Finite size effects on the helical edge states on the Lieb lattice

Abstract

For a two-dimensional Lieb lattice, that is, a line-centered square lattice, the inclusion of the intrinsic spin-orbit (ISO) coupling opens a topologically nontrivial gap, and gives rise to the quantum spin Hall (QSH) effect characterized by two pairs of gapless helical edge states within the bulk gap. Generally, due to the finite size effect in QSH systems, the edge states on the two sides of a strip of finite width can couple together to open a gap in the spectrum. In this paper, we investigate the finite size effect of helical edge states on the Lieb lattice with ISO coupling under three different kinds of boundary conditions, i.e., the straight, bearded and asymmetry edges. The spectrum and wave function of edge modes are derived analytically for a tight-binding model on the Lieb lattice. For a strip Lieb lattice with two straight edges, the ISO coupling induces the Dirac-like bulk states to localize at the edges to become the helical edge states with the same Dirac-like spectrum. Moreover, it is found that in the case with two straight edges the gapless Dirac-like spectrum remains unchanged with decreasing the width of the strip Lieb lattice, and no gap is opened in the edge band. It is concluded that the finite size effect of QSH states is absent in the case with the straight edges. However, in the other two cases with the bearded and asymmetry edges, the energy gap induced by the finite size effect is still opened with decreasing the width of the strip. It is also proposed that the edge band dispersion can be controlled by applying an on-site potential energy on the outermost atoms.

Exciton Polaritons in a Two-Dimensional Lieb Lattice with Spin-Orbit Coupling

“…Two-dimensional lattices with flat energy bands attract keen research interest as a platform to study exotic many-body effects including itinerant ferromagnetism [1], Wigner crystallization [2] and fractional quantum Hall phases [3]. A notable example of a flat-band system is the Lieb lattice [4], a decorated square lattice found in nature in the cuprates exhibiting high-T c superconductivity [5] and studied extensively in recent years for its topologically nontrivial phases [6][7][8][9][10][11][12][13][14][15][16]. In bosonic systems, models of particles in two-dimensional Lieb lattice potentials with flat energy bands are a highly valuable tool for researchers, having recently been experimentally realized in photonic waveguide arrays [17][18][19][20] and ultracold atoms in optical lattices [21].…”

Finite-size effects in non-Hermitian topological systems

“…To solve the Schrödinger equation, we take the ansatz 43,72,[93][94][95] , where the quantity ρ L is a complex number. With this, Eq.…”

Exotic States Emerged By Spin-Orbit Coupling, Lattice Modulation and Magnetic Field in Lieb Nano-ribbons

“…Nano-ribons are constructed by allowing the periodic boundaries in the x-axis and open boundaries in the y-axis. The open boundaries of the nano-ribons may have three different types: straight-straight, straight-zigzag and zigzag-zigzag boundaries [25]. The edge modes may depend on the type of the open boundaries [25].…”

“…The open boundaries of the nano-ribons may have three different types: straight-straight, straight-zigzag and zigzag-zigzag boundaries [25]. The edge modes may depend on the type of the open boundaries [25]. However, the boundary condition does not significantly change the topological properties of the ground state [25].…”

“…The edge modes may depend on the type of the open boundaries [25]. However, the boundary condition does not significantly change the topological properties of the ground state [25]. In this paper we only consider the straight-straight open boundaries in the y-axis, as shown in Fig.…”