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arXiv:1707.03749 · Physical review. B./Physical review. B

Nonequilibrium phase diagram of a one-dimensional quasiperiodic system with a single-particle mobility edge

Archak Purkayastha, Abhishek Dhar, Manas Kulkarni

Published Nov 27, 2017• 58 citations
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Abstract

We investigate and map out the nonequilibrium phase diagram of a generalization of the well known Aubry-Andr\'e-Harper (AAH) model. This generalized AAH (GAAH) model is known to have a single-particle mobility edge which also has an additional self-dual property akin to that of the critical point of the AAH model. By calculating the population imbalance, we get hints of a rich phase diagram. We also find a fascinating connection between single particle wave functions near the mobility edge of the GAAH model and the wave functions of the critical AAH model. By placing this model far from equilibrium with the aid of two baths, we investigate the open system transport via system size scaling of nonequilibrium steady state (NESS) current, calculated by fully exact nonequilibrium Green's function (NEGF) formalism. The critical point of the AAH model now generalizes to a `critical' line separating regions of ballistic and localized transport. Like the critical point of the AAH model, current scales subdiffusively with system size on the `critical' line ($I\ensuremath{\sim}{N}^{\ensuremath{-}2\ifmmode\pm\else\textpm\fi{}0.1}$). However, remarkably, the scaling exponent on this line is distinctly different from that obtained for the critical AAH model (where $I\ensuremath{\sim}{N}^{\ensuremath{-}1.4\ifmmode\pm\else\textpm\fi{}0.05}$). All these results can be understood from the above-mentioned connection between states near the mobility edge of the GAAH model and those of the critical AAH model. A very interesting high temperature nonequilibrium phase diagram of the GAAH model emerges from our calculations.

