  We study nonequilibrium quantum phase transitions in XY spin 1/2 chain using
the $C^*$ algebra. We show that the well-known quantum phase transition at
magnetic field $h = 1$ persists also in the nonequilibrium setting as long as
one of the reservoirs is set to absolute zero temperature. In addition, we find
nonequilibrium phase transitions associated to imaginary part of the
correlation matrix for any two different temperatures of the reservoirs at $h =
1$ and $h = h_{\rm c} \equiv|1-\gamma^2|$, where $\gamma$ is the anisotropy and
$h$ the magnetic field strength. In particular, two nonequilibrium quantum
phase transitions coexist at $h=1$. In addition we also study the quantum
mutual information in all regimes and find a logarithmic correction of the area
law in the nonequilibrium steady state independent of the system parameters. We
use these nonequilibrium phase transitions to test the utility of two models of
reduced density operator, namely Lindblad mesoreservoir and modified Redfield
equation. We show that the nonequilibrium quantum phase transition at $h = 1$
related to the divergence of magnetic susceptibility is recovered in the
mesoreservoir approach, whereas it is not recovered using the Redfield master
equation formalism. However none of the reduced density operator approaches
could recover all the transitions observed by the $C^*$ algebra. We also study
thermalization properties of the mesoreservoir approach.

Quantum Physics;Mesoscale and Nanoscale Physics;Statistical Mechanics