Non-equilibrium transport properties of quasiperiodic models

Abstract

The advances in the manipulation of ultracold atoms trapped by optical lattices has established this set-up as a promising analogue simulator of quantum Hamiltonians. The extended coherence times allow for the monitoring of relaxation processes from non-equilibrium initial configurations, driven exclusively by intrinsic mechanisms. Furthermore, two-terminal transport measurements analogous to solid state physics are now routinely performed. This has opened an era of renewed intense activity towards the topics of non-equilibrium dynamics and transport, and fundamental questions on localisation and thermalisation in isolated many-body systems. Quasiperiodic models, long been known, have recently gained relevance due to their realisation in experiments on the same platform, which fall in this large body of investigations. Quasiperiodic potentials are incommensurate with the underlying periodicity of the lattice, neither periodic nor disordered, yet deterministic. This can lead, even in one-dimension and absence of interactions, to a localisation transition (in the Aubry-Andr{\'e}-Harper or AAH model), an energy-dependent transition or \ov mobility edge'' (generalised AAH or GAAH model), and critical states yielding anomalous diffusion (Fibonacci model). The majority of previous studies have been focused on spin or particle currents, we dedicate most of the thesis to extend the characterisation of their transport properties, with an emphasis on their thermal features. We first explore the capability of quasiperiodic models as working mediums in two-terminal quantum autonomous thermal machines, that convert heat to work through non-equilibrium steady-state currents of microscopic particles. In particular, we show that the mobility edge in the GAAH model can function as an energy filter, and demonstrate large thermoelectric effects, exceeding existing predictions by several orders of magnitude. We further investigate the interplay with dephasing noise from incoherent scattering. Heat and electric currents in the Fibonacci model turn from anomalous to standard diffusive. However, the conductivities exhibit a non-trivial dependence on the dephasing strength, which can be exploited to enhance the performance of the device. These findings open the route to a new class of efficient and versatile quasiperiodic steady-state thermoelectric engines. Quasiperiodic models further provide a testbed to investigate the possibility of many-body localisation (MBL). While the AAH model displays single-particle localisation and signatures of a possible MBL phase have been observed in presence of interactions, the lack of parallel experiments leaves to debate whether the anomalous diffusion survives in the interacting Fibonacci model. We contribute by studying real-time spread of density-density correlations at infinite temperature via dynamical quantum typicality, an approach not previously applied to quasiperiodic systems. Our findings suggest a possible crossover to MBL preceded by a regime of anomalous subdiffusive transport. In the last part of the thesis, we perform a similar numerical study in frequency rather than time domain. Inspired by further experimental results on ultracold atoms, we probe the transport properties of the XXZ model in presence of a integrability-breaking term with initial spin helix states, characterised by a winding magnetisation profile. Within the eigenstate thermalisation hypothesis framework, we evaluate correlation functions on statistical ensembles and on single states using the kernel polynomial method.