Complexity of waves in nonlinear disordered media

The statistical properties of the phases of several modes nonlinearly coupled in a random system are investigated by means of a Hamiltonian model with disordered couplings. The regime in which the modes have a stationary distribution of their energies and in which the phases are coupled is studied for arbitrary degrees of randomness and energy. The complexity versus temperature and strength of nonlinearity is calculated. A phase diagram is derived in terms of the stored energy and amount of disorder. Implications in random lasing, nonlinear wave propagation, and finite-temperature Bose-Einstein condensation are discussed.

Figure 1

Phase diagram in the $\mathcal{P},R_J$ plane. Three phases are present: paramagnetic (PM) (low $\mathcal{P}$), ferromagnetic (FM) (high $\mathcal{P}$/weak disorder), and spin glass (SG) (high $\mathcal{P}$/strong disorder). The solid lines are thermodynamic transitions; the dashed line represents the dynamic PM/SG transition.

Figure 2

Phase diagram in the plane $J_0,T$ in ${\sigma }_J$ units. Also negative $J_0$ are considered. Three phases are found: PM (high $T$, low $J_0$), FM (low $T$/large $J_0$). and SG (low $T$/low or negative $J_0$). The solid lines are thermodynamic transitions: random first order between PM and SG and standard first order between PM and FM and between SG and FM. The dashed line represents the dynamic PM/SG transition.

Figure 3

Detail of the $J_0,T$ phase diagram around the tricritical point. Solid lines are thermodynamic transitions. Also the transition between the SG (1RSB) and the approximated RS solution for the FM phase is displayed (double-dotted line) showing no appreciable difference with the exact one. The dashed line represents the dynamic PM/SG transition. The dotted bold line represents the FM spinodal lines inside both the PM and the SG phases. The spinodal of the RS FM phase is shown as well (smaller dots).

Figure 4

Complexity $\Sigma (\mathcal{P})$ of the lowest lying glassy states in free energy between the values of the pumping rate corresponding to the dynamic and static transitions from the PM to the SG phase at $R_J=0.5$.

Figure 5

Left panel: Complexity of the lowest lying glassy states in free energy $\Sigma (\mathcal{P})$ between dynamic and static transition from the PM phase to the SG phase along the $R_J=0.3$, 0.4, and $0.5$ lines. The qualitative behavior is identical for any $R_J\gtrsim 0.3$. Right panel: $\Sigma$ vs the effective temperature $T$ in ${\sigma }_J$ units.

Figure 6

$\Sigma (f)$ (left panel) and $\Sigma (m)$ (right panel) in the glassy phase at the static transition effective temperature, $T=0.140\hspace{0.5em}99$. This is the picture that holds for any $R_J\gtrsim 0.3$.

Figure 7

$\Sigma (f)$ (left panel) and $\Sigma (m)$ (right panel) in the glassy phase at the static transition effective temperature, $T=0.15<T_d$. The lowest state free energy of metastable glassy states is denoted by $f_1$ (i.e., corresponding to $m=1$ in the right panel, see text). The free energy of maximum complexity is denoted by $f^{*}$ (correspondingly $m^{*}$ in the right-hand plot).

Figure 8

Discontinuity of the order parameters at the transition points for three values of $R_J$. Top left panel: Jump in $q_{0,1}$, at the PM/FM transition in $\mathcal{P}$ for small disorder, $R_J\simeq 0.1$. Top right panel: Discontinuities in $r_{0,1}$, $\mathrm{r̃}$, and $\mathrm{m̃}$ at the same transition. For such small $R_J$ the replica symmetry breaking is practically invisible: $q_1\simeq q_0$, $r_1\simeq r_0$ [to the precision of our computation, $\mathcal{O}({10}^{-5})$]. Middle left panel (across tricritical region in Fig. 1: $q_{0,1}$ vs $\mathcal{P}$ at $R_J\simeq 0.26$ where, increasing the pumping rate, first a PM/FM transition occurs followed by a FM/SG one. Middle right panel: $r_{0,1}$, $\mathrm{r̃}$, and $\mathrm{m̃}$ vs $\mathcal{P}$ for the same interval. First-order transition points are signaled by vertical lines. Left bottom panel: $q_{0,1}$ vs $\mathcal{P}$ for large disorder, $R_J=0.4$ across the PM/SG random first-order transition. Right bottom panel: $r_{0,1}$, $\mathrm{r̃}$, and $\mathrm{m̃}$ are always zero in the SG phase and in the PM phase.

Figure 9

Complexity vs free energy curve is plotted in the SG phase (left) at $T/{\sigma }_J=0.1$ and in the FM phase (right).

Figure 10

Complexity curves of the FM phase at $R_J=3.54$ at temperatures between $T=0.082\hspace{0.5em}{\sigma }_J$ (right most) and $T=0.139\hspace{0.5em}{\sigma }_J$ (left most). Both the magnitude of the maximal complexity and the free energy interval in which $\Sigma (f)>0$ decreases. Notice that the equilibrium free energy decreases as the temperature increases.

Figure 11

Top panel panel: Free energy of the FM and SG phases vs $R_J$ at $T=0.0785$. As the degree of disorder increases the system undergoes a first-order phase transition from a ferromagnetic phase to a spin-glass phase. Middle panel: Order parameters $r_1$, $r_0$, and $\mathrm{r̃}$ and the magnetization $\mathrm{m̃}$ are shown vs $R_J$. Beyond the transition point their values drop to zero in the SG phase. Bottom panel: $q$ order parameters for the FM phase and the SG phase.