A note on primes in short intervals

This paper is concerned with the number of primes in short intervals. We prove that for every $\theta>1/2$ the intervals $[x, x+x^{\theta}]$ contain the expected number of primes for $x\rightarrow \infty$, with the assumption of an heuristic hypothesis weaker than the Lindel\"{o}f hypothesi

Prime numbers between squares

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is out of reach at present, even under the assumption of the Riemann Hypothesis. The aim of this paper is to provide a conditional proof of the conjecture assuming a hypothesis about the behavior of Selberg's integral in short interval

Majorana fermion description of the Kondo lattice: Variational and Path integral approach

All models of interacting electrons and spins can be reformulated as theories of interacting Majorana fermions. We consider the Kondo lattice model that admits a symmetric representation in terms of Majorana fermions. In the first part of this work we study two variational states, which are natural in the Majorana formulation. At weak coupling a state in which three Majorana fermions tend to propagate together as bound objects is favored, while for strong coupling a better description is obtained by having deconfined Majorana fermions. This way of looking at the Kondo lattice offers an alternative phenomenological description of this model. In the second part of the paper we provide a detailed derivation of the discretized path integral formulation of any Majorana fermion theory. This general formulation will be useful as a starting point for further studies, such as Quantum Monte Carlo, perturbative expansions, and Renormalization Group analysis. As an example we use this path integral formalism to formulate a finite temperature variational calculation, which generalizes the ground state variational calculation of the first part. This calculation shows how the formation of three-body bound states of Majorana fermions can be handled in the path integral formalism.Comment: 12 pages, 3 figure

Ferromagnetism in the one-dimensional Kondo lattice: mean-field approach via Majorana fermion canonical transformation

Using a canonical transformation it is possible to faithfully represent the Kondo lattice model in terms of Majorana fermions. Studying this representation we discovered an exact mapping between the Kondo lattice Hamiltonian and a Hamiltonian describing three spinless fermions interacting on a lattice. We investigate the effectiveness of this three fermion representation by performing a zero temperature mean-field study of the phase diagram at different couplings and fillings for the one-dimensional case, focusing on the appearance of ferromagnetism. The solutions agree in many respects with the known numerical and analytical results. In particular, in the ferromagnetic region connected to the solution at zero electron density, we have a quantitative agreement on the value of the commensurability parameter discovered in recent DMRG (in one dimension) and DMFT (in infinite dimensions) simulations; furthermore we provide a theoretical justification for it, identifying a symmetry of the Hamiltonian. This ferromagnetic phase is stabilized by the emergence of a spin-selective Kondo insulator that is described quite conveniently by the three spinless fermions. We discovered also a different ferromagnetic phase at high filling and low couplings. This phase resembles the RKKY ferromagnetic phase existing at vanishing filling, but it incorporates much more of the Kondo effect, making it energetically more favorable than the typical spiral (spin ordered) mean field ground states. We believe that this second phase represents a prototype for the strange ferromagnetic tongue identified by numerical simulations inside the paramagnetic dome. At the end of the work we also provide a discussion of possible orders different from the ferromagnetic one. In particular at half-filling, where we obtain as ground state at high coupling the correct Kondo insulating state.Comment: 21 pages, 10 figure