Many-body mobility edges in a one-dimensional system of interacting fermions

Abstract

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...

Many-body localization in the presence of long-range interactions and long-range hopping

“…We calculate the return probability defined as where |l is the l th eigenstate of the Hamiltonian (1), and n i (t) is the density operator at site i at time t. For a given disorder configuration, we average over all the sites followed by averaging over various independent disorder configurations to get the average value of the autocorrelation function C(t). Return probability is a direct measure of the extent of localized states in the manybody spectrum [17,22]. For a fully localized system, C(t) quickly saturates to a large value (close to one) after the initial rapid decay.…”

“…1 with parameter chosen such that the non interacting system has single particle mobility edges. We further chose V such that the system with nearest neighbour interaction has a fraction of many-body states localized [22]. But as shown in our earlier work [22], in order for the system to show MBL in this parameter regime, the system must be away from half filling.…”

“…In this work, we study a model of spinless fermions in one dimension in the presence of an aperiodic deterministic potential [22,[55][56][57], which can also be viewed as a generalized Aubry-Andre potential [17,58]. The virtue of this model over a fully random disorder potential is that by changing parameters in this model, one can realize a system with all single particle states localized/delocalized as well as the system that has single particle mobility edges.…”

Many-body localization of spinless fermions with attractive interactions in one dimension

“…As a consequence, the low-energy density region at V < 0 maps to the high-energy density at the V > 0 side. Using this argument and by inspection of existing studies of the energy-density versus disorder phase diagram (see, e.g., [3,6,26,44,45]), one can already draw some conclusions on the structure of the phase diagram on the attractive side. The special point of V = 2t is the most studied one, for which there are also full energydensity versus disorder strength phase diagrams available (see, e.g., [26]).…”

Investigating many-body mobility edges in isolated quantum systems

“…In constrast, localization renders uncorrelated energy levels whose r α ≈ 0.39. A proper scaling analysis [15,72,73], shown in Fig. 2(b), suggests a ME, separating ergodic from nonthermal behavior at energy densities ε 0.2.…”