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The energy spectra for the tight-binding models on the Lieb and kagom\'e lattices both exhibit a flat band. We present a model which continuously interpolates between these two limits. The flat band located in the middle of the three-band spectrum for the Lieb lattice is distorted, generating two pairs of Dirac points. While the upper pair evolves into graphene-like Dirac cones in the kagom\'e limit, the low energy pair evolves until it merges producing the band-bottom flat band. The topological characterization of the Dirac points is achieved by projecting the Hamiltonian on the two relevant bands in order to obtain an effective Dirac Hamiltonian. The low energy pair of Dirac points is particularly interesting in this respect: when they emerge, they have opposite winding numbers, but as they merge, they have the same winding number. This apparent paradox is due to a continuous rotation of their states in pseudo-spin space, characterized by a winding vector. This simple, but quite rich model, suggests a way to a systematic characterization of two-band contact points in multiband systems., Comment: 9 pages, 6 figures

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, France

The merging or emergence of a pair of Dirac points may be classified according to whether the winding numbers which characterize them are opposite ($+-$ scenario) or identical ($++$ scenario). From the touching point between two parabolic bands (one of them can be flat), two Dirac points with the {\it same} winding number emerge under appropriate distortion (interaction, etc), following the $++$ scenario. Under further distortion, these Dirac points merge following the $+-$ scenario, that is corresponding to {\it opposite} winding numbers. This apparent contradiction is solved by the fact that the winding number is actually defined around a unit vector on the Bloch sphere and that this vector rotates during the motion of the Dirac points. This is shown here within the simplest two-band lattice model (Mielke) exhibiting a flat band. We argue on several examples that the evolution between the two scenarios is general., Comment: 5 pages, 6 figures

We reconsider the thermodynamic derivation by L. Boltzmann of the Stefan law and we generalize it for various different physical systems {\it whose chemical potential vanishes}. Being only based on classical arguments, therefore independent of the quantum statistics, this derivation applies as well to the saturated Bose gas in various geometries as to "compensated" Fermi gas near a neutrality point, such as a gas of Weyl Fermions. It unifies in the same framework the thermodynamics of many different bosonic or fermionic non-interacting gases which were until now described in completely different contexts., Comment: 16 pages

We experimentally reveal the emergence of edge states in a photonic lattice with orbital bands. We use a two-dimensional honeycomb lattice of coupled micropillars whose bulk spectrum shows four gapless bands arising from the coupling of $p$-like photonic orbitals. We observe zero-energy edge states whose topological origin is similar to that of conventional edge states in graphene. Additionally, we report novel dispersive edge states that emerge not only in zigzag and bearded terminations, but also in armchair edges. The observations are reproduced by tight-binding and analytical calculations. Our work shows the potentiality of coupled micropillars in elucidating some of the electronic properties of emergent 2D materials with orbital bands.

The orbital magnetic susceptibility of an electron gas in a periodic potential depends not only on the zero field energy spectrum but also on the geometric structure of cell-periodic Bloch states which encodes interband effects. In addition to the Berry curvature, we explicitly relate the orbital susceptibility of two-band models to a quantum metric tensor defining a distance in Hilbert space. Within a simple tight-binding model allowing for a tunable Bloch geometry, we show that interband effects are essential even in the absence of Berry curvature. We also show that for a flat band model, the quantum metric gives rise to a very strong orbital paramagnetism., Comment: 13 pages, 3 figures

We present a theory of quantum oscillations in insulators that are particle-hole symmetric and non-topological but with arbitrary band dispersion, at both zero and non-zero temperature. At temperatures $T$ less than or comparable to the gap, the dependence of oscillations on $T$ is markedly different from that in metals and depends crucially on the position of the chemical potential $\mu$ in the gap. If $\mu$ is in the middle of the gap, oscillations do not change with $T$; however, if $\mu$ is asymmetrically positioned in the gap, surprisingly, oscillations go to zero at a critical value of the inverse field determined by $T$ and $\mu$ and then change their phase by $\pi$ and grow again. Additionally, the temperature dependence is different for quantities derived from the grand canonical potential, such as magnetization and susceptibility, and those derived from the density of states, such as resistivity. However, the non-trivial features arising from asymmetric $\mu$ are present in both.

