Controlling energy spectra and skin effect via boundary conditions in non-Hermitian lattices

Abstract

Non-Hermitian systems exhibit unique spectral properties, including the non-Hermitian skin effect and exceptional points, often influenced by boundary conditions. The modulation of these phenomena by generalized boundary conditions remains unexplored and not understood. Here, we analyze the Hatano-Nelson model with generalized boundary conditions induced by complex hopping amplitudes at the boundary. Using similarity transformations, we determine the conditions yielding real energy spectra and skin effect, and identify the emergence of exceptional points where spectra transition from real to complex. We demonstrate that tuning the boundary hopping amplitudes precisely controls the non-Hermitian skin effect, i.e., the localization of eigenmodes at the lattice edges. These findings reveal the sensitivity of spectral and localization properties to boundary conditions, providing a framework for engineering quantum lattice models with tailored spectral and localization features, with potential applications in quantum devices.

Figures (4)

• Figure 1: A sketch of the two isospectral Hamiltonians $H$ in \ref{['eq:Hamiltonian']} and $\widetilde{H}$ in \ref{['eq:TransformedHamiltonian']}. The non-Hermitian Hamiltonian $H$ coincides with the Hatano-Nelson model with generalized boundary conditions or, alternatively, in the presence of a defective link. The Hermitian Hamiltonian $\widetilde{H}$ coincides with that of a lattice fermion with generalized boundary conditions or in the presence of a defective link.
• Figure 2: The energy spectra with exceptional points of the two isospectral Hamiltonians $H$ and $\widetilde{H}$ with $\alpha_{\rm L}=r e^ {\frac{1}{2} q N }$$\alpha_{\rm R}= e^ {-\frac{1}{2} q N }$ as a function of $r$ for lattices with $N=4$ (a) and $N=10$ (b), with $q=4$.
• Figure 3: Energy spectra of the two isospectral Hamiltonians $H$ and $\widetilde{H}$ in \ref{['eq:Hamiltonian', 'eq:TransformedHamiltonian']}) on a lattice with $N=10$ lattice sites and with $\alpha_{\rm L}$ and $\alpha_{\rm R}$ as in \ref{['eq:alphas']} as a function of $\rho$ varying continuously in the interval $[0,q/2]$ and for different choices of $\phi$ and $q$. We take $\phi=0$ [first row, (a), (b)], $\phi=\pi/2$ [second row, (c), (d)], $\phi=\pi$ [third row, (e), (f)], with $q=4$ [first column, (a), (c), (e)], $q=4+{\mathrm i}\pi$ [second column, (b), (d), (f)]. Note that 4 out of $N=10$ eigenvalues are doubly degenerate. The highlighted points correspond to $\rho=0,1,2$, with the exceptional point $\rho=1$ corresponding to a purely real energy spectrum, in agreement with the condition in \ref{['eq:specialBC']}. The trajectories followed by the complex eigenvalues as a function of $\rho$ swirl about the origin with a rotation angle that increases with $\Im(q)$, and their topology is affected by the choice of $\phi$.
• Figure 4: Average probability density for the right and left eigenmodes of the Hamiltonians $H$ and $\widetilde{H}$ in \ref{['eq:Hamiltonian', 'eq:TransformedHamiltonian']} on a lattice with $N=10$ lattice sites, with $\alpha_{\rm L}$ and $\alpha_{\rm R}$ as in \ref{['eq:alphas']} as a function of $\rho$ varying continuously in the interval $[0,2]$ with $q=4$ and $\phi=0$. (a) Right eigenmode and (b) left eigenmode of $H$. The eigenmodes of $H$ exhibit skin effect with exponential localization at the boundary for any $\rho\neq0$. (c) Right eigenmode and (d) left eigenmode of $\widetilde{H}$. In this case, the eigenmodes of $\widetilde{H}$ exhibit skin effect with exponential localization at the boundary for any $\rho\neq1$. The right and left eigenmodes of the Hamiltonians $H$ and $\widetilde{H}$ are given by \ref{['eq:eigenmodes']}. Different choices of the parameters $q$, $\phi$, and of the lattice size $N$ lead to qualitatively similar outcomes.