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.
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.
Sistemas quasiperiódicos unidimensionais tem sido um extenso objeto de pesquisa, tanto para físicos quanto para matemáticos. Esse interesse é movido, principalmente, pelas interessantes propriedades de localização desses sistemas e a natureza fractal dos espectros de energia. O mais paradigmático desses sistemas é o modelo de Aubry-André-Harper (AAH), que é bem conhecido por apresentar um transição de localização quando a força do potencial é aumentada, em que seus autoestados vão de estendidos para localizados. As propriedades de localização dos autoestados provocam consequências dramáticas no regime de transporte do sistema. Nesta dissertação, utilizamos simulações numéricas para estudar as propriedades de transporte de cadeias fermiônicas acopladas a banhos térmicos nas extremidades, sujeitas a duas escolhas de potenciais quasiperiódicos: o modelo AAH e o modelo de Fibonacci. O regime de transporte é classificado verificando-se a dependência da corrente de partícula com o tamanho do sistema no estado estacionário de não-equilíbrio (NESS), que calculamos numericamente. O trabalho focou-se principalmente na interação entre os potenciais quasiperiódicos e dephasing. Mostramos que apesar da ação do ruído de dephasing sempre induzir um comportamente difusivo, as propriedades de transporte do sistema ainda são visivelmente afetadas pelas propriedades de localização dos autoestados. Essa influência é refletida na dependência da constante de difusão com a constante de acoplamente com os banhos de dephasing. Em particular, mostramos que, na fase localizada do modelo AAH e na fase subdifusiva do modelo de Fibonacci, a ação do ruído de dephasing pode levar a um aumento do transporte, isto é, um valor de corrente maior em comparação com o caso sem dephasing.
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.
In this study, we investigate the localization transition and quantum criticality {in the ground state of the} disordered Aubry-André-Harper (AAH) model, where a quasiperiodic potential is hybridized with a disordered potential. In the clean limit, the AAH model undergoes a localization transition from an extended phase to a localized phase via an intermediate critical phase as the strength of the quasiperiodic potential is varied. While the staggered potential merely shifts the critical point to a lower value, Fibonacci and Thue-Morse potentials induce immediate localization. This contrast reveals the sensitivity of localization behavior to the structural complexity of the potential, with the onset of localization correlating with the sequence's complexity. More specifically, the system follows a hierarchy defined by the complexity measures of the applied potentials. In addition, the typical fidelity susceptibility exhibits a power-law scaling behavior at the localization transition, enabling reliable extraction of the critical exponent. We focus on the AAH model with the Fibonacci potential due to its minimal finite-size effects compared to other cases. For the disordered AAH model with the Fibonacci potential, we determine critical exponents that differ from those of the AAH model without disorder and the Anderson model. Moreover, despite differences in localization behavior, we find that the disordered AAH models with the staggered potential and the Fibonacci potential share the same correlation-length critical exponent. These findings provide a unified framework for understanding localization transitions in quasiperiodic systems and are amenable to experimental validation using emerging techniques.
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.
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.
Anderson localization and the mobility edge (ME) have been extensively studied in isolated aperiodic systems. Conventional theory suggests that dephasing and decoherence should disrupt localization and facilitate transport. In this work, we investigate localization behaviors in a dimerized off-diagonal Aubry-Andre-Harper (AAH) quasicrystal subject to on-site pure dephasing. In the strong-dephasing limit, we apply adiabatic elimination within the Lindblad master equation framework to derive an effective classical Markov transition matrix that governs the dissipative relaxation dynamics. Counterintuitively, we demonstrate that pure dephasing can induce distinct MEs, including both conventional MEs separating extended and localized states and anomalous MEs separating multifractal critical states from localized states, even when all eigenstates of the original closed coherent system are delocalized or multifractal. Using fractal dimension finite-size scaling, wave-packet spreading dynamics, and energy spectrum statistics, we numerically verify the coexistence of fully extended, multifractal critical, and localized regions within the relaxation spectrum of the dissipative system, and construct the global dissipative phase diagram. These findings reveal that dephasing can see as a powerful mechanism for controlling localization transitions, thus enhancing our understanding of dissipative quasicrystal systems.