Quasiperiodic moiré materials provide a new platform for realizing critical electronic states, yet a direct and experimentally practical method to characterize this criticality has been lacking. We show that a multifractal analysis of the local density of states (LDOS), accessible via scanning tunneling microscopy, offers an unambiguous signature of criticality from a single experimental sample. Applying this approach to a one-dimensional quasiperiodic model, a stringent test case due to its fractal energy spectrum, we find a clear distinction between the broad singularity spectra $f\left(α\right)$ of critical states and the point-like spectra of extended states. We further demonstrate that these multifractal signatures remain robust over a wide range of energy broadenings relevant to experiments. Our results establish a model-independent, experimentally feasible framework for identifying and probing multifractality in the growing family of quasiperiodic and moiré materials.
Quasiperiodic moiré materials provide a new platform for realizing critical electronic states, yet a direct and experimentally practical method to characterize this criticality has been lacking. We show that a multifractal analysis of the local density of states (LDOS), accessible via scanning tunneling microscopy, offers an unambiguous signature of criticality from a single experimental sample. Applying this approach to a one-dimensional quasiperiodic model, a stringent test case due to its fractal energy spectrum, we find a clear distinction between the broad singularity spectra f(α) of critical states and the point-like spectra of extended states. We further demonstrate that these multifractal signatures remain robust over a wide range of energy broadenings relevant to experiments. Our results establish a model-independent, experimentally feasible framework for identifying and probing multifractality in the growing family of quasiperiodic and moiré materials.
We demonstrate that a mean field approximation can be confidently employed in quasiperiodic moiré systems to treat interactions and quasiperiodicity on equal footing. We obtain the mean field phase diagram for an illustrative one-dimensional moiré system that exhibits narrow bands and a regime with non-interacting multifractal critical states. By systematically comparing our findings with existing exact results, we identify the regimes where the mean field approximation provides an accurate description. Interestingly, in the critical regime, we obtain a quasifractal charge density wave, consistent with the exact results. To complement this study, we employ a real-space implementation of the time-dependent Hartree-Fock, enabling the computation of the excitation spectrum and response functions at the RPA level. These findings indicate that a mean field approximation to treat systems hosting multifractal critical states, as found in two-dimensional quasiperiodic moiré systems, is an appropriate methodology.
A reentrant localization transition was predicted recently in a one-dimensional quasiperiodic lattice with dimerized hopping between the nearest-neighbor sites [Roy et al., Phys. Rev. Lett. 126, 106803 (2021)]. It was shown that the interplay between the hopping dimerization and a staggered quasiperiodic disorder manifests two localization transitions through two intermediate phases, resulting in four critical points as a function of the quasiperiodic potential. In this paper, we study the critical nature of the states across the localization transitions by computing the mass exponents and the corresponding fractal dimensions of the states through a multifractal analysis. Moreover, we analyze the phenomenon of this reentrant localization transition by examining the spectral properties of the eigenstates. By performing a systematic finite-size scaling analysis for a fixed value of the hopping dimerization, we obtain accurate critical disorder strengths for different transitions and the associated critical exponents. Further, we complement the critical nature of the energy spectrum by computing the Hausdorff dimensions.
We unveil a mechanism for quasicondensation of hard-core bosons in the presence of quasiperiodicity-induced multifractal single-particle states. The observed phase, here dubbed a fractal quasicondensate, is characterized by natural orbitals with multifractal properties and by an occupancy of the lowest natural orbital, ${\ensuremath{\lambda}}_{0}\ensuremath{\simeq}{L}^{\ensuremath{\gamma}}$, which grows with system size but with a nonuniversal scaling exponent, $\ensuremath{\gamma}<1/2$. In contrast to fractal quasicondensates obtained when the chemical potential lies in a region of multifractal single-particle states, placing the chemical potential in regions of localized or delocalized states yields, respectively, no condensation or the usual 1D quasicondensation with $\ensuremath{\gamma}=1/2$. Our findings are established by studying one-dimensional hard-core bosons subjected to various quasiperiodic potentials, including the well-known Aubry-Andr\'e model, employing a mapping to noninteracting fermions that allows for numerically exact results. We discuss how to test our findings in state-of-the-art ultracold atom experiments.
Abstract We demonstrate that quasiperiodicity can radically change the ground state properties of 1D moiré systems with respect to their periodic counterparts. By studying an illustrative example we show that while narrow bands play a significant role in enhancing interactions both for commensurate and incommensurate structures, only quasiperiodicity is able to extend the ordered phase down to an infinitesimal interaction strength. In this regime, the quasiperiodic-enabled state has contributions from infinitely many wave vectors. This quasi-factal regime cannot be stabilized in the commensurate case even in the presence of a narrow band. These findings suggest that quasiperiodicity may be a critical factor in stabilizing non-trivial ordered phases in interacting moiré structures and signal out multifractal non-interacting phases, recently found in 2D incommensurate moiré systems, as particularly promising parent states.
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.
We study the Anderson orthogonality catastrophe, and the corresponding X-ray edge problem, in systems that are at a localization transition driven by a deterministic quasiperiodic potential. Specifically, we address how the ground state of the Aubry-Andre model, at its critical point, responds to an instantaneous local quench. At this critical point, both the single-particle wavefunctions and the density of states are fractal. We find, numerically, that the overlap between post-quench and pre-quench wavefunctions, as well as the core-hole Green function, evolve in a complex, non-monotonic way with system size and time respectively. We interpret our results in terms of the fractal density of states at this critical point. In a given sample, as the post-quench time increases, the system resolves increasingly finely spaced minibands, leading to a series of alternating temporal regimes in which the response is flat or algebraically decaying. In addition, the fractal critical wavefunctions give rise to a quench response that varies strongly from site to site across the sample, which produces broad distributions of many-body observables. Upon averaging this broad distribution over samples, we recover coarse-grained power-laws and dynamical exponents characterizing the X-ray edge singularity. We discuss how these features can be probed in ultra-cold atomic gases using radio-frequency spectroscopy and Ramsey interference.