Engineering flat electronic bands in quasiperiodic and fractal loop geometries

Abstract

Exact construction of one electron eigenstates with flat, non-dispersive bands, and localized over clusters of various sizes is reported for a class of quasi-one-dimensional looped networks. Quasiperiodic Fibonacci and Berker fractal geometries are embedded in the arms of the loop threaded by a uniform magnetic flux. We work out an analytical scheme to unravel the localized single particle states pinned at various atomic sites or over clusters of them. The magnetic field is varied to control, in a subtle way, the extent of localization and the location of the flat band states in energy space. In addition to this we show that an appropriate tuning of the field can lead to a re-entrant behavior of the effective mass of the electron in a band, with a periodic flip in its sign.Geometrically frustrated lattices (GFLs) supporting flat, dispersionless bands in their energy spectrum with macroscopically degenerate eigenstates have drawn great interest in recent times [1][2][3][4][5][6][7][8]. Initial interest in antiferromagnetic Heisenberg model on frustrated lattices [9][10][11][12][13][14] has evolved into extensive studies of the gapped flat band states to gapless chiral modes in graphenes [15], in optical lattices of ultracold atoms [16], waveguide arrays [17], or in microcavities having exciton-polaritons [18]. The quenched kinetic energy of an electron in a flat band state (FBS) leads to the possibility of achieving strongly correlated electronic states, topologically ordered phases, such as the lattice versions of fractional quantum Hall states [19]. Recently, the controlled growth of artificial lattices with complications such as in the kagome class has added excitement to such studies [20,21].Spinless fermions are easily trapped in flat bands [6]. The nondispersive character of the energy (E)-wave vector (k) curve implies an infinite effective mass of the electron, leading to practically its immobility in the lattice. Such states are therefore strictly localized either on special sets of vertices, or in a finite cluster of atomic sites spanning finite areas of the underlying lattice. Recently it has been shown that an infinity of such cluster-localized single particle states can be exactly constructed even in a class of deterministic fractals [22]. Apart from its interest in direct relation to the study of GFLs, this work provides an example where eigen-* Corresponding author. (A. Nandy), arunava_chakrabarti@yahoo.co.in (A. Chakrabarti). values corresponding to localized eigenstates in an infinite fractal geometry can be exactly evaluated, a task that is a non-trivial one if one remembers that these fractal systems are free from translational invariance of any kind.In this communication we unravel and analyze groups of flat, dispersionless energy bands in some tailor made GFLs. The lattices display an interesting competition between long range translational order along the horizontal (x−) axis and an aperiodic growth in the transverse directions. In each case, the skeleton is an infinite array of diamond shaped...

Artificial flat band systems: from lattice models to experiments

“…In 2017 Ramachandran et al [39] introduced the most general flat band generator for bipartite lattices with chiral flat bands, where the first example (to our knowledge) of a chiral flat band in a three-dimensional network was obtained. Other novel approaches for obtaining flat band lattices involve origami rules [42], the repetition of oligomers [43], local symmetries [44], and self-similar (fractal) constructions [45,46].…”

Landau-Zener Bloch Oscillations with Perturbed Flat Bands

“…FBs have been studied in a number of lattice models in threedimensional, two-dimensional, and even one-dimensional (1D) settings [7][8][9], and recently realized experimentally with photonic waveguide arrays [10-12], excitonpolariton condensates [13,14], and ultra-cold atomic condensates [15].FB networks rely on the existence of compact localized eigenstates (CLS) due to destructive interference, enabled by the local symmetries of the network. FB networks are constructed using graph theory [7,[16][17][18][19], CLS [7,20,21], and can be perturbed e.g. by disorder to arrive at unexected new scaling laws [21,22], and correlated potentials to arrive at diverging densitites of states, gaps, and designable mobility edges [23,24].…”

“…FB networks rely on the existence of compact localized eigenstates (CLS) due to destructive interference, enabled by the local symmetries of the network. FB networks are constructed using graph theory [7,[16][17][18][19], CLS [7,20,21], and can be perturbed e.g. by disorder to arrive at unexected new scaling laws [21,22], and correlated potentials to arrive at diverging densitites of states, gaps, and designable mobility edges [23,24].…”

Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems

“…Exact calculations on hierarchical lattices are also currently widely used in a variety of statistical mechanics [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], finance [37], and, most recently, DNA-binding [38] problems.…”