Flat bands in lattices with non-Hermitian coupling

We study non-Hermitian photonic lattices that exhibit competition between conservative and non-Hermitian (gain/loss) couplings. A bipartite sublattice symmetry enforces the existence of non-Hermitian flat bands, which are typically embedded in an auxiliary dispersive band and give rise to nondiffracting “compact localized states”. Band crossings take the form of non-Hermitian degeneracies known as exceptional points. Excitations of the lattice can produce either diffracting or amplifying behaviors. If the non-Hermitian coupling is fine-tuned to generate an effective $\pi$ flux, the lattice spectrum becomes completely flat, a non-Hermitian analog of Aharonov-Bohm caging in which the magnetic field is replaced by balanced gain and loss. When the effective flux is zero, the non-Hermitian band crossing points give rise to asymmetric diffraction and anomalous linear amplification.

Figure 1

Schematic of trimers with diagonal and off-diagonal gain/loss and their energy spectra. (a) Diagonal gain/loss $V$: red (blue) sites have net gain (loss). Complex energy eigenvalues $E$ emerge at $V_c\approx 0.42C$ when neighboring modes coalesce at EP. (b) Non-Hermitian coupling $\gamma$: red (blue) links indicate net gain (loss) occurs when neighboring sites are in phase, and vice versa when out of phase. Complex energy eigenvalues emerge at ${\gamma }_c\approx 1.22C$ via ${\mathrm{EP}}_2$ only after first crossing ${\mathrm{EP}}_1$. Here $\mathrm{\Delta }=2C$.

Figure 2

(a) Diamond chain of resonators with antisymmetric non-Hermitian couplings, which form a non-Hermitian Aharonov-Bohm cage. Compact localized eigenmode amplitudes with phases $\theta =2\arg (C+i\gamma )$ are indicated by shaded sites. (b) Plot of band energies, given by Eq. (5), versus the non-Hermiticity parameter $\gamma /C$. The spectrum flattens with increasing $\gamma /C$; complex eigenvalues with nonzero $\mathrm{Im}\left(E\right)$ (red curves) appear only after the two dispersive bands merge at ${\mathrm{EP}}_2, \gamma /C\approx 1.1$. The other model parameters are $\alpha =1/2, \mathrm{\Delta }=2C$, and lattice size $N=20$ cells, with periodic boundary conditions. (c),(d) Intensity profile $|{\mathrm{\Psi }}_n{\left(z\right)|}^2$ of a localized $b$ or $a$ sublattice excitation, when the lattice is tuned to ${\mathrm{EP}}_1$. (c) The $b$ input excites an eigenstate that oscillates periodically. (d) The $a$ input leads to quadratically growing total intensity.

Figure 3

(a) Stub lattice hosting a non-Hermitian flat band embedded in a dispersive band. Compact localized eigenmode amplitudes with phase $\theta =\arg (C+i\gamma )$ are indicated by shaded sites (here, $r^2\equiv {[1+{\gamma }^2/C^2]}^{-1}$). (b) Plot of band energies, given by Eq. (5), versus the non-Hermiticity parameter $\gamma /C$. Complex eigenvalues [nonzero $\mathrm{Im}\left(E\right)$ in red] emerge only when the two dispersive bands merge at ${\mathrm{EP}}_2$, first for modes at the Brillouin zone edge ($k=\pm \pi$) and progressing towards $k=0$. We take $\mathrm{\Delta }=2C$, and a lattice size of $N=20$ cells with periodic boundary conditions. (c),(d) Intensity profile $|{\mathrm{\Psi }}_n{\left(z\right)|}^2$ of a localized $b$ or $a$ sublattice excitation, when the lattice is tuned to $\gamma =2C/3$. (c) The $b$ excitation excites dispersive bands, producing discrete diffraction. (d) The $a$ excitation excites both flat and dispersive bands, producing asymmetric diffraction and linear intensity amplification.

Figure 4

Disordered non-Hermitian Aharonov-Bohm cage. (a) Mean eigenvalue shifts $\langle \left|\delta E\right|\rangle$ versus disorder strength $W$, averaging over 200 disorder samples. The non-Hermitian ($E=0$; blue) and Hermitian-like ($E=2C$; red) flat bands scale as $\sim \sqrt{W}$ and $\sim W$, respectively. Shaded regions denote the standard deviation. (b) Mean participation number. All the eigenmodes are strongly localized, $P\sim P_{\mathrm{CLS}}$, and the localization is insensitive to $W$. Here $\gamma =C$ and $\mathrm{\Delta }=2C$, as in Fig. 2, and the lattice size is $N=40$ unit cells with periodic boundary conditions.

Figure 5

Mean participation number versus disorder strength $W$ for the disordered embedded flat band lattice, showing the eigenstates at the flat band energy ($E=0$; blue) and in the center of the upper dispersive band ($E\approx 2.8C$). The results are averaged over 20 disorder samples, and shaded regions denote the standard deviation. Here $\gamma =2C/3$ and $\mathrm{\Delta }=2C$, as in Fig. 3, and the lattice size is $N=500$ unit cells with periodic boundary conditions.