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arXiv:1703.05844 · Physical review. E

Fractality in nonequilibrium steady states of quasiperiodic systems

Vipin Kerala Varma, Clélia de Mulatier, Marko Žnidarič

Published Sep 19, 2017• 58 citations
arXivHTMLPDF

Abstract

We investigate the nonequilibrium response of quasiperiodic systems to boundary driving. In particular, we focus on the Aubry-André-Harper model at its metal-insulator transition and the diagonal Fibonacci model. We find that opening the system at the boundaries provides a viable experimental technique to probe its underlying fractality, which is reflected in the fractal spatial dependence of simple observables (such as magnetization) in the nonequilibrium steady state. We also find that the dynamics in the nonequilibrium steady state depends on the length of the chain chosen: generic length chains harbour qualitatively slower transport (different scaling exponent) than Fibonacci length chains, which is in turn slower than in the closed system. We conjecture that such fractal nonequilibrium steady states should arise in generic driven critical systems that have fractal properties.

DOI
10.1103/physreve.96.032130

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We investigate the nonequilibrium response of quasiperiodic systems to boundary driving. In particular, we focus on the Aubry-André-Harper model at its metal-insulator transition and the diagonal Fibonacci model. We find that opening the system at the boundaries provides a viable experimental technique to probe its underlying fractality, which is reflected in the fractal spatial dependence of simple observables (such as magnetization) in the nonequilibrium steady state. We also find that the dynamics in the nonequilibrium steady state depends on the length of the chain chosen: generic length chains harbour qualitatively slower transport (different scaling exponent) than Fibonacci length chains, which is in turn slower than in the closed system. We conjecture that such fractal nonequilibrium steady states should arise in generic driven critical systems that have fractal properties.
We study dephasing enhanced transport in boundary-driven quasiperiodic systems. Specifically, we consider dephasing modeled by current-preserving Lindblad dissipators acting on the noninteracting Aubry-Andr\'e-Harper and Fibonacci bulk systems. The former is known to undergo a critical localization transition with a suppression of ballistic transport above a critical value of the potential. At the critical point, the presence of nonergodic extended states yields anomalous subdiffusion. The Fibonacci model, on the other hand, yields anomalous transport with a continuously varying exponent depending on the potential strength. By computing the covariance matrix in the nonequilibrium steady state, we show that sufficiently strong dephasing always renders the transport diffusive. The interplay between dephasing and quasiperiodicity gives rise to a maximum of the diffusion coefficient for finite dephasing, which suggests that the combination of quasiperiodic geometries and dephasing can be used to control noise enhanced transport.
One dimensional quasiperiodic systems have been a vibrant topic of theorical research for physicists and mathematicians alike, owing to their highly non-trivial localization properties and the fractal nature of their energy spectrum.The most paradigmatic of such systems, the Aubry-André-Harper (AAH) model, is well known for suffering a localization transition when the potential strength is increased, in which its eigenstates go from extended to localized, similarly to Anderson phase transition which occurs in disordered three-dimensional systems.The localization of the eigenstates has dramatic effects in the transport regime of the system.In this dissertation, we used numerical simulations to study the transport properties of boundary-driven fermionic quantum chains subject to two choices for the on-site potential: the aforementioned AAH model and the Fibonacci model.The transport regime is classified via the scaling of the particle current with the system size in the non-equilibrium steady state (NESS), which we calculate numerically.Our main focus was the interplay between the quasiperiodic potential and dephasing noise.We showed that even though dephasing always leads to diffusive behaviour, the transport properties of the system are still visibly affected by localization properties of the eigenstates when the coupling to dephasing is small.This influence is reflected in the scaling of the diffusion constant with the coupling.In particular, we showed that, in the localized phase of the AAH model and in the subdifusive phase of the Fibonacci model, dephasing noise may actually lead to an enhanced transport, that is, a larger value of the current compared to the zero dephasing case.
We systematically investigate the entanglement and information dynamics of quasiperiodic systems across their extended, critical, and localized phases, aiming to identify dynamical signatures that can reveal the multifractal spatial structure of critical states and distinguish critical phases from the extended and localized regimes. Focusing on the generalized Aubry-André-Harper model, we complement the half-chain entanglement entropy with the spatially resolved subsystem information capacity (SIC) and demonstrate that critical states exhibit pronounced spatial heterogeneity absent in the extended and localized phases. In the steady state, the SIC reveals a stepwise ramp as a function of subsystem size, reflecting an underlying fragmentation of the chain into weakly connected subregions. Dynamically, information initially localized within such a subregion can undergo coherent long-lived oscillations, dubbed subregion echoes, whose period scales with the subregion length, in quantitative agreement with a quasiparticle picture of confined quasiparticle reflections. We trace this internal fragmentation to the incommensurately distributed zeros (IDZs) in the off-diagonal hopping terms of the Hamiltonian. To establish the generality of the SIC as a diagnostic tool, we further apply it to a mobility-edge phase with coexisting extended and localized states and to a critical phase that does not originate from IDZ fragmentation, and show that the SIC can cleanly distinguish these scenarios through their distinct steady-state profiles, initial-site sensitivities, and the presence or absence of subregion echoes. Our results demonstrate that the SIC serves as a powerful real-space probe for diagnosing critical phases and uncovering the bottlenecked connectivity that underlies the multifractal structure of critical states.
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.
Driven many-body quantum systems where some parameter in the Hamiltonian is varied quasiperiodically in time may exhibit nonequilibrium steady states that are qualitatively different from their periodically driven counterparts. Here we consider a prototypical integrable spin system, the spin-1/21/21/2 transverse field Ising model in one dimension, in a pulsed magnetic field. The time dependence of the field is taken to be quasiperiodic by choosing the pulses to be of two types that alternate according to a Fibonacci sequence. We show that a steady state emerges after an exponentially long time when local properties (or equivalently, reduced density matrices of subsystems with size much smaller than the full system) are considered. We use the temporal evolution of certain coarse-grained quantities in momentum space to understand this nonequilibrium steady state in more detail and show that unlike the previously known cases, this steady state is neither described by a periodic generalized Gibbs ensemble nor by an infinite temperature ensemble. Finally, we study a toy problem with a single two-level system driven by a Fibonacci sequence; this problem shows how sensitive the nature of the final steady state is to the different parameters.
We study the effect of quasiperiodic perturbations on one-dimensional all-bands-flat lattice models. Such networks can be diagonalized by a finite sequence of local unitary transformations parameterized by angles ${\ensuremath{\theta}}_{i}$. Without loss of generality, we focus on the case of two bands with bandgap $\mathrm{\ensuremath{\Delta}}$. Weak perturbations lead to an effective Hamiltonian with both on- and off-diagonal quasiperiodic terms that depend on ${\ensuremath{\theta}}_{i}$. For some angle values, the effective model coincides with the extended Harper model. By varying the parameters of the quasiperiodic potentials, we observe localized insulating states and an entire parameter range hosting critical states with subdiffusive transport. For finite quasiperiodic potential strength, the critical-to-insulating transition becomes energy dependent with what we term fractality edges separating localized from critical states.