Abstract
Non-Hermitian (NH) Hamiltonians can be used to describe dissipative systems, 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 propose a generalization of a perviously-developed method with which to find all the bulk states of system with open boundaries to NH models. Using the exact solutions in combination with variational states, we elucidate genuinely NH aspects of the interplay between bulk and boundary at the phase transitions.
Highlights
One of the fundamental postulates of quantum mechanics is the assumption that observables are described by Hermitian operators, which ensures realness of the measured eigenvalues
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
Summary
One of the fundamental postulates of quantum mechanics is the assumption that observables are described by Hermitian operators, which ensures realness of the measured eigenvalues. Alleviating the Hermiticity condition may introduce effects that, at first glance, seem surprising or unintuitive, such as the possible breakdown of the conventional bulk-boundary correspondence (BBC) [30,31,32,33,34,35,36,37,38]. The breakdown of the conventional BBC as well as the emergence of the NH skin effect has been experimentally verified in mechanical systems [20,21], topoelectrical circuits [16], and optical [61] systems This phenomenology has been suggested to be of practical use in sensors whose sensitivity increases exponentially with the size of the system [66].
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