Sublattice signatures of transitions in a $\mathcal{P}\mathcal{T}$-symmetric dimer lattice

Lattice models with non-hermitian, parity and time-reversal ($\mathcal{P}\mathcal{T}$) symmetric Hamiltonians, realized most readily in coupled optical systems, have been intensely studied in the past few years. A $\mathcal{P}\mathcal{T}$-symmetric dimer lattice consists of dimers with intra-dimer coupling $\nu$, inter-dimer coupling ${\nu }^{\mathrm{\prime }}$, and balanced gain and loss potentials $\pm i\gamma$ within each dimer. This model undergoes two independent transitions, namely a $\mathcal{P}\mathcal{T}$-breaking transition and a topological transition. We numerically and analytically investigate the signatures of these transitions in the time-evolution of states that are initially localized on the gain-site or the loss-site.