Figure - available from: Applied Sciences
This content is subject to copyright.
The pentakis dodecahedron has 12 icosahedral vertices (red dots) and 20 dodecahedral vertices (yellow dots). There are five distinct ways to choose eight yellow dots forming a cube—then, the other 12 dodecahedral vertices are said to form a “co-cube”. The short edges of the PD mesh are colored in grey and the long edges in blue. The couplings entering the model Hamiltonian (1) are indicated.
Source publication
The main objective of a statistical mechanical calculation is drawing the phase diagram of a many-body system. In this respect, discrete systems offer the clear advantage over continuum systems of an easier enumeration of microstates, though at the cost of added abstraction. With this in mind, we examine a system of particles living on the vertices...
Citations
... To avoid having to be concernedwith boundary effects, a natural choice is a (two-dimensional) system of particles confined on a surface with the topology of a sphere. A further simplification is obtained by discretizing the space, e.g., by taking particles to lie at the nodes/sites of a dense, quasi-regular triangular mesh on the sphere [1]. The combination of a lower dimensionality and discrete particle positions can make the model amenable to an exact treatment, either analytic or numerical. ...
Discrete statistical systems offer a significant advantage over systems defined in the continuum, since they allow for an easier enumeration of microstates. We introduce a lattice-gas model on the vertices of a polyhedron called a pentakis icosidodecahedron and draw its exact phase diagram by the Wang–Landau method. Using different values for the couplings between first-, second-, and third-neighbor particles, we explore various interaction patterns for the model, ranging from softly repulsive to Lennard-Jones-like and SALR. We highlight the existence of sharp transitions between distinct low-temperature “phases”, featuring, among others, regular polyhedral, cluster-crystal-like, and worm-like structures. When attempting to reproduce the equation of state of the model by Monte Carlo simulation, we find hysteretic behavior near zero temperature, implying a bottleneck issue for Metropolis dynamics near phase-crossover points.
... This gives us the opportunity to revisit the conclusions reached in [5] for a number of triangular-lattice-gas models, for which a more careful analysis of the liquid-vapor equilibrium is planned for the next future. Other directions of research development may concern the clustering of two-dimensional lattice particles with overlapping cores (i.e., the discrete-space counterpart of the study in [40]) or the finite-size phases of particles living on the nodes of a dense polyhedral mesh (much denser than considered in [41]). ...
We reconsider model II of Orban et al. (J. Chem. Phys. 1968, 49, 1778–1783), a two-dimensional lattice-gas system featuring a crystalline phase and two distinct fluid phases (liquid and vapor). In this system, a particle prevents other particles from occupying sites up to third neighbors on the square lattice, while attracting (with decreasing strength) particles sitting at fourth- or fifth-neighbor sites. To make the model more realistic, we assume a finite repulsion at third-neighbor distance, with the result that a second crystalline phase appears at higher pressures. However, the similarity with real-world substances is only partial: Upon closer inspection, the alleged liquid–vapor transition turns out to be a continuous (albeit sharp) crossover, even near the putative triple point. Closer to the standard picture is instead the freezing transition, as we show by computing the free-energy barrier relative to crystal nucleation from the “liquid”.
... By increasing quantum fluctuations, in fact, a structural transition from quasicrystal to cluster triangular crystal featuring the properties of a supersolid is observed [19,66,67]. We point out that the discussed methodology is also useful to analyze the superfluid character of further peculiar inhomogeneous systems such as, for instance, bosons enclosed within spherical traps or subject to a polyhedral-symmetric substrate potential [68][69][70]. ...
In this work, we explore the relevant methodology for the investigation of interacting systems with contact interactions, and we introduce a class of zonal estimators for path-integral Monte Carlo methods, designed to provide physical information about limited regions of inhomogeneous systems. We demonstrate the usefulness of zonal estimators by their application to a system of trapped bosons in a quasiperiodic potential in two dimensions, focusing on finite temperature properties across a wide range of values of the potential. Finally, we comment on the generalization of such estimators to local fluctuations of the particle numbers and to magnetic ordering in multi-component systems, spin systems, and systems with nonlocal interactions.
