Symmetric non-Hermitian skin effect with emergent nonlocal correspondence

The non-Hermitian skin effect (NHSE) refers to the extensive number of eigenstates of a non-Hermitian system that are localized in open boundaries. Here, we predict a universal phenomenon that the local particle-hole(-like) symmetry (PHS) leads to nonlocal pairs of skin modes distributed on different boundaries, manifesting a nonlocalization of the local PHS, which is unique to non-Hermitian systems. We develop a generic theory for the emergent nonlocal symmetry-protected NHSE by connecting the non-Hermitian system to an extended Hermitian Hamiltonian in a quadruplicate Hilbert space, which maps the skin modes to the topological zero modes, and the PHS to an emergent nonlocal symmetry in the perspective of many body physics. The predicted nonlocal NHSE is robust against perturbations. We propose optical Raman lattice models to observe the predicted phenomena in all physical dimensions, which are accessible with cold-atom experiments.

Figure 1

The mapping between the skin modes of non-Hermitian Hamiltonian and the topological zero modes of the extended Hermitian Hamiltonian in the 1D and 2D cases. In (a1) and (b1) the blue (red) regions denote the right (left) eigenstates' wavefunction distributions of the skin modes. The right eigenvector ${|E\rangle }_{\mathrm{skin}}$ and the left eigenvector ${\langle \langle -E|}_{\mathrm{skin}}$ (the subscripts are omitted in the figures) are locally related by the PHS $S=UP$ and so do ${\langle \langle -E|}_{\mathrm{skin}}$ and ${|-E\rangle }_{\mathrm{skin}}$. By the construction of the extended Hamiltonian in Eq. (2), these right (left) eigenvectors are mapped to the topological zero modes $|{\chi }_3{\rangle }_{\mathrm{topo}}$ and $|{\chi }_1{\rangle }_{\mathrm{topo}}$ ($|{\chi }_2{\rangle }_{\mathrm{topo}}$ and $|{\chi }_4{\rangle }_{\mathrm{topo}}$) in (a2) and (b2). And the PHS $S$ is mapped to the emergent nonlocal symmetry ${\tilde{S}}^{\prime }$ that relates the spatially separated zero modes when all negative energy bands of the Hermitian Hamiltonian are filled [denoted by the colored background in (b2)].

Figure 2

The spectra, symmetric NHSE, localized states, and the robustness of skin modes of the 1D model in Eq. (8). (a) The OBC (blue, larger dots) and PBC (red, smaller dots) spectra of the model in Eq. (8) with $t_0=t_{so}=1.0,m_z=0.3,{\gamma }_{\downarrow }=0.6,K=2\pi /3$ and the size of the system $L=50$. (b) Density profile of eigenstates in the real space, summed over all eigenstates and spin degree of freedom, in which the inset plots the profile of the two bands' eigenstates respectively. [(c1),(c2)] The wavefunction distribution of the pair of topological zero modes' right eigenvectors ${\psi }_{\mathrm{topo}1/2,\mathrm{R}}$ (c1) and left eigenvectors ${\psi }_{\mathrm{topo}1/2,\mathrm{L}}$ (c2). [(c3),(c4)] The wavefunction distribution of a pair of skin modes' right eigenvectors ${\psi }_{\pm E,R}$ and left eigenvectors ${\psi }_{\pm E,L}$ with eigenenergy $\pm E=\pm (0.27-0.05\mathrm{I})$ (blue and red dashed respectively). (d) The OBC spectrum of the same model but with spin-dependent spin-conserved hopping coefficients $t_{0\uparrow }=0.5,t_{0\downarrow }=1.0$ breaking the PHS. The inset plots the density profile in this case, illustrating the skin modes distributed in opposite ends.

Figure 3

The numerical results and experimental scheme of the 2D model in Eq. (9). (a) The OBC (blue, larger dots) and PBC (red, smaller dots) spectra of the model in Eq. (9) with $t_0^x=t_0^y=1.0,t_{so}^x=-i{t}_{so}^y=0.4,m_z=0.3,{\gamma }_{\downarrow }=0.6,\vec{K}=\pi /a(cos\theta ,sin\theta ),\theta ={50}^{\circ }$ and the size of the system $L_x=L_y=30$. (b) Density profile in the real space, summed over all eigenstates and the spin degree of freedom. [(c1),(c3)] The real space distribution of right eigenvectors $|{\psi }_{\pm E,R}{|}^2$ of a pair of skin modes with eigenenergies $\pm E=\mp (1.63+0.17\mathrm{i})$ respectively, and [(c2),(c4)] the distribution of corresponding left eigenvectors $|{\psi }_{\pm E,L}{|}^2$. (d) The scheme of Raman optical lattice experiment to realize the 2D model in Eq. (9). The green beam propagates along $z$ direction and is ${\sigma }^{+}$ polarized, inducing the effective damping of the ground state. The Raman beam's (black running wave) electric field ${\vec{E}}_R\propto (({\vec{e}}_x-{\vec{e}}_y)/\sqrt{2}+i{\vec{e}}_z)e^{i\vec{K}·\vec{j}}$ is circular polarized and the two linear components of it couple with the two optical lattice beams (red and blue) respectively, inducing the spin-flipped hoppings in the two directions.