DNLS model (e.g., $N=100$, $10^{-2} \leq \beta \leq 10$) demonstrate:

• In the $(a, h)$ plane, MF isotherms $h(a; T)$ almost coincide with simulation data for all $T$.
• In $(\mu, a)$ and $(\mu, h)$ representations, MF correctly reproduces the curves’ shapes at high $T$; at low $T$ a near-constant shift $\mu_{\rm exact} \approx \mu_{\rm MF} - 1$ is present, which vanishes as $\beta \to 0$.
• Approaching $h \to 2a^2$ (the critical line), the MF prediction becomes essentially exact: $h_{int} \to 0$, spatial correlations vanish, and the solution matches the factorized site-level case.
• The small-$w$ expansion for the single-site partition function reproduces numerical integrals to within $10^{-4}$ for $|w| < 0.1$, regardless of the sign of $\beta$.

These facts confirm that MF provides explicit, integral-based thermodynamics and a quantitatively faithful account of the DNLS equilibrium structure across both positive and negative absolute temperature regimes. The mean-field approach smoothly interpolates across the infinite-temperature transition and matches transfer-operator and Monte Carlo results in all qualitative and semi-quantitative respects.