Parity-time symmetry in a flat-band system

Abstract

In this paper we introduce Parity-Time (PT ) symmetric perturbation to a one-dimensional Lieb lattice, which is otherwise P-symmetric and has a flat band. In the flat band there are a multitude of degenerate dark states, and the degeneracy N increases with the system size. We show that the degeneracy in the flat band is completely lifted due to the non-Hermitian perturbation in general, but it is partially maintained with the half-gain-half-loss perturbation and its "V" variant that we consider. With these perturbations, we show that both randomly positioned states and pinned states at the symmetry plane in the flat band can undergo thresholdless PT breaking. They are distinguished by their different rates of acquiring non-Hermicity as the PT -symmetric perturbation grows, which are insensitive to the system size. Using a degenerate perturbation theory, we derive analytically the rate for the pinned states, whose spatial profiles are also insensitive to the system size. Finally, we find that the presence of weak disorder has a strong effect on modes in the dispersive bands but not on those in the flat band. The latter respond in completely different ways to the growing PT -symmetric perturbation, depending on whether they are randomly positioned or pinned.

Anomalous Minimum and Scaling Behavior of Localization Length Near an Isolated Flat Band

“…The latter approach was used in Ref. [7] where E A in one unit cell of the Lieb lattice is detuned by G/10, and the resulting defect state confirms an evanescent tail given by Eq. (6).…”

“…Systems that exhibit flat bands have attracted considerable interest in the past decades, including optical [1,2] and photonic lattices [3][4][5][6][7], graphene [8,9], superconductors [10][11][12][13], fractional quantum Hall systems [14][15][16] and exciton-polariton condensates [17,18]. As the name suggests, a flat band is dispersionless, or in other words, its density of states (DOS) diverges at a particular energy, known as the flat band energy.…”

Localized gap modes in nonlinear dimerized Lieb lattices

“…However, the presence of Kerr nonlinearity in the system may exhibit conical diffraction at the Dirac cone [30,34]. Due to their simple geometry, Lieb lattices are in the focus of research in ultracold systems as an optical trap for fermions [35], providing existence of magnon Hall effect in spite of the presence of inversion symmetry [18], protection and formation of robust zero modes localized at point defects [36], optimization of the BCS critical temperature and superfluid weight [37], and lifting the flat band modes by PT-symmetric perturbation due to thresholdless PT-symmetry breaking [38]. Moreover, it has been shown that the Lieb lattice acts as a good platform for analyzing various topological transitions in Chern insulators [20,[39][40][41].…”

Flat bands in lattices with non-Hermitian coupling

“…Recently, Ramezani has shown that a flat band with completely real eigenvalues can exist in a quasi-1D PT photonic lattice 36 . This PT symmetric flat band was realized by fine-tuning gain/loss levels, and it is not obvious what features of this model, compared to earlier models [33][34][35] , make it possible. Here, we show that PT symmetric flat bands generically occur in non-Hermitian lattices with a bipartite sublattice symmetry hosting a differing number of sites per sublattice 37,38 .…”

“…An important question is whether the existence of flat bands is compatible with non-Hermiticity, and if so whether the presence of EPs might alter the behavior of the flat band states. Previous studies have analyzed how Hermitian flat bands are changed by the application of non-Hermitian perturbations [33][34][35] , finding either that the symmetries protecting the flat band states are spoilt, or that the flat bands simply acquire nonzero gain or loss (breaking PT symmetry). Recently, Ramezani has shown that a flat band with completely real eigenvalues can exist in a quasi-1D PT photonic lattice 36 .…”