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The freezing phase transition in hard core lattice gases on triangular lattice with exclusion up to seventh next-nearest neighbor

This paper employs a flat histogram algorithm with cluster moves to demonstrate that hard core lattice gas models on a triangular lattice with exclusion up to the seventh next-nearest neighbor ( $4 \leq k \leq 7$ ) undergo a single, discontinuous freezing phase transition from a fluid to a sublattice-ordered phase, providing precise determinations of their critical parameters.

Original authors: Asweel Ahmed A Jaleel, Dipanjan Mandal, Jetin E. Thomas, R. Rajesh

Published 2026-09-14

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CC BY 4.0

Original authors: Asweel Ahmed A Jaleel, Dipanjan Mandal, Jetin E. Thomas, R. Rajesh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded room where everyone is trying to find a seat, but with a strict rule: no two people can sit too close to each other. If you sit down, you effectively claim not just your own chair, but also a surrounding zone of empty space that no one else can occupy. This simple constraint, applied to millions of tiny particles on a grid, creates a fascinating puzzle for physicists. They are not studying how heat or cold changes the behavior of these particles; in this specific scenario, temperature is irrelevant. Instead, the entire drama is driven by density. As more and more particles are forced into the same space, they eventually run out of room to move freely. At a certain point, the chaos of a fluid suddenly snaps into an orderly, solid-like pattern. This is known as an entropy-driven phase transition, a phenomenon where order emerges purely because the particles are trying to maximize their available space. Scientists have long studied this on simple grids, like squares or honeycombs, but the triangular grid presents a unique challenge. It is a shape that more closely mimics the continuous, unbroken space of the real world, yet it has proven difficult to predict exactly how the particles will arrange themselves when they are packed tightly together.

A team of researchers recently tackled this problem by simulating hard-core lattice gases on a triangular grid, pushing the limits of how far a particle's "exclusion zone" reaches. In their model, a particle can block its immediate neighbors, or it can block neighbors further away, depending on a parameter they call $k$ . For years, scientists had only been able to map out the behavior for cases where the exclusion zone was relatively small. The new study extends this exploration to much larger exclusion zones, covering cases where a particle blocks up to seven layers of neighbors. Using a powerful computer simulation technique that allows the system to jump between different arrangements efficiently, the team mapped out the exact moment when the disordered fluid turns into an ordered solid. They found that for these larger exclusion zones, the system undergoes a single, sharp transition. It does not gradually slow down or change in a smooth curve; instead, it flips abruptly from a messy, fluid state to a highly organized state where particles settle into a specific, repeating pattern.

The researchers discovered that this sudden shift happens at a very specific density and pressure for each type of exclusion zone. By analyzing the mathematical properties of their simulation data, they were able to pinpoint the exact chemical potential—the driving force that pushes particles into the system—where this change occurs. For the case where a particle blocks up to four layers of neighbors, the transition happens at a chemical potential of approximately 2.87. When the exclusion zone grows to five layers, this value jumps to about 4.72. The team also calculated the precise densities of the fluid just before the switch and the solid just after. For the four-layer exclusion, the fluid exists at a density of roughly 0.74, while the resulting solid packs in at a density of about 0.91. These numbers are not just estimates; the researchers used multiple independent methods to verify them, including looking at how the system's energy fluctuates and examining the mathematical "zeros" of the system's partition function, which act like a fingerprint for the type of transition.

One of the most significant findings of this work is that it corrects previous misunderstandings about how these systems behave. Earlier studies using a different computational method had suggested that for the four-layer exclusion case, the transition might be continuous, meaning the change from fluid to solid would happen gradually. The new simulations, which are better equipped to handle the complex, crowded states near full packing, show clearly that this is not the case. The transition is discontinuous, or "first-order," meaning there is a distinct gap between the fluid and solid phases where the two can coexist. This distinction is crucial because it changes our fundamental understanding of how matter organizes itself under pressure. The researchers also noted that as the exclusion zone gets larger, the density of the fluid phase tends to stabilize around 0.8, while the solid phase gets closer and closer to being completely full, or a density of 1.0.

The study also serves as a benchmark for the tools used to explore these physical systems. The team compared their results with those from a popular method called the tensor renormalization group, which had been used to study similar problems. While that method worked well for simpler cases, it struggled to predict the correct behavior for the larger exclusion zones, often missing the sharp nature of the transition or predicting the wrong critical values. The new approach, which combines cluster moves with a flat histogram technique, proved to be far more accurate for these difficult, high-density scenarios. This suggests that for systems where particles have a large "personal space" requirement, the new simulation method is the superior tool for uncovering the truth.

Ultimately, this work provides a clearer picture of how order emerges from chaos in crowded environments. While the models used are abstract, they serve as a proxy for understanding real-world phenomena, such as how molecules adsorb onto surfaces or how certain materials freeze. The researchers found that even as they increased the complexity of the exclusion rules up to seven layers, the system still followed a single, sharp path to order. They did not find evidence of the intermediate, "hexatic" phases that are sometimes seen in the continuous limit of hard sphere systems. This implies that for the range of exclusion zones they studied, the path from fluid to solid is direct and abrupt. The results offer a definitive map for these specific triangular lattice systems, filling in a gap in our knowledge and providing a solid foundation for future studies into how particles behave when they are forced to share space.

