Phase transitions in number theory: from the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed

Analysis of SnS2 hyperdoped with V proposed as efficient absorber material

Intermediate-band materials can improve the photovoltaic efficiency of solar cells through the absorption of two subband-gap photons that allow extra electron-hole pair formations. Previous theoretical and experimental findings support the proposal that the layered SnS2 compound, with a band-gap of around 2 eV, is a candidate for an intermediate-band material when it is doped with a specific transition-metal. In this work we characterize vanadium doped SnS2 using density functional theory at the dilution level experimentally found and including a dispersion correction combined with the site-occupancy-disorder method. In order to analyze the electronic characteristics that depend on geometry, two SnS2 polytypes partially substituted with vanadium in symmetry-adapted non-equivalent configurations were studied. In addition the magnetic configurations of vanadium in a SnS2 2H-polytype and its comparison with a 4H-polytype were also characterized. We demonstrate that a narrow intermediate-band is formed, when these dopant atoms are located in different layers. Our theoretical predictions confirm the recent experimental findings in which a paramagnetic intermediate-band material in a SnS2 2H-polytype with 10% vanadium concentration is obtained.