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We study the geometric phase of a two-level system under the presence of astructured environment, particularly analysing its correction with the ohmicityparameter $s$ and the onset of non-Markovianity. We firstly examine the systemcoupled to a set of harmonic oscillators and studied the decoherence factor asfunction of the environment's ohmicity parameter. Secondly, we propose thetwo-level system coupled to a non-equilibrium environment, and show that theseenvironments display non-Markovian effects for all values of the ohmicityparameter. The geometric phase of the two-level system is therefore computedunder the presence of both types of environment. The correction to the unitarygeometric phase is analysed in both, Markovian and non-Markovian regimes. UnderMarkovian environments, the correction induced on the system's phase is mainlyruled by the coupling constant between the system and the environment, while inthe non-Markovian regime, memory effects seem to trigger a significantcorrection to the unitary geometric phase. The result is significant to thequantum information processing based on the geometric phase in quantum opensystems

Correction to the geometric phase by structured environments: the onset of non-Markovian effects