Inspired by recent experiments with cold atoms in optical lattices, we consider a St\"uckelberg interferometer for a particle performing Bloch oscillations in a tight-binding model on the honeycomb lattice. The interferometer is made of two avoided crossings at the saddle points of the band structure (i.e. at M points of the reciprocal space). This problem is reminiscent of the double Dirac cone St\"uckelberg interferometer that was recently studied in the continuum limit [Phys. Rev. Lett. 112, 155302 (2014)]. Although the two problems share similarities -- such as the appearance of a geometric phase shift -- lattice effects, not captured by the continuum limit, make them truly different. The particle dynamics in the presence of a force is described by the Bloch Hamiltonian $H(\boldsymbol{k})$ defined from the tight-binding Hamiltonian and the position operator. This leads to many interesting effects for the lattice St\"uckelberg interferometer: a twisting of the two Landau-Zener tunnelings, saturation of the inter-band transition probability in the sudden (infinite force) limit and extended periodicity or even non-periodicity beyond the first Brillouin zone. In particular, St\"uckelberg interferometry gives access to the overlap matrix of cell-periodic Bloch states thereby allowing to fully characterize the geometry of Bloch states, as e.g. to obtain the quantum metric tensor., Comment: 20 pages, 19 figures, added references, minor corrections

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"

We study the time evolution of a two-dimensional quantum particle exhibiting an energy spectrum, made of two bands, with two Dirac cones, as e.g. in the band structure of a honeycomb lattice. A force is applied such that the particle experiences two Landau-Zener transitions in succession. The adiabatic evolution between the two transitions leads to St\"uckelberg interferences, due to two possible trajectories in energy space. In addition to well-known dynamical and Stokes phases, the interference pattern reveals a geometric phase which depends on the chirality (winding number) and the mass sign associated to each Dirac cone, as well as on the type of trajectory (parallel or diagonal with respect to the two cones) in parameter space. This geometric phase reveals the coupling between the bands encoded in the structure of the wavefunctions., Comment: 20 pages, 12 figures

This review aims at a theoretical discussion of Dirac points in two-dimensional systems. Whereas Dirac points and Dirac fermions are prominent low-energy electrons in graphene (two-dimensional graphite), research on Dirac fermions in low-energy physics has spread beyond condensed-matter systems. In these alternative systems, a large versatility in the manipulation of the relevant band parameters can be achieved. This allows for a systematic study of the motion and different possible fusions of Dirac points, which are beyond the physical limits of graphene. We introduce the basic properties of Dirac fermions and the motion of Dirac points here and aim at a topological classification of these motions. The theoretical concepts are illustrated in particular model systems., Comment: 26 pages, 7 figures; review presented at Institut Henri Poincar\'e, S\'eminaire Poincar\'e "Mati\`ere de Dirac", June 2014

Edge states are one important ingredient to understand transport properties of graphene nanoribbons. We study experimentally the existence and the internal structure of edge states under uniaxial strain of the three main edges: zigzag, bearded, and armchair. The experiments are performed on artificial microwave graphene flakes, where the wavefunctions are obtained by direct imaging. We show that uniaxial strain can be used to manipulate the edge states: a single parameter controls their existence and their spatial extension into the ribbon. By combining tight-binding approach and topological arguments, we provide an accurate description of our experimental findings. A new type of zero-energy state appearing at the intersection of two edges, namely the corner state, is also observed and discussed., Comment: 15 pages, 9 figures

We show that a St\"{u}ckelberg interferometer made of two massive Dirac cones can reveal information on band eigenstates such as the chirality and mass sign of the cones. For a given spectrum with two gapped cones, we propose several low-energy Hamiltonians differing by their eigenstates properties. The corresponding inter-band transition probability is affected by such differences in its interference fringes being shifted by a new phase of geometrical origin. This phase can be a useful bulk probe for topological band structures realized with artificial crystals., Comment: 5 pages, 4 figures, 1 table; accepted for publication in Phys. Rev. Lett

