Flat bands and nontrivial topological properties in an extended Lieb lattice

Abstract

We report the appearance of multiple numbers of completely flat band states in an extended Lieb lattice model in two dimensions with five atomic sites per unit cell. We also show that this edge-centered square lattice can host intriguing topologically nontrivial phases when intrinsic spinorbit (ISO) coupling is introduced in the microscopic description of the corresponding tight-binding Hamiltonian of the system. This ISO coupling strength acts like a complex next-nearest-neighbor hopping term for this model and can be, in principle, tuned in a real-life experimental setup. In the presence of this ISO coupling, the band spectrum of the system gets gapped out, leading to nonzero integer values of the spin Chern number for different bands, indicating the nontrivial topological properties of the system. Furthermore, we show that for certain values of the ISO coupling, nearly flat bands with nonzero Chern numbers emerge in this lattice model. This opens up the possibility of realizing interesting fractional quantum spin Hall physics in this model when interaction is taken into account. This study might be very useful in an analogous optical lattice experimental setup. A possible application of our results can also be anticipated in the field of photonics using single-mode photonic waveguide networks. * ; Present address: Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel.

Localization, phases, and transitions in three-dimensional extended Lieb lattices

“…Systems that exhibit flat-band physics correspond usually to specially "engineered" lattice structures such as quasi-1D lattices [6,17,18], diamond-type lattices [19], and so-called Lieb lattices, [7,[20][21][22][23][24]. Indeed, the Lieb lattice, a two-dimensional (2D) extension of a simple cubic lattice, was the first where the flat band structure was recognized and used to enhance magnetic effects in model studies [2, 25,26].…”

“…Most other flat-band systems cited above are also of the Lieb type and exists as either 2D, quasi-1D or 1D lattices [27]. Less attention has been given to 3D flat-band systems [19] or extended Lieb lattices [24,28]. Furthermore, while disorder in quasi-1D [29][30][31][32] and 2D [33] has previously received some attention, comparatively little work has investigated the influence of disorder on 3D flat-band systems [17,34,35].…”

Photonic flat-band lattices and unconventional light localization

“…sHCL was first studied more than two decades ago in the context of electronic systems [95,115]. Recently Dirac cones and conical diffraction in such superlattices were also theoretically investigated [104,116,117] in cold atoms and optics, but experimental study is lacking because of the difficulty in constructing such sophisticated lattices. sHCL has five lattice sites (a-e) per unit cell, as shown in Figure 7A, and its tight-binding band structure consists of five bands, including a completely flatband touching two dispersive conical bands at the center (Γ point) of the first BZ, which resembles the pseudospin-1 Dirac cone in the Lieb lattice.…”

Disorder effects in the two-dimensional Lieb lattice and its extensions

“…This is equivalent to effectively having zero kinetic energy in these bands and hence to promoting the remaining terms in the Hamiltonian such as potential and interaction terms. Usually, such flat bands emerge only for specially "engineered" lattice structures such as quasi-1D lattices [13][14][15], diamond-type lattices [16], and Lieb lattices [17][18][19][20][21][22]. Lieb lattices were originally used to enhance magnetic effects in model studies [9,23,24] but have now been shown to be of experimental relevance in Wigner crystals [12], high-temperature superconductivity [10,18], photonic wave guide arrays [8,[25][26][27][28], Bose-Einstein condensates [29,30], ultra-cold atoms in optical lattices [31] and electronic systems [32].…”