Technical Summary: Freezing Phase Transition in Hard Core Lattice Gases on a Triangular Lattice

Problem Statement
Hard core lattice gas (HCLG) models serve as minimal frameworks for studying entropy-driven phase transitions, specifically fluid-to-solid freezing transitions induced by increasing density. In the $k$ -NN model, a particle excludes all sites up to the $k$ -th next-nearest neighbor from occupation. While the phase diagrams for $k$ -NN models on square and honeycomb lattices are relatively well-established, the triangular lattice case for $k \ge 4$ remains less understood. Previous studies on the triangular lattice were limited to $k \le 3$ . For $k=4$ and $k=5$ , Tensor Renormalization Group (TRG) methods yielded inconclusive results regarding the nature of the transition (continuous vs. discontinuous) and provided a wide range of values for critical parameters. Furthermore, standard Monte Carlo simulations of HCLG models with large excluded volumes often suffer from equilibration issues near full packing due to the system becoming trapped in metastable states when only local moves are permitted.

Methodology
The authors investigate the $k$ -NN model on a triangular lattice for $1 \le k \le 7$ using the Strip Cluster Wang Landau (SCWL) algorithm. This method combines a flat histogram approach with rejection-free cluster moves to efficiently sample the density of states, $g(N, L)$ , where $N$ is the number of particles and $L$ is the linear system size.

Key algorithmic features include:

Cluster Moves: The algorithm updates configurations on strips spanning the lattice. It selects a random row (orientation dependent on $k$ ) and identifies open segments between excluded sites. Particles within a segment are removed and re-deposited according to probabilities derived from the current estimate of the density of states.
Flat Histogram Protocol: The simulation employs the Wang Landau protocol to flatten the histogram of visited particle numbers. The modification factor $f$ is iteratively reduced (starting from $f=1$ down to $f_{min} = 2^{-22}$ ) until the histogram is sufficiently flat ( $H_{min} > 0.8 H_{max}$ ).
Thermodynamic Analysis: Once $g(N, L)$ $g (N, L)$ is determined, the grand canonical partition function, pressure, and average observables are calculated. The nature of the phase transition is analyzed using:
- Partition Function Zeros (PFZ): The loci of zeros in the complex fugacity plane ( $z = e^\mu$ ) are examined.
- Non-Convexity (NC) of Entropy: The entropy $S(N) = \ln g(N, L)$ is analyzed for non-convex regions, which indicate first-order transitions.
- Finite Size Scaling: Critical exponents and parameters are extrapolated to the thermodynamic limit ( $L \to \infty$ ) using scaling laws appropriate for first-order transitions in two dimensions ( $\nu = 1/2$ ).

Key Contributions and Results

Benchmarking and Validation:
- For $k=1$ (Hard Hexagon model), the SCWL results for critical chemical potential ( $\mu_c \approx 2.4064$ ) and density ( $\rho_c \approx 0.826$ ) agree well with the exact solution, validating the algorithm's accuracy.
- For $k=2$ , the study provides improved estimates for the critical density ( $\rho_c = 0.7419(5)$ ), which aligns closely with transfer matrix results and differs from some prior Monte Carlo estimates.
New Results for $4 \le k \le 7$ :
- Single Phase Transition: Contrary to some expectations for large excluded volumes, the system undergoes a single phase transition from a low-density disordered fluid phase to a high-density sublattice-ordered phase for all studied $k$ values ($4, 5, 6, 7$).
- Discontinuous Nature: The transition is established as discontinuous (first-order) for $k \ge 4$ $k \geq 4$ . This is evidenced by:
  - The PFZ forming a circle that pinches the real axis at a single point with an angle of $\pi/2$ .
  - The presence of non-convex regions in the numerically obtained entropy.
- Critical Parameters: The authors provide accurate estimates for the critical chemical potential ( $\mu_c$ $μ_{c}$ ), coexistence densities ( $\rho_f$ $ρ_{f}$ and $\rho_s$ $ρ_{s}$ ), and critical pressure ( $P_c$ $P_{c}$ ) for $k=4$ $k = 4$ through $7$.
  - Example ( $k=4$ ): $\mu_c = 2.8696(2)$ , $\rho_f = 0.7404(2)$ , $\rho_s = 0.9067(2)$ .
  - Example ( $k=5$ ): $\mu_c = 4.720(3)$ , $\rho_f = 0.916(3)$ , $\rho_s = 0.988(3)$ .
Sublattice Ordering:
- The high-density phase corresponds to the occupation of specific sublattices. The number of sublattices ( $N_{sub}$ ) increases with $k$ : $N_{sub}=9$ for $k=4$ , $12$ for $k=5$ , $13$ for $k=6$ , and $16$ for $k=7$ .

Significance and Comparison with TRG
The paper highlights a significant discrepancy between the SCWL results and previous TRG studies for $k=4$ and $k=5$ .

TRG Limitations: Previous TRG studies suggested a continuous transition for $k=4$ and provided a critical chemical potential range ( $2.65 < \mu_c < 2.7$ ) that does not overlap with the SCWL result ( $\mu_c \approx 2.87$ ). For $k=5$ , TRG suggested a range of $4.4 < \mu_c < 4.5$ , whereas SCWL finds $\mu_c \approx 4.72$ .
SCWL Advantage: The authors attribute the superior accuracy of the SCWL algorithm to its ability to efficiently access high-density states and overcome equilibration issues near full packing, which are problematic for local move algorithms and potentially for TRG in these specific regimes.

Conclusion
The study establishes that for $k$ -NN models on the triangular lattice with $4 \le k \le 7$ , the system exhibits a single, discontinuous phase transition. The SCWL algorithm proves to be a robust tool for determining the phase diagram and critical parameters of hard core lattice gases with large excluded volumes, offering more precise results than TRG methods in this regime. The authors note that while the continuum limit (hard sphere gas) is expected to show intermediate phases (like the hexatic phase) for very large $k$ , such phases are not observed for the computationally tractable range of $k \le 7$ studied here.

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