Higher-Order Topological Corner States Induced by Gain and Loss

Higher-order topological insulators and superconductors are topological phases that exhibit novel boundary states on corners or hinges. Recent experimental advances in controlling dissipation such as gain and loss in atomic and optical systems provide a powerful tool for exploring non-Hermitian topological phases. Here we show that higher-order topological corner states can emerge by introducing staggered on-site gain and loss to a Hermitian system in a trivial phase. For such a non-Hermitian system, we establish a general bulk-corner correspondence by developing a biorthogonal nested-Wilson-loop and edge-polarization theory, which can be applied to a wide class of non-Hermitian systems with higher-order topological orders. The theory gives rise to topological invariants characterizing the non-Hermitian topological multipole moments (i.e., corner states) that are protected by reflection or chiral symmetry. Such gain- and loss-induced higher-order topological corner states can be experimentally realized using photons in coupled cavities or cold atoms in optical lattices.

Figure 1

(a) Lattice representation of the non-Hermitian model in Eq. (1). All sites in blue (red) have particle loss (gain) with a rate $\gamma$. $\varphi$ is the magnetic flux for each plaquette, and $J_{h,v}$ (${\lambda }_{h,v}$) are the tunneling amplitudes between sites in different color (shape) along the horizontal and vertical directions, respectively. (b) Experimental implementation of the lattice model in (a) using coupled arrays of microring cavities.

Figure 2

[(a) and (b)] Energy spectra of the non-Hermitian Hamiltonian Eq. (1) with open boundaries in both directions. The bulk energy gap closes at $|\gamma |={\gamma }_c$, where a topological phase transition occurs and in-gap corner states (red curves with fourfold degeneracy) emerge at $|\gamma |>{\gamma }_c$. (c) Typical density distributions $|{\mathrm{\Psi }}_{\text{corner}}^R(x,y){|}^2$ of the four corner states, with $|{\mathrm{\Psi }}_{\text{corner}}^R(x,y)⟩$ being the right eigenstate. The inset shows the corresponding eigenenergies around the four corner states with $\mathrm{Re}[E]=0$. (d) Typical density distributions of the bulk states. $\gamma =2$ in (c) and (d). Common parameters are system size $N_h=N_v=20$ (unit cells), $J=\sqrt{2}$, and ${\lambda }_v=1$ (leading to ${\gamma }_c=\sqrt{2}$). We set ${\lambda }_h=1$ as the energy unit.

Figure 3

(a) Phase diagram in the $\gamma \text{−}{\lambda }_v$ plane for $J=\sqrt{2}{\lambda }_h$, with one trivial phase [yellow area with $(Q_1,Q_2)=(0,0)$] and two topological phases [$T\text{−}\mathrm{I}$, orange area with (1, 0); and $T\text{−}\mathrm{II}$, green area with (0,1)]. The patterned region has a vanishing Wannier-band gap [98]. (b) Wannier-band structures with ${\lambda }_v={\lambda }_h$ and $\gamma =2$ [the blue square in (a)]. The imaginary parts are locked at 0. (c) Wannier-sector (green circles) and edge (purple circles) polarizations as well as topological invariant $Q_1$ (red squares) and winding number $W$ (blue stars), with ${\lambda }_v={\lambda }_h$ [the thin green line in (a)]. (d) Complex energy spectra with open boundaries [$N_h=N_v=20$ (unit cells)] and ${\lambda }_v=0.6{\lambda }_h$ [the thin blue line in (a)]. The bulk gap persists upon the phase transition. We set ${\lambda }_h=1$ as the energy unit.

Figure 4

[(a) and (b)] The horizontal (blue circles) and vertical (red crosses) polarizations for ${\lambda }_v=0.6{\lambda }_h$ [along the thin blue line in Fig. 3]. The polarization $p_h^{{ϵ}_v}$ is ill defined in the patterned region due to the vanishing gap between Wannier sectors. The phase boundaries are given by the solid lines. [(c) and (d)] The Wannier values (${ϵ}_{h,j}$, ${ϵ}_{v,j}$) and edge-polarization distributions ($p_h^{i_v}$, $p_v^{i_h}$) for the cylindrical geometry with ${\lambda }_v=0.6{\lambda }_h$ and $\gamma =1.9$ [indicated by the red cross in Fig. 3]. Blue circles (red crosses) are the results for open boundary along the horizontal (vertical) direction. The in-gap Wannier values at $\frac{1}{2}$ are responsible for the edge polarization. Other parameters are the same as in Fig. 3.