Non-Hermitian systems with a real spectrum and selective skin effect

Abstract

In this work we first show a simple approach to constructing non-Hermitian Hamiltonians with a real spectrum, which are \textit{not} obtained by a non-unitary transformation such as the imaginary gauge transformation. They are given, instead, by the product of a Hermitian Hamiltonian $H_0$ and a positive semi-definite matrix $A$. Depending on whether $A$ has zero eigenvalue(s), the resulting $H$ can possess an exceptional point at zero energy. When $A$ is only required to be Hermitian instead, the resulting $H$ is pseudo-Hermitian that can have real and complex conjugate energy levels. In the special case where $A$ is diagonal, we compare our approach to an imaginary gauge transformation, which reveals a selective non-Hermitian skin effect in our approach, i.e., only the zero mode is a skin mode and the non-zero modes reside in the bulk. We further show that this selective non-Hermitian skin mode has a much lower lasing threshold than its counterpart in the standard non-Hermitian skin effect with the same spatial profile, when we pump at the boundary where they are localized. The form of our construction can also be found, for example, in dynamical matrices describing coupled frictionless harmonic oscillators with different masses.

arXiv comment7 pages, 5 figures

References in corpus (17)

• Localization transitions in non-Hermitian quantum mechanics
• Higher-order topological insulators and semimetals on the breathing Kagome and pyrochlore lattices
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Vortex Pinning and Non-Hermitian Quantum Mechanics
• Enhanced sensitivity operation of an optical gyroscope near an exceptional point
• Observation of higher-order non-Hermitian skin effect
• Non-Hermitian Morphing of Topological Modes
• Defect states emerging from a non-Hermitian flat band of photonic zero modes
• Symmetry-protected zero-mode laser with a tunable spatial profile
• Non-Hermitian lattices with a flat band and polynomial power increase
• Imaginary gauge transformation in momentum space and Dirac exceptional point
• Chiral symmetry in non-Hermitian systems: product rule and Clifford algebra
• Pseudo-chirality: a manifestation of Noether's theorem in non-Hermitian systems
• Analysis of Dirac exceptional points and their isospectral Hermitian counterparts
• Non-Hermitian gauged laser arrays with localized excitations: Anomalous threshold and generalized principle of selective pumping
• Linear localization of zero modes in weakly coupled non-Hermitian reservoirs
• Non-unitary transformation approach to $\mathcal{P}\mathcal{T}$ dynamics

Cited by in corpus (1)

• Exceptional points in SSH-like models with hopping amplitude gradient