### Nonlocal properties of quantum transport in metallic networks

Possible experimental realizations of graphs (networks) use either semiconducting or metallic wires. The elastic scattering of electrons over imputiries induces a diffusive motion (elastic scattering is much more efficient in metallic wires which have smaller elastic mean free path) responsible for the Ohm’s law for the classical resistance of a wire. Additionnally, in phase coherent conductors, the quantum interferences bring a small contribution to the transport coefficients: the so-called weak localization correction. In a wire, these interferences are responsible of a coherent backscattering that increases the return probability of an electron, and diminishes the conductance of the wire.

Weak localization has been investigated experimentally in networks, which provide arrangements particularly well suited to study interference effects. In 1984, the group of B.Pannetier *et al* observed beautiful Altshuler-Aronov-Spivak oscillations on honeycomb lattices. These experiments were very well fitted by the theory of B.Douçot and R.Rammal (1985) which assumes a uniform integration of the cooperon over the network. This theory was later made more efficient by M.Pascaud and G.Montambaux (1999) [this is discussed in the first article]. However it appears that this theory does not hold in general, but only applies to regular networks. For arbitrary networks, the integration of the cooperon over a wire must be properly weighted. The weight of each wire depends on the whole network and on the way it is connected to external contacts. The second article provides the general theory to describe quantum transport in networks of diffusive wires. It is shown that this is not a simple refinement of the Douçot and Rammal theory, since it predicts results differing both quantitatively and qualitatively: the most spectacular case is a change in sign of the weak localization of some transmissions for multiterminal networks.

- Christophe Texier and Gilles Montambaux,

**Weak localization in multiterminal networks of diffusive wires**

Phys. Rev. Lett.**92**(18), 186801 (2004).

cond-mat/0312060 or ccsd-00000925.

- Christophe Texier and Gilles Montambaux,

**How to increase a transmission with weak localization ? A geometrical effect**

in « Quantum information and decoherence in nanosystems », ed. by C. Glattli, M. Sanquer and J. Trân Thanh Vân, The Gioi publishers, 2005, Vietnam, p. 279.Proceedings of the XXXIXth Moriond conference, La Thuile, Italy, january 2004.

cond-mat/0404716, April 2004.

- Christophe Texier and Gilles Montambaux,

**Quantum oscillations in mesoscopic rings and anomalous diffusion**

J. Phys. A: Math. Gen.**38**, 3455-3471, (2005).

cond-mat/0411147.

### Conductance and resistance fluctuations/correlations

- Christophe Texier and Gilles Montambaux,

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

Physica E**75**, 33-46 (2016)

cond-mat arXiv:1506.08224

## Experiments

The phase coherence length of a large square network is extracted from magnetoconductance (MC) oscillations. The low temperature phase coherence length (25mK < T < 1K) is due to electron-electron interaction. A direct measurement is performed by a carefull analysis of the harmonics of the MC. **These experiments were realised in Meydi Ferrier’s PhD**.

- Meydi Ferrier, Lionel Angers, Alistair C.H. Rowe, Sophie Guéron, Hélène Bouchiat, Christophe Texier, Gilles Montambaux and Dominique Mailly,

**Direct measurement of the phase coherence length in a GaAs/GaAlAs square network**

Phys. Rev. Lett.**93**, 246804, (2004).

cond-mat/0402534 or ccsd-00001168.

The following article presents measurements of weak localization (AAS oscillations) and universal conductance fluctuations (AB oscillations) in networks with anisotropic (fixed) aspect ratio. We analyze the dependence of WL and UCF as a function of the size of the system and demonstrate dimensional crossovers.

- Félicien Schopfer, François Mallet, Dominique Mailly, Christophe Texier, Gilles Montambaux, Christopher Bäuerle and Laurent Saminadayar,

**Dimensional crossover in quantum networks: from macroscopic to mesoscopic Physics**

Phys. Rev. Lett.**98**, 026807 (2007).

cond-mat/0611127.