We experimentally study the propagation of microwaves in an artificial honeycomb lattice made of dielectric resonators. This evanescent propagation is well described by a tight-binding model, very much like the propagation of electrons in graphene. We measure the density of states, as well as the wave function associated with each eigenfrequency. By changing the distance between the resonators, it is possible to modulate the amplitude of next-(next-)nearest-neighbor hopping parameters and to study their effect on the density of states. The main effect is the density of states becoming dissymmetric and a shift of the energy of the Dirac points. We study the basic elements: An isolated resonator, a two-level system, and a square lattice. Our observations are in good agreement with analytical solutions for corresponding infinite lattice., Comment: 10 pages, 9 figures

By means of a microwave tight-binding analogue experiment of a graphene-like lattice, we observe a topological transition between a phase with a point-like band gap characteristic of massless Dirac fermions and a gapped phase. By applying a controlled anisotropy on the structure, we investigate the transition directly via density of states measurements. The wave function associated with each eigenvalue is mapped and reveals new states at the Dirac point, localized on the armchair edges. We find that with increasing anisotropy, these new states are more and more localized at the edges., Comment: Physical Review Letters (2013) XXX

Motivated by a recent experiment in a tunable graphene analog [L. Tarruell et al., Nature 483, 302 (2012)], we consider a generalization of the Landau-Zener problem to the case of a quadratic crossing between two bands in the vicinity of the merging transition of Dirac cones. The latter is described by the so-called universal hamiltonian. In this framework, the inter-band tunneling problem depends on two dimensionless parameters: one measures the proximity to the merging transition and the other the adiabaticity of the motion. Under the influence of a constant force, the probability for a particle to tunnel from the lower to the upper band is computed numerically in the whole range of these two parameters and analytically in different limits using (i) the Stueckelberg theory for two successive linear band crossings, (ii) diabatic perturbation theory, (iii) adiabatic perturbation theory and (iv) a modified Stueckelberg formula. We obtain a complete phase diagram and explain the presence of unexpected probability oscillations in terms of interferences between two poles in the complex time plane. We also compare our results to the above mentioned experiment., Comment: 14 pages, 19 figures, accepted in Phys. Rev. A

We report on the measurement of phase coherence length in a high mobility two-dimensional electron gas patterned in two different geometries, a wire and a ring. The phase coherence length is extracted both from the weak localization correction in long wires and from the amplitude of the Aharonov-Bohm oscillations in a single ring, in a low temperature regime when decoherence is dominated by electronic interactions. We show that these two measurements lead to different phase coherence lengths, namely $L_{\Phi}^\mathrm{wire}\propto T^{-1/3}$ and $L_{\Phi}^\mathrm{ring}\propto T^{-1/2}$. This difference reflects the fact that the electrons winding around the ring necessarily explore the whole sample (ergodic trajectories), while in a long wire the electrons lose their phase coherence before reaching the edges of the sample (diffusive regime)., Comment: LaTeX, 5 pages, 4 pdf figures ; v2: revised version

In this brief report, we show the equivalence between the tight-binding descriptions of the monolayer and bilayer honeycomb lattices. With appropriate value of the third nearest neighbors coupling, the Hamiltonian for a monolayer is equivalent to the low energy effective Hamiltonian for bilayer in the presence of trigonal warping. A simple physical argument is provided to explain this correspondance., Comment: 4 pages, 4 figures

Bloch oscillations are a powerful tool to investigate spectra with Dirac points. By varying band parameters, Dirac points can be manipulated and merged at a topological transition towards a gapped phase. Under a constant force, a Fermi sea initially in the lower band performs Bloch oscillations and may Zener tunnel to the upper band mostly at the location of the Dirac points. The tunneling probability is computed from the low energy universal Hamiltonian describing the vicinity of the merging. The agreement with a recent experiment on cold atoms in an optical lattice is very good., Comment: 4 pages, 4 figures; (v2) added refs and Supplementary Material with 1 figure

