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arXiv:2208.11930 · Physical review. B./Physical review. B

Critical-to-insulator transitions and fractality edges in perturbed flat bands

Sanghoon Lee, Alexei Andreanov, Sergej Flach A

Published Jan 17, 2023• 33 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.107.014204

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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 \ensuremathθi{\ensuremath{\theta}}_{i}\ensuremathθi​. Without loss of generality, we focus on the case of two bands with bandgap \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ. Weak perturbations lead to an effective Hamiltonian with both on- and off-diagonal quasiperiodic terms that depend on \ensuremathθi{\ensuremath{\theta}}_{i}\ensuremathθi​. 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.
We study the effect of infinitesimal onsite disorder on ddd-dimensional all bands flat lattices. The lattices are generated from diagonal Hamiltonians by a sequence of (d+1)(d+1)(d+1) local unitary transformations parametrized by angles ${\ensuremath{\theta}}_{i}$. Without loss of generality, we consider the case of two flat bands separated by a finite gap, $\mathrm{\ensuremath{\Delta}}$. The perturbed states originating from the flat bands are described by an effective tight-binding network with finite on- and off-diagonal disorder strength which depends on the manifold angles ${\ensuremath{\theta}}_{i}$. The original infinitesimal on-site disorder strength WWW is only affecting the overall scale of the effective Hamiltonian. Upon variation of the manifold angles for d=1d=1d=1 and d=2d=2d=2 we find that localization persists for any choice of local unitaries, and the localization length can be maximized for specific values of ${\ensuremath{\theta}}_{i}$. Instead, in d=3d=3d=3 we identify a nonperturbative metal-insulator transition upon varying all bands' flat manifold angles.
In recent years, there has been a growing interest in flatband systems which exhibit macroscopic degeneracies. These systems offer a valuable mathematical framework for the extreme sensitivity to perturbations and interactions. This sensitivity unveils a wide variety of exotic and unconventional physical phenomena. Moreover, the progress in their experimental realization contributes to the expanding landscape of exploration in this field. This thesis aims to summarize all the works throughout the Ph.D. program. Firstly, an in-depth exploration was conducted on the impact of weak quasiperiodic perturbations on one-dimensional two-band all-bands-flat lattices. These tight-binding Hamiltonian are diagonalized through a sequence of local unitary transformations. By adjusting the quasiperiodic potential parameters, the key achievement involves finding a critical-to-insulator transition and identifying fractality edges in the flatband systems with quasiperiodic perturbations. Next, the investigation delved into the effects of on-site interactions among hard-core bosons in one- and two-dimensional cross-stitch lattices. One key finding is that groundstate energy primarily depends on compact localized states. Moreover, the presence of barriers of compact localized states trap bosons, leading to the emergence of non-ergodic excitation and Hilbert space fragmentation. Lastly, a compact localized eigenstate of the one-dimensional diamond chain using an electric circuit was successfully generated via local (linear and non-linear) driving. This achievement opens up a versatile circuit platform for the generation of flatbands and holds promise for potential applications in the field of quantum information. I hope these collective efforts have expanded the frontiers of the field and made a meaningful contribution to the scientific community.
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 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.
We propose a general framework for constructing self-dual one-dimensional quasiperiodic lattice models with arbitrary-range hoppings and multifractal behaviors. Our framework generates a broad spectrum of one-dimensional quasicrystals, ranging from the off-diagonal Aubry-Andr\'e-Harper models on one end, to those featuring long-range hoppings with varied quasiperiodic modulations on another. Focusing on models with off-diagonal quasiperiodic hoppings with power-law decay, we exploit the fact that, when the self-dual condition is satisfied, the system must be in the critical state with multifractal properties. This enables the engineering of models with competing extended, critical, and localized phases, with richly structured mobility edges separating them. As an outstanding example, we show that a limiting case of our family of self-dual quasicrystals can be implemented using Rydberg-atom arrays. Our work offers a systematic route toward critical phases from self-duality considerations, and would facilitate the experimental simulation of these exotic states.
We investigate the zero-temperature metal-insulator transition in a one-dimensional two-component Fermi gas in the presence of a quasiperiodic potential resulting from the superposition of two optical lattices of equal intensity but incommensurate periods. A mobility edge separating (low-energy) Anderson localized and (high-energy) extended single-particle states appears in this continuous-space model beyond a critical intensity of the quasiperiodic potential. To discern the metallic phase from the insulating phase in the interacting many-fermion system, we employ unbiased quantum Monte Carlo (QMC) simulations combined with the many-particle localization length familiar from the modern theory of the insulating state. In the noninteracting limit, the critical optical-lattice intensity for the metal-insulator transition predicted by the QMC simulations coincides with the Anderson localization transition of the single-particle eigenstates. We show that weak repulsive interactions induce a shift of this critical point towards larger intensities, meaning that repulsion favors metallic behavior. This shift appears to be linear in the interaction parameter, suggesting that even infinitesimal interactions can affect the position of the critical point.