The mechanistic origin of branching-driven nucleation in abrupt phase transitions

Abstract

Phase transitions are the macroscopic manifestation of microscopic processes that drive a system towards a new state. The detailed evolution of these processes, particularly in abrupt phase transitions, are currently not fully understood. Here, we introduce a theoretical framework based on internal node dependencies within a single-layer lattice. Crucially, we demonstrate that the fundamental mechanism underlying abrupt transitions is nucleation propagation preceded by a slow cascading process which scales with the range of dependencies. Our findings show that the synergy between these two distinct stages is essential for the occurrence of an abrupt transition. The first stage of a slow cascading mechanism was recently observed experimentally in superconducting layered materials, where heat acts as the dependency links, for the limit of infinite dependency range. Our model thus generalizes the framework to include finite dependency ranges, revealing previously unobserved mechanisms that could be experimentally verified through controlling the range of thermal diffusion in the material. As a universal mechanism, our model provides a robust method to test nucleation-controlled phase transitions in multiple systems, providing a path to discover and understand microscopic mechanisms in phase transitions.

Figures (9)

• Figure 1: Distinct phase transitions in the internally dependent lattice.a, Schematic of the model, where dependency links are shown in orange. Initially, we remove a fraction of nodes, marked with cyan crosses at the top panel. Following this removal, we repeatedly iterate two stages. At the percolation failure stage, we remove nodes that become disconnected from the GCC, marked with a red cross in the second panel. This is followed by the dependency failure stage, where we remove nodes that depend on the lost nodes (yellow crosses). The process continues by alternating the percolation and dependency failure stages until a steady state is reached. b, The final size (after cascading stops) of the GCC, $P_{\infty}(p)$, as a function of the initial non-removed probability, $p$, for small values of $r\leq 7$, as indicated in the plot. All the transitions are continuous and the inset shows that the power-law exponent, $\beta$, in the vicinity of the transition has the same value of $\beta=5/36$ as in regular two-dimension percolation. c, The final size of the GCC, $P_{\infty}(p)$, as a function of the initial non-removed probability, $p$, for larger values of $r\geq 8$, as indicated in the plot. All the transitions are abrupt. For larger $r$ values, of the order of the system size, the curvature close to the critical point indicates a mixed-order transition with critical exponents. As shown in the inset, the exponent for the mixed-order transition at $r=5000$ has a value $\beta=1/2$, see Eq. \ref{['Eq:beta']}. d, The phase diagram and the critical threshold $p_c$ versus the dependency length $r$. All results in this Figure are for networks with size $N=L \times L=5000\times 5000$.
• Figure 2: The time evolution of the spontaneous failure process shows how localized branching can initiate nucleation propagation.a, Snapshots of the largest cluster (yellow), failed nodes in previous steps (blue), and failed nodes at the current step (red), for $r=50$ at four different times during the cascade at $p_c$, as shown in panel (b). The system size is $L=5000$, but for clarity we zoom in a window of $L=2000$. b, The size of the GCC, $P_\infty(t)$, as a function of time (iterations) for $r=50$, $r=500$, $r=5000$ at the critical point $p_c$ during the spontaneous cascade which leads to the abrupt collapse. The dashed vertical lines indicate the point where the plateau branching process ends and the nucleation propagation process starts (there is no nucleation propagation at $r=5000$). Notice that the duration of branching, $\tau_b$, increases with $r$ (Eq. \ref{['Eq:27']}), while the duration of nucleation, $\tau_n$, decreases accordingly. c, Fraction of failing nodes at each time step for the same $r$ values as in panel (b). d, Size of the failed-GCC, $P^f_\infty(t)$, as a function of time. The failed-GCC starts with zero size, until a nucleus is formed and steadily propagates radially through the system at different rates. The system size is $L = 5000$. Similar plots for smaller $r$ values are presented in the Supplementary Information, Figs. S1 and S2.
• Figure 3: Scaling behavior of abrupt cascade failures during the branching process (top row) and during the nucleation process (bottom row).a, Scaling collapse of the distribution $P(\tau_b)$ of the branching plateau duration, $\tau_b$, at $p_c$. b, Scaling of the mean and standard deviation for the branching duration $\tau_b$ with $r$. Note that the scaling with $r$ is the generalization of the scaling with $L$ for $r=L$, found experimentally and theoretically see bonamassa2023interdependentzhou2014simultaneous. c, The dependence of $\langle \tau_b \rangle$ and $\sigma(\tau_b)$ on $L$, for fixed $r$, e.g. $r=50$, shows independence on $L$. This shows that the emergence of nucleation does not depend on the size of the system. Inset: The distribution of $\tau_b$ remains the same across different system sizes. d, Scaling collapse of the nucleation duration distribution $P(\tau_n)$. e, Scaling of the nucleation duration, $\tau_n$, with $r$. f, Scaling of $\langle \tau_n \rangle$ with $L$ for $r=50$. Each point represents an average over 1000 runs. The system size is $L=1000$, except for panels (c) and (f).
• Figure 4: The branching factor controls the system evolution near criticality. Average branching factor $\langle \eta \rangle$ as a function of $\Delta p= p-p_c$, for a,$r=L$, and b, for finite-range dependency, $r = 50, 70, 90$. The inset shows that $\langle \eta \rangle(p_c)$ increases linearly with $r$. Note that in contrast to $r=L$, where $\eta=1$ at criticality, for finite $r$, $\eta$ is surprisingly below 1 and approaches 1 for $r=L$. c, The distribution of distance, $P(d)$, between failed nodes and the nucleus center at a given step during the branching process, for different $r$, showing that most of failures occur near the microscopic nucleation center. Inset: The 25th percentile value of the distribution of distances between failed nodes and the nucleus center, measured at steps during the branching process at criticality showing linearity with $r$. The points represent the average of 10,000 independent runs for a system size of $L=1000$.
• Figure S1: a, Time evolution of the GCC, for different values of $r$. b, The number of failed nodes per time step, $t$, c, The size of the failed-GCC, as a function of time steps, $t$. All results are for $L=5000$.