We report a theoretical low-field magnetotransport study unveiling the effect of pseudospin in realistic models of weakly disordered graphene-based materials. Using an efficient Kubo computational method, and simulating the effect of charges trapped in the oxide, different magnetoconductance fingerprints are numerically obtained in system sizes as large as 0.3 micronmeter squared, containing tens of millions of carbon atoms. In two-dimensional graphene, a strong valley mixing is found to irreparably yield a positive magnetoconductance (weak localization), whereas crossovers from positive to a negative magnetoconductance (weak antilocalization) are obtained by reducing disorder strength down to the ballistic limit. In sharp contrast, graphene nanoribbons with lateral size as large as 10nm show no sign of weak antilocalization, even for very small disorder strength. Our results rationalize the emergence of a complex phase diagram of magnetoconductance fingerprints, shedding some new light on the microscopical origin of pseudospin effects., Comment: 8 pages, 5 figures

We investigate the fine structure of the edge states energy spectrum for zigzag and armchair ribbons of graphene in a strong magnetic field. At low energy, the spectra can be described by an effective Schrodinger Hamiltonian with a double well potential, symmetric in the zigzag case and asymmetric in the armchair case. We develop a semiclassical formalism based on the WKB approximation to calculate analytically the energy spectrum for the two types of edges, including regions which were not studied earlier. Our results are in very good quantitative agreement with numerical calculations. This approach leads to a qualitative description of the spectra in terms of the quantization of unusual classical orbits in the real space., Comment: 20 pages, 25 figures

We propose a simple description of the spectrum of edge states in the quantum Hall regime, in terms of semiclassical quantization of skipping orbits along hard wall boundaries, ${\cal A}=2 \pi (n+\gamma) \ell_B^2$, where ${\cal A}$ is the area enclosed between a skipping orbit and the wall and $\ell_B$ is the magnetic length. Remarkably, this description provides an excellent quantitative agreement with the exact spectrum. We discuss the value of $\gamma$ when the skipping orbits touch one or two edges, and its variation when the orbits graze the edges and the semiclassical quantization has to be corrected by diffraction effects. The value of $\gamma$ evolves continuously from $1/2$ to $3/4$. We calculate the energy dependence of the drift velocity along the different Landau levels. We compare the structure of the semiclassical cyclotron orbits, their position with respect to the edge, to the wave function of the corresponding eigenstates., Comment: 9 pages, 16 figures

We have studied the quantum oscillations of the conductance for arrays of connected mesoscopic metallic rings, in the presence of an external magnetic field. Several geometries have been considered: a linear array of rings connected with short or long wires compared to the phase coherence length, square networks and hollow cylinders. Compared to the well-known case of the isolated ring, we show that for connected rings, the winding of the Brownian trajectories around the rings is modified, leading to a different harmonics content of the quantum oscillations. We relate this harmonics content to the distribution of winding numbers. We consider the limits where coherence length $L_\varphi$ is small or large compared to the perimeter $L$ of each ring constituting the network. In the latter case, the coherent diffusive trajectories explore a region larger than $L$, whence a network dependent harmonics content. Our analysis is based on the calculation of the spectral determinant of the diffusion equation for which we have a simple expression on any network. It is also based on the hypothesis that the time dependence of the dephasing between diffusive trajectories can be described by an exponential decay with a single characteristic time $\tau_\varphi$ (model A) . At low temperature, decoherence is limited by electron-electron interaction, and can be modelled in a one-electron picture by the fluctuating electric field created by other electrons (model B). It is described by a functional of the trajectories and thus the dependence on geometry is crucial. Expressions for the magnetoconductance oscillations are derived within this model and compared to the results of model A. It is shown that they involve several temperature-dependent length scales., Comment: 35 pages, revtex4, 25 figures (34 pdf files)