DOI
10.1103/physrevb.96.180204

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•Oct 6, 2020•0 citations
We investigate and map out the nonequilibrium phase diagram of a generalization of the well known Aubry-Andr'e-Harper (AAH) model. This generalized AAH (GAAH) model is known to have a single-particle mobility edge which also has an additional self-dual property akin to that of the critical point of the AAH model. By calculating the population imbalance, we get hints of a rich phase diagram. We also find a fascinating connection between single particle wave functions near the mobility edge of the GAAH model and the wave functions of the critical AAH model. By placing this model far from equilibrium with the aid of two baths, we investigate the open system transport via system size scaling of nonequilibrium steady state (NESS) current, calculated by fully exact nonequilibrium Green's function (NEGF) formalism. The critical point of the AAH model now generalizes to a critical' line separating regions of ballistic and localized transport. Like the critical point of the AAH model, current scales subdiffusively with system size on the critical' line (I\ensuremath∼N\ensuremath−2\ifmmode±\else\textpm\fi0.1I\ensuremath{\sim}{N}^{\ensuremath{-}2\ifmmode\pm\else\textpm\fi{}0.1}I\ensuremath∼N\ensuremath−2\ifmmode±\else\textpm\fi0.1). However, remarkably, the scaling exponent on this line is distinctly different from that obtained for the critical AAH model (where I\ensuremath∼N\ensuremath−1.4\ifmmode±\else\textpm\fi0.05I\ensuremath{\sim}{N}^{\ensuremath{-}1.4\ifmmode\pm\else\textpm\fi{}0.05}I\ensuremath∼N\ensuremath−1.4\ifmmode±\else\textpm\fi0.05). All these results can be understood from the above-mentioned connection between states near the mobility edge of the GAAH model and those of the critical AAH model. A very interesting high temperature nonequilibrium phase diagram of the GAAH model emerges from our calculations.
In this paper, we look at four generalizations of the one-dimensional Aubry-Andr\'e-Harper (AAH) model which possess mobility edges. We map out a phase diagram in terms of population imbalance and look at the system size dependence of the steady-state imbalance. We find nonmonotonic behavior of imbalance with the system parameters, which contradicts the idea that the relaxation of an initial imbalance is fixed only by the ratio of the number of extended states to the number of localized states. We propose that there exist dimensionless parameters, which depend on the fraction of single-particle localized states, single-particle extended states, and the mean participation ratio of these states. These ingredients fully control the imbalance in the long time limit and we present numerical evidence of this claim. Among the four models considered, three of them have interesting duality relations and their locations of mobility edges are known. One of the models (next-nearest-neighbor hopping) has no known duality but a mobility edge exists and the model has been experimentally realized. Our findings are an important step forward to understanding nonequilibrium phenomena in a family of interesting models with incommensurate potentials.
Quasiperiodic systems host exotic transport regimes that are distinct from those found in periodic or disordered lattices. In this work, we study quantum transport in the Aubry-André-Harper lattice in a two-terminal setup coupled to zero-temperature reservoirs, where the conductance is evaluated via the nonequilibrium Green's function method. In the extended phase, we uncover a universal subdiffusive transport when the bath chemical potential aligns with the band edges. Specifically, the typical conductance displays a scaling of Gtyp∼L−2\mathcal{G}_{\text{typ}}\sim L^{-2}Gtyp​∼L−2 with system size LLL. We attribute this behavior to the emergence of an exceptional point (Jordan normal form) in the transfer matrix in the thermodynamic limit. In the localized phase, the conductance shows exponential decay governed by the Lyapunov exponent. Intriguingly, in the critical phase, we identify pronounced log-periodic oscillations of the conductance as a function of system size, arising from the discrete scale invariance inherent to the singular-continuous spectrum. We further extend our analysis to the generalized Aubry-André-Harper model and provide numerical evidence suggesting that the exact mobility edge resides within a finite spectral gap. This results in a counter-intuitive exponential suppression of conductance precisely at the mobility edge. Our work highlights the distinct transport behaviors in quasiperiodic systems and elucidates how they are rigorously dictated by the underlying local spectral structure.
We develop a theoretical scheme to perform a readout of the properties of a quasiperiodic system by coupling it to one or two qubits. We show that the decoherence dynamics of a single qubit coupled via a pure dephasing type term to a one-dimensional quasiperiodic system with a potential given by the Andr\'e-Aubry-Harper (AAH) model and its generalized versions (GAAH model) is sensitive to the nature of the single-particle eigenstates (SPEs). More specifically, we can use the non-Markovianity of the qubit dynamics as quantified by the backflow of information to clearly distinguish the localized, delocalized, and mixed regimes with a mobility edge of the AAH and GAAH models and evidence the transition between them. By attaching two qubits at distinct sites of the system, we demonstrate that the transport property of the quasiperiodic system is encoded in the scaling of the threshold time to develop correlations between the qubits with the distance between the qubits. This scaling can also be used to distinguish and infer different regimes of transport such as ballistic, diffusive, and no transport engendered by SPEs that are delocalized, critical, and localized respectively. In addition, the localization length of the SPEs can also be gleaned from the exponential decay of correlations at long times as a function of distance between qubits. When there is a mobility edge allowing the coexistence of different kinds of SPEs in the spectrum, such as the coexistence of localized and delocalized states in the GAAH models, we find that the transport behavior and the scaling of the threshold time with qubit separation are governed by the fastest spreading 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.
The phase diagram of the one-dimensional hexagonal Harper (HH) model reveals the presence of two metallic phases and one insulating phase, separated by critical lines and a bicritical point. In our work, we investigate transport in the different phases by considering both the isolated and open system scenario. For the case of the isolated system, we study the single-particle dynamics at the bicritical point and along the critical lines. We find that the single-particle wave packet dynamics is superdiffusive in the critical regions with the transport at the bicritical point faster than that along the critical lines. In addition, we study domain wall (DW) dynamics via unconventional multiparticle states in the two metallic phases. The DW state is constructed by partially filling half of the chain while the other half is kept empty; the dynamics of this state reveals distinct behavior in the deep metallic regime of the two metallic phases. Interestingly, the distinct behavior is absent if one instead considers a fully filled half chain. For the open system scenario, we study transport in the nonequilibrium steady state (NESS). We observe that in the critical regions, transport is subdiffusive in nature. Moreover, we find that the transport scaling exponents for the open system scenario are the same at the bicritical point and along the critical lines, unlike the closed system case. In addition, we observe an even-odd size effect on the NESS density and current and on the optimal system-bath coupling parameter (corresponding to the maximum current) in the deep metallic regime of the HH model.
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.