Nature of protected zero-energy states in Penrose quasicrystals

Day-Roberts, Ezra; Fernandes, Rafael M.; Kamenev, Alex

doi:10.1103/physrevb.102.064210

Cited by 20 publications

(11 citation statements)

References 46 publications

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Section: Discussionsupporting

confidence: 77%

Statistical mechanics of dimers on quasiperiodic tilings

Lloyd,

Biswas,

Simon

et al. 2021

Preprint

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We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Despite the lack of periodicity we prove that each infinite tiling admits 'perfect matchings' in which every vertex is touched by one dimer. We introduce an auxiliary 'AB * ' tiling obtained from the AB tiling by deleting all 8-fold coordinated vertices. The AB * tiling is again two-dimensional, infinite, and quasiperiodic. The AB * tiling has a single connected component, which admits perfect matchings. We find that in all perfect matchings, dimers on the AB * tiling lie along disjoint onedimensional loops and ladders, separated by 'membranes', sets of edges where dimers are absent. As a result, the dimer partition function of the AB * tiling factorizes into the product of dimer partition functions along these structures. We compute the partition function and free energy per edge on the AB * tiling using an analytic transfer matrix approach. Returning to the AB tiling, we find that membranes in the AB * tiling become 'pseudomembranes', sets of edges which collectively host at most one dimer. This leads to a remarkable discrete scale-invariance in the matching problem. The structure suggests that the AB tiling should exhibit highly inhomogenous and slowly decaying connected dimer correlations. Using Monte Carlo simulations, we find evidence supporting this supposition in the form of connected dimer correlations consistent with power law behaviour. Within the set of perfect matchings we find quasiperiodic analogues to the staggered and columnar phases observed in periodic systems.

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Section: Discussionsupporting

confidence: 77%

Statistical mechanics of dimers on quasiperiodic tilings

Lloyd,

Biswas,

Simon

et al. 2021

Preprint

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“…Here, the number of monomers hosted in a maximum matching is associated with the number of zero modes, while the monomer-confining regions are associated with wave functions whose support is confined within a compact subgraph. The computed monomer densities and geometry of monomer-confining regions on the Penrose tiling [60] are consistent with the density of zero modes and the nature of confined states obtained from investigations of hopping problems on the Penrose tiling [107][108][109]. Reference [73] recently computed a finite density of confined zero modes on the AB tiling; naively, this appears to be in conflict with our results that demonstrate that the AB tiling can be perfectly matched with vanishing monomer density in the thermodynamic limit.…”

Section: Discussionsupporting

confidence: 75%

Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings

Lloyd

Biswas²,

Simon³

et al. 2022

Phys. Rev. B

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We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Using the discrete scale-symmetry of quasiperiodic tilings, we prove that each infinite tiling admits "perfect matchings", where every vertex is touched by one dimer. We show the appearance of so-called monomer pseudomembranes. These are sets of edges, which collectively host exactly one dimer, which bound certain eightfold-symmetric regions of the tiling. Regions bounded by pseudomembranes are matched together in a way that resembles perfect matchings of the tiling itself. These structures emerge at all scales, suggesting the preservation of collective dimer fluctuations over long distances. We provide numerical evidence, via Monte Carlo simulations, of dimer correlations consistent with power laws over a hierarchy of different lengthscales. We also find evidence of rich monomer correlations, with monomers displaying a pattern of attraction and repulsion to different regions within pseudomembranes, along with signatures of deconfinement within certain annular regions of the tiling.

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“…Here, N is the number of sites. Note that a peculiar electronic structure, called confined state [6,46,47], is present at half filling (n = 1) in the non-interacting system. To make a general statement, we avoid these states, focusing the fillings away from the half filling.…”

Section: A Extended Hubbard Model On Penrose Tilingmentioning

confidence: 99%

Hyperuniform electron distributions controlled by electron interactions in quasicrystal

Sakai,

Arita,

Ohtsuki

2021

Preprint

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We study how the electron-electron interactions influence the charge distributions in the metallic state of quasicrystals. As a simple theoretical model, we introduce an extended Hubbard model on the Penrose lattice, and numerically solve the model (up to ∼ 1.4 million sites) within the Hartree-Fock approximation. Because each site on the quasiperiodic lattice has a different local geometry, the Coulomb interaction, in particular the intersite one, works in a site-dependent way, leading to a nontrivial redistribution of the charge. The resultant charge distribution patterns are not multifractal but characterized by hyperuniformity, which offers a measure to distinguish various inhomogeneous but ordered distributions. We clarify how the electron interactions alter the order metric of the hyperuniformity, revealing that the intersite interaction considerably affects the hyperuniformity in particular on the electron-rich side.

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Nature of protected zero-energy states in Penrose quasicrystals

Cited by 20 publications

References 46 publications

Statistical mechanics of dimers on quasiperiodic tilings

Statistical mechanics of dimers on quasiperiodic tilings

Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings

Hyperuniform electron distributions controlled by electron interactions in quasicrystal

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