Peierls instability of the Lieb lattice

It is shown that the energy of the electron system in the two-dimensional Lieb lattice decreases owing to displacements of the edge atoms from the lattice sites along the edges. This decrease in the electron energy gives rise to soft phonon modes, anharmonic phonons, and to a lattice instability. Under certain conditions, the decrease in the electron energy can exceed the increase in the elastic energy of the ion lattice, and the total energy as a function of the displacements of edge atoms takes the form of a double-well potential. As a result, in the case of a pronounced instability, a partially ordered sublattice of edge atoms arises with the number of equilibrium positions twice as large as the number of atoms. The quantum tunneling of edge atoms between equilibrium positions results in the formation of quantum tunneling modes. The possible experimental manifestations of such instability and the extension of the model under study to the three-dimensional lattices are discussed.