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arXiv:2305.03800 · Nature Physics

Incommensurability enabled quasi-fractal order in 1D narrow-band moiré systems

Miguel Gonçalves, Bruno Amorim, Flavio Elias Riche, Eduardo V. Castro, Pedro Ribeiro

Published Oct 11, 2024• 25 citations
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Abstract

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.

DOI
10.1038/s41567-024-02662-2

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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 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.
We study the interplay between electronic interactions and quasiperiodicity in a one-dimensional narrow-band system, focusing on ground-state and low-energy excitation properties. Using band projection as low-energy effective approach, we show that a projection restricted to first order in the interaction strength fails to reproduce the correlated phase diagram. This contrasts with the standard success of first-order band projection in translationally invariant flatband systems and highlights the essential role of virtual processes involving remote bands in quasiperiodic settings. By incorporating second-order interband contributions perturbatively, we obtain an effective Hamiltonian that quantitatively reproduces the exact phase iagram previously obtained using density matrix renormalization group calculations, including the transition between a Luttinger liquid and a charge-density-wave phase and the crossover to a quasifractal charge-density-wave regime at strong quasiperiodicity. We further use this controlled framework to investigate low-energy neutral excitations and the optical conductivity, identifying clear dynamical signatures distinguishing the different phases. Our results establish second-order band projection as a reliable tool for correlated quasiperiodic narrow-band systems and suggest a promising route for studying interacting quasiperiodic and moiré materials beyond one dimension.
We show that quasiperiodicity can enhance superconductivity in one-dimensional narrow-band systems with s-wave pairing. Using a generalized Aubry-André-Harper model featuring quasiperiodic modulations in both the on-site potential and the hoppings, we study superconductivity across the extended, critical, and localized phases present in the noninteracting limit. Our results show that within the critical and localized phases, the superconducting critical temperature exhibits an algebraic scaling with the interaction strength, in contrast to the conventional BCS scaling observed in periodic approximants with a similar density of states. This direct comparison establishes that the enhancement originates from the nature of the single-particle eigenstates, rather than from band flattening alone. Furthermore, we find that the superfluid weight remains finite across all phases, including the localized regime, highlighting that superconducting phase coherence is retained throughout. The effects of quasiperiodicity on the superfluid weight are most pronounced in the weak-coupling regime, indicating the nontrivial role of the localization properties of the parent eigenstates in shaping the superconducting phases.
Abstract Recent studies of 2D moiré materials have opened opportunities for advancing condensed matter physics. However, the effect of 1D moiré potentials on topological and correlated phases remains largely unexplored. Here we reveal a sequence of trivial-to-topological transitions and periodic-moiré-spin density waves induced by the 1D commensurate moiré potentials for spin-1/2 fermionic atoms. Such a reentrant topology from a trivial phase is absent without the moiré potential and can be understood as the renormalization of topological parameters by the moiré strength. We then unveil the critical exponent and localization properties of the single-particle eigenstates. The periodic spin density wave of many-body ground states is contributed by the moiré potential, and is enhanced by on-site interactions but suppressed by nearest-neighbor interactions. Our results enrich the topological physics with multiple transitions and spin-density orders in 1D moiré systems, and the realization of the proposed model is promising in near-future ultracold atom setups.
Inspired by the rich physics of twisted 2D bilayer moiré systems, we study Coulomb interacting systems subjected to two overlapping finite 1D lattice potentials of unequal periods through exact numerical diagonalization. Unmatching underlying lattice periods lead to a 1D bichromatic "moiré" superlattice with a large unit cell and consequently a strongly flattened band, exponentially enhancing the effective dimensionless electron-electron interaction strength and manifesting clear signatures of enhanced Mott gaps at discrete fillings. An important nonperturbative finding is a remarkable fine-tuning effect of the precise lattice commensuration, where slight variations in the relative lattice periods may lead to a suppression of the correlated insulating phase, in qualitative agreement with the observed fragility of the correlated insulating phase in twisted bilayer graphene. Our predictions, which should be directly verifiable in bichromatic optical lattices, establish that the competition between interaction and incommensuration is a key element of the physics of moiré superlattices.
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)f\left(α\right)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.