Spontaneous symmetry-breaking in equilibrium tree-packing configurations of a kinetically constrained cubic-lattice system

Abstract: We explore kinetic-constraint induced thermodynamic phase transition in the cubic lattice, employing the Fredrikson-Anderson spin model with hyperparameter $K=2$ as a representative kinetic system. Each lattice site may flip its binary occupation state if at most one of its six nearest neighbors is currently occupied. The whole set of microscopic configurations that are kinetically connected with the fully empty one is described by an equilibrium partition function with a single global constraint, that is, the occupied sites do not form closed loops but instead organize into different tree components in the lattice. We discover a continuous thermodynamic gas--crystal phase transition in the cubic system and determine the critical chemical potential $μ^* \approx -3.252$, at which the occupied sites of the equilibrium tree-packing configurations start to prefer one of the two nested cubic sublattices. This thermodynamic phase transition is absent in the two-dimensional square lattice.

• The paper establishes a continuous gas–crystal transition at chemical potential μ* = −3.25160(8), where loop-free occupied sites spontaneously favor one of two cubic-lattice sublattices.
• Finite-size scaling gives β = 0.2864(1) and ν = 0.5969(2), values distinct from three-dimensional Ising estimates but requiring larger systems and alternative fits to confirm a new universality class.
• The two-dimensional model shows no thermodynamic ordering and approaches density 2/3 at high occupation bias, identifying three dimensions as the minimum dimension for this equilibrium transition in the K=2 system.

Overview and motivation

Kinetically constrained spin models such as the Fredrickson–Andersen (FA) model are standard minimal systems for glassy dynamics, yet their equilibrium thermodynamics has received comparatively little attention. This paper by Hai-Jun Zhou (2608.17308) studies the FA model with kinetic threshold $K=2$ on periodic cubic lattices, where each site flips its binary occupation state only if fewer than two of its six nearest neighbors are occupied. Starting from the fully empty configuration, this rule implies a global topological invariant: occupied sites can never form closed loops. The kinetically reachable configuration space — the "fully unfrozen kinetic cluster" $\mathcal{C}_1$ — therefore coincides exactly with the set of loop-free configurations, in which occupied sites organize into mutually disconnected tree components and empty sites form a feedback vertex set.

The equilibrium statistical mechanics of $\mathcal{C}_1$ is encoded in a partition function with a single global constraint,

$$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$

where $\mu < 0$ is the chemical potential controlling the occupation density $\rho$. Crucially, unlike lattice glass models or the $K=1$ (hard-core) case, the constraint is non-local and does not restrict the number of occupied neighbors of any individual site; effective interactions between sites are strongly non-local and non-reciprocal. This makes analytical treatment in finite dimensions very difficult, motivating the numerical approach adopted here.

Monte Carlo methodology

The author employs an equilibrium Markov-chain Monte Carlo scheme obeying detailed balance, combining single-site flips with pairwise state-swapping (with swap probability $p_{\text{swap}} = 1 - e^{\mu}$). The central algorithmic device is a dynamically maintained set $\Gamma_0$ of flippable empty sites: an empty site belongs to $\Gamma_0$ if occupying it would not close a loop. Tree indices are assigned to occupied sites to distinguish tree components, so that an empty site touching two or more occupied sites of the same component is temporarily blocked. Tree indices are updated after every configuration change.

For each $\mathcal{C}_1$0 point, 64–65 independent trajectories are run on lattices with even side lengths $\mathcal{C}_1$1 (even $\mathcal{C}_1$2 preserves the bipartition into two nested sublattices $\mathcal{C}_1$3 and $\mathcal{C}_1$4), with at least $\mathcal{C}_1$5 equilibrium configurations sampled per point. Autocorrelation times of the order parameter remain below $\mathcal{C}_1$6 time units throughout, indicating that equilibration is achieved.

