Phase behaviour and ordering in hard core lattice gas models[HBNI Th109]

Abstract

Systems of particles that interact only through excluded volume interactions are minimal models to study entropy-driven phase transitions. In this thesis we study in detail two hard core lattice gas models on the square lattice, (1) The k-NN model which is the discrete version of the hard disc system in two dimensions and (2) hard rectangles and squares. In the k-NN model the first k nearest neighboring lattice points of a particle are excluded from being occupied by another particle. In the 4-NN model, using Monte Carlo simulations we find the existence of two continuous transitions with increasing density – the first from disordered to sublattice-ordered phase and the second from sublattice-ordered to columnar-ordered phase. We further analytically rationalize the existence of multiple transitions by high-activity series expansion. Extending the argument, we conjecture that if the model satisfies (i) the high density phase is columnar and (ii) sliding instability is present in only a fraction of the sublattices, then the system will show multiple transitions. We verify the same in k-NN models with k=6, 7, 8, 9, 10, 11 using Monte Carlo simulations. The conjecture reduces to determining, for a given k greater than 5, whether the high density phase has columnar order at close packing. Finding out close packing structure for k-NN with k up to 820302, we show that there are only eighteen values of k, all less than k = 4134, that show columnar order, while the others show sublattice order. We further analytically study the columnar phase in classical hard-square lattice gas and mxd hard rectangles on square lattice. By deriving the exact expression for the first d + 2 terms in the free energy expansion, we obtain lower bounds for the critical density and activity for nematic-columnar transition. To obtain a better estimate of the critical pa- rameters, we estimate the interfacial tension between two phases with different columnar order in a system of 2xd hard rectangles. Setting the interfacial tension to zero, we obtain a condition for the limit of stability of the columnar ordered phase. For all values of d, the critical parameters obtained are in good agreement with numerical data.