Order of the entropy-driven order-disorder (fluid-crystal) transition in non-sliding hard-core lattice gases

Non-sliding hard-core lattice gases - lattice particle systems with finitely many periodic perfect coverings whose defects always trap proportional empty space, including the hard diamond, hard cross and hard hexagon models - are known to undergo an entropy-driven order-disorder transition: Jauslin and Lebowitz (2018) proved a convergent high-fugacity expansion around each crystalline sublattice and $\tau$ crystalline Gibbs states, which, combined with the classical low-fugacity cluster expansion, forces a transition to exist, and they confined the Lee-Yang zeros to an annulus. The paper explicitly stopped short of determining the order of this transition. The problem posed here is to rigorously determine the number and order (first-order versus continuous) of the fluid-crystal transition(s) in these systems, equivalently the way the thermodynamic-limit Lee-Yang zero distribution pinches the positive real fugacity axis. The flagship instance is the hard cross model (third-nearest-neighbor exclusion on the two-dimensional square lattice), for which Eisenberg and Baram conjectured a first-order transition with a metastable super-cooled fluid. The only model in the class whose transition order is known is the exactly solved hard hexagon, which is continuous, so the answer is expected to be model-dependent; no rigorous result exists for any other non-sliding model.

Background

Hard-core lattice gases are the simplest models of entropy-driven ordering. Identical particles each occupy a fixed finite set of lattice sites (a shape, for example a two-site diamond, a five-site cross, or a seven-site hexagon on the triangular lattice), admissible configurations are those in which no site is covered twice, all admissible configurations have the same energy, and the only control parameter is the fugacity (activity) $z$, with respect to which the grand-canonical partition function $\Xi_\Lambda(z)=\sum_X z^{|X|}$ is a polynomial over admissible particle sets $X$ in a finite box $\Lambda$. In the thermodynamic limit the pressure $p(z)=\lim_{|\Lambda|\to\infty}|\Lambda|^{-1}\log\Xi_\Lambda(z)$ is analytic at small $z$ by the low-fugacity (Mayer) cluster expansion. Because the interaction is pure exclusion, any transition between a low-density fluid and a high-density crystal is driven entirely by entropy. Lee and Yang proposed to study phase transitions through the zeros of $\Xi_\Lambda(z)$ in the complex $z$-plane (Lee-Yang zeros): in the thermodynamic limit the pressure is singular precisely where the limiting distribution of zeros reaches the positive real axis, the only place a physical transition can occur since physical fugacity is positive.

The paradigmatic examples sit on two-dimensional lattices. For the hard diamond model (nearest-neighbor exclusion on the square lattice $\mathbb{Z}^2$), crystallization was first proved rigorously by Dobrushin, while Gaunt and Fisher had earlier computed low- and high-fugacity series and estimated the transition. For the hard cross model (third-nearest-neighbor exclusion on $\mathbb{Z}^2$), Heilmann and Praestgaard identified the crystalline high-density phase, and Eisenberg and Baram, from low- and high-fugacity series and transfer-matrix data, conjectured a first-order transition with a metastable super-cooled fluid. For the hard hexagon model (nearest-neighbor exclusion on the triangular lattice), Baxter's exact solution, analyzed further through the theory of modular functions by Joyce, yields a continuous transition with critical behavior in the universality class of the three-state Potts model (a standard exactly solved spin model) - to date the only exactly solved case.

Jauslin and Lebowitz introduced the class of non-sliding hard-core lattice particle systems: models whose lattice admits finitely many periodic perfect coverings (maximal-density tilings) $\mathcal{L}_1,\dots,\mathcal{L}_\tau$, in which every defect (deviation from a covering) necessarily traps an amount of empty space proportional to its size, and in which a particle configuration is determined by its defects. The diamond, cross, hexagon and a large family of polyomino models (particles whose shapes are connected unions of elementary lattice cells) are non-sliding, while 'sliding' models (for example long rods on a lattice) can have pathological high-density behavior. Building on Gaunt and Fisher's high-density expansion idea and on Pirogov-Sinai theory - the standard machinery for proving phase diagrams of lattice models with several competing periodic ground states - Jauslin and Lebowitz reorganized the dilute defect gas around each covering into a gas of Gaunt-Fisher configurations (defect clusters together with the particles surrounding them) and proved, for these models in $d\ge 2$ dimensions, that the pressure and correlation functions admit a convergent expansion in the inverse fugacity $y=z^{-1}$, that there are $\tau$ distinct crystalline extremal Gibbs states (pure phases) at high fugacity, and hence, combined with the low-fugacity expansion, that an order-disorder transition exists; the Lee-Yang zeros lie in an annulus bounded away from both $0$ and $\infty$.

All of this establishes existence only. The paper states verbatim that it will 'stop short of proving the order of the phase transition, for which new techniques would need to be developed'. The number of transition points, their fugacities $z_t$, and the order - first order, meaning a discontinuous jump of the density with a fluid and a crystalline Gibbs state coexisting at $z_t$, or continuous, meaning the crystalline order parameter vanishes continuously and the correlation length diverges at a critical point - are all undetermined; later work of He and Jauslin states explicitly that these results give no value for the critical fugacity 'or even on the number of phase transitions'. In Lee-Yang terms, the annulus bound says nothing about whether, where, and with what density the zeros pinch the positive real axis. The difficulty is intrinsic: the two convergent expansions live on opposite sides of the transition, the coexistence region has no small parameter, and unlike standard applications of Pirogov-Sinai theory there is no energy contrast between the competing phases - the fluid and the crystal differ only entropically.

Problem Statement

For non-sliding hard-core lattice particle systems on $\mathbb{Z}^d$ with $d\ge 2$, rigorously determine the fluid-crystal (order-disorder) transition that separates the analytic low-fugacity fluid regime from the analytic high-fugacity crystalline regime. Concretely, for a given model of the class: (a) prove the number of transition points $z_t$ - in particular, whether there is a single transition; (b) determine the order of each transition, that is, whether it is first order (a discontinuous jump of the density at $z_t$, with a translation-invariant fluid Gibbs state and sublattice-periodic crystalline Gibbs states coexisting at $z_t$) or continuous (the density continuous through $z_t$, the crystalline order parameter - for example the excess of the occupation density on the preferentially occupied covering sublattice over the fluid density, in one of the crystalline Gibbs states - vanishing continuously, and the correlation length of the truncated two-point correlations diverging), or of another singularity class, as encoded in the local behavior of the pressure $p(z)$ at $z_t$; and (c) equivalently, determine the thermodynamic-limit distribution of the Lee-Yang zeros: whether and where it pinches the positive real axis and with what local density; the expected dictionary - a positive pinch density accompanying a first-order transition, a zero-density touch a continuous one - is a heuristic guide, not an established theorem. The minimal complete target is the source's motivating model, the hard cross model (third-nearest-neighbor exclusion on $\mathbb{Z}^2$): determine the number and order of its fluid-crystal transition(s) - that is, resolve (a) and (b), equivalently (c), for the hard cross - thereby proving or refuting the Eisenberg-Baram conjecture that its transition is first order. A class-level solution would be a criterion, in terms of the particle shape and lattice geometry, that determines the number and order of the transitions for every non-sliding model; a rigorous determination of the order for another individual model of the class is partial progress toward that criterion. All claims must hold in the thermodynamic limit; finite-size numerical evidence alone does not answer the question. The companion open threads acknowledged by the source paper - continuum crystallization, sharpness of the convergence constants, and the classification of sliding models - are adjacent problems outside this problem's scope.

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