Hexatic order and two-step melting at large exclusion range in the triangular-lattice k-NN hard-core lattice gas, and the algorithms to reach it

In the $k$-NN hard-core lattice gas on the triangular lattice, each particle excludes occupation of all sites up to its $k$-th nearest neighbors; the model interpolates between Baxter's exactly solved hard-hexagon model ($k=1$) and, as $k\to\infty$, the continuum two-dimensional hard-disk gas. Flat-histogram simulations with strip-cluster Wang-Landau moves have established that for each $k$ from 3 to 7 studied so far the model freezes in a single first-order transition from a fluid to a sublattice-ordered solid, with no intermediate phase ($k=1$ and $k=2$ undergo continuous Potts-universality transitions), whereas continuum hard disks melt in two steps through a hexatic phase. Whether the lattice model develops continuum-like two-step melting at some finite, computationally accessible $k$ — and the methodological advances (tensor-network constructions with a characterized reliability domain for large excluded volumes, or cluster flat-histogram algorithms scaling beyond $k=7$) needed to reach that regime, including reconciliation of the known tensor-renormalization-group versus flat-histogram discrepancies at $k=4$ and $k=5$ — remains open.

Background

Hard-core lattice gases (HCLGs) are lattice models whose only interaction is excluded volume: an occupied site forbids occupation of a fixed set of neighboring sites. They are the minimal models of entropy-driven ordering, relevant to adsorbed monolayers, colloids on templated surfaces, and the theory of freezing. On the triangular lattice, the natural one-parameter family is the $k$-NN model: a particle excludes all sites up to and including its $k$-th nearest neighbors. The exclusion region is a discrete disk of radius $k$ lattice spacings, so as $k$ grows with the exclusion radius held fixed, the lattice spacing measured in disk diameters tends to zero: the $k\to\infty$ limit is the two-dimensional hard-disk gas. The small-$k$ members are classics. The $k=1$ model is Baxter's hard-hexagon model, the only known exactly solved hard-core lattice gas with a nontrivial phase transition: at a critical activity it undergoes a single continuous transition (in the universality class of the three-state Potts spin model) between a fluid and a phase in which one of the three triangular sublattices is preferentially occupied. The $k=2$ model has a single continuous transition in the four-state Potts universality class (the $q=4$ member of the same spin-model family), and flat-histogram studies show that $k=3$ through $k=7$ each freeze in a single first-order transition to a sublattice-ordered solid. These results were obtained with the strip-cluster Wang-Landau (SCWL) algorithm, a rejection-free flat-histogram Monte Carlo method that determines the density of states $g(n)$ (the number of configurations at each particle number $n$) by emptying whole strips of the lattice and reoccupying them with appropriately biased probabilities, reaching the low-entropy near-close-packed states that local-move algorithms cannot sample. First-order character is established in two standard ways: from the zeros of the grand-canonical partition function in the complex activity plane, whose locus pinches the positive real axis at $\mathrm{e}^{\mu_c}$ at angle $\pi/2$ for a discontinuous transition, and from non-convexity of the microcanonical entropy $s(\rho)$, whose convex-hull segments give the coexistence densities through a Maxwell construction. The continuum endpoint of the family behaves very differently: two-dimensional hard disks melt in two steps — a first-order transition from liquid to a hexatic phase (quasi-long-range sixfold bond-orientational order without translational order), then a transition to the solid — in accord with the Kosterlitz-Thouless-Nelson-Halperin-Young (KTHNY) theory of defect-mediated melting. A hexatic phase has never been confirmed in any hard-core lattice gas. The single claimed observation, a two-step liquid-hexatic-solid transition for the triangular-lattice 3-NN model, was refuted by detailed flat-histogram entropies showing one first-order transition. On the square lattice the $k$-NN model does show multiple transitions for $k=4,10,11$ and beyond, but these are driven by sliding instabilities of columnar high-density phases and involve no hexatic order; the honeycomb lattice shows varied sublattice and columnar phenomenology for $k\le 5$. The lattice thus appears to pin commensurate crystalline order and mask the continuum melting scenario. This raises the central question of the source paper (Jaleel, Mandal, Thomas and Rajesh, Phys. Rev. E 106, 044136 (2022)): is the single first-order fluid-to-sublattice-solid freezing seen for all $3\le k\le 7$ replaced, at some finite computationally accessible $k$, by the two-step melting of the hard-disk continuum? The $k\le 7$ data do not extrapolate to the continuum limit. Answering this is bottlenecked on method: SCWL works up to $k=7$ but its cluster moves grow costlier as the excluded volume grows (the number of sites excluded per particle rises roughly quadratically with $k$: 6, 12, 18, 30, 36, 42, and 54 for $k=1$ through $7$), and the tensor renormalization group (TRG) — coarse-graining contraction of the partition function written as a tensor network, in principle the natural tool at large $k$ — is unreliable precisely there: already at $k=4,5$ it disagreed with SCWL, predicting a continuous 4-NN transition and critical chemical potentials outside the SCWL values. The source therefore poses both the scientific question (two-step melting at large $k$) and the enabling methodological problem (improve TRG or characterize its reliability for large excluded volumes; scale flat-histogram cluster algorithms beyond $k=7$).

Problem Statement

Determine the phase behavior of the triangular-lattice $k$-NN hard-core lattice gas (in which each occupied site excludes occupation by other particles of all sites up to and including its $k$-th nearest neighbors on the triangular lattice) at exclusion ranges $k>7$, and thereby establish whether this family of lattice discretizations of hard disks recovers the two-step (liquid-hexatic-solid) melting of the continuum at some finite, computationally accessible $k$, or whether commensurate sublattice ordering precludes an intermediate phase for all tractable $k$, where a $k$ counts as computationally accessible/tractable when a finite-size-converged phase diagram can be obtained at that exclusion range within a stated lattice-size and computational budget. Concretely, an answer should provide: (1) the phase diagram for $k>7$, carried to the largest $k$ for which the chosen method is demonstrably finite-size converged and at least through $k=10$ — the number, location, and order of all phase transition(s), the coexistence densities and critical chemical potentials and pressures, and the symmetry (sublattice, columnar, or intermediate) and orientational character of every phase; (2) a determination, from finite-size-controlled evidence (partition-function zeros, entropy non-convexity, and direct bond-orientational correlation functions), of whether any intermediate phase with quasi-long-range hexatic order appears between the fluid and the solid; and (3) the methodological component that makes (1)-(2) possible at large $k$: either an improved tensor-network/tensor-renormalization-group construction with a characterized regime of reliability for large excluded volumes, or cluster flat-histogram algorithms whose computational cost scales beyond $k=7$; either route suffices, and whichever route is taken must include a reconciliation or characterization of the TRG-SCWL discrepancies observed at $k=4$ and $k=5$ and must demonstrate reproduction of the known $k\le 7$ benchmarks. These three components jointly serve one scientific objective: establishing whether, and at what exclusion range, lattice hard-core models approach the continuum hard-disk melting phenomenology. The separable extensions flagged in the source's outlook — multi-component mixtures and hard-core gases with attractive interactions via multi-dimensional densities of states — lie outside this completion boundary.

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