Asymptotic quantum many-body localization from thermal disorder

We consider a quantum lattice system with infinite-dimensional on-site Hilbert space, very similar to the Bose-Hubbard model. We investigate many-body localization in this model, induced by thermal fluctuations rather than disorder in the Hamiltonian. We provide evidence that the Green-Kubo conductivity $\kappa(\beta)$, defined as the time-integrated current autocorrelation function, decays faster than any polynomial in the inverse temperature $\beta$ as $\beta \to 0$. More precisely, we define approximations $\kappa_{\tau}(\beta)$ to $\kappa(\beta)$ by integrating the current-current autocorrelation function up to a large but finite time $\tau$ and we rigorously show that $\beta^{-n}\kappa_{\beta^{-m}}(\beta)$ vanishes as $\beta \to 0$, for any $n,m \in \mathbb{N}$ such that $m-n$ is sufficiently large.Comment: 53 pages, v1-->v2, revised version accepted in Comm.Math.Phys. We added an extensive outline of proofs, a glossary of symbols and more explanations in Section

Glassy dynamics in strongly anharmonic chains of oscillators

We review the mechanism for transport in strongly anharmonic chains of oscillators near the atomic limit where all oscillators are decoupled. In this regime, the motion of most oscillators remains close to integrable, i.e. quasi-periodic, on very long time scales, while a few chaotic spots move very slowly and redistribute the energy across the system. The material acquires several characteristic properties of dynamical glasses: intermittency, jamming and a drastic reduction of the mobility as a function of the thermodynamical parameters. We consider both classical and quantum systems, though with more emphasis on the former, and we discuss also the connections with quenched disordered systems, which display a similar physics to a large extent.Comment: Review paper. Invited submission to the CRAS (special issue on Fourier's legacy). 16 pages, 3 figure

Asymptotic localization of energy in non-disordered oscillator chains

We study two popular one-dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete non-linear Schr\"odinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter $\epsilon >0$, is small. We rigorously establish that the thermal conductivity of the chains has a non-perturbative origin, with respect to the coupling constant $\epsilon$, and we provide strong evidence that it decays faster than any power law in $\epsilon$ as $\epsilon \rightarrow 0$. The weak coupling regime also translates into a high temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature.Comment: v1 -> v2: minor corrections, references added. 33 pages, 1 figure. To appear in Comm. Pure Appl. Mat