Capital to topological insulators, the bulk-boundary correspondence ties a topological invariant computed from the bulk (extended) states with those at the boundary, which are hence robust to disorder. Here we put forward an ordering unique to non-Hermitian lattices, whereby a pristine system becomes devoid of extended states, a property which turns out to be robust to disorder. This is enabled by a peculiar type of non-Hermitian degeneracy where a macroscopic fraction of the states coalesce at a single point with geometrical multiplicity of 1, that we call a phenomenal point.

Recently, the search for topological states of matter has turned to non-Hermitian systems, which exhibit a rich variety of unique properties without Hermitian counterparts. Lattices modeled through non-Hermitian Hamiltonians appear in the context of photonic systems, where one needs to account for gain and loss, circuits of resonators, and also when modeling the lifetime due to interactions in condensed matter systems. Here we provide a brief overview of this rapidly growing subject, the search for topological states and a bulk-boundary correspondence in non-Hermitian systems.

Topological phases of matter have attracted much attention over the years. Motivated by analogy with photonic lattices, here we examine the edge states of a one-dimensional trimer lattice in the phases with and without inversion symmetry protection. In contrast to the Su-Schrieffer-Heeger model, we show that the edge states in the inversion-symmetry broken phase of the trimer model turn out to be chiral, i.e., instead of appearing in pairs localized at opposite edges they can appear at a single edge. Interestingly, these chiral edge states remain robust to large amounts of disorder. In addition, we use the Zak phase to characterize the emergence of degenerate edge states in the inversion-symmetric phase of the trimer model. Furthermore, we capture the essentials of the whole family of trimers through a mapping onto the commensurate off-diagonal Aubry-André-Harper model, which allow us to establish a direct connection between chiral edge modes in the two models, including the calculation of Chern numbers. We thus suggest that the chiral edge modes of the trimer lattice have a topological origin inherited from this effective mapping. Also, we find a nontrivial connection between the topological phase transition point in the trimer lattice and the one in its associated two-dimensional parent system, in agreement with results in the context of Thouless pumping in photonic lattices. arXiv:1810.05566v2 [cond-mat.mes-hall]

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The pseudogap effects and the expected quantum phase transitions (QPTs) in cuprate materials are yet unclear in nature. A single band tight-binding (TB) model for the CuO 2 planes of these materials had predicted the existence of definite pseudogap states at half-filling, after considering that a crystal symmetry breaking and noncollinear spin orientations of the single particle states are allowed. Here we show that after including hole doping in the model, a QPT which lies beneath the superconducting dome exists and is a second-order one. In it, an antiferromagntic-insulator (AFI) ground state, showing strong spin fluctuations at low doping, coalesce with an excited paramagnetic pseudogap (PPG) state, exhibiting a broken lattice symmetry at the critical hole density. A critical doping value xc = 0.2 resulted, which surprisingly coincided with the experimentally measured one, in spite of the fact that the model parameters were not yet optimized. Above this value the system becomes a paramagnetic metal. The band structures and Fermi surface with doping are evaluated and their evolution show a close resemblance with the experimental observations, including the topological change in structure at the critical hole density.