Analysis of edge-state energies in the integer quantum Hall effect is carried out within the semiclassical approximation. When the system is wide so that each edge can be considered separatly, this problem is equivalent to that of a one dimensional harmonic oscillator centered at x=x_c and an infinite wall at x=0, and appears in numerous physical contexts. The eigenvalues E_n(x_c) for a given quantum number n are solutions of the equation S(E,x_c)=\pi [n+ \gamma(E,x_c)] where S is the WKB action and 0<\gamma<1 encodes all the information on the connection procedure at the turning points. A careful implication of the WKB connection formulae results in an excellent approximation to the exact energy eigenvalues. The dependence of \gamma [E_n(x_c),x_c] \equiv \gamma_c (x_c) on x_c is analyzed between its two extreme values 1/2 as x_c goes to -infinity far inside the sample and 3/4 as x_c goes to infinity far outside the sample. The edge-state energies E_n(x_c) obey an almost exact scaling law of the form E_n(x_c)=4 [n+\gamma_n(x_c)] f(x_c/4 n +3) and the scaling function f(y) is explicitly elucidated, Comment: 9 pages, 9 figures

We address a simple but fundamental issue arising in the study of graphene, as well as of other systems that have a crystalline structure with more than one atom per unit cell. For these systems, the choice of the tight-binding basis is not unique. For monolayer graphene two bases are widely used in the literature. While the expectation values of operators describing physical quantities should be independent of basis, the form of the operators may depend on the basis, especially in the presence of disorder or of an applied magnetic field. Using the inappropriate form of certain operators may lead to erroneous physical predictions. We discuss the two bases used to describe monolayer graphene, as well as the form of the most commonly used operators in the two bases. We repeat our analysis for the case of bilayer graphene., Comment: 15 pages, 4 figures

We calculate the modulation by a magnetic field of the critical current of a long disordered Josephson junction in the diffusive limit, i.e. when the dimensions of the junction are larger that the elastic mean free path, and when the length $L$ is much larger than the width $w$. Due to the averaging of the gauge invariant phase factor over diffusive trajectories, the well-known oscillations of the Fraunhofer pattern are smoothed out and replaced by an exponential decay at large field. The predicted pattern is universal, i.e., it is independent of the disorder strength. We point out an interesting relation with the physics of speckle correlations in optics of turbid media., Comment: 4 pages, 2 figures

We consider a tight-binding model on the honeycomb lattice in a magnetic field. For special values of the hopping integrals, the dispersion relation is linear in one direction and quadratic in the other. We find that, in this case, the energy of the Landau levels varies with the field B as E_n(B) ~ [(n+\gamma)B]^{2/3}. This result is obtained from the low-field study of the tight-binding spectrum on the honeycomb lattice in a magnetic field (Hofstadter spectrum) as well as from a calculation in the continuum approximation at low field. The latter links the new spectrum to the one of a modified quartic oscillator. The obtained value $\gamma=1/2$ is found to result from the cancellation of a Berry phase., Comment: 4 pages, 4 figures

We study weak localization in chains of metallic rings. We show than nonlocality of quantum transport can drastically affect the behaviour of the harmonics of magnetoconductance oscillations. Two different geometries are considered: the case of rings separated by long wires compared to the phase coherence length and the case of contacted rings. In a second part we discuss the role of decoherence due to electron-electron interaction in these two geometries., Comment: 4 pages, LaTeX, 2 eps figures, proceedings of the 6th rencontres du Vietnam "Nanophysics: from fundamentals to applications", Hanoi, 6-12 August 2006

We consider the correction $\Delta\sigma_\mathrm{ee}$ due to electron-electron interaction to the conductivity of a weakly disordered metal (Al'tshuler-Aronov correction). The correction is related to the spectral determinant of the Laplace operator. The case of a large square metallic network is considered. The variation of $\Delta\sigma_\mathrm{ee}(L_T)$ as a function of the thermal length $L_T$ is found very similar to the variation of the weak localization $\Delta\sigma_\mathrm{WL}(L_\phi)$ as a function of the phase coherence length. Our result for $\Delta\sigma_\mathrm{ee}$ interpolates between the known 1d and 2d results, but the interaction parameter entering the expression of $\Delta\sigma_\mathrm{ee}$ keeps a 1d behaviour. Quite surprisingly, the result is very close to the 2d logarithmic behaviour already for $L_T\sim{a}/2$, where $a$ is the lattice parameter., Comment: 6 pages, RevTex4, 3 eps figures

