Glass phase of two-dimensional triangular elastic lattices with disorder

We study two dimensional triangular elastic lattices in a background of point disorder, excluding dislocations (tethered network). Using both (replica symmetric) static and (equilibrium) dynamic renormalization group for the corresponding $N=2$ component model, we find a transition to a glass phase for $T < T_g$, described by a plane of perturbative fixed points. The growth of displacements is found to be asymptotically isotropic with $u_T^2 \sim u_L^2 \sim A_1 \ln^2 r$, with universal subdominant anisotropy $u_T^2 - u_L^2 \sim A_2 \ln r$. where $A_1$ and $A_2$ depend continuously on temperature and the Poisson ratio $\sigma$. We also obtain the continuously varying dynamical exponent $z$. For the Cardy-Ostlund $N=1$ model, a particular case of the above model, we point out a discrepancy in the value of $A_1$ with other published results in the litterature. We find that our result reconciles the order of magnitude of the RG predictions with the most recent numerical simulations.Comment: 25 pages, RevTeX, uses epsf,multicol and amssym

Price decomposition in large-scale stochastic optimal control

We are interested in optimally driving a dynamical system that can be influenced by exogenous noises. This is generally called a Stochastic Optimal Control (SOC) problem and the Dynamic Programming (DP) principle is the natural way of solving it. Unfortunately, DP faces the so-called curse of dimensionality: the complexity of solving DP equations grows exponentially with the dimension of the information variable that is sufficient to take optimal decisions (the state variable). For a large class of SOC problems, which includes important practical problems, we propose an original way of obtaining strategies to drive the system. The algorithm we introduce is based on Lagrangian relaxation, of which the application to decomposition is well-known in the deterministic framework. However, its application to such closed-loop problems is not straightforward and an additional statistical approximation concerning the dual process is needed. We give a convergence proof, that derives directly from classical results concerning duality in optimization, and enlghten the error made by our approximation. Numerical results are also provided, on a large-scale SOC problem. This idea extends the original DADP algorithm that was presented by Barty, Carpentier and Girardeau (2010)

Topological Weyl Semi-metal from a Lattice Model

We define and study a three dimensional lattice model which displays a Weyl semi-metallic phase. This model consists of coupled layers of quantum (anomalous) Hall insulators. The Weyl semi-metallic phase appears between a resulting quantum Hall insulating phase and a normal insulating phase. Weyl fermions in this Weyl semi-metal, similar to Dirac fermions in graphene, have their lattice pseudo-spin locked to their momenta. We investigate surface states and Fermi arcs, and their evolution for different phases, by exactly diagonalizing the lattice model as well as by analyzing their topological origins.Comment: Accepted version for publication in EPL. 6 pages, 4 figure

Dynamic consistency for Stochastic Optimal Control problems

For a sequence of dynamic optimization problems, we aim at discussing a notion of consistency over time. This notion can be informally introduced as follows. At the very first time step $t_0$, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time step $t_0, t_1, ..., T$; at the next time step $t_1$, he is able to formulate a new optimization problem starting at time $t_1$ that yields a new sequence of optimal decision rules. This process can be continued until final time $T$ is reached. A family of optimization problems formulated in this way is said to be time consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of time consistency, well-known in the field of Economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (2007) and studied in the Stochastic Programming framework by Shapiro (2009) and for Markov Decision Processes (MDP) by Ruszczynski (2009). We here link this notion with the concept of "state variable" in MDP, and show that a significant class of dynamic optimization problems are dynamically consistent, provided that an adequate state variable is chosen

Minimal conductivity, topological Berry winding and duality in three-band semimetals

The physics of massless relativistic quantum particles has recently arisen in the electronic properties of solids following the discovery of graphene. Around the accidental crossing of two energy bands, the electronic excitations are described by a Weyl equation initially derived for ultra-relativistic particles. Similar three and four band semimetals have recently been discovered in two and three dimensions. Among the remarkable features of graphene are the characterization of the band crossings by a topological Berry winding, leading to an anomalous quantum Hall effect, and a finite minimal conductivity at the band crossing while the electronic density vanishes. Here we show that these two properties are intimately related: this result paves the way to a direct measure of the topological nature of a semi-metal. By considering three band semimetals with a flat band in two dimensions, we find that only few of them support a topological Berry phase. The same semimetals are the only ones displaying a non vanishing minimal conductivity at the band crossing. The existence of both a minimal conductivity and a topological robustness originates from properties of the underlying lattice, which are encoded not by a symmetry of their Bloch Hamiltonian, but by a duality

Probing (topological) Floquet states through DC transport

We consider the differential conductance of a periodically driven system connected to infinite electrodes. We focus on the situation where the dissipation occurs predominantly in these electrodes. Using analytical arguments and a detailed numerical study we relate the differential conductances of such a system in two and three terminal geometries to the spectrum of quasi-energies of the Floquet operator. Moreover these differential conductances are found to provide an accurate probe of the existence of gaps in this quasi-energy spectrum, being quantized when topological edge states occur within these gaps. Our analysis opens the perspective to describe the intermediate time dynamics of driven mesoscopic conductors as topological Floquet filters.Comment: 8 pages, 6 figures, invited contribution to the special issue of Physica E on "Frontiers in quantum electronic transport" in memory of Markus Buttike

Time Blocks Decomposition of Multistage Stochastic Optimization Problems

Multistage stochastic optimization problems are, by essence, complex because their solutions are indexed both by stages (time) and by uncertainties (scenarios). Their large scale nature makes decomposition methods appealing.The most common approaches are time decomposition --- and state-based resolution methods, like stochastic dynamic programming, in stochastic optimal control --- and scenario decomposition --- like progressive hedging in stochastic programming. We present a method to decompose multistage stochastic optimization problems by time blocks, which covers both stochastic programming and stochastic dynamic programming. Once established a dynamic programming equation with value functions defined on the history space (a history is a sequence of uncertainties and controls), we provide conditions to reduce the history using a compressed "state" variable. This reduction is done by time blocks, that is, at stages that are not necessarily all the original unit stages, and we prove areduced dynamic programming equation. Then, we apply the reduction method by time blocks to \emph{two time-scales} stochastic optimization problems and to a novel class of so-called \emph{decision-hazard-decision} problems, arising in many practical situations, like in stock management. The \emph{time blocks decomposition} scheme is as follows: we use dynamic programming at slow time scale where the slow time scale noises are supposed to be stagewise independent, and we produce slow time scale Bellman functions; then, we use stochastic programming at short time scale, within two consecutive slow time steps, with the final short time scale cost given by the slow time scale Bellman functions, and without assuming stagewise independence for the short time scale noises

Tunable thermopower in a graphene-based topological insulator

Following the recent proposal by Weeks et al., which suggested that indium (or thallium) adatoms deposited on the surface of graphene should turn the latter into a quantum spin Hall (QSH) insulator characterized by a sizeable gap, we perform a systematic study of the transport properties of this system as a function of the density of randomly distributed adatoms. While the samples are, by construction, very disordered, we find that they exhibit an extremely stable QSH phase with no signature of the spatial inhomogeneities of the adatom configuration. We find that a simple rescaling of the spin-orbit coupling parameter allows us to account for the behaviour of the inhomogeneous system using a homogeneous model. This robustness opens the route to a much easier experimental realization of this topological insulator. We additionally find this material to be a very promising candidate for thermopower generation with a target temperature tunable from 1 to 80K and an efficiency ZT close to 1.Comment: 7 pages, 5 figure