Van Hove bound states in the continuum: Localized subradiant states in finite open lattices

We show that finite lattices with arbitrary boundaries may support large degenerate subspaces, stemming from the underlying translational symmetry of the lattice. When the lattice is coupled to an environment, a potentially large number of these states remains weakly or perfectly uncoupled from the environment, realizing a new kind of bound states in the continuum. These states are strongly localized along particular directions of the lattice, which, in the limit of strong coupling to the environment, leads to spatially localized subradiant states.

Figure 1

(a) Number of degenerate states at the VH energy, $n_{\text{VH}}$, for a triangular lattice with $L_x$ columns and $L_y$ rows. (b) Same as (a) for a triangular lattice with a central hole of radius 1/6 of the smaller size (Sinai billiard). (c) Even-odd effect in $n_{\text{VH}}$ for $L_x\times L_x$ lattices without (blue squares) and with (orange circles) a circular hole at the center. (d) Example $L_x\times L_y=20\times 20$ lattice with a circular hole of radius $R=3.25$. Each dot is a lattice site, while the dashed line indicates the boundary of the central hole.

Figure 2

Properties of the eigenstates of a $23\times 23$ Sinai lattice with losses in the 12 leftmost sites of the bottom row, with loss rate $\gamma =0.102$ (left column), $\gamma =1.002$ (central column), and $\gamma =10.002$ (right column). Top row: IPR vs $\left|\text{Im}\right(E\left)\right|$. Middle row: IPR vs $\text{Re}\left(E\right)$. Bottom row: $\left|\text{Im}\right(E\left)\right|$ vs $\text{Re}\left(E\right)$. Labels (A), (B), (C) in panels (a1)–(a3) refer to the subspaces defined in Sec. 2. Curves at the top and right of each panel show the marginal distributions of the data. Note panel (c3) is a semilogarithmic plot.

Figure 3

(a) DOS of a $9\times 9$ square with connections to fibers on three sites at the top and bottom boundaries as shown in the left inset. The right inset shows the DOS divergence close to $E_{\text{VH}}=2t$. (b) Transmittance of the same system, normalized to the input flux on a single fiber. The inset highlights the absence of resonant transmission features at $E_{\text{VH}}$.

Figure 4

Examples of VH lattice scars. The intensity of red color is proportional to the probability density at each lattice site (circles) after a long time evolution of a Gaussian wave packet, subject to localized losses at the sites marked by blue diamonds. (a) $19\times 19$ square triangular lattice with losses in the 17 bottom-left sites. (b) $19\times 19$ Sinai triangular lattice with losses in four sites in the upper right corner. The dimensions of the VHD for each closed system are (a) $n_{\text{VH}}=18$ and (b) 5; only one BIC remains in each system after coupling to the environment.