This lecture is a tutorial introduction to coherent effects in disordered electronic systems. Avoiding technicalities as most as possible, I present some personal points of view to describe well-known signatures of phase coherence like weak localization correction or universal conductance fluctuations. I show how these physical properties of phase coherent conductors can be simply related to the classical return probability for a diffusive particle. The diffusion equation is then solved in various appropriate geometries and in the presence of a magnetic field. The important notion of quantum crossing is developed, which is at the origin of the quantum effects. The analogy with optics is exploited and the relation between universal conductance fluctuations and speckle fluctuations in optics is explained. The last part concerns the effect of electron-electron interactions. Using the same simple description, I derive qualitatively the expressions of the Altshuler-Aronov anomaly of the density of states, and of the correction to the conductivity. The last part, slightly more technical, adresses the question of the lifetime of a quasiparticle in a disordered metal., Comment: 45 pages, 15 figures, Lectures notes for the "International School on Physics of Zero and One Dimensional Nanoscopic Systems", Saha Institute of Nuclear Physics, Calcutta, India, 1-9 February, 2006

We report on magnetoconductance measurements of metallic networks of various sizes ranging from 10 to $10^{6}$ plaquettes, with anisotropic aspect ratio. Both Altshuler-Aronov-Spivak (AAS) $h/2e$ periodic oscillations and Aharonov-Bohm (AB) $h/e$ periodic oscillations are observed for all networks. For large samples, the amplitude of both oscillations results from the incoherent superposition of contributions of phase coherent regions. When the transverse size becomes smaller than the phase coherent length $L_\phi$, one enters a new regime which is phase coherent (mesoscopic) along one direction and macroscopic along the other, leading to a new size dependence of the quantum oscillations., Comment: 4 pages, to appear in Phys. Rev. Lett

We study the effect of the electron-electron interaction on the weak localization correction of a ring pierced by a magnetic flux. We compute exactly the path integral giving the magnetoconductivity for an isolated ring. The results are interpreted in a time representation. This allows to characterize the nature of the phase coherence relaxation in the ring. The nature of the relaxation depends on the time regime (diffusive or ergodic) but also on the harmonics $n$ of the magnetoconductivity. Whereas phase coherence relaxation is non exponential for the harmonic $n=0$, it is always exponential for harmonics $n\neq0$. Then we consider the case of a ring connected to reservoirs and discuss the effect of connecting wires. We recover the behaviour of the harmonics predicted recently by Ludwig & Mirlin for a large perimeter (compared to the Nyquist length). We also predict a new behaviour when the Nyquist length exceeds the perimeter., Comment: 21 pages, RevTeX4, 8 eps figures; version of 10/2006 : eqs.(100-102) of section V.C corrected

We consider the weak localization correction to the conductance of a ring connected to a network. We analyze the harmonics content of the Al'tshuler-Aronov-Spivak (AAS) oscillations and we show that the presence of wires connected to the ring is responsible for a behaviour different from the one predicted by AAS. The physical origin of this behaviour is the anomalous diffusion of Brownian trajectories around the ring, due to the diffusion in the wires. We show that this problem is related to the anomalous diffusion along the skeleton of a comb. We study in detail the winding properties of Brownian curves around a ring connected to an arbitrary network. Our analysis is based on the spectral determinant and on the introduction of an effective perimeter probing the different time scales. A general expression of this length is derived for arbitrary networks. More specifically we consider the case of a ring connected to wires, to a square network, and to a Bethe lattice., Comment: 17 pages, 7 eps figures

