Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis)appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in non-Hermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries. We illustrate our general insights by deriving the phase diagram for several microscopic open boundary models, including exactly solvable non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators.

Non-Hermitian Hamiltonians, which describe a wide range of dissipative systems, and higher-order topological phases, which exhibit novel boundary states on corners and hinges, comprise two areas of intense current research. Here we investigate systems where these frontiers merge and formulate a generalized biorthogonal bulk-boundary correspondence, which dictates the appearance of boundary modes at parameter values that are, in general, radically different from those that mark phase transitions in periodic systems. By analyzing the interplay between corner/hinge, edge/surface and bulk degrees of freedom we establish that the non-Hermitian extensions of higher-order topological phases exhibit an even richer phenomenology than their Hermitian counterparts and that this can be understood in a unifying way within our biorthogonal framework. Saliently this works in the presence of the non-Hermitian skin effect, and also naturally encompasses genuinely non-Hermitian phenomena in the absence thereof. PACS numbers:Introduction. Topological phases of matter are at the forefront of condensed-matter research with a recent focus on higher-order topological phases [1][2][3][4][5][6][7][8][9][10][11][12][13], where a subtle interplay between topology and crystalline symmetry results in the appearance of boundary states on boundaries with a codimension higher than one, i.e., corners or hinges. Another increasingly popular direction of research revolves around studying topology in the context of non-Hermitian physics, which is a relevant approach for describing a wide range of dissipative systems [14? -39]. Saliently these models feature a breakdown of the conventional bulk-boundary correspondence [16][17][18][19][20][21][22], which is intimately linked to the piling up of "bulk" states at the boundaries known as the non-Hermitian skin effect [14,16,21]. These models can be understood with open boundaries directly by defining a biorthogonal bulk-boundary correspondence [16], which combines the right and left wave functions of the boundary modes to each other to form a "biorthogonal state." By studying the behavior of this state, it is then possible to reconcile the physics of open non-Hermitian systems.Here we show that the concept of a biorthogonal bulk-boundary correspondence can be generalized to capture non-Hermitian extensions of higher-order topological phases. Indeed, such phases have very recently been studied in a number of works resulting in the observation of variations to the skin effect and the suggestion of topological invariants [40][41][42][43]. Here, unlike Refs. 40-43, we focus on the biorthogonal properties of the open boundary systems, and show that this provides a comprehensive and transparent interpretation of the physical features of non-Hermitian extensions of higher-order topological phases. In particular, it unravels a subtle interplay between crystalline lattice symmetries, sample geometry, and boundary/bulk states that goes qualitatively beyond that of the Hermitian realm.To elucidate these results we...

Non-Hermitian (NH) Hamiltonians can be used to describe dissipative systems, notably including systems with gain and loss, and are currently intensively studied in the context of topology. A salient difference between Hermitian and NH models is the breakdown of the conventional bulk-boundary correspondence, invalidating the use of topological invariants computed from the Bloch bands to characterize boundary modes in generic NH systems. One way to overcome this difficulty is to use the framework of biorthogonal quantum mechanics to define a biorthogonal polarization, which functions as a real-space invariant signaling the presence of boundary states. Here, we generalize the concept of the biorthogonal polarization beyond the previous results to systems with any number of boundary modes and show that it is invariant under basis transformations as well as local unitary transformations. Additionally, we focus on the anisotropic Su-Schrieffer-Heeger chain and study gap closings analytically. We also propose a generalization of a previously developed method with which to find all the bulk states of the system with open boundaries to NH models. Using the exact solutions for the bulk and boundary states, we elucidate genuinely NH aspects of the interplay between the bulk and boundary at the phase transitions.