Shortcomings of the Bond Orientational Order Parameters for the Analysis of Disordered Particulate Matter

Local structure characterization with the bond-orientational order parameters q4, q6, ... introduced by Steinhardt et al. has become a standard tool in condensed matter physics, with applications including glass, jamming, melting or crystallization transitions and cluster formation. Here we discuss two fundamental flaws in the definition of these parameters that significantly affect their interpretation for studies of disordered systems, and offer a remedy. First, the definition of the bond-orientational order parameters considers the geometrical arrangement of a set of neighboring spheres NN(p) around a given central particle p; we show that procedure to select the spheres constituting the neighborhood NN(p) can have greater influence on both the numerical values and qualitative trend of ql than a change of the physical parameters, such as packing fraction. Second, the discrete nature of neighborhood implies that NN(p) is not a continuous function of the particle coordinates; this discontinuity, inherited by ql, leads to a lack of robustness of the ql as structure metrics. Both issues can be avoided by a morphometric approach leading to the robust Minkowski structure metrics ql'. These ql' are of a similar mathematical form as the conventional bond-orientational order parameters and are mathematically equivalent to the recently introduced Minkowski tensors [Europhys. Lett. 90, 34001 (2010); Phys. Rev. E. 85, 030301 (2012)]

Liquid-solid transitions in the three-body hard-core model

We determine the phase diagram for a generalisation of two-and three-dimensional hard spheres: a classical system with three-body interactions realised as a hard cut-off on the mean-square distance for each triplet of particles. Quantum versions of this model are important in the context of the unitary Bose gas, which is currently under close theoretical and experimental scrutiny. In two dimensions, the three-body hard-core model possesses a conventional atomic liquid phase and a peculiar solid phase formed by dimers. These dimers interact effectively as hard disks. In three dimensions, the solid phase consists of isolated atoms that arrange in a simple-hexagonal lattice.Comment: 6 pages, 8 figures; reorganized introduction, expanded 3D sectio

Thermodynamic phases in two-dimensional active matter

Active matter has been intensely studied for its wealth of intriguing properties such as collective motion, motility-induced phase separation (MIPS), and giant fluctuations away from criticality. However, the precise connection of active materials with their equilibrium counterparts has remained unclear. For two-dimensional (2D) systems, this is also because the experimental and theoretical understanding of the liquid, hexatic, and solid equilibrium phases and their phase transitions is very recent. Here, we use self-propelled particles with inverse-power-law repulsions (but without alignment interactions) as a minimal model for 2D active materials. A kinetic Monte Carlo (MC) algorithm allows us to map out the complete quantitative phase diagram. We demonstrate that the active system preserves all equilibrium phases, and that phase transitions are shifted to higher densities as a function of activity. The two-step melting scenario is maintained. At high activity, a critical point opens up a gas-liquid MIPS region. We expect that the independent appearance of two-step melting and of MIPS is generic for a large class of two-dimensional active systems.Comment: 14 pages, 4 figure

A kinetic-Monte Carlo perspective on active matter

We study non-equilibrium phases for interacting two-dimensional self-propelled particles with isotropic pair-wise interactions using a persistent kinetic Monte Carlo (MC) approach. We establish the quantitative phase diagram, including the motility-induced phase separation (MIPS) that is a commonly observed collective phenomena in active matter. In addition, we demonstrate for several different potential forms the presence of two-step melting, with an intermediate hexatic phase, in regions far from equilibrium. Increased activity can melt a two-dimensional solid and the melting lines remain disjoint from MIPS. We establish this phase diagram for a range of the inter-particle potential stiffnesses, and identify the MIPS phase even in the hard-disk limit. We establish that the full description of the phase behavior requires three independent control parameters.Comment: 11 pages, 10 figure