Topological flat Wannier-Stark bands

Abstract

We analyze the spectrum and eigenstates of a quantum particle in a bipartite two-dimensional tight-binding dice network with short range hopping under the action of a dc bias. We find that the energy spectrum consists of a periodic repetition of one-dimensional energy band multiplets, with one member in the multiplet being strictly flat. The corresponding macroscopic degeneracy invokes eigenstates localized exponentially perpendicular to the dc field direction, and super-exponentially along the dc field direction. We also show that the band multiplet is characterized by a topological winding number (Zak phase), which changes abruptly if we vary the dc field strength. These changes are induced by gap closings between the flat and dispersive bands, and reflect the number of these closings.Recently much attention has been paid to flat bands in one-, two-and three-dimensional lattices with short range hoppings and non-trivial geometry [1]. Flat bands with finite range hoppings exist due to destructive interference leading to a macroscopic number of degenerate compact localized eigenstates (CLS) which have strictly zero amplitudes outside a finite region of the lattice [2]. Flat band networks have been proposed in one, two, and three dimensions and various flat band generators were identified [3][4][5][6]. Experimental observations of flat bands and CLS are reported in photonic waveguide networks [7][8][9][10][11][12][13][14][15], exciton-polariton condensates [16][17][18], and ultracold atomic condensates [19,20]. The tight binding network equations correspond to an eigenvalue problem EΨ l = − m t lm Ψ m . For bipartite lattices, the existence of flat bands and CLS is ensured through a proper usage of the protecting chiral symmetry [6]. For example, for the dice lattice shown in Fig. 1(a) and t lm = 1 the CLS consists of an empty C site which is surrounded by six excited A and B sites with alternating amplitudes ±1/ √ 6.When a dc field is added, quantum particles start to experience well studied Bloch oscillations (see e.g. [21,22]). Our results are therefore applicable for ultracold atoms in optical lattices where the electric field is substituted by a tilt of the lattice in the gravitational field [23] or accelerating the whole lattice [24]. The same type of perturbations can be arranged in optical waveguide arrays where the electric field is modeled by a curved geometry of the waveguides [25].In the present letter we report on a new family of flat bands which exist in dc biased bipartite lattices and which are not supported by CLS, despite of the short range hopping. We consider the dice lattice in the presence of a dc field oriented along the y-direction (Fig.1(a)). The absence of CLS indicates nontrivial topology. We compute winding numbers (Zak phase) which indeed show abrupt changes through conical intersection point degeneracies upon variations of the dc field strength. We also generalize to different orientations of the field F and other lattice geometries.

Nontrivial topological flat bands in a diamond-octagon lattice geometry

“…The physics of flat band (FB) systems has drawn a lot of research attention in recent years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. One of the main reasons why such dispersionless flat bands are of great interest to the physics community is that, they give rise to highly degenerate manifold of single-particle states, which can act as a good platform to study rich, strongly correlated phenomena.…”

Compact discrete breathers on flat-band networks

“…The effects of different types of perturbations have been studied in several examples of flat band networks [9,10], as well as the effects of disorder and nonlinearity and interaction between them [11]. Further studies focused on non-Hermitian flat band networks [12], topological flat Wannier-Stark bands [13], Bloch oscillations [14], Fano resonances [15], fractional charge transport [16] and the existence of nontrivial superfluid weights [17]. Chiral flat band networks revealed that CLS and their macroscopic degeneracy can be protected under any perturbation which does not lift the bi-partiteness of the network [18].…”

Observation of quincunx-shaped and dipole-like flatband states in photonic rhombic lattices without band-touching

“…However, in a two-dimensional dice lattice, Kolovsky et.al. showed that CLSs stop being compact in presence of a dc electric field, and instead they turn into exponentially (super-exponentially) localized in the perpendicular (parallel) direction of the field 53 . Despite those theoretical studies, the intrinsic mechanism played by external fields in flatband systems is still unclear, and in many cases not supported by experimental observation.…”