Subdiffusion in One-Dimensional Hamiltonian Chains with Sparse Interactions

Abstract

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 manybody 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.

Localisation and Sub-Diffusive Transport in Quantum Spin Chains With Dilute Disorder

“…In the following I shall refer to this regime as 'many-body localised', and to the onset of these slow dynamics as the 'manybody localisation transition'. In particular, there has been a great deal of work studying the transport properties close to the many-body localisation transition, with a large body of evidence pointing towards the presence of a Griffths-like subdiffusive regime and other anomalous transport properties [38][39][40][41][42][43][44][45][46][47], and gaining further insights this region could lead to an improved understanding of the many-body localisation transition itself.…”

Subdiffusion in a one-dimensional Anderson insulator with random dephasing: Finite-size scaling, Griffiths effects, and possible implications for many-body localization

“…Despite widespread interest [13,[17][18][19][20][21][22][23][24][25], the microscopic origin of this subdiffusion is still uncertain. The prevailing theory, first proposed in Ref.…”

Sub-diffusive phases in open clean long-range systems

“…This often occurs in quantum systems at the critical region (or point) separating localized and delocalized phases. For example, this is the case for disordered interacting one-dimensional systems where the sub-diffusion is thought to originate from rare occurrence of ordered regions, also known as the Griffiths effect [8][9][10][11][12][13]. This picture however does not carry over to quasiperiodic systems which lacks disorder.…”