Quantum Master Equation for Electronic Structure in Open Graphene: Non‐Hermitian Hamiltonians and Types of Dissipative Dynamics

A reduced density operator approach is proposed for electron hopping between the nearest neighboring carbon atoms in the monolayer graphene lattice in the presence of the environment. To analytically describe at least the simplest of dissipative effects which can occur, we consider the tight‐binding approximation and propose the minimal non‐Hermitian extension of the band Hamiltonian, which models spontaneous emission or injection in bands as collective states of electrons in the presence of lattice atoms. We find that two types of dissipative evolution can exist in the open graphene's electronic structure, depending on whether it is sustainable (e.g. supported by particle injections of some kind) or undergoing an irreversible decay. In the first case, the master equation is nonlinear with respect to the density operator; whereas in the second case it is linear but does not conserve the trace norm. We also consider the evolution of pure states, which allows us to separate quantum‐mechanical properties from statistical ones, and derive the effective Hamiltonian and the Schrödinger equation which turns out to be nonlinear. The evolution equations which result indicate that electronic transport in graphene becomes significantly complex in the presence of even weak dissipative effects.