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Physical review. B./Physical review. B

Seeking a many-body mobility edge with matrix product states in a quasiperiodic model

Nicholas Pomata, Sriram Ganeshan, Tzu-Chieh Wei

Published Sep 1, 2023• 1 citation
Publisher

Abstract

We investigate the possibility of a many-body mobility edge in the generalized Aubry-Andr\'e (GAA) model with interactions using the Shift-Invert Matrix Product States (SIMPS) algorithm [Phys. Rev. Lett. 118, 017201 (2017)]. The noninteracting GAA model is a one-dimensional quasiperiodic model with a self-duality-induced mobility edge. To search for a many-body mobility edge in the interacting case, we exploit the advantages of SIMPS that it targets many-body states in an energy-resolved fashion and does not require all many-body states to be localized for some to converge. Our analysis indicates that the targeted states in the presence of the single-particle mobility edge match neither ``MBL-like'' (where MBL denotes many-body localization) fully converged localized states nor the fully delocalized case in which SIMPS fails to converge. We benchmark the algorithm's output both for parameters that give fully converged, ``MBL-like'' localized states and for delocalized parameters where SIMPS fails to converge. In the intermediate cases, where the parameters produce a single-particle mobility edge, we find many-body states that develop entropy oscillations as a function of cut position at larger bond dimensions. These oscillations at larger bond dimensions, which are also found in the fully localized benchmark but not the fully delocalized benchmark, occur both at the band edge and center and may indicate convergence to a nonthermal state (either localized or critical).

DOI
10.1103/physrevb.108.094201

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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.

•Jan 1, 2017•21 citations
We investigate the possibility of a many-body mobility edge in the generalized Aubry-Andr'e (GAA) model with interactions using the Shift-Invert Matrix Product States (SIMPS) algorithm [Phys. Rev. Lett. 118, 017201 (2017)]. The noninteracting GAA model is a one-dimensional quasiperiodic model with a self-duality-induced mobility edge. To search for a many-body mobility edge in the interacting case, we exploit the advantages of SIMPS that it targets many-body states in an energy-resolved fashion and does not require all many-body states to be localized for some to converge. Our analysis indicates that the targeted states in the presence of the single-particle mobility edge match neither MBL-like'' (where MBL denotes many-body localization) fully converged localized states nor the fully delocalized case in which SIMPS fails to converge. We benchmark the algorithm's output both for parameters that give fully converged, MBL-like'' localized states and for delocalized parameters where SIMPS fails to converge. In the intermediate cases, where the parameters produce a single-particle mobility edge, we find many-body states that develop entropy oscillations as a function of cut position at larger bond dimensions. These oscillations at larger bond dimensions, which are also found in the fully localized benchmark but not the fully delocalized benchmark, occur both at the band edge and center and may indicate convergence to a nonthermal state (either localized or critical).
We investigate the possibility of a many-body mobility edge in the generalized Aubry-André (GAA) model with interactions using the Shift-Invert Matrix Product States (SIMPS) algorithm [Phys. Rev. Lett. 118, 017201 (2017)]. The non-interacting GAA model is a one-dimensional quasiperiodic model with a self-duality-induced mobility edge. To search for a many-body mobility edge in the interacting case, we exploit the advantages of SIMPS that it targets many-body states in an energy-resolved fashion and does not require all many-body states to be localized for some to converge. Our analysis indicates that the targeted states in the presence of the single-particle mobility edge match neither `MBL-like' fully-converged localized states nor the fully delocalized case where SIMPS fails to converge. We benchmark the algorithm's output both for parameters that give fully converged, `MBL-like' localized states and for delocalized parameters where SIMPS fails to converge. In the intermediate cases, where the parameters produce a single-particle mobility edge, we find many-body states that develop entropy oscillations as a function of cut position at larger bond dimensions. These oscillations at larger bond dimensions, which are also found in the fully-localized benchmark but not the fully-delocalized benchmark, occur both at the band edge and center and may indicate convergence to a non-thermal state (either localized or critical).
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.
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.
We experimentally study many-body localization (MBL) with ultracold atoms in a weak one-dimensional quasiperiodic potential, which in the noninteracting limit exhibits an intermediate phase that is characterized by a mobility edge. We measure the time evolution of an initial charge density wave after a quench and analyze the corresponding relaxation exponents. We find clear signatures of MBL when the corresponding noninteracting model is deep in the localized phase. We also critically compare and contrast our results with those from a tight-binding Aubry-André model, which does not exhibit a single-particle intermediate phase, in order to identify signatures of a potential many-body intermediate phase.
We study many-body localization (MBL) in the quasiperiodic ${t}_{1}\text{\ensuremath{-}}{t}_{2}$ model, focusing on the role of next-nearest-neighbor (NNN) hopping t2{t}_{2}t2​, which introduces a single-particle mobility edge. The calculated phase diagram can be divided into three distinct regimes, depending on the strength of the short-range interaction UUU. For weak interactions ($U\ensuremath{\ll}{t}_{1}$), this model is always nonthermal. For intermediate interactions ($U\ensuremath{\sim}{t}_{1}$), the thermal-MBL phase transition in this model is qualitatively the same as that of the Aubry-Andre (AA) model, which is consistent with existing experimental observations. For strong interactions $(U\ensuremath{\gg}{t}_{1})$, the NNN hopping produces qualitatively new physics because it breaks down the Hilbert space fragmentation present in the AA model. The NNN hopping is thus irrelevant when the interaction is intermediate but relevant for strong 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.