Recent advances and open challenges in percolation

Percolation is the paradigm for random connectivity and has been one of the most applied statistical models. With simple geometrical rules a transition is obtained which is related to magnetic models. This transition is, in all dimensions, one of the most robust continuous transitions known. We present a very brief overview of more than 60 years of work in this area and discuss several open questions for a variety of models, including classical, explosive, invasion, bootstrap, and correlated percolation

Magnetic and superconducting instabilities in the periodic Anderson model: an RPA stud

We study the magnetic and superconducting instabilities of the periodic Anderson model with infinite Coulomb repulsion U in the random phase approximation. The Neel temperature and the superconducting critical temperature are obtained as functions of electronic density (chemical pressure) and hybridization V (pressure). It is found that close to the region where the system exhibits magnetic order the critical temperature T_c is much smaller than the Neel temperature, in qualitative agreement with some T_N/T_c ratios found for some heavy-fermion materials. In our study, all the magnetic and superconducting physical behaviour of the system has its origin in the fluctuating boson fields implementing the infinite on-site Coulomb repulsion among the f-electrons.Comment: 9 pages, 2 figure

Phase diagram and magnetic collective excitations of the Hubbard model in graphene sheets and layers

We discuss the magnetic phases of the Hubbard model for the honeycomb lattice both in two and three spatial dimensions. A ground state phase diagram is obtained depending on the interaction strength U and electronic density n. We find a first order phase transition between ferromagnetic regions where the spin is maximally polarized (Nagaoka ferromagnetism) and regions with smaller magnetization (weak ferromagnetism). When taking into account the possibility of spiral states, we find that the lowest critical U is obtained for an ordering momentum different from zero. The evolution of the ordering momentum with doping is discussed. The magnetic excitations (spin waves) in the antiferromagnetic insulating phase are calculated from the random-phase-approximation for the spin susceptibility. We also compute the spin fluctuation correction to the mean field magnetization by virtual emission/absorpion of spin waves. In the large $U$ limit, the renormalized magnetization agrees qualitatively with the Holstein-Primakoff theory of the Heisenberg antiferromagnet, although the latter approach produces a larger renormalization