We establish rigorously that transport is slower than diffusive for a class of disordered one-dimensional Hamiltonian chains. This is done by deriving quantitative bounds on the variance in equilibrium of the energy or particle current, as a function of time. The slow transport stems from the presence of rare insulating regions (Griffiths regions). In many-body disordered quantum chains, they correspond to regions of anomalously high disorder, where the system is in a localized phase. In contrast, we deal with quantum and classical disordered chains where the interactions, usually referred to as anharmonic couplings in classical systems, are sparse. The system hosts thus rare regions with no interactions and, since the chain is Anderson localized in the absence of interactions, the non-interacting rare regions are insulating. Part of the mathematical interest of our model is that it is one of the few non-integrable models where the diffusion constant can be rigorously proven not to be infinite.
We establish rigorously that transport is slower than diffusive for a class of disordered one-dimensional Hamiltonian chains. This is done by deriving quantitative bounds on the variance in equilibrium of the energy or particle current, as a function of time. The slow transport stems from the presence of rare insulating regions (Griffiths regions). In many-body disordered quantum chains, they correspond to regions of anomalously high disorder, where the system is in a localized phase. In contrast, we deal with quantum and classical disordered chains where the interactions, usually referred to as anharmonic couplings in classical systems, are sparse. The system hosts thus rare regions with no interactions and, since the chain is Anderson localized in the absence of interactions, the non-interacting rare regions are insulating. Part of the mathematical interest of our model is that it is one of the few non-integrable models where the diffusion constant can be rigorously proven not to be infinite.
Abstract We study anomalous transport arising in disordered one-dimensional spin chains, specifically focusing on the subdiffusive transport typically found in a phase preceding the many-body localization transition. Different types of transport can be distinguished by the scaling of the average resistance with system’s length. We address the following question: what is the distribution of resistance over different disorder realizations, and how does it differ between transport types? In particular, an often evoked so-called Griffiths picture, that aims to explain slow transport as being due to rare regions of high disorder, would predict that the diverging resistivity is due to fat power-law tails in the resistance distribution. Studying many-particle systems with and without interactions we do not find any clear signs of fat tails. The data is compatible with distributions that decay faster than any power law required by the fat tails scenario. Among the distributions compatible with the data, a simple additivity argument suggests a Gaussian distribution for a fractional power of the resistance.
We introduce and study a toy model for anomalous transport and Griffiths effects in one-dimensional (1D) quantum disordered isolated systems near the many-body localization (MBL) transition. The model is constituted by a collection of 1D tight-binding chains with on-site random energies, locally coupled to a weak perturbation, corresponding to a random matrix which belongs to the Gaussian orthogonal ensemble (GOE) and mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in the absence of such a coupling the model is localized as expected for the one-dimensional Anderson model, increasing the coupling with the GOE perturbation we find a transition to a conducting phase driven by the proliferation of quantum avalanches, which does not fit the standard paradigm of the Anderson transition. In particular an intermediate Griffiths region emerges, where exponentially distributed insulating segments coexist with a few rare resonances. Typical correlations decay exponentially fast, while average correlations decay as stretched exponentials and diverge with the length of the chain, indicating that the conducting inclusions have a fractal structure and that the localization length is broadly distributed at the critical point. This behavior is consistent with a Kosterlitz-Thouless-like criticality of the MBL transition. Transport and relaxation are dominated by rare resonances and rare strong insulating regions, and show anomalous behaviors strikingly similar to those observed in recent simulations and experiments in the bad metal delocalized phase preceding MBL. In particular, we find subdiffusive transport and power-law decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region. Concomitantly, the ac conductivity vanishes near zero frequency with an anomalous power law.
We study transport in the boundary-driven XX spin chain with onsite disorder and randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. We then present an exactly solvable semiclassical model of conductors and insulators, which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The subdiffusion in these models is a consequence of rare insulating regions, and they therefore exhibit, in a more controlled setup, the physics of which have been conjectured to cause the subdiffusive transport observed in interacting many-body localizable systems. For the quantum model we show that finite-size effects come from the interplay of three characteristic lengths: one associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. The latter grows logarithmically with system size, and we conjecture that this may be the reason why the heavy-tailed distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.
We study transport in a one-dimensional boundary-driven Anderson insulator (the XX spin chain with onsite disorder) with randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. This model is intended to mimic the passage of an excitation through (many-body) insulating regions or ergodic bubbles, therefore providing a toy model for the diffusion-subdiffusion transition observed in the disordered Heisenberg model by \ifmmode \check{Z}\else \v{Z}\fi{}nidari\ifmmode \check{c}\else \v{c}\fi{} et al. [Phys. Rev. Lett. 117, 040601 (2016)]. We also present the exact solution of a semiclassical model of conductors and insulators introduced by Agarwal et al. [Phys. Rev. Lett. 114, 160401 (2015)], which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The critical properties of both models, when passing from diffusion to subdiffusion, are interpreted in terms of ``Griffiths effects.'' We show that the finite-size scaling comes from the interplay of three characteristic lengths: one associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. We conjecture that the latter, which grows logarithmically with system size, may potentially be responsible for the fact that heavy-tailed resistance distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.
Using a numerically exact technique we study spin transport and entanglement growth in the delocalized phase of a disordered spin chain with long-range interactions, decaying as a power law, ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ with distance. For all considered $\ensuremath{\alpha}$'s and disorder strengths we find that the entanglement entropy grows sublinearly, and the underlying transport is subdiffusive. Since rare-blocking regions, which are central to the Griffiths theory of transport in disordered interacting systems, can be easily circumvented by long-range hops across the lattice, they cannot explain the mechanism of slow transport in long-range systems. Specifically, we show that for long-range systems the Griffiths theory predicts diffusive transport, which is inconsistent with our results.
It is widely believed that many-body localization in one dimension is fragile and can be easily destroyed by thermal inclusions; however, there are still many open questions regarding the stability of the localized phase and under what conditions it breaks down. Here I construct models with dilute disorder, which interpolate between translationally invariant and fully random models, in order to study the breakdown of localization. This opens up the possibility to controllably increase the density of thermal regions and examine the breakdown of localization as this density is increased. At strong disorder, the numerical results are consistent with commonly used diagnostics for localization even when the concentration of thermalizing regions is high. At moderate disorder, I present evidence for slow dynamics and subdiffusive transport across a large region of the phase diagram, suggestive of a ``bad metal'' phase. This suggests that dilute disorder may be a useful effective model for studying Griffiths effects in many-body localization, and perhaps also in a wider class of disordered systems.