We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of long-range interactions or long-range hopping. Based on perturbative arguments there is a common belief that MBL can exist only in systems with short-range interactions and short-range hopping. We analyze effects of power-law interactions and power-law hopping, separately, on a system which has all the single particle states localized in the absence of interactions. Since delocalization is driven by proliferation of resonances in the Fock space, we mapped this model to an effective Anderson model on a complex graph in the Fock space, and calculated the probability distribution of the number of resonances up to third order. Though the most-probable value of the number of resonances diverge for the system with long-range hopping (t(r) ∼ t0/r α with α < 2), there is no enhancement of the number of resonances as the range of power-law interactions increases. This indicates that the long-range hopping delocalizes the many-body localized system but in contrast to this, there is no signature of delocalization in the presence of long-range interactions. We further provide support in favor of this analysis based on dynamics of the system after a quench starting from a charge density wave ordered state, level spacing statistics, return probability, participation ratio and Shannon entropy in the Fock space. We demonstrate that MBL persists in the presence of long-range interactions though long-range hopping with 1 < α < 2 delocalizes the system partially, with almost all the states extended for α ≤ 1. Even in a system which has single-particle mobility edges in the non-interacting limit, turning on long-range interactions does not cause delocalization.

We study many-body localization (MBL) in an interacting one-dimensional system with a deterministic aperiodic potential. Below the threshold potential h < hc, the non-interacting system has single particle mobility edges (MEs) at ±Ec while for h > hc all the single particle states are localized. We demonstrate that even in the presence of single particle MEs, interactions do not always delocalise the system and the interacting system can have MBL. Our numerical calculation of energy level spacing statistics, participation ratio in the Fock space and Shannon entropy shows that for some regime of particle densities, even for h < hc many-body states at the top (with E > E2) and the bottom of the spectrum (with E < E1) remain localized though states in the middle of the spectrum are delocalized. Variance of entanglement entropy (EE) also diverges at E1,2 indicating a transition from MBL to delocalized regime though transition from volume to area law scaling for EE and from thermal to non-thermal behavior occur inside the MBL regime much below E1 and above E2. Interplay of disorder and interactions in quantum systems is a topic of great interest in condensed matter physics. In a non-interacting system with random disorder, any small amount of disorder is sufficient to localize all the single particle states in one and two dimensions [1][2][3], except in systems where back scattering is suppressed e.g. in graphene [4,5], while in three dimensions (3-d) there occurs a single particle mobility edge (ME) leading to a metal-Anderson Insulator transition. The question of immense interest, that has remained unanswered for decades, is what happens to Anderson localization when both disorder and interactions are present in a system. Recently based on perturbative treatment of interactions for the case where all the single particle states are localised, it has been established that Anderson localization can survive interactions and disordered many-body eigenstates can be localized resulting in a many-body localized (MBL) phase, provided that interactions are sufficiently weak [6]. The question we want to answer in this work is what happens in the presence of interactions when the non-interacting system has single particle MEs? Conventional wisdom says that in the presence of interactions, localised states will get hybridised with the extended states resulting in delocalization. In this work based on exact diagonalisation (ED) study of an interacting model of spin-less fermions in the presence of a deterministic aperiodic potential, where the non-interacting system has MEs, we demonstrate that for some parameter regimes, many-body states at the top and the bottom of the spectrum remain localised even in the presence of interactions.The MBL phase and the MBL transition are unique for several reasons and challenge the basic foundations of quantum statistical physics [7,8]. In the MBL phase the system explores only an exponentially small fraction of the configuration space and local observables do not thermalize leading to violati...

We analyze many-body localization (MBL) to delocalization transition in Sherrington-Kirkpatrick (SK) model of Ising spin glass (SG) in the presence of a transverse field Γ. Based on energy resolved analysis, which is of relevance for a closed quantum system, we show that the quantum SK model has many-body mobility edges separating MBL phase which is non-ergodic and non-thermal from the delocalized phase which is ergodic and thermal. The range of the delocalized regime increases with increase in the strength of Γ and eventually for Γ larger than ΓCP the entire many-body spectrum is delocalized. We show that the Renyi entropy is almost independent of the system size in the MBL phase, hinting towards an area law in this infinite range model while the delocalized phase shows volume law scaling of Renyi entropy. We further obtain spin glass transition curve in energy density ǫ-Γ plane from the collapse of eigenstate spin susceptibility. We demonstrate that in most of the parameter regime SG transition occurs close to the MBL transition indicating that the SG phase is non-ergodic and non-thermal while the paramagnetic phase is delocalized and thermal.

We analyze the nature of the single particle states, away from the Dirac point, in the presence of long-range charge impurities in a tight-binding model for electrons on a two-dimensional honeycomb lattice which is of direct relevance for graphene. For a disorder potential V ( r) = V0 exp(−| r − rimp| 2 /ξ 2 ), we demonstrate that not only the Dirac state but all the single particle states remain extended for weak enough disorder. Based on our numerical calculations of inverse participation ratio, dc conductivity, diffusion coefficient and the localization length from time evolution dynamics of the wave packet, we show that the threshold V th required to localize a single particle state of energy E( k) is minimum for the states near the band edge and is maximum for states near the band center, implying a mobility edge starting from the band edge for weak disorder and moving towards the band center as the disorder strength increases. This can be explained in terms of the low energy Hamiltonian at any point k which has the same nature as that at the Dirac point. From the nature of the eigenfunctions it follows that a weak long range impurity will cause weak anti localization effects, which can be suppressed, giving localization if the strength of impurities is sufficiently large to cause inter-valley scattering. The inter valley spacing 2| k| increases as one moves in from the band edge towards the band center, which is reflected in the behavior of V th and the mobility edge.