We show that, in low dimensional conductors, the quasiparticle decay and the relaxation of the phase are not exponential processes. In quasi-one dimension, they scale as $e^{- (t/\tau_N)^{3/2}}$ where the characteristic time $\tau_{in}$, identical for both processes, is a power $T^{2/3}$ of the temperature. This result implies a distribution of relaxation times., Comment: 6 pages, LaTeX, 1 eps figure, contribution to the proceedings of the XXXIXth Moriond conference, La Thuile, Italy, January 2004

We study the quantum transport in multiterminal networks of quasi-one-dimensional diffusive wires. When calculating the weak localization correction to the conductances, we show that the Cooperon must be properly weighted over each wire. This can even change the sign of the weak localization correction in certain geometries., Comment: 4 pages, LaTeX, 7 eps figures, contribution to the proceedings of the XXXIXth Moriond conference, La Thuile, Italy, January 2004

We show that in low dimensional disordered conductors, the quasiparticle decay and the relaxation of the phase are not exponential processes. In the quasi-one dimensional case, both behave at small time as $e^{- (t/\tau_{in})^{3/2}}$ where the inelastic time $\tau_{in}$, identical for both processes, is a power $T^{2/3}$ of the temperature. This result implies the existence of an unusual distribution of relaxation times that we obtain., Comment: 4 pages, 1 figure

The low temperature magnetoconductance of a large array of quantum coherentloops exhibits Altshuler-Aronov-Spivak oscillations which periodicitycorresponds to 1/2 flux quantum per loop.We show that the measurement of the harmonics content in a square networkprovides an accurate way to determine the electron phase coherence length$L\_{\phi}$ in units of the lattice length without any adjustableparameters.We use this method to determine $L\_{\phi}$ in a network realised from a 2Delectron gas (2DEG) in a GaAS/GaAlAs heterojunction. The temperaturedependence follows a power law $T^{-1/3}$ from 1.3 K to 25 mK with nosaturation, as expected for 1D diffusive electronic motion andelectron-electron scattering as the main decoherence mechanism., Comment: Additional experimental data in version 2

We study the quantum transport through networks of diffusive wires connected to reservoirs in the Landauer-B\"uttiker formalism. The elements of the conductance matrix are computed by the diagrammatic method. We recover the combination of classical resistances and obtain the weak localization corrections. For arbitrary networks, we show how the cooperon must be properly weighted over the different wires. Its nonlocality is clearly analyzed. We predict a new geometrical effect that may change the sign of the weak localization correction in multiterminal geometries., Comment: 4 pages, LaTeX, 4 figures, 8 eps files

Recent experiments on quantum dots in the Coulomb Blockade regime have shown how adding successive electrons into a dot modifies the energy spectrum and the wave functions of the electrons already present in the dot. Using a microscopic model, we study the importance of electron-electron interaction on these ``scrambling'' effects. We compute the Hartree-Fock single particle properties as function of the number $p$ of added electrons. We define parametric correlation functions that characterize the scrambling properties of the Hartree-Fock wave functions and energy spectra. We find that each of these correlation functions exhibit a universal behavior in terms of the ratio $p/p^{\ast}$ where $p^{\ast}$ is a characteristic number that decreases with increasing either the interaction strength, the disorder strength or the system size, Comment: proceedings Recontres de Moriond, les Arcs 2001, 2 figures

We consider the scattering theory for the Schr\"odinger operator $-\Dc_x^2+V(x)$ on graphs made of one-dimensional wires connected to external leads. We derive two expressions for the scattering matrix on arbitrary graphs. One involves matrices that couple arcs (oriented bonds), the other involves matrices that couple vertices. We discuss a simple way to tune the coupling between the graph and the leads. The efficiency of the formalism is demonstrated on a few known examples., Comment: 21 pages, LaTeX, 10 eps figures

