## A. Decoherence by electronic interactions

From the experimental point of view, the great interest in weak localization is to provide a powerful tool to probe decoherence mechanisms in weakly disordered metals or semiconductors. In the works listed above, the dephasing is described by introducing a phenomenological parameter in the cooperon. This procedure is correct for some dephasing mechanisms spin-flip, spin-orbit scattering, effect of penetration of a magnetic field in the wires) however it does not describe correctly the effect of electron-electron interaction, which is the dominant mechanism at low temperature (below 1K).

A description of electron-electron interaction was provided in a pioneering paper by Al’tshuler, Aronov&Khmel’nitskii in 1982: these authors modeled the dephasing in a one-particle picture through the interaction of the electron with a classically fluctuating electric field whose fluctuations are given by the fluctuation-dissipation theorem. They showed that in a wire, this leads to a phase coherence length behaving like L_{φ}∝ T ^{-1/3}. It is only very recently that Ludwig&Mirlin (PRB, 2004) noticed that the effect of the nontrivial geometry of a ring and the effect of electron-electron interaction combine in a nontrivial way and lead to unexpected behaviour of the magnetoconductance harmonics. We re-examine this question in the following paper, correct Ludwig&Mirlin’s magnetoconductance’s prefactor, and predict a new behaviour in a regime not studied by Ludwig&Mirlin.

- Christophe Texier and Gilles Montambaux,

**Dephasing due to electron-electron interaction in a diffusive ring**

Phys. Rev. B**72**, 115327 (2005).

cond-mat/0505199. - Erratum Phys. Rev. B
**74**, 209902(E) (2006).

- Christophe Texier and Gilles Montambaux,

**Comment on « Interaction-induced dephasing of Aharonov-Bohm oscillations » [Ludwig and Mirlin, Phys. Rev.~B****69**, 193306 (2004).]

pdf file.

In this*unpublished comment*, we show that Ludwig & Mirlin did a mistake when evaluating the preexponential factor of their path integral within semiclassical method. This affects the temperature dependence of the preexponential factor of AB/AAS harmonics of conductance. - Christophe Texier,

**Effect of connecting wires on the decoherence due to electron-electron interaction in a metallic ring**

Phys. Rev. B**76**, 153312 (2007).

cond-mat arXiv:0707.2916.

In this short paper I re-examine in detail the effect of the connecting arms in the problem of decoherence due to electron-electron interaction in a metallic ring. The two results of Ludwig & Mirlin (2004), corrected according to our comment, and [us, PRB72, 115327 (2005)] are rediscussed. The conclusion is that connecting wires do not affect decoherence in realistic situations. - Christophe Texier and Gilles Montambaux,

**Magnetoconductance oscillations in metallic rings and decoherence due to electron-electron interaction**

proceedings of the 6th rencontres du Vietnam « Nanophysics: from fundamentals to applications », Hanoi, 6-12 August 2006.

cond-mat arXiv:0704.0742.

In this article, decoherence due to electronic interactions is studied in chain of rings. When rings are separated by long (>>L_{φ}) arms of short arms (<< L_{φ}).

- Christophe Texier, Pierre Delplace and Gilles Montambaux

**Quantum oscillations and decoherence due to electron-electron interaction in metallic networks and hollow cylinders**

Phys. Rev. B**80**, 205413 (2009).

cond-mat arXiv:0907.3133 - Maximilian Treiber, Christophe Texier, Oleg M. Yevtushenko, Jan von Delft and Igor V. Lerner

**Thermal noise and dephasing due to electron interactions in non-trivial geometries**

Phys. Rev. B**84**, 054204 (2011).

cond-mat arXiv:1105.0554

We propose a real-space derivation of the fluctuation-dissipation theorem in metals. The result is used in order to obtain a**decoherence rate formula**describing**electronic interactions**in complex geometries (decoherence by e-e interaction must be accounted for through a*functional*of the electronic trajectories). This result generalizes the result of (Marquardt, von Delft*et al*, Phys. Rev. B 76, 195331 (2007) ; von Delft, Marquardt*et al*, Phys. Rev. B 76, 195332 (2007) ; von Delft, Int. J. Mod. Phys. B 22, 727 (2008) ; Treiber*et al*, Phys. Rev. B 80, 201305(R) (2009)) providing a description of decoherence accounting for classical and quantum noise regimes in simple geometries, and our results (Phys. Rev. B**72**, 115327 (2005) ; Phys. Rev. B**80**, 205413 (2009)) accounting for classical Johnson-Nyquist noise in arbitrary geometries.

## Experiments

The temperature dependences of the various length scales have been **observed in experiments** in the team of Hélène Bouchiat, Sophie Guéron and Meydi Ferrier (**weak localization of large square networks**):

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

**Geometrical dependence of decoherence by electronic interactions in a GaAs/GaAlAs square network**

Phys. Rev. Lett.**100**, 146802 (2008).

cond-mat arXiv:0707.3890

Experimental evidence of result of Ludwig&Mirlin (2004) and us (PRB72, 2005).

And also in the **experimental analysis of Aharonov-Bohm oscillations of a single mesoscopic ring** :

- Thibaut Capron, Christophe Texier, Gilles Montambaux, Dominique Mailly, Andreas Wieck and Laurent Saminadayar,

**Ergodic***vs*diffusive decoherence in mesoscopic devices

Phys. Rev. B**87**, 041307(R) (2013).

cond-mat arXiv:1206.0757

Experimental evidence of result of Ludwig&Mirlin (2004) and us (PRB72, 2005).

## B. Other interaction effects

### Al’tshuler-Aronov correction in a large square network

All previous works on quantum transport in networks have discussed the weak localization correction (WK) or the universal conductance fluctuations (UCF). Both are phase coherent quantities. Another quantum correction is the Al’tshuler-Aronov (AA) correction, which is due to the electron-electron interaction. Contrary WL and UCF, this quantum correction is not sensitive to phase coherence and does not feel the magnetic field, what allows to distinguish the two quantities experimentally. AA correction is computed in a large square network. Interpolation between 1d and 2d results is obtained. Comparison to some experimental results is provided.

- Christophe Texier and Gilles Montambaux,

**Al’tshuler-Aronov correction to the conductivity of a large metallic square network**

Phys. Rev. B**76**, 094202 (2007).

cond-mat arXiv:0704.0741.

The Altshuler-Aronov correction of large planar networks has been **studied experimentally** in the team of Christopher Bäuerle & Laurent Saminadayar (Institut Néel, Grenoble), during François Mallet’s PhD.

### Non-linear transport in diffusive metal and magnetic field asymmetry

Büttiker & Sanchéz and Spivak & Zyuzin discovered in 2004 an interesting effect of electronic interaction, which induces an asymmetry of the nonlinear conductance under magnetic field reversal. We have studied the effect in diffusive sample

- Christophe Texier and Johannes Mitscherling

**Non-linear conductance in mesoscopic weakly disordered wires — Interaction and magnetic field asymmetry**,

cond-mat arXiv:1510.02214