Geometric phases, arising from cyclic evolutions in a curved parameter space, appear in a wealth of physical settings. Recently, and largely motivated by the need of an experimentally realistic definition for quantum computing applications, the quant
um geometric phase was generalized to open systems. The definition takes a kinematical approach, with an initial state that is evolved cyclically but coupled to an environment --- leading to a correction of the geometric phase with respect to the uncoupled case. We obtain this correction by measuring the nonunitary evolution of the reduced density matrix of a spin one-half coupled to an environment. In particular, we consider a bath that can be tuned near a quantum phase transition, and demonstrate how the criticality information imprinted in the decoherence factor translates into the geometric phase. The experiments are done with a NMR quantum simulator, in which the critical environment is modeled using a one-qubit system.
We study the quantum fidelity approach to characterize thermal phase transitions. Specifically, we focus on the mixed-state fidelity induced by a perturbation in temperature. We consider the behavior of fidelity in two types of second-order thermal p
hase transitions (based on the type of non-analiticity of free energy), and we find that usual fidelity criteria for identifying critical points is more applicable to the case of $lambda$ transitions (divergent second derivatives of free energy). Our study also reveals limitations of the fidelity approach: sensitivity to high temperature thermal fluctuations that wash out information about the transition, and inability of fidelity to distinguish between crossovers and proper phase transitions. In spite of these limitations, however, we find that fidelity remains a good pre-criterion for testing thermal phase transitions, which we use to analyze the non-zero temperature phase diagram of the Lipkin-Meshkov-Glick model.
