We examine the effect of adding $\mathcal{PT}$-symmetric gain and loss terms to quasi-one-dimensional lattices (ribbons) that possess flat bands. We focus on three representative cases: the Lieb ribbon, the kagome ribbon, and the stub ribbon. In general, we find that the effect on the flat band depends strongly on the geometrical details of the lattice being examined. One interesting result that emerges from an analytical calculation of the band structure of the Lieb ribbon including gain and loss is that its flat band survives the addition of $\mathcal{P}\mathcal{T}$ symmetry for any amount of gain and loss and also survives the presence of anisotropic couplings. For the other two lattices, any presence of gain and loss destroys their flat bands. For all three ribbons, there are finite stability windows whose size decreases with the strength of the gain and loss parameter. For the Lieb and kagome cases, the size of this window converges to a finite value. The existence of finite stability windows plus the constancy of the Lieb flat band are in marked contrast to the behavior of a pure one-dimensional lattice.

• Received 7 October 2015

DOI:https://doi.org/10.1103/PhysRevA.92.063813