Phase diagram of hard $k\times k\times k$ cubes on the cubic lattice: the fate of sublattice order for $k\ge 3$, the near-full-packing regime, the continuum limit, and the failure of lattice density-functional theory

The lattice gas of parallel hard cubes, each occupying a $k\times k\times k$ block of sites of the simple cubic lattice, is a canonical model of entropy-driven ordering in three dimensions. For $k=2$, grand-canonical Monte Carlo simulations with cluster moves (Vigneshwar, Mandal, Damle, Dhar and Rajesh, 2019) found four phases with increasing density — disordered, layered, sublattice-ordered and columnar — while the lattice fundamental-measure density functional of Lafuente and Cuesta predicts only a disordered $\to$ layered $\to$ columnar sequence and misses the sublattice-ordered phase. That study left three linked questions open: why the approximate density functional fails for $k=2$ and how it must be modified; what the phase diagram is for larger cubes, in particular whether sublattice order survives for $k\ge 3$ (preliminary $3\times 3\times 3$ data suggest it may be absent) and how the finite-$k$ phase boundaries extrapolate to the freezing transition of continuum parallel hard cubes; and what the true equilibrium behavior is arbitrarily close to full packing, where simulations become trapped in long-lived metastable states.

Background

Hard-core lattice gases are the simplest models of entropy-driven ordering: particles interact only through the rule that no two of them may overlap, yet at high density they can break translational symmetry because ordered arrangements leave more room for collective fluctuations. A standard family is obtained by placing parallel axis-aligned cubes of integer edge length $k$ on the simple cubic lattice, each cube covering a $k\times k\times k$ block of sites, with no site shared by two cubes; the control parameter is the packing fraction $\eta$ (the fraction of lattice sites covered), which reaches $\eta=1$ at full packing. In two dimensions the analogous $k\times k$ hard-square gas has been studied since the 1960s: for $k=2$ the close-packed state is a columnar phase (particles concentrated in stripes that are ordered along one direction and disordered along the other), and high-activity expansions for $k\times k$ hard rectangles (Nath, Kundu and Rajesh, 2015) show that the packing fraction at the ordering transition approaches a limiting value strictly below 1 as $k$ grows. In the limit of large $k$ the lattice model approaches the fluid of parallel hard cubes in $\mathbb{R}^3$, which undergoes an entropy-driven freezing transition from a disordered fluid to a positionally ordered solid (Groh and Mulder, 2001, and earlier work cited there). How the discrete-lattice phase diagram converges to this continuum behavior is not understood.

The reference simulation study for cubes restricted to the lattice is Vigneshwar, Mandal, Damle, Dhar and Rajesh (2019), who studied $k=2$ using grand-canonical Monte Carlo simulations (in which the particle number fluctuates at fixed fugacity) supplemented by cluster moves of the evaporation–deposition type, in which all cubes whose reference corners lie in a randomly chosen one-dimensional row are removed and the row is refilled with exactly computed equilibrium probabilities; these moves equilibrate dense hard-particle systems far better than single-particle updates. On lattices of linear size up to $L=200$ they found four phases with increasing $\eta$: a disordered fluid; a layered (smectic-like) phase in which the system splits into slabs of size $2\times L\times L$ preferentially occupied by cubes, breaking translational symmetry along one axis; a sublattice-ordered, solid-like phase in which cubes preferentially occupy one of the eight sublattices of fixed coordinate parity, breaking translational symmetry along all three axes; and a columnar phase of weakly interacting parallel columns of cross-section $2\times 2$, breaking symmetry along two axes. Finite-size scaling (extrapolation of results on increasing system sizes to the thermodynamic limit) indicates that the disordered–layered transition is continuous and consistent with the O(3) universality class (the critical behavior of the three-dimensional Heisenberg model) perturbed by cubic anisotropy, while the layered–sublattice and sublattice–columnar transitions are first order. These results overturned earlier, much smaller simulations of the same model by Panagiotopoulos (2005, system sizes $L=18$ and $24$), which had found no transitions at all; the discrepancy is attributed to system size.

