Chiral symmetry in non-Hermitian systems: Product rule and Clifford algebra

Chiral symmetry provides the symmetry protection for a large class of topological edge states. It exists in non-Hermitian systems as well, and the same anticommutation relation between the Hamiltonian and a linear chiral operator, i.e., $\{H,\mathrm{\Pi }\}=0$, now warrants a symmetric spectrum about the origin of the complex energy plane. Utilizing two general approaches to identify and generate chiral symmetry, we first show that its symmetry operator in non-Hermitian systems can go beyond simple spatial transformations such as parity or rotation and include imaginary gauge transformations in a systematic way. Furthermore, we reveal hidden non-Hermitian chiral symmetry and its associated particle-hole symmetry, where their operators take unfamiliar forms due to the presence of energy nonconserving elements. Finally, our implementation of non-Hermitian chiral symmetry in a topological lattice leads to an edge state with “folded” localization, where its tail is reflected by the opposite edge and resides on a separate sublattice.

Figure 1

Generating non-Hermitian chiral symmetry by the product rule. (a), (d) Schematics of two tight-binding lattices. Solid and dashed links indicate couplings $t\in \mathbb{R}$ and $t^{\prime }=0.3t$. (b), (c) Real and imaginary parts of the complex spectrum for the lattice in (a) as a function of the gain and loss strength $\tau$. The black dot marks its exceptional point. (e), (f) Same as (b), (c) but for the lattice in (d).

Figure 2

Localized non-Hermitian zero mode at an EP. (a) Schematic of a square lattice before an imaginary gauge transformation is applied. Horizontal and vertical couplings are given by $t$ and $1.1t$, respectively. (b)–(d) Spatial profile of a zero mode at an EP with a modified coupling $t^{\prime }=0.7t$, $0.5t$, and $0.4t$, respectively. See the main text for the application of this imaginary gauge transformation.

Figure 3

A hidden non-Hermitian chiral symmetry. (a) Schematic of a four-cavity ring with complex couplings $g_1\equiv g_{1r}+i{g}_{1i}$, $g_2\equiv g_{2r}+i{g}_{2i}$, and their complex conjugates. Blue and orange denote the on-site detunings $\pm \mathrm{\Delta }$. (b) Cross section of four coupled waveguides embedded in a silica substrate that emulate the tight-binding model shown schematically in (a). The four waveguides are denoted by ${\mathcal{D}}_{1–4}$, and the rectangular regions between two nearest neighbors are denoted by ${\mathcal{M}}_{13},{\mathcal{M}}_{14},{\mathcal{M}}_{23},{\mathcal{M}}_{24}$. Complex couplings $g_1$ and $g_2$ are achieved by locally introducing gain ($-ir$ in the dielectric constant) in ${\mathcal{M}}_{13},{\mathcal{M}}_{23}$, while $g_1^{*}$ and $g_2^{*}$ are implemented by loss ($ir$ in the dielectric constant) in ${\mathcal{M}}_{14},{\mathcal{M}}_{24}$. (c), (d) Trajectories of the complex eigenvalues $\lambda$ in the paraxial equation when $r$ is increased from 0 to 0.009 with (c) $\mathrm{\Delta }=1.1216{\mathrm{nm}}^{-1}$ and (d) $\mathrm{\Delta }=1.1216+0.6990i{\mathrm{nm}}^{-1}$. Black dots show finite-difference simulations of (b), which are captured qualitatively by the tight-binding model (thick solid lines). The arrows indicate the motion of $\lambda$'s with increasing $r$, and the real and imaginary axes are shown as thin solid lines. The parameters of the tight-binding model used are $g_{1i}=-2.3895r{\mathrm{nm}}^{-1}$, $g_{2i}=-18.648r{\mathrm{nm}}^{-1}$, $g_{1r}=0.4107{\mathrm{nm}}^{-1}$, $g_{2r}=2.1054{\mathrm{nm}}^{-1}$. They give the best fit for the trajectories of the finite-difference solutions of the paraxial equation.

Figure 4

Non-Hermitian extension of a topological lattice. (a) Schematic of $H_s$ given by Eq. (20) (top) and its equivalent form (middle). The bottom shows a similar lattice but with different detunings, and its complex band structure (solid lines) is shown in (b). Shaded areas represent the projection of the bulk band gaps on the real energy axis. Solid and open dots in (b) indicate the energy of the edge states in (c) and (d), respectively. They are finite in length with 64 and 63 sites, where purple, orange, and gray denote the on-site detunings $\pm \beta$ and 0. Inset in (c): Amplitude of this edge state in the first (solid) and last (dashed) cavities of all 16 unit cells. $g_2/g_3=0.8$ and $\beta /g_3=-0.3+0.01i$ are used.

Figure 5

Hermitian vs non-Hermitian chiral symmetry. (a) Schematic of a finite-size honeycomb lattice with only nearest-neighbor couplings. The two sublattices $A$ and $B$ are marked by solid and open dots. (b) Its non-Hermitian counterpart with also NHPH and $\mathcal{P}\mathcal{T}$ symmetries. (c), (d) Complex spectra for (a) and (b) at $\tau =\sqrt{2g}$. Dots of increasing sizes indicate nondegenerate, doubly degenerate, and triply degenerate states. (e), (f) Spatial profiles of the zero modes in (c) and (d).

Figure 6

Intensity patterns of two propagation modes in Fig. 3. The insets show their eigenvalues of the paraxial equation in the unit of ${\mathrm{nm}}^{-1}$.

Figure 7

Non-Hermitian chiral symmetry of a 3D pyramid. (a) Schematic of the system. Purple and orange denote the on-site detunings $\pm \beta$. (b) Its eigenvalue spectrum with $g_2=g_1\in \mathbb{R}$, $g_3=0.8g_1$, and detuning $\alpha e^{i\pi /4}{\gamma }^3{\gamma }^5$, where $\alpha$ is increased from 0 to 2. (c) Same as (d) but with an additional detuning $\alpha e^{i\pi /3}{\gamma }^1{\gamma }^2$.