  We study the quantum phase transition of the one-dimensional phase model in
the presence of dissipative frustration, provided by an interaction of the
system with the environment through two non-commuting operators. Such a model
can be realized in Josephson junction chains with shunt resistances and
resistances between the chain and the ground. Using a self-consistent harmonic
approximation, we determine the phase diagram at zero temperature which
exhibits a quantum phase transition between an ordered phase, corresponding to
the superconducting state, and a disordered phase, corresponding to the
insulating state with localized superconducting charge. Interestingly, we find
that the critical line separating the two phases has a non monotonic behavior
as a function of the dissipative coupling strength. This result is a
consequence of the frustration between (i) one dissipative coupling that
quenches the quantum phase fluctuations favoring the ordered phase and (ii) one
that quenches the quantum momentum (charge) fluctuations leading to a vanishing
phase coherence. Moreover, within the self-consistent harmonic approximation,
we analyze the dissipation induced crossover between a first and second order
phase transition, showing that quantum frustration increases the range in which
the phase transition is second order. The non monotonic behavior is reflected
also in the purity of the system that quantifies the degree of correlation
between the system and the environment, and in the logarithmic negativity as
entanglement measure that encodes the internal quantum correlations in the
chain.
