Thermodynamic phase transitions in lattice spin systems with severe kinetic constraints: Numerical simulation results

Abstract: The Fredrickson-Andersen model with hyperparameter $K=1$ is a severely constrained kinetic lattice spin system, such that any site is temporarily blocked from changing its packing state (empty or occupied) if there is one or more occupied nearest neighbors. Starting from a completely random initial configuration with a fraction $ρ$ of sites being occupied, some of the sites may be permanently frozen to their initial state under this severe kinetic constraint. The remaining sites can switch states at least occasionally, and they form the unfrozen subsystem associated with the given initial configuration. In the present work we investigate thermodynamic phase transitions in such unfrozen subsystems of the two-dimensional square lattice and the three-dimensional cubic lattice by extensive numerical simulations. We demonstrate that the giant connected component of the unfrozen subsystem collapses at certain critical value $ρ_{c}$ of initial packing density, with $ρ_c = 0.2475$ for the square lattice and $ρ_c = 0.2809$ for the cubic lattice. This phase transition belongs to the same universality class of the conventional site percolation. We also observe that the ground states (densest packing configurations) experience a continuous crystal-to-glass phase transition at the critical value $ρ^* = 0.1423$ of initial packing density for the cubic lattice. For the two-dimensional square lattice we argue that long-range crystalline order is destroyed in the ground states as long as the initial packing density $ρ$ is positive.

• The paper demonstrates that the unfrozen subsystem undergoes continuous site-percolation transitions at ρc = 0.2475 in two dimensions and ρc = 0.2809 in three dimensions, using large-scale simulations and finite-size scaling.
• The paper finds that ground states lose crystalline order at any positive density in two dimensions, while three-dimensional systems exhibit a continuous crystal-to-glass transition at ρ* = 0.14226(4).
• The paper uses maximum matching, exact ground-state classification, and ergodic sampling to show that kinetic constraints alone can generate thermodynamic transitions resembling random-field and percolation phenomena without static interactions.

Overview

This paper reports extensive numerical simulations of the Fredrickson–Andersen (FA) kinetic spin model in its most severely constrained case, $K=1$, on two- and three-dimensional hypercubic lattices with periodic boundary conditions (2607.06205). Each site carries a binary packing state, and a site may flip only if it has fewer than $K$ occupied nearest neighbors. Starting from a random initial configuration with packing density $\rho$, any occupied site adjacent to another occupied site is permanently frozen occupied; empty sites adjacent to frozen occupied sites are permanently frozen empty. The remaining unfrozen sites form a subsystem whose thermodynamics is the subject of the study. The central findings are twofold: first, the giant connected component of the unfrozen subsystem undergoes a continuous collapse transition at $\rho_c = 0.2475$ (square lattice) and $\rho_c = 0.2809$ (cubic lattice), belonging to the conventional site percolation universality class; second, the ground states of the unfrozen subsystem exhibit a crystal-to-glass transition at $\rho^* \approx 0.1423$ in three dimensions, while in two dimensions long-range order is destroyed at any positive $\rho$, implying $\rho^* = 0$.

Model and mapping to combinatorial problems

The energy of a configuration is $E = -\sum_i c_i$, so there are no static interactions; all interaction effects enter through the kinetic rule $b_i(t) < K$. For $K$0, the set of unfrozen occupied sites must form an independent set of the lattice, and the unfrozen empty sites form a vertex cover. The ground-state problem at chemical potential $K$1 is therefore exactly the maximum independent set (equivalently minimum vertex cover) problem on the unfrozen subgraph — an NP-hard problem in general but polynomially solvable on bipartite graphs via maximum matching.

The authors exploit this structure fully. A single maximum matching MM of the bipartite unfrozen graph yields the ground-state energy through König's theorem, $K$2. Moreover, a fix-and-propagate procedure classifies every unfrozen site as type-$K$3 (occupied in all ground states), type-$K$4 (empty in all ground states), or type-$K$5 (state-flexible). After cycle simplification, the state-flexible subgraph contains no leaf-removal core, so its ground states can be sampled ergodically by single-site flips. This exact coarse-graining is what makes the subsequent finite-size scaling analyses tractable at large system sizes ($K$6 up to 32000 in 2D).

Collapse transition of the unfrozen subsystem

As $K$7 increases, frozen domains proliferate and fragment the unfrozen network. The relative size $K$8 of the giant connected component drops sharply near $K$9 in two dimensions and $\rho$0 in three dimensions. Finite-size scaling of $\rho$1 against $\rho$2 gives:

Lattice	$\rho$3	$\rho$4	$\rho$5	$\rho$6
Square ($\rho$7)	0.247521(2)	1.3375(27)	0.1395(16)	≈1.02
Cubic ($\rho$8)	0.280942(4)	0.9053(33)	0.4482(19)	≈0.97

The 2D exponents agree closely with the exact percolation values $\rho$9 and $\rho_c = 0.2475$0; refitting with these values fixed produces a data collapse of essentially identical quality ($\rho_c = 0.2475$1). In 3D, the estimates are consistent with modern site-percolation values ($\rho_c = 0.2475$2, $\rho_c = 0.2475$3), and the hyperscaling-derived Fisher exponent $\rho_c = 0.2475$4 matches the accepted $\rho_c = 0.2475$5. The fractal dimension extracted independently, $\rho_c = 0.2475$6 in 2D, agrees with $\rho_c = 0.2475$7. The implication is notable: although the kinetically induced defects are spatially correlated and form frozen domains of arbitrary shape, these correlations do not alter the universality class of the connectivity transition.

