Abstract
We present an alternative approach to studying topology in open quantum systems, relying directly on Green’s functions and avoiding the need to construct an effective non-Hermitian (nH) Hamiltonian. We define an energy-dependent Chern number based on the eigenstates of the inverse Green’s function matrix of the system which contains, within the self-energy, all the information about the influence of the environment, interactions, gain or losses. We explicitly calculate this topological invariant for a system consisting of a single 2D Dirac cone and find that it is half-integer quantized when certain assumptions about the self-energy are made. Away from these conditions, which cannot or are not usually considered within the formalism of nH Hamiltonians, we find that such a quantization is usually lost and the Chern number vanishes, and that in special cases, it can change to integer quantization.
Highlights
To study the properties of quantum systems and understand how they manifest themselves in our macroscopic everyday world, it is usually necessary to take into account the interaction of such systems with their surroundings
In most cases the Chern number will vanish, revealing that in these cases the damping renders the topology of the system trivial
In this work we have presented an alternative approach for the study and topological classification of open quantum systems
Summary
To study the properties of quantum systems and understand how they manifest themselves in our macroscopic everyday world, it is usually necessary to take into account the interaction of such systems with their surroundings. A related topological invariant based on Green’s function has been recently proposed by Kawabata et al [34] They showed that in one and three spatial dimensions, nH topological systems can be described by effective Chern-Simons field theories and are characterized by quantized topological response functions. We will propose an analogous definition for a energydependent nH Chern number that can be obtained directly from the Green’s functions, and show that this topological invariant is quantized in many scenarios If this number is constructed based on the retarded Green’s function and if certain assumptions about the energy and momentum dependence of the self-energy are fulfilled, we show that this invariant coincides with the nH Chern number defined in Ref. We shall delineate more precisely the conditions under which such quantization is lost, allowing us to shed some light onto the potential limitations of the nH topology
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