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arXiv:1811.01912 · SciPost Physics

Many-body localization in a quasiperiodic Fibonacci chain

Nicolas Macé, Nicolas Laflorencie, Fabien Alet

Published Apr 29, 2019• 64 citations
arXivHTMLPDF

Abstract

We study the many-body localization (MBL) properties of a chain of interacting fermions subject to a quasiperiodic potential such that the non-interacting chain is always delocalized and displays multifractality. Contrary to naive expectations, adding interactions in this systems does not enhance delocalization, and a MBL transition is observed. Due to the local properties of the quasiperiodic potential, the MBL phase presents specific features, such as additional peaks in the density distribution. We furthermore investigate the fate of multifractality in the ergodic phase for low potential values. Our analysis is based on exact numerical studies of eigenstates and dynamical properties after a quench.

DOI
10.21468/scipostphys.6.4.050

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We study the many-body localization (MBL) properties of a chain of interacting fermions subject to a quasiperiodic potential such that the non-interacting chain is always delocalized and displays multifractality. Contrary to naive expectations, adding interactions in this systems does not enhance delocalization, and a MBL transition is observed. Due to the local properties of the quasiperiodic potential, the MBL phase presents specific features, such as additional peaks in the density distribution. We furthermore investigate the fate of multifractality in the ergodic phase for low potential values. Our analysis is based on exact numerical studies of eigenstates and dynamical properties after a quench.
In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in Hilbert space, the support of the eigenstates. In this Letter, using exact diagonalization techniques we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases for up to L=24 spin-1/2 particles (i.e., Hilbert space of size N≃2.7×10^{6}). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions.
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 investigate the localization properties of a spin chain with an antiferromagnetic nearest-neighbor coupling, subject to an external quasiperiodic on-site magnetic field. The quasiperiodic modulation interpolates between two paradigmatic models, namely the Aubry-Andr\'e and the Fibonacci models. We find that stronger many-body interactions extend the ergodic phase in the former, whereas they shrink it in the latter. Furthermore, the many-body localization transition points at the two limits of the interpolation appear to be continuously connected along the deformation of the quasiperiodic modulation. As a result, the position of the many-body localization transition depends on the interaction strength for an intermediate degree of deformation. Moreover, in the region of parameter space where the single-particle spectrum contains both localized and extended states, many-body interactions induce an anomalous effect: weak interactions localize the system, whereas stronger interactions enhance ergodicity. We map the model's localization phase diagram using the decay of the quenched spin imbalance in relatively long chains. This is accomplished employing a time-dependent variational approach applied to a matrix product state decomposition of the many-body state. Our model serves as a rich playground for testing many-body localization under tunable potentials.
We investigate the localization properties of a spin chain with an antiferromagnetic nearest-neighbor coupling, subject to an external quasiperiodic on-site magnetic field. The quasiperiodic modulation interpolates between two paradigmatic models, namely the Aubry-André and the Fibonacci models. We find that stronger many- body interactions extend the ergodic phase in the former, whereas they shrink it in the latter. Furthermore, the many-body localization transition points at the two limits of the interpolation appear to be continuously connected along the deformation of the quasiperiodic modulation. As a result, the position of the many-body localization transition depends on the interaction strength for an intermediate degree of deformation. Moreover, in the region of parameter space where the single-particle spectrum contains both localized and extended states, many-body interactions induce an anomalous effect: weak interactions localize the system, whereas stronger interactions enhance ergodicity. We map the model’s localization phase diagram using the decay of the quenched spin imbalance in relatively long chains. This is accomplished employing a time-dependent variational approach applied to a matrix product state decomposition of the many-body state. Our model serves as a rich playground for testing many-body localization under tunable potential
We study a quasiperiodic model in one dimension, namely the extended Aubry-Andr\'e-Harper (EAAH) chain, that realizes a critical phase comprising entirely single-particle critical states in the noninteracting limit. In the presence of short-range interactions, the noninteracting critical phase transforms to a many-body critical (MBC) phase, separated by lines of MBC-ergodic, MBC-many-body localized (MBL) and ergodic-MBL phase transitions that meet at a triple point. We elucidate the unusual characteristics of the MBC phase compared to the ergodic and MBL phases through the localization properties of the excitations in real space and Fock space (FS), and the eigenstate inverse participation ratio (IPR). We show that the MBC phase, like the MBL phase, is well described by a multifractal scaling of the IPR and a linear finite-size scaling ansatz near the transition to the ergodic and MBL phases. However, the MBC phase, at the same time, exhibits delocalization of all single-particle excitations and a system-size-dependent Fock-space localization length, analogous to the ergodic phase. Remarkably, we find evidence of unusual Widom lines on the phase diagram in the form of lines of pronounced peaks or dips in the FS localization properties inside the MBC and MBL phases. These Widom lines either emerge as a continuation of the precursor phase transition line terminating at the triple point, or originate from a phase boundary.
Localization in one-dimensional interacting systems can be caused by disorder potentials or non-Hermiticity. The former phenomenon is the many-body localization (MBL), and the latter is the many-body non-Hermitian skin effect (NHSE). In this work, we numerically investigate the interplay between these two kinds of localization, where the energy-resolved MBL arises from a deterministic quasiperiodic potential in a fermionic chain. We propose a set of eigenstate properties and long-time dynamics that collectively distinguish the two localization mechanisms in the presence of non-Hermiticity. By computing the proposed diagnostics, we show that the thermal states are vulnerable to the many-body NHSE while the MBL states remain resilient up to a strong non-Hermiticity. Finally, we discuss experimental observables that probe the difference between the two localizations in a non-Hermitian quasiperiodic fermionic chain. Our results pave the way toward experimental observations on the interplay of interaction, quasiperiodic potential, and non-Hermiticity.