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arXiv:1702.04349 · Europhysics Letters (EPL)

Transport in quasiperiodic interacting systems: From superdiffusion to subdiffusion

Yevgeny Bar Lev, Dante Marvin Kennes, C. Klöckner, David R. Reichman, Christoph Karrasch

Published Aug 1, 2017• 118 citations
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Abstract

Using a combination of numerically exact and renormalization-group techniques we study the nonequilibrium transport of electrons in a one-dimensional interacting system subject to a quasiperiodic potential. For this purpose we calculate the growth of the mean-square displacement as well as the melting of domain walls. While the system is nonintegrable for all studied parameters, there is no finite region of parameters for which we observe diffusive transport. In particular, our model shows a rich dynamical behavior crossing over from superdiffusion to subdiffusion. We discuss the implications of our results for the general problem of many-body localization, with a particular emphasis on the rare region Griffiths picture of subdiffusion.

DOI
10.1209/0295-5075/119/37003

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Using a combination of numerically exact and renormalization-group techniques we study the nonequilibrium transport of electrons in a one-dimensional interacting system subject to a quasiperiodic potential. For this purpose we calculate the growth of the mean-square displacement as well as the melting of domain walls. While the system is nonintegrable for all studied parameters, there is no finite region of parameters for which we observe diffusive transport. In particular, our model shows a rich dynamical behavior crossing over from superdiffusion to subdiffusion. We discuss the implications of our results for the general problem of many-body localization, with a particular emphasis on the rare region Griffiths picture of subdiffusion.
We study high-temperature magnetization transport in a many-body spin-1/2 chain with on-site quasiperiodic potential governed by the Fibonacci rule. In the absence of interactions it is known that the system is critical with the transport described by a continuously varying dynamical exponent (from ballistic to localized) as a function of the on-site potential strength. Upon introducing weak interactions, we find that an anomalous noninteracting dynamical exponent becomes diffusive for any potential strength. This is borne out by a boundary-driven Lindblad dynamics as well as unitary dynamics, with agreeing diffusion constants. This must be contrasted to a random potential where transport is subdiffusive at such small interactions. Mean-field treatment of the dynamics for small UUU always slows down the noninteracting dynamics to subdiffusion, and is therefore unable to describe diffusion in an interacting quasiperiodic system. Finally, briefly exploring larger interactions we find a regime of interaction-induced subdiffusive dynamics, despite the on-site potential itself having no ``rare regions.''
We theoretically study transport properties in one-dimensional interacting quasiperiodic systems at infinite temperature. We compare and contrast the dynamical transport properties across the many-body localization (MBL) transition in quasiperiodic and random models. Using exact diagonalization we compute the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and the return probability $R(\ensuremath{\tau})$ and study their average low-frequency and long-time power-law behavior, respectively. We show that the low-energy transport dynamics is markedly distinct in both the thermal and MBL phases in quasiperiodic and random models and find that the diffusive and MBL regimes of the quasiperiodic model are more robust than those in the random system. Using the distribution of the dc conductivity, we quantify the contribution of sample-to-sample and state-to-state fluctuations of $\ensuremath{\sigma}(\ensuremath{\omega})$ across the MBL transition. We find that the activated dynamical scaling ansatz works poorly in the quasiperiodic model but holds in the random model with an estimated activation exponent $\ensuremath{\psi}\ensuremath{\approx}0.9$. We argue that near the MBL transition in quasiperiodic systems, critical eigenstates give rise to a subdiffusive crossover regime on finite-size systems.
Recent studies of quantum many-body dynamics uncover many intriguing phenomena of thermalization and localization in the presence of a quasiperiodic potential. In this paper, the authors study nonequilibrium steady states of the one-dimensional interacting Aubry-Andre-Harper model by taking advantage of tensor network-based algorithms. In a weak potential regime, they find a superdiffusive to subdiffusive transition in spin transport within the thermal phase. The correlation structure of the steady states derived from the quantum mutual information shows an unusual oscillation pattern before reaching the localization transition. Our results suggest a rich phase diagram for the model.
Integrable models form pillars of theoretical physics because they allow for full analytical understanding. Despite being rare, many realistic systems can be described by models that are close to integrable. Therefore, an important question is how small perturbations influence the behavior of solvable models. This is particularly true for many-body interacting quantum systems where no general theorems about their stability are known. Here, we show that no such theorem can exist by providing an explicit example of a one-dimensional many-body system in a quasiperiodic potential whose transport properties discontinuously change from localization to diffusion upon switching on interaction. This demonstrates an inherent instability of a possible many-body localization in a quasiperiodic potential at small interactions. We also show how the transport properties can be strongly modified by engineering potential at only a few lattice sites.
We study transport in a one-dimensional boundary-driven Anderson insulator (the XX spin chain with onsite disorder) with randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. This model is intended to mimic the passage of an excitation through (many-body) insulating regions or ergodic bubbles, therefore providing a toy model for the diffusion-subdiffusion transition observed in the disordered Heisenberg model by \ifmmode \check{Z}\else \v{Z}\fi{}nidari\ifmmode \check{c}\else \v{c}\fi{} et al. [Phys. Rev. Lett. 117, 040601 (2016)]. We also present the exact solution of a semiclassical model of conductors and insulators introduced by Agarwal et al. [Phys. Rev. Lett. 114, 160401 (2015)], which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The critical properties of both models, when passing from diffusion to subdiffusion, are interpreted in terms of ``Griffiths effects.'' We show that the finite-size scaling comes from the interplay of three characteristic lengths: one associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. We conjecture that the latter, which grows logarithmically with system size, may potentially be responsible for the fact that heavy-tailed resistance distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.
We study the transport properties and the spectral statistics of a one-dimensional closed quantum system of interacting spinless fermions in a quasiperiodic potential which produces a single-particle mobility edge in the absence of interaction. For such systems, it has been shown that the many-body eigenstates can be of three different kinds: extended and eigenstate thermalization hypothesis (ETH) obeying (thermal), localized and ETH violating (many body localized), and extended and ETH violating (nonergodic extended). Here we investigate the nonergodic extended phase from the point of view of level spacing statistics and charge transport. We calculate the dc conductivity and the low-frequency conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and show that both are consistent with subdiffusive transport. This is contrasted with diffusive transport in the thermal phase and blocked transport in the MBL phase.