Tom Kennedy — Page 3

arXiv1997

Phase separation in the neutral Falicov-Kimball model

The Falicov-Kimball model consists of spinless electrons and classical particles (ions) on a lattice. The electrons hop between nearest neighbor sites while the ions do not. We consider the model with equal numbers of ions and electrons and with a large on-site attractive force between ions and electrons. For densities 1/4 and 1/5 the ion configuration in the ground state had been proved to be periodic. We prove that for density 2/9 it is periodic as well. However, for densities between 1/4 and 1/5 other than 2/9 we prove that the ion configuration in the ground state is not periodic. Instead there is phase separation. For densities in (1/5,2/9) the ground state ion configuration is a mixture of the density 1/5 and 2/9 ground state ion configurations. For the interval (2/9,1/4) it is a mixture of the density 2/9 and 1/4 ground states.

cond-mat↗

arXiv1996

Majority Rule at Low Temperatures on the Square and Triangular Lattices

We consider the majority rule renormalization group transformation applied to nearest neighbor Ising models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the transformation is not defined. We use the methods of van Enter, Fernandez and Sokal, who proved the renormalized measure is not Gibbsian for 7 by 7 blocks if the temperature is too low. For the triangular lattice we prove that a zero temperature majority rule transformation may be defined. The resulting renormalized Hamiltonian is local with 14 different types of interactions.

cond-mat↗