Pseudochirality: A Manifestation of Noether’s Theorem in Non-Hermitian Systems

Noether’s theorem relates constants of motion to the symmetries of the system. Here we investigate a manifestation of Noether’s theorem in non-Hermitian systems, where the inner product is defined differently from quantum mechanics. In this framework, a generalized symmetry that we term pseudochirality emerges naturally as the counterpart of symmetries defined by a commutation relation in quantum mechanics. Using this observation, we reveal previously unidentified constants of motion in non-Hermitian systems with parity-time and chiral symmetries. We further elaborate the disparate implications of pseudochirality induced constant of motion: It signals the pair excitation of a generalized “particle” and the corresponding “hole” but vanishes universally when the pseudochiral operator is antisymmetric. This disparity, when manifested in a non-Hermitian topological lattice with the Landau gauge, depends on whether the lattice size is even or odd. We further discuss previously unidentified symmetries of this non-Hermitian topological system, and we reveal how its constant of motion due to pseudochirality can be used as an indicator of whether a pure chiral edge state is excited.

Figure 1

Constants of motion due to pseudochirality in a non-Hermitian dimer. (a),(b) Intensity evolution in the dimer as a function of time. $b_{1,2}=1$, 1 in (a) and $-\sqrt{2},0$ in (b), reflected by the asymmetric and symmetric couplings in the insets. In both cases ${\psi }_1=2$, ${\psi }_2=1$ at $t=0$ and $b_3=1+0.1i$. Spatial Gaussian mode profiles are imposed along $x$. (c),(d) Constant(s) of motion in the two cases.

Figure 2

Non-Hermitian symmetries in a topological lattice with the Landau gauge. (a) Schematic showing the couplings and the gain (left) and loss (right) edges. (b) Its band structure showing one quarter of the first Brillouin zone. The lattice constant is set to 1. (c) Eigenvalues of $H$ in an $11\times 11$ lattice. Clustered dots in BD1 give the CW chiral edge band, and the large dot shows the mode plotted in Fig. 3. (d) Same as (c) but with gain on both edges. High-gain modes are enlarged.

Figure 3

Constant of motion in a topological lattice with the Landau gauge. (a),(b) A CW chiral edge mode and the only zero mode (${\omega }_{\mu }=0$) in the system with $n=11$, respectively. (c) Constant of motion ($A$) for a pure CW chiral edge state (solid). Dashed and dash-dotted lines show, respectively, its values when the zero mode and the CCW chiral edge state is also excited. (d) Same as (c) but for $n=10$ where a zero mode does not exist.