The microscopic theory for such lattice fluids is the lattice version of fundamental-measure theory (FMT), a density-functional approximation for hard particles that builds the excess free-energy functional from geometric measures of the particle shapes. Classical density-functional theory (DFT) replaces the equilibrium many-body problem by minimization of a free-energy functional of the one-body density profile, so an accurate functional yields phase diagrams and structure without configuration sampling. The lattice construction of Lafuente and Cuesta (2002; 2003) is exact on zero-dimensional cavities — regions small enough to hold at most one particle — which fixes its behavior in severely confined geometries. Applied to the pure cube fluid it predicts, for $k=2$, only a disordered $\to$ layered $\to$ columnar sequence (the fluid becomes unstable to layering at packing fraction $\eta\approx 0.57$, followed by a narrow first-order layered–columnar transition), and for $k=6$ a disordered $\to$ solid $\to$ two-columnar sequence (here 'solid' denotes the functional's positionally ordered phase, which breaks translational symmetry along all three axes — the symmetry class of the lattice's sublattice-ordered phase). In particular it produces no sublattice-ordered phase for $k=2$, in direct conflict with the simulations. A phenomenological Landau theory (a free energy written directly in terms of the layering and columnar order parameters with symmetry-allowed couplings) constructed in the 2019 paper does describe all four simulated phases and the transition orders, so what fails is specifically the quantitative microscopic functional, and the reason for its failure is not known.

Problem Statement

Establish the equilibrium phase behavior of the hard $k\times k\times k$ cube lattice gas on the simple cubic lattice as a function of cube size $k$ and packing fraction, together with a microscopic theory that predicts it. The 2019 study of $k=2$ poses this as four interlinked deliverables:

(1) Phase diagram for $k\ge 3$. Determine, by systematic simulation and/or analysis, the equilibrium phases of $3\times3\times3$ cubes — and the trend for general $k\ge3$ — as the packing fraction increases from the disordered fluid to full packing: which of the disordered, layered, sublattice-ordered and columnar phases occur, in what sequence, and with what orders of transition. In particular, does the sublattice-ordered phase persist for $k\ge3$ (preliminary $3\times3\times3$ simulations suggest it may be absent), and do the Lafuente–Cuesta predictions for $k=6$ (a disordered $\to$ solid $\to$ columnar sequence with two types of columnar phase) hold?

(2) True equilibrium near full packing. Establish the equilibrium behavior arbitrarily close to maximum packing — for $k=2$ beyond packing fraction $\eta\approx0.96$, where current algorithms get trapped in long-lived metastable states with misaligned planes, and correspondingly for larger $k$ — whether by new equilibration algorithms, high-density expansions, or other analytical or numerical means.

(3) Repairing the density functional. Explain why the Lafuente–Cuesta lattice fundamental-measure functional fails to produce the sublattice-ordered phase for $k=2$, and construct a modified lattice density functional (or demonstrably equivalent microscopic theory) that reproduces the correct phase sequence and transition orders for $k=2$ and makes testable predictions for $k\ge3$.

(4) Continuum limit. Extrapolate the finite-$k$ phase boundaries to large $k$ and connect the discrete-lattice phase diagram to the continuum limit of parallel hard cubes, which simulations and density-functional theory indicate undergoes a single continuous freezing transition from a disordered fluid to a positionally ordered solid; clarify how the layered and columnar phases that the lattice results suggest persist at large $k$ reconcile with this single fluid–solid transition.

These deliverables jointly serve one overarching objective — the phase diagram of hard cubes on the cubic lattice as a function of size and packing fraction, and its theoretical prediction. Deliverables (1) and (2) determine the phase diagram, (4) is its large-$k$ asymptote, and (3) is the microscopic theory that should predict all of them; the simulations in (1)–(2) also supply the test data for (3). A complete solution must address all four deliverables, none of which contains another: (1) requires both the $3\times3\times3$ phase sequence with its transition orders and the trend for general $k\ge3$; a phase sequence alone does not settle the true equilibrium behavior arbitrarily close to full packing, which is the separate subject of (2); (3) counts as complete only when it both explains the functional's failure and supplies a modified functional (or demonstrably equivalent microscopic theory) that reproduces the $k=2$ phase sequence; and the finite-$k$ trend of (1) does not settle the convergence to the continuum transition, which is the separate subject of (4). Progress on only a subset of these deliverables leaves the overarching objective unresolved.

References

1. N. Vigneshwar, D. Mandal, K. Damle, D. Dhar, and R. Rajesh, 'Phase diagram of a system of hard cubes on the cubic lattice', Phys. Rev. E 99, 052129 (2019), DOI: 10.1103/PhysRevE.99.052129, arXiv:1902.06408, https://arxiv.org/abs/1902.06408
2. L. Lafuente and J. A. Cuesta, 'Phase behavior of hard-core lattice gases: A fundamental measure approach', J. Chem. Phys. 119, 10832 (2003), DOI: 10.1063/1.1615511, arXiv:cond-mat/0306221, https://arxiv.org/abs/cond-mat/0306221
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