Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification

Abstract

Non-Hermitian (NH) lattice Hamiltonians display a unique kind of energy gap and extreme sensitivity to boundary conditions. Due to the NH skin effect, the separation between edge and bulk states is blurred and the (conventional) bulk-boundary correspondence is lost. Here, we restore the bulk-boundary correspondence for the most paradigmatic class of NH Hamiltonians, namely those with one complex band and without symmetries. We obtain the desired NH Hamiltonian from the mean-field evolution of driven-dissipative cavity arrays, in which NH terms-in the form of non-reciprocal hopping amplitudes, gain and loss-are explicitly modeled via coupling to (engineered and non-engineered) reservoirs. This approach removes the arbitrariness in the definition of the topological invariant, as point-gapped spectra differing by a complex-energy shift are not treated as equivalent; the origin of the complex plane provides a common reference (base point) for the evaluation of the topological invariant. This implies that topologically non-trivial Hamiltonians are only a strict subset of those with a point gap and that the NH skin effect does not have a topological origin. We analyze the NH Hamiltonians so obtained via the singular value decomposition, which allows to express the NH bulk-boundary correspondence in the following simple form: an integer value \nu <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ν</mml:mi> </mml:math> of the topological invariant defined in the bulk corresponds to \vert \nu\vert <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="postfix">|</mml:mo> <mml:mi>ν</mml:mi> <mml:mo stretchy="false" form="postfix">|</mml:mo> </mml:mrow> </mml:math> singular vectors exponentially localized at the system edge under open boundary conditions, in which the sign of \nu <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ν</mml:mi> </mml:math> determines which edge. Non-trivial topology manifests as directional amplification of a coherent input with gain exponential in system size. Our work solves an outstanding problem in the theory of NH topological phases and opens up new avenues in topological photonics.