Ground states on the square lattice: no long-range order for any $\rho_c = 0.2475$8

At $\rho_c = 0.2475$9 the ground state is the perfect checkerboard crystal with energy density $\rho_c = 0.2809$0. With increasing $\rho_c = 0.2809$1, the energy density departs smoothly from $\rho_c = 0.2809$2, and example ground states at $\rho_c = 0.2809$3 show mosaic patterns in which different regions occupy opposite sublattices. Frozen sites act as quenched local biases favoring one or the other checkerboard sublattice, and accommodating them via domain walls lowers the total energy below that of any single-sublattice crystal.

An imbalance order parameter $\rho_c = 0.2809$4, measuring the excess of type-$\rho_c = 0.2809$5 over type-$\rho_c = 0.2809$6 sites between the two sublattices, decreases with $\rho_c = 0.2809$7 and drifts leftward with system size. A finite-size scaling fit yields $\rho_c = 0.2809$8 with $\rho_c = 0.2809$9 and $\rho^* \approx 0.1423$0, but the authors are explicit that this is an extrapolation below the sampled density window and that $\rho^* \approx 0.1423$1 indicates only weak support for the scaling ansatz; they treat it strictly as an upper bound on the true critical density.

The decisive argument follows Imry–Ma reasoning adapted to this system. Numerically, across dimensions $\rho^* \approx 0.1423$2–$\rho^* \approx 0.1423$3, the number imbalance $\rho^* \approx 0.1423$4 within a region of side $\rho^* \approx 0.1423$5 satisfies $\rho^* \approx 0.1423$6. Flipping a region costs boundary energy $\rho^* \approx 0.1423$7 but gains $\rho^* \approx 0.1423$8 internally when $\rho^* \approx 0.1423$9. In $\rho$0 the probability of net energy gain is $\rho$1, which is $\rho$2 independent of $\rho$3: arbitrarily large favorable fluctuations exist, so no ground state of the infinite square lattice can sustain long-range order at any positive defect density. For $\rho$4, $\rho$5 vanishes with $\rho$6 and ordered phases can survive to positive $\rho$7. This dimension-dependent conclusion is consistent with rigorous results on the random-field Ising model (Aizenman–Wehr, Bricmont–Kupiainen, Ding–Zhuang), and the authors state that a rigorous proof of $\rho$8 for the square lattice will appear in a separate paper.

Ground states on the cubic lattice: a continuous crystal-to-glass transition

In three dimensions the situation changes qualitatively. For $\rho$9, nearly all type-$\rho^* = 0$0 sites lie on one sublattice, indicating crystalline order; above $\rho^* = 0$1, type-$\rho^* = 0$2 sites distribute equally between sublattices and form interpenetrating locally ordered domains separated by irregular boundaries — a glassy ground-state structure. Finite-size scaling of $\rho^* = 0$3 collapses cleanly onto a single curve with:

$\rho^* = 0$4

Unlike the 2D case, the fitted critical point lies well inside the sampled density range, so no extrapolation is involved. The exponents are close to those reported for the 3D random-field Ising model ($\rho^* = 0$5, $\rho^* = 0$6), supporting the physical mapping in which frozen domains play the role of random fields and domain-wall avoidance plays the role of ferromagnetic coupling. Whether the present model lies in the RFIM universality class remains explicitly unresolved, since the frozen sites are correlated rather than independent; the authors also note that fixing $\rho^* = 0$7 worsens the fit substantially and that the $\rho^* = 0$8 distribution is unimodal, so they interpret the small positive $\rho^* = 0$9 as evidence of a highly cooperative but still continuous transition. The extreme smallness of $E = -\sum_i c_i$0 itself signals that further work is needed to characterize this transition fully.

Limitations and open questions

Several caveats are stated plainly by the authors. The 2D critical point $E = -\sum_i c_i$1 from FSS is an extrapolation outside the data range with a poor chi-square, and the true value is argued to be exactly zero. The universality class of the 3D ground-state transition relative to the RFIM is undetermined, given correlated quenched defects. All results pertain to $E = -\sum_i c_i$2; whether the crystal-to-glass transition persists as a true transition or becomes a crossover at finite chemical potential is open, as is the full phase diagram including the critical lines $E = -\sum_i c_i$3 and $E = -\sum_i c_i$4 and a possible tricritical point among crystalline, glass, and gas phases. Dynamical questions — facilitation, growing relaxation times, dynamical heterogeneity, and distinct dynamics of type-$E = -\sum_i c_i$5/type-$E = -\sum_i c_i$6 versus type-$E = -\sum_i c_i$7 sites at finite $E = -\sum_i c_i$8 — are entirely unaddressed. Finally, everything here concerns $E = -\sum_i c_i$9; extending to $b_i(t) < K$0, where frozen occupied sites form closed loops and unfrozen configurations become loop-free tree packings, is identified as substantially more challenging.

Conclusion

The paper establishes that severe kinetic constraints alone, without any static interaction energy, generate genuine thermodynamic phase transitions in finite-dimensional FA models. The unfrozen subsystem undergoes a percolation-class collapse at $b_i(t) < K$1 (2D) and $b_i(t) < K$2 (3D), while its ground states undergo a crystal-to-glass transition at $b_i(t) < K$3 in 3D and at $b_i(t) < K$4 in 2D, the latter following from an Imry–Ma-type instability argument. The combination of exact matching-based ground-state enumeration, uniform sampling of the state-flexible sector, and high-quality finite-size scaling provides a clean quantitative foundation, and the demonstrated correspondence with the random-field Ising model offers a concrete route toward theoretical characterization of these kinetics-induced transitions.