Universality class of interactingk-mers in a two-dimensional lattice system

Abstract

Monte Carlo simulations and finite-size scaling analysis have been carried out to study the critical behavior in a submonolayer two-dimensional gas of repulsive linear $k$-mers on a square lattice at half coverage. A $(2k\ifmmode\times\else\texttimes\fi{}2)$ ordered phase, characterized by alternating files of $k$-mers separated by $k$ adjacent empty sites, is separated from the disordered state by a order-disorder phase transition occurring at a finite critical temperature. Based on the strong axial anisotropy of the low-temperature phase for $k\ensuremath{\geqslant}2$, an order parameter measuring the orientation of the particles has been introduced. Taking advantage of its definition, an accurate determination of the critical exponents has been obtained for three adsorbate sizes. Namely, $\ensuremath{\nu}=0.53(1)$, $\ensuremath{\beta}=0.02(1)$, $\ensuremath{\gamma}=1.14(3)$, and $\ensuremath{\alpha}=0.93(3)$ for $k=2$ (dimers); $\ensuremath{\nu}=0.54(1)$, $\ensuremath{\beta}=0.03(1)$, $\ensuremath{\gamma}=1.16(3)$, and $\ensuremath{\alpha}=0.89(3)$ for $k=3$ (trimers); and $\ensuremath{\nu}=0.53(2)$, $\ensuremath{\beta}=0.02(1)$, $\ensuremath{\gamma}=1.14(3)$, and $\ensuremath{\alpha}=0.89(4)$ for $k=4$ (tetramers). In the studied cases, the results reveal that the system does not belong to the universality class of the two-dimensional Ising model ($k=1$, monomers). We pointed out that the breaking of the orientational symmetry characterizing the low-temperature phase for particles occupying more than one site is the main source of this change in the universality class. Consequently, we suggested that the behavior observed for dimers, trimers, and tetramers could be generalized to include larger particle sizes $(k\ensuremath{\geqslant}2)$. Finally, hyperscaling relations have been validated, leading to independent controls and consistency checks of the values of all the critical exponents.