Anomalous minimum and scaling behavior of localization length near an isolated flat band

Using one-dimensional tight-binding lattices and an analytical expression based on the Green's matrix, we show that anomalous minimum of the localization length near an isolated flat band, previously found for evanescent waves in a defect-free photonic crystal waveguide, is a generic feature and exists in the Anderson regime as well, i.e., in the presence of disorder. Our finding reveals a scaling behavior of the localization length in terms of the disorder strength, as well as a summation rule of the inverse localization length in terms of the density of states in different bands. Most interesting, the latter indicates the possibility of having two localization minima inside a band gap, if this band gap is formed by two flat bands such as in a double-sided Lieb lattice.Comment: 8 pages, 9 figure

Parity-time symmetry in a flat band system

In this paper we introduce Parity-Time ($\cal PT$) symmetric perturbation to a one-dimensional Lieb lattice, which is otherwise $\cal P$-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 $\cal PT$ breaking. They are distinguished by their different rates of acquiring non-Hermicity as the $\cal PT$-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 $\cal PT$-symmetric perturbation, depending on whether they are randomly positioned or pinned.Comment: 8 pages, 8 figure

Three-dimensional quantum phase diagram of the exact ground states of a mixture of two species of spin-1 Bose gases with interspecies spin exchange

We find nearly all the exact ground states of a mixture of two species of spin-1 atoms with both interspecies and intraspecies spin exchanges in absence of a magnetic field. The quantum phase diagram in the three-dimensional parameter space and its two-dimensional cross sections are described. The boundaries where the ground states are either continuous or discontinuous are determined, with the latter identified as where quantum phase transitions take place. The two species are always disentangled if the interspecies spin coupling is ferromagnetic or zero. Quantum phase transitions occur when the interspecies spin coupling varies between antiferromagtic and zero or ferromagnetic while the two intraspecies spin couplings both remain ferromagnetic. On the other hand, by tuning the interspecies spin coupling from zero to antiferromagnetic and then back to zero, one can circumvent the quantum phase transition due to sign change of the intraspecies spin coupling of a single species, which is spin-decoupled with the other species with ferromagnetic intraspecies spin coupling. Overall speaking, interplay among interspecies and two intraspecies spin exchanges significantly enriches quantum phases of spinor atomic gases.Comment: 16 pages, received by PRA on 23 August 201

Optical Reciprocity Induced Symmetry of the Scattering Eigenstates in Non-PT\cal PT-Symmetric Heterostructures

The scattering matrix $S$ obeys the unitary relation $S^\dagger S=1$ in a Hermitian system and the symmetry property ${\cal PT}S{\cal PT}=S^{-1}$ in a Parity-Time (${\cal PT}$) symmetric system. Here we report a different symmetry relation of the $S$ matrix in a one-dimensional heterostructure, which is given by the amplitude ratio of the incident waves in the scattering eigenstates. It originates from the optical reciprocity and holds independent of the Hermiticity or $\cal PT$ symmetry of the system. Using this symmetry relation, we probe a quasi-transition that is reminiscent of the spontaneous symmetry breaking of a $\cal PT$-symmetric $S$ matrix, now with unbalanced gain and loss and even in the absence of gain. We show that the additional symmetry relation provides a clear evidence of an exceptional point, even when all other signatures of the $\cal PT$ symmetry breaking are completely erased. We also discuss the existence of a final exceptional point in this correspondence, which is attributed to asymmetric reflections from the two sides of the heterostructure.Comment: 5 pages, 4 figure