# Séminaires de l’année 2011

14 décembre à 14h | Nicolas Dupuis (LPTMC) |

Séminaire "Fluides quantiques" du LPTMS | Quanutum criticality of a Bose gas near the Mott transition |

We discuss the quantum criticality of a Bose gas near the Mott transition using a non-perturbative renormalization-group approach to the Bose-Hubbard model (bosons hopping on a lattice with an onsite repulsion). This approach reproduces the phase diagram of the model (in very good quantitative agreement with the numerically exact Quantum Monte Carlo result) and captures the two universality classes of the Mott transition. We show how the universal character of the density-driven Mott transition manifests itself in the pressure P(mu,T), a quantity which can now be measured in cold atomic gases in an optical lattice. Finally, we compare our theoretical results with recent experiments on the thermodynamics of a two-dimensional Bose gas, with or without an optical lattice. |

13 décembre à 11h | Vahagn Poghosyan (Université Catholique de Louvain) |

The Discrete Laplacian in the Theory of Exactly Solvable Stochastic Lattice Models | |

Following the recent proposal made by [J. Bouttier et al., Phys. Rev. E 76, 041140,(2007)], we study analytically the mobility properties of a single vacancy in the close-packed dimer model on the square lattice. Using the spanning web representation, we find determinantal expressions for various observable quantities. In the limiting case of large lattices, they can be reduced to the calculation of Toeplitz determinants and minors thereof. The probability for the vacancy to be strictly jammed and other diffusion characteristics are computed exactly. Single site height probabilities in the Abelian sandpile model, and the corresponding mean height , are directly related to the probability P_{ret} that a loop-erased random walk passes through a nearest neighbour of the starting site (return probability). The exact values of these quantities on the square lattice have been conjectured, in particular = 25/8 and P_{ret} = 5/16. We provide a rigorous proof of this conjecture by using a local monomer–dimer formulation of these questions. The detailed calculations of the asymptotics of two-site corrélation functions for height variables in the two-dimensional Abelian sandpile model is presented. Combinatorial methods for the enumeration of spanning trees are used. We extend the well-known result for the corrélation \sigma_{1,1} of minimal height h_1=h_2=1 to \sigma_{1,h}=P_{1,h}-P_1P_h for height values h=2,3,4. These results confirm the dominant logarithmic behaviour \sigma_{1,h} \simeq (c_h\log r + d_h)/r^4 + {\cal O}(r^{-5}) for large r, predicted by logarithmic conformal field theory based on field identifications obtained previously. We obtain, from our lattice calculations, the explicit values for the coefficients c_h and d_h. |

9 décembre à 14h30 | Vladimir Gritsev (University of Fribourg) |

Séminaire exceptionnel | Dynamical deviation from integrability |

When classical many-body integrable system is weakly perturbed by the non-integrable perturbation, its properties remain similar to the unperturbed one, the statement known as a Kolmogorov-Arnold-Moser theorem. What happens when quantum integrable many-body system is perturbed? Recently these questions have been raised by several experiments with ultracold atomic systems. I am going to discuss our approach to deviation from integrability in quantum many-body systems. Our understanding is based on a finding that behind every quantum integrable many-body system there is an effective hidden classical one which for small deviation from integrability determines the behavior of a quantum system. |

6 décembre à 11h | Arul Lakshminarayan (IIT Madras) |

On the spectra of partial transposed density matrices | |

The spectrum of the partial transposes of random density matrices from the measure induced by the Haar measure on pure states is studied. It is shown how a simple random matrix models predicts an observed transition from dominantly Negative Partial Transpose (NPT) phase to a dominantly Positive Partial Transpose (PPT) one. We use the theory of extreme eigenvalues of random matrices, and the Tracy-Widom distribution in particular, to find the fraction of NPT states in the critical cases. A model of coupled rotors is studied as a dynamical example. Substantial deviations from random states are found in the critical cases and form a very stringent test of the Bohigas-Giannoni-Schmit conjecture that random matrices describe fluctuation properties of quantized chaotic systems. |

29 novembre à 11h | Luca Peliti (Universita di Napoli) |

The fate of beneficial mutations in a range-expansion wave | |

Recent theoretical and experimental studies have shown that range expansions can have severe consequences for the gene pool of the expanding population. Due to strongly enhanced genetic drift at the advancing frontier, neutral and weakly deleterious mutations can reach large frequencies in the newly colonized regions, as if they were surfing the front of the range expansion. These findings raise the question under which conditions beneficial mutations may be able to prevail at shifting range margins, thereby promoting adaptation. Here, we study, by means of individual-based simulations, wave-like range expansions in linear habitats, and show that the surfing probability of non-neutral mutations becomes substantial only when the mutation appears further than a certain distance ahead of the bulk of the wave. This characteristic distance is proportional to the inverse fitness of the mutant type, and only weakly (logarithmically) dependent on the carrying capacity. Moreover, we show that the surfing of beneficial mutations is to an excellent approximation captured by a branching process within a moving field of growth rates. In order to quantify the rate of adaptation, our results are finally used to predict, for a given mutation rate, how frequently substitutions by beneficial mutations occur at invasion fronts. Importantly, we find that small amounts of genetic drift increase the fixation rate of beneficial mutations at the advancing front, and thus should be important for adaptation during species invasions. |

28 novembre à 16h | Francis Corson (The Ro |