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arXiv:1701.00236 · Physical review. B./Physical review. B

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

Sabyasachi Nag, Arti Garg

Published Aug 28, 2017• 53 citations
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Abstract

We study many-body localization (MBL) in an interacting one-dimensional system with a deterministic aperiodic potential. Below the threshold potential $h<{h}_{c}$, the noninteracting system has single particle mobility edges (MEs) at $\ifmmode\pm\else\textpm\fi{}{E}_{c}$ while for $h>{h}_{c}$ all the single particle states are localized. We demonstrate that even in the presence of single particle MEs, interactions do not always delocalize 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 $h<{h}_{c}$, many-body states at the top (with $E>{E}_{2}$) and the bottom of the spectrum (with $E<{E}_{1}$) remain localized, though states in the middle of the spectrum are delocalized. Variance of entanglement entropy (EE) also diverges at ${E}_{1,2}$, indicating a transition from MBL to a delocalized regime, though transitions from volume to area law scaling for EE and from thermal to nonthermal behavior occur inside the MBL regime much below ${E}_{1}$ and above ${E}_{2}$.

DOI
10.1103/physrevb.96.060203

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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 and 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 [$t(r)\ensuremath{\sim}{t}_{0}/{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}<2$], 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 $1<\ensuremath{\alpha}<2$ delocalizes the system partially, with almost all the states extended for $\ensuremath{\alpha}\ensuremath{\le}1$. Even in a system which has single-particle mobility edges in the noninteracting limit, turning on long-range interactions does not cause delocalization.
We investigate many-body localization in the presence of a single-particle mobility edge. By considering an interacting deterministic model with an incommensurate potential in one dimension we find that the single-particle mobility edge in the noninteracting system leads to a many-body mobility edge in the corresponding interacting system for certain parameter regimes. Using exact diagonalization, we probe the mobility edge via energy resolved entanglement entropy (EE) and study the energy resolved applicability (or failure) of the eigenstate thermalization hypothesis (ETH). Our numerical results indicate that the transition separating area and volume law scaling of the EE does not coincide with the nonthermal to thermal transition. Consequently, there exists an extended nonergodic phase for an intermediate energy window where the many-body eigenstates violate the ETH while manifesting volume law EE scaling. We also establish that the model possesses an infinite temperature many-body localization transition despite the existence of a single-particle mobility edge. We propose a practical scheme to test our predictions in atomic optical lattice experiments which can directly probe the effects of the mobility edge.
In one dimension, noninteracting particles can undergo a localization-delocalization transition in a quasiperiodic potential. Recent studies have suggested that this transition transforms into a many-body localization (MBL) transition upon the introduction of interactions. It has also been shown that mobility edges can appear in the single particle spectrum for certain types of quasiperiodic potentials. Here, we investigate the effect of interactions in two models with such mobility edges. Employing the technique of exact diagonalization for finite-sized systems, we calculate the level spacing distribution, time evolution of entanglement entropy, optical conductivity, and return probability to detect MBL. We find that MBL does indeed occur in one of the two models we study, but the entanglement appears to grow faster than logarithmically with time unlike in other MBL systems.
Noninteracting fermions in one dimension can undergo a localization-delocalization transition in the presence of a quasiperiodic potential as a function of that potential. In the presence of interactions, this transition transforms into a many-body localization (MBL) transition. Recent studies have suggested that this type of transition can also occur in models with quasiperiodic potentials that possess single-particle mobility edges. Two such models were studied by Modak and Mukerjee [Phys. Rev. Lett. 115, 230401 (2015)] but only one was found to exhibit an MBL transition in the presence of interactions while the other one did not. In this work we investigate the occurrence of MBL in the presence of weak interactions in five different models with single-particle mobility edges in one dimension with a view to obtaining a criterion for the same. We find that not all such models undergo a thermal-MBL phase transition in the presence of weak interactions. We propose a criterion to determine whether MBL is likely to occur in the presence of interaction based only on the properties of the noninteracting models. The relevant quantity $\ensuremath{\epsilon}$ is a measure of how localized the localized states are relative to how delocalized the delocalized states are in the noninteracting model. We also study various other features of the noninteracting models such as the divergence of the localization length at the mobility edge and the presence or absence of ``ergodicity'' and localization in their many-body eigenstates. However, we find that these features cannot be used to predict the occurrence of MBL upon the introduction of weak interactions.
We review the physics of many-body localization in models with incommensurate potentials. In particular, we consider one-dimensional quasiperiodic models with single-particle mobility edges. A conventional perspective suggests that delocalized states act as a thermalizing bath for the localized states in the presence of of interactions. However, contrary to this intuition there is evidence that such systems can display non-ergodicity. This is in part due to the fact that the delocalized states do not have any kind of protection due to symmetry or topology and are thus susceptible to localization. A study of such incommensurate models, in the non-interacting limit, shows that they admit extended, partially extended, and fully localized many-body states. Non-interacting incommensurate models cannot thermalize dynamically and remain localized upon the introduction of interactions. In particular, for a certain range of energy, the system can host a non-ergodic extended (i.e. metallic) phase in which the energy eigenstates violate the eigenstate thermalization hypothesis (ETH) but the entanglement entropy obeys volume-law scaling. The level statistics and entanglement growth also indicate the lack of ergodicity in these models. The phenomenon of localization and non-ergodicity in a system with interactions despite the presence of single-particle delocalized states is closely related to the so-called many-body proximity effect and can also be observed in models with disorder coupled to systems with delocalized degrees of freedom. Many-body localization in systems with incommensurate potentials (without single-particle mobility edges) have been realized experimentally, and we show how this can be modified to study the the effects of such mobility edges. Demonstrating the failure of thermalization in the presence of a single-particle mobility edge in the thermodynamic limit would indicate a more robust violation of the ETH.
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