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Register a new userFermion doubling theorems in 2D non-Hermitian systems for Fermi points and exceptional points
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The fermion doubling theorem plays a pivotal role in Hermitian topological materials. It states, for example, that Weyl points must come in pairs in three-dimensional semimetals. Here, we present an extension of the doubling theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional non-Hermitian systems without any symmetry constraints, which can host two different types of topological point nodes, namely, (i) Fermi points and (ii) exceptional points. We show that these two types of protected point nodes obey doubling theorems, which require that the point nodes come in pairs. To prove the doubling theorem for exceptional points, we introduce a generalized winding number invariant, which we call the discriminant number. Importantly, this invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with exceptional points of arbitrary order, and moreover can also be used to characterize non-defective degeneracy points. Furthermore, we show that a surface of a three-dimensional system can violate the non-Hermitian doubling theorems, which implies unusual bulk physics.
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The usual concepts of topological physics, such as the Berry curvature, cannot be applied directly to non-Hermitian systems. We show that another object, the quantum metric, which often plays a secondary role in Hermitian systems, becomes a crucial quantity near exceptional points in non-Hermitian systems, where it diverges in a way that fully controls the description of wavepacket trajectories. The quantum metric behaviour is responsible for a constant acceleration with a fixed direction, and for a non-vanishing constant velocity with a controllable direction. Both contributions are independent of the wavepacket size.
Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode selective losses, and minimal quantum systems, and the meteoric research on them has mainly focused on the wide range of novel functionalities they demonstrate. Here, we address the following questions: Does anything remain constant in the dynamics of such open systems? What are the consequences of such conserved quantities? Through spectral-decomposition method and explicit, recursive procedure, we obtain all conserved observables for general $mathcal{PT}$-symmetric systems. We then generalize the analysis to Hamiltonians with other antilinear symmetries, and discuss the consequences of conservation laws for open systems. We illustrate our findings with several physically motivated examples.
An astroid-shaped loop of exceptional points (EPs), comprising four cusps, is found to spawn from the triple degeneracy point in the Brillouin zone (BZ) of a Lieb lattice with nearest-neighbor hoppings when non-Hermiticity is introduced. The occurrence of the EP loop is due to the realness of the discriminant which is guaranteed by the non-Hermitian chiral symmetry. The EPs at the four cusps involve the coalescence of three eigenstates, which is the combined result of the non-Hermitian chiral symmetry and mirror-T symmetry. The EP loop is exactly an astroid in the limit of an infinitesimal non-Hermiticity. The EP loop expands from the $M$ point with increasing non-Hermiticity and splits into two EP loops at a critical non-Hermiticity. The further increase of non-Hermiticity contracts the two EP loops towards and finally to two EPs at the $X$ and $Y$ points in the BZ, accompanied by the emergence of Dirac-like cones. The two EPs vanish at a larger non-Hermiticity. The EP loop disappears and several discrete EPs are found to survive when next-nearest hoppings are introduced to break the non-Hermitian chiral symmetry. A topological invariant called the discriminant number is used to characterize their robustness against perturbations. Both discrete EPs and those on the EP loop(s) are found to show anisotropic asymptotic behaviors. Finally, the experimental realization of the Lieb lattice using a coupled waveguide array is discussed.
We investigate the effects of non-Hermiticity on topological pumping, and uncover a connection between a topological edge invariant based on topological pumping and the winding numbers of exceptional points. In Hermitian lattices, it is known that the topologically nontrivial regime of the topological pump only arises in the infinite-system limit. In finite non-Hermitian lattices, however, topologically nontrivial behavior can also appear. We show that this can be understood in terms of the effects of encircling a pair of exceptional points during a pumping cycle. This phenomenon is observed experimentally, in a non-Hermitian microwave network containing variable gain amplifiers.
Topological semimetals feature a diversity of nodal manifolds including nodal points, various nodal lines and surfaces, and recently novel quantum states in non-Hermitian systems have been arousing widespread research interests. In contrast to Hermitian systems whose bulk nodal points must form closed manifolds, it is fascinating to find that for non-Hermitian systems exotic nodal manifolds can be bounded by exceptional points in the bulk band structure. Such exceptional points, at which energy bands coalesce with band conservation violated, are iconic for non-Hermitian systems. In this work, we show that a variety of nodal lines and drumheads with exceptional boundary can be realized on 2D and 3D honeycomb lattices through natural and physically feasible non-Hermitian processes. The bulk nodal Fermi-arc and drumhead states, although is analogous to, but should be essentially distinguished from the surface counterpart of Weyl and nodal-line semimetals, respectively, for which surface nodal-manifold bands eventually sink into bulk bands. Then we rigorously examine the bulk-boundary correspondence of these exotic states with open boundary condition, and find that these exotic bulk states are thereby undermined, showing the essential importance of periodic boundary condition for the existence of these exotic states. As periodic boundary condition is non-realistic for real materials, we furthermore propose a practically feasible electrical-circuit simulation, with non-Hermitian devices implemented by ordinary operational amplifiers, to emulate these extraordinary states.
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