We study the influence of boundary conditions transverse to the transport direction for disordered mesoscopic conductors both at the Anderson metal-insulator transition and in the metallic regime. We show that the boundary conditions strongly influence the conductance distribution exactly at the metal-insulator transition and we discuss implications for the standard picture of one-parameter scaling. We show in particular that the scaling function that describes the change of conductance with system size depends on the boundary conditions from the metallic regime up to the metal-insulator transition. An experiment is proposed that might test the correctness of the one-parameter scaling theory., Comment: 6 revtex pages, 2 figures

In this brief tutorial review, I show how phase coherent properties of disordered conductors can be described in a simple and unified way. These properties include transport properties like weak-localization correction and universal conductance fluctuations, and thermodynamic properties like orbital magnetism and persistent currents. They can be related to the classical return probability for a diffusive particle. For a network with $N$ nodes, the return probability can be related to the determinant of a $N \times N$ "connectivity" matrix $M$ so that the magnetization and the transport quantities can be {\it directly} written in term of $\det M$., Comment: 12 pages, Invited talk, International Workshop on superconductivity, magnetoresistive materials and correlated quantum systems, Rencontres du Vietnam, Hanoi 1999, p. 207 (Vietnam National University Press)

We show how the orbital magnetization of an interacting disordered diffusive electron gas can be simply related to the magnetization of the non-interacting system having the same geometry. This result is applied to the persistent current of a mesoscopic ring and to the relation between Landau diamagnetism and the interaction correction to the magnetization of diffusive systems. The field dependence of this interaction contribution can be deduced directly from the de Haas-van Alphen oscillations of the free electron gas. Known results for the free orbital magnetism of finite systems can be used to derive the interaction contribution in the diffusive regime in various geometries., Comment: 4 pages, 2 figures

We study the spectral determinant of the Laplacian on finite graphs characterized by their number of vertices V and of bonds B. We present a path integral derivation which leads to two equivalent expressions of the spectral determinant of the Laplacian either in terms of a V x V vertex matrix or a 2B x 2B link matrix that couples the arcs (oriented bonds) together. This latter expression allows us to rewrite the spectral determinant as an infinite product of contributions of periodic orbits on the graph. We also present a diagrammatic method that permits us to write the spectral determinant in terms of a finite number of periodic orbit contributions. These results are generalized to the case of graphs in a magnetic field. Several examples illustrating this formalism are presented and its application to the thermodynamic and transport properties of weakly disordered and coherent mesoscopic networks is discussed., Comment: 33 pages, submitted to Ann. Phys

We discuss some recent results on the statistics of the Coulomb Blockade in disordered quantum dots containing spinless interacting fermions using the self-consistent Hartree-Fock approximation. We concentrate on the regime r_s >~1, with finite dimensionless conductance g. We present significantly different results for the cases of a Coulomb and a nearest-neighbour bare interaction. We show that neglecting rearrangements when the particle number is changed (Koopmans' approximation) can lead to large errors, and show how the spectral statistics of the self-consistent single particle orbitals evolve with the interaction strength., Comment: Contribution for proceedings of Moriond 99, 5 pages, incl. moriond.sty. More details given in cond-mat/9902258

We investigate the addition spectrum of disordered quantum dots containing spinless interacting fermions using the self-consistent Hartree-Fock approximation. We concentrate on the regime r_s >~1, with finite dimensionless conductance g. We find that in this approximation the peak spacing fluctuations do not scale with the mean single particle level spacing for either Coulomb or nearest neighbour interactions when r_s >~1. We also show that Koopmans' approximation to the addition spectrum can lead to errors that are of order the mean level spacing or larger, both in the mean addition spectrum peak spacings, and in the peak spacing fluctuations., Comment: 35 pages including 22 figures (eps)

We analyse the ground state of spinless fermions on a lattice in a weakly disordered potential, interacting via a nearest neighbour interaction, by applying the self-consistent Hartree-Fock approximation. We find that charge density modulations emerge progressively when r_s >1, even away from half-filling, with only short-range density correlations. Classical geometry dependent "magic numbers" can show up in the addition spectrum which are remarkably robust against quantum fluctuations and disorder averaging., Comment: 4 pages, 3 eps figures

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 figures