Artificial graphenes: Dirac matter beyond condensed matter

After the discovery of graphene and its many fascinating properties, there has been a growing interest for the study of "artificial graphenes". These are totally different and novel systems which bear exciting similarities with graphene. Among them are lattices of ultracold atoms, microwave or photonic lattices, "molecular graphene" or new compounds like phosphorene. The advantage of these structures is that they serve as new playgrounds for measuring and testing physical phenomena which may not be reachable in graphene, in particular: the possibility of controlling the existence of Dirac points (or Dirac cones) existing in the electronic spectrum of graphene, of performing interference experiments in reciprocal space, of probing geometrical properties of the wave functions, of manipulating edge states, etc. These cones, which describe the band structure in the vicinity of the two connected energy bands, are characterized by a topological "charge". They can be moved in reciprocal space by appropriate modification of external parameters (pressure, twist, sliding, stress, etc.). They can be manipulated, created or suppressed under the condition that the total topological charge be conserved. In this short review, I discuss several aspects of the scenarios of merging or emergence of Dirac points as well as the experimental investigations of these scenarios in condensed matter and beyond.Comment: 16 pages, 26 figures. To appear in Comptes-rendus de l'Acad\'emie des Sciences, Franc

Mesoscopic Charge Density Wave in a Magnetic Flux

The stability of a Charge Density Wave (CDW) in a one-dimensional ring pierced by a Aharonov-Bohm flux is studied in a mean-field picture. It is found that the stability depends on the parity of the number $N$ of electrons. When the size of the ring becomes as small as the coherence length $\xi$, the CDW gap increases for even $N$ and decreases for odd $N$. Then when $N$ is even, the CDW gap decreases with flux but it increases when $N$ is odd. The variation of the BCS ratio with size and flux is also calculated. We derive the harmonics expansion of the persistent current in a presence of a finite gap.Comment: Latex, 7 pages, 10 figure

Persistent Currents for Interacting Electrons: a Simple Hartree-Fock Picture

The average persistent current of diffusive electrons in the Hartree-Fock approximation is derived in a simple non-diagrammatic picture. The Fourier expansion directly reflects the winding number decomposition of the diffusive motion around the ring. One recovers the results of Ambegaokar and Eckern, and Schmid. Moreover one finds an expression for which is valid beyond the diffusive regime.Comment: 7 pages, latex, no figure

Four-terminal resistances in mesoscopic networks of metallic wires: Weak localisation and correlations

We consider the electronic transport in multi-terminal mesoscopic networks of weakly disordered metallic wires. After a brief description of the classical transport, we analyze the weak localisation (WL) correction to the four-terminal resistances, which involves an integration of the Cooperon over the wires with proper weights. We provide an interpretation of these weights in terms of classical transport properties. We illustrate the formalism on examples and show that weak localisation to four-terminal conductances may become large in some situations. In a second part, we study the correlations of four-terminal resistances and show that integration of Diffuson and Cooperon inside the network involves the same weights as the WL. The formulae are applied to multiconnected wire geometries.Comment: 20 pages, contribution to a special issue in Physica E "Frontiers in quantum electronic transport - in memory of Markus B\"uttiker