Continuous gas–crystal transition and sublattice symmetry-breaking

The density curve $\mathcal{C}_1$7 exhibits a change of trend near $\mathcal{C}_1$8, which microscopic inspection attributes to spontaneous breaking of the symmetry between the two nested cubic sublattices. A staggered magnetization-like order parameter

$\mathcal{C}_1$9

vanishes as $\mathcal{C}_1$0 in the symmetric phase but approaches a nonzero constant once one sublattice is preferentially occupied. The mean $\mathcal{C}_1$1 decays to zero for $\mathcal{C}_1$2 and saturates to a positive value for $\mathcal{C}_1$3, and the probability distribution $\mathcal{C}_1$4 evolves from a single peak at $\mathcal{C}_1$5 ($\mathcal{C}_1$6, $\mathcal{C}_1$7) to two symmetric peaks at $\mathcal{C}_1$8 separated by a deep valley at $\mathcal{C}_1$9 — behavior characteristic of a continuous transition analogous to the onset of ferromagnetism in the three-dimensional Ising model.

Finite-size-scaling analysis using 47 data points and a sixth-order polynomial scaling function yields

• Critical chemical potential: $$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$0
• Critical exponents: $$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$1, $$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$2
• Fit quality: rescaled chi-square $$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$3

Notably, these exponents are clearly distinct from those of the three-dimensional Ising universality class ($$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$4, $$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$5). The author accordingly suggests the transition may belong to a different universality class, while explicitly conceding that more work is needed to settle this question.

In the crystalline phase the two sublattices play structurally distinct roles within the tree components. At $$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$6, an occupied site on the dense sublattice is most likely a leaf (one occupied neighbor), whereas an occupied site on the sparse sublattice is typically a junction of 4–6 branches; correspondingly, empty sites on the dense sublattice usually have 2–4 occupied neighbors, while empty sites on the sparse sublattice are most often fully surrounded by six occupied neighbors. This asymmetry directly implies that empty sites on the sparse sublattice are kinetically harder to flip than those on the dense one.

Absence of a transition in two dimensions

Applying the identical methodology to periodic square lattices yields a qualitatively different result: no symmetry-breaking occurs at any chemical potential. The density $$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$7 is smooth and approaches the maximum value $$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$8 as $$Z(\mu) = \sum_{\vec{c}} I(\text{no occupied loop}) \prod_i e^{-\mu c_i},$$9, $\mu < 0$0 reflects only finite-size fluctuations around zero, and site-resolved occupation fractions $\mu < 0$1 concentrate around the mean density ($\mu < 0$2 at $\mu < 0$3) as trajectory length grows. The two-dimensional system thus remains in a homogeneous disordered gas phase at all $\mu < 0$4, establishing that dimension three is the minimum for a thermodynamic phase transition in the $\mu < 0$5 FA system. This sharp dimensional contrast is the paper's most striking structural finding, and it raises the open question of whether it carries over to qualitative differences in dynamical behavior under the FA kinetic rule.

Limitations and open questions

Several caveats attach to the results. The claim of a new universality class rests on exponent estimates from systems up to $\mu < 0$6 with polynomial FSS fits; corrections to scaling or alternative fitting forms could shift the exponents toward Ising values, and the author explicitly flags this as unresolved. All configurations are sampled from random initial conditions equivalent to the fully empty state; whether the crystalline phase is kinetically accessible under genuine single-site-flipping FA dynamics at sufficiently negative $\mu < 0$7 remains undetermined — if it is avoided, the system would fall out of equilibrium. Other directions left open include extension to other lattice geometries and higher dimensions, the possible emergence of a thermodynamic glass phase from random initial conditions with a positive occupied fraction, and the cases $\mu < 0$8, whose equilibrium problem maps onto the considerably harder $\mu < 0$9-core attack problem.

Conclusion

This work establishes that the $\rho$0 Fredrickson–Andersen model on the cubic lattice undergoes a continuous gas–crystal equilibrium phase transition at $\rho$1, driven by spontaneous breaking of the symmetry between nested sublattices within globally constrained tree-packing configurations, with critical exponents apparently distinct from the three-dimensional Ising class. The absence of any such transition in two dimensions delineates a sharp dimensional boundary for ordering in kinetically constrained systems and motivates comparative dynamical studies of the square and cubic cases.