Spectral and transport properties of the two-dimensional Lieb lattice

Abstract

The specific topology of the line centered square lattice (known also as the Lieb lattice) induces remarkable spectral properties as the macroscopically degenerated zero energy flat band, the Dirac cone in the low energy spectrum, and the peculiar Hofstadter-type spectrum in magnetic field. We study here the properties of the finite Lieb lattice with periodic and vanishing boundary conditions. We find out the behavior of the flat band induced by disorder and external magnetic and electric fields. We show that in the confined Lieb plaquette threaded by a perpendicular magnetic flux there are edge states with nontrivial behavior. The specific class of twisted edge states, which have alternating chirality, are sensitive to disorder and do not support IQHE, but contribute to the longitudinal resistance. The symmetry of the transmittance matrix in the energy range where these states are located is revealed. The diamagnetic moments of the bulk and edge states in the Dirac-Landau domain, and also of the flat states in crossed magnetic and electric fields are shown.

Ground state spin and excitation energies in half-filled Lieb lattices

“…For our case, the mid-spectrum level degeneracy is g = N + 1 [15] and the interesting problem here is that the spin values predicted by the above theorems at half-filling are different. They are s L = (N − 1)/2 for the Lieb state and s H = (N + 1)/2 for the Hund one [37], being related by the formula s L = s H − 1 and suggesting a single spin-flip process between them.…”

“…In the finite Lieb lattice there is a degenerate level ǫ = 0 at mid spectrum having one of the degenerate states located on A sites and all of the other states located on B and C sites [15]. For one cell the degeneracy of zero energy level is g = 2 and, following the introduced notation, Φ 4 is localized on A lattice sites and Φ 5 on B, C lattice sites, thus making them spatially disjoint.…”

“…1. We shall use the notations from [15], the eight eigenstates being grouped in three branches: the states Φ…”

“…the Lieb lattice, was first proposed in [1] as a rigorous example of itinerant ferromagnetism in the presence of on-site Hubbard interactions at half-filling. Recently, the Lieb lattice received renewed attention in the context of optical and photonic lattices [2][3][4][5][6][7], 2D superconductivity [8,9], or for its specific topological properties [10][11][12][13][14][15]. In particular, the artificial lattice realization offers the advantage of parameters controlling, leading to various regimes not available in real-atoms lattices, so that one can test a vast spectrum of theoretical prediction.…”

“…Also, for the case of nano-systems, one is typically interested not as much in the ground states configurations, but especially in the excitation energies which are the experimentally measurable quantities. Moreover we shall give insights on the mechanisms that impose the Lieb state as the ground state using the nonoverlapping property of one of the states in the mid-spectrum [15] and the electron-hole symmetry of the Hamiltonian [38][39][40].…”