Higher-dimensional quasicrystalline approach to the Hofstadter butterfly topological-phase band conductances: Symbolic sequences and self-similar rules at all magnetic fluxes

Abstract

The topological properties of the quantum Hall effect in a crystalline lattice, described by Chern numbers of the Hofstadter butterfly quantum phase diagram, are deduced by using a geometrical method to generate the structure of quasicrystals: the cut and projection method. Based on this, we provide a geometric unified approach to the Hofstadter topological phase diagram at all fluxes. Then we show that for any flux, the bands conductance follow a two letter symbolic sequence . As a result, bands conductance at different fluxes obey inflation/deflation rules as the ones observed to build quasicrystals. The bands conductance symbolic sequences are given by the Sturmian coding of the flux and can be found by considering a circle map, a billiard or trajectories on a torus. Simple and fast techniques are thus provided to obtain Chern numbers at any magnetic flux. This approach rationalize the previously observed topological equivalences between the Fibonacci and Harper potentials (also known as the almost Mathieu operator problem) or with other trigonometric potential, as well as the relationship with Farey sequences and trees. arXiv:1906.02632v1 [cond-mat.mes-hall] 6 Jun 2019 2

Hofstadter Topology: Non-crystalline Topological Materials at High Flux

“…q − 1 because of the periodicity in the BZ as established by Eq. (4). Because a multiple of q gap closings separate H φ from the trivial atomic limit at large M where the Chern number is zero, it must be that C φ= 2πp q =0 ∈ qZ, zero included.…”

“…When a two dimensional crystalline lattice in which electrons have a trivial band structure is pierced by a uniform magnetic field, translational symmetry is broken and the energy spectrum develops a complex, fractal structure known as the Hofstadter Butterfly [1]. This system is host to a wealth of nontrivial Chern number topology despite the triviality of the original band structure [2][3][4][5][6][7]. In this work, we show that the Hofstadter problem acquires new properties when the initial band structure is already topological and demonstrate new phases not possible in crystalline insulators of the same spatial dimension.…”

“…However, we can diagnose the topology directly with the nested Wilson loop and Kramers' theorem for (U C 2z T ) 2 = −1 (see App. E 3), showing there are no 4 The nontrivial phase with θ = π is a "strong" symmetry-protected topological phase with corner state pumping. If both w φ=0 2 = w φ=Φ/2 2 = 1, then θ = 0 but both H φ=0 and H φ=Φ/2 have nontrivial corner states, which is a "weak" 3D fragile state.…”

Mechanical, electronic, optical, piezoelectric and ferroic properties of strained graphene and other strained monolayers and multilayers: an update

“…A powerful method to describe the structure of twisted 2D materials relies on the cut and projection method used to generate quasicrystals [294,[318][319][320]. There, the projected structure is interpreted as the set of points of best fit between the two rotated structures [294].…”

Topological states in quasicrystals

“…As the 2D lattice is not necessarily periodic, it is possible to generalize the IQHE to 2D quasicrystals. There has been some research exploring the IQHE in 2D quasicrystals [34,[36][37][38]. For example, Tran et al [34] investigate the topological properties of a 2D quasicrystal subjected to a uniform magnetic field.…”