Exceptional topology of non-Hermitian systems

The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. In particular, how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm is discussed. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open Fermi-Seifert surfaces, where conventional band-touching points are replaced by the aforementioned exceptional degeneracies. Furthermore, new notions of gapped phases including topological phases in single-band systems are detailed, and the manner in which a given physical context may affect the symmetry-based topological classification is clarified. A unique property of NH systems with relevance beyond the field of topological phases consists of the anomalous relation between bulk and boundary physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, a picture of intriguing phenomena such as the NH bulk-boundary correspondence and the NH skin effect is put together. Finally, applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits, and mechanical systems and genuine quantum systems such as electronic transport settings at material junctions and dissipative cold-atom setups are reviewed.

Figure 1

(a) Schematic depiction of the Hatano-Nelson model; see Eq. (5). (b) Sum of absolute squares of amplitudes per site of all right eigenstates for the Hamiltonian in Eq. (5) with OBCs for 100 sites and $|J_L|/|J_R|=2$. Inset: energy in the complex plane, which winds around the base point $E_B$ in the complex plane with winding number $w=1$ when $|J_L|>|J_R|$ (in this case $|J_L|/|J_R|=2$). (c)–(g) Absolute value of Eq. (8) for different values of $\mu$, as indicated: (c) $\mu =0$. (d) $\mu =0.5$. (e) $\mu =1$. (f) $\mu =1.5$. (g) $\mu =2$. The plotting axes are shown in (c), and the color corresponds to the argument of $E_{\mu }$ with the color bar in (g). For $\mu$ increasing from zero, two vortices ($E_{\mu }=0$) split and merge again when $\mu \to 2$. The vortices are shown by red (gray) spheres. There is an additional zero-energy solution when $\mu =0$, which is shown with a black sphere in (c), at $(k_x,k_y)=(-\pi /2,0)$, which is gapped rather than split upon increasing $\mu$. Note that if we were to decrease $\mu$ from 0 to $-2$, it would be this (black) zero that splits into vortices, and the other zero (red) would be gapped out.

Figure 2

(a) Solutions to $d_R^2-d_I^2=0$ (solid line) and ${\mathbf{d}}_R·{\mathbf{d}}_I=0$ (dashed line) [cf. Eq. (12)] form closed loops in a two-dimensional parameter space. Exceptional points appear when both equations are satisfied simultaneously, i.e., when the two loops intersect. (b) Exceptional points are connected by (imaginary) Fermi arcs: When ${\mathbf{d}}_R·{\mathbf{d}}_I=0$ and $d_R^2-d_I^2<0$ (gray region), $\mathrm{Re}[E]=0$ [green (light gray) line], while $\mathrm{Im}[E]=0$ [dark green (dark gray)] when ${\mathbf{d}}_R·{\mathbf{d}}_I=0$ and $d_R^2-d_I^2>0$ (outside the gray region).

Figure 3

Schematic plot of the energy spectrum of a Hermitian, two-dimensional Weyl node $H=k_x{\sigma }_x+k_y{\sigma }_y$. Upon adding a mass term $\epsilon {\sigma }_z$, a gap opens in the spectrum (here shown for $\epsilon =0.1$). When instead an imaginary term $i{b}_z{\sigma }_z$ is added to the Hamiltonian, a ring of exceptional points, i.e., an exceptional ring, appears (here shown in red for $b_z=0.3$). The addition of an imaginary term $i{b}_x{\sigma }_x$ leads to the appearance of exceptional points (here shown in red for $b_x=0.3$). The orange plots represent the absolute value of the energy $\pm |E|$, which for the Hermitian case simply corresponds to the energy $E$, whereas the pink and blue plots show the real and imaginary parts of the energy $\mathrm{Re}[E]$ and $\mathrm{Im}[E]$, respectively.

Figure 4

Exceptional rings or knots where the energy is degenerate [in red (dark gray)] and Seifert surfaces where $\mathrm{Re}[E]=0$ [in green (light gray)] appearing in the spectra of short-range hopping models resulting in (a) an exceptional ring, (b) a trefoil knot, (c) two exceptional rings, and (d) a Hopf link. See [41] for Hamiltonian details.

Figure 5

Schematic depiction of (a) gaps in Hermitian models, (b) point gaps and (c) line gaps in NH models, with the bulk bands shown in red (gray).

Figure 6

(a) Schematic depiction of the NH-SSH model (top panel) [see Eq. (22)] and the Lee model (bottom panel) ([149]) and their unitary equivalence. (b) Sum of absolute squares of amplitudes per site of all right eigenstates for the Hamiltonian in Eq. (22) with OBCs for $t_1=2.2$, $t_2=1$, $\gamma =1.5$, and 30 unit cells. For this choice of parameters the magnitude of hopping to the left ($t_1+\gamma =3.7$) is larger than hopping to the right ($t_1-\gamma =0.7$), and we observe a piling up of states at the left end. Inset: absolute value of the eigenvalues as a function of $t_1$ for the same parameter choice with OBCs and PBCs in blue (dark gray) and light gray, respectively, and with the in-gap end states in the OBC case in red (dashed lines). The nonzero value of the winding number is explicitly indicated by green shaded areas and the black line corresponds to the value of $t_1$ for which the wave-function localization is plotted. We note that because the PBC and OBC Hamiltonians for the NH-SSH model and Lee’s model are related via a unitary transform $U_N$, the PBC and OBC spectra, respectively, are identical for Lee’s model.

Figure 7

(a) Domain-wall geometry between a NH-SSH domain [cf. Eq. (22)] [top half circle with lattice sites in red and blue (gray and black)] and a Hermitian region [bottom half circle with sites in green and purple (light and dark gray)]. The NN hopping parameters for the NH-SSH chain are $t_1$, $t_2$, and $\gamma$, while in the Hermitian part they are $t_1^{\prime }$ and $t_2^{\prime }$. The two chains are coupled to each other via the hopping parameter $t_2^{\prime }$. (b)–(d) Absolute value of the eigenvalues as a function of $t_1$ for $t_2=1$, $\gamma =1.5$, $t_2^{\prime }=0.5$, and $N=18$ unit cells in both chains, and (b) $t_1^{\prime }=3$, (c) $t_1^{\prime }=1$, and (d) $t_1^{\prime }=1/2$. The spectrum in the Hermitian SSH chain is (b) gapped, (c) gapped with a smaller gap, and (d) gapless. The black, dashed grid lines correspond to the gap closings in the PBC spectrum of the NH-SSH chain, showing that even though the NH model is coupled to a Hermitian chain via a domain wall, the anomalous physics persists when the gap in the Hermitian chain is large enough.