Linking topological features of the Hofstadter model to optical diffraction figures

Colandrea, Francesco Di; D’Errico, Alessio; Maffei, Maria; Price, Hannah M.; Lewenstein, Maciej; Marrucci, Lorenzo; Cardano, Filippo; Dauphin, Alexandre; Massignan, Pietro

doi:10.1088/1367-2630/ac4126

Cited by 8 publications

(4 citation statements)

References 70 publications

(121 reference statements)

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“…Insightful predictions on other rich physics in 1D quasi-periodic lattices are proposed, for example, topology, non-Hermeticity, superconductivity and superfluidity [25][26][27][28]. Some interesting experimental realizations of quasicrystalline physics have also been discussed recently [29,30].…”

Section: Introductionmentioning

confidence: 99%

Transport through quantum anomalous Hall bilayers with lattice mismatch

Zhang

Wang

et al. 2022

New J. Phys.

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We theoretically investigate quantum transport properties of quantum anomalous Hall bilayers, with arbitrary ratio of lattice constants, i.e., with lattice mismatch. In the simplest case of ratio 1 (but with different model parameters in two layers), the inter-layer coupling results in resonant traversing between forward propagating waves in two layers. In the case of generic ratios, there is a quantized conductance plateau originated from two Chern numbers associated with two layers. However, the phase boundary of this quantization plateau consists of a fractal transitional region (instead of a clear transition line) of interpenetrating edge states (with quantized conductance) and bulk states (with unquantized conductance). We attribute these bulk states as mismatch induced in-gap bulk states. Different from in-gap localized states induced by random disorder, these in-gap bulk states are extended in the limit of vanishing random disorder. However, the detailed fine structure of this transitional region is sensitive to disorder, lattice structure, sample size, and even the configuration of leads connecting to it, due to the bulk and topologically trivial nature of these in-gap bulk states.

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Section: Introductionmentioning

confidence: 99%

Transport through quantum anomalous Hall bilayers with lattice mismatch

Zhang

Wang

et al. 2022

New J. Phys.

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“…For example, the Peierls phases that arise in the Hamiltonian due to the magnetic field (Peierls, 1933) can also be generated using artificial gauge fields (Goldman et al, 2014) or Floquet modulation (Eckardt, 2017). Moreover, the full scope of the Hofstadter model is still being revealed, with papers on its application to hyperbolic lattices (Stegmaier et al, 2022), higher-dimensional crystals (Colandrea et al, 2022), and synthesized materials (Bodesheim et al, 2023), all published within the last couple of years.…”

Section: Statement Of Needmentioning

confidence: 99%

“…The 𝑟-th gap is colored with respect to 𝑡 = ∑ 𝑟 𝑖=0 𝐶 𝑖 , where 𝐶 𝑖 is the Chern number of band 𝑖. The size of the points in the Wannier diagram is proportional to the size of the gaps (Colandrea et al, 2022).…”

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confidence: 99%

HofstadterTools: A Python package for analyzing the Hofstadter model

Andrews

2024

JOSS

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The Hofstadter model successfully describes the behavior of non-interacting quantum particles hopping on a lattice coupled to a gauge field, and hence is ubiquitous in many fields of research, including condensed matter, optical, and atomic physics. Motivated by this, we introduce HofstadterTools (https://hofstadter.tools), a Python package that can be used to analyze the energy spectrum of a generalized Hofstadter model, with any combination of hoppings on any regular Euclidean lattice. The package can be applied to compute key properties of the band structure, such as quantum geometry and topology, as well as plot Hofstadter butterflies and Wannier diagrams that are colored according to their Chern numbers.

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“…For instance, the fourth space-like dimensional can be emulated by employing synthetic dimension [80,81], periodic driving (Floquet states) [82,83], quantum quench [84][85][86][87] or generalized Thouless pumping [54]. For these reasons, we envisage that 4D Euler insulators can be also engineered in several artificial systems ranging from ultracold atoms [53,55,88,89], photonics [56,[90][91][92] and meta-materials [93,94] to acoustic systems [95,96], trapped-ion simulators [97], superconducting systems [98,99] and electric circuits [100][101][102].…”

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confidence: 99%

Second Euler number in four dimensional synthetic matter

Bouhon¹,

Zhu²,

Slager³

et al. 2023

Preprint

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Two-dimensional Euler insulators are novel kind of systems that host multi-gap topological phases, quantified by a quantised first Euler number in their bulk. Recently, these phases have been experimentally realised in suitable two-dimensional synthetic matter setups. Here we introduce the second Euler invariant, a familiar invariant in both differential topology (Chern-Gauss-Bonnet theorem) and in four-dimensional Euclidean gravity, whose existence has not been explored in condensed matter systems. Specifically, we firstly define two specific novel models in four dimensions that support a non-zero second Euler number in the bulk together with peculiar gapless boundary states. Secondly, we discuss its robustness in general spacetime-inversion invariant phases and its role in the classification of topological degenerate real bands through real Grassmannians. In particular, we derive from homotopy arguments the minimal Bloch Hamiltonian form from which the tight-binding models of any second Euler phase can be generated. Considering more concretely the gapped Euler phase associated with the tangent bundle of the four-sphere, we show that the bulk band structure of the nontrivial 4D Euler phase necessarily exhibits triplets of linked nodal surfaces (where the three types of nodal surfaces are formed by the crossing of the three successive pairs of bands within one four-band subspace). Finally, we show how to engineer these new topological phases in a four-dimensional ultracold atom setup. Our results naturally generalize the second Chern and spin Chern numbers to the case of four-dimensional phases that are characterised by real Hamiltonians and open doors for implementing such unexplored higher-dimensional phases in artificial engineered systems, ranging from ultracold atoms to photonics and electric circuits.

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Linking topological features of the Hofstadter model to optical diffraction figures

Cited by 8 publications

References 70 publications

Transport through quantum anomalous Hall bilayers with lattice mismatch

Transport through quantum anomalous Hall bilayers with lattice mismatch

HofstadterTools: A Python package for analyzing the Hofstadter model

Second Euler number in four dimensional synthetic matter

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