  Symmetry is a powerful concept in physics, and its recent application to
understand nonequilibrium behavior is providing deep insights and
groundbreaking exact results. Here we show how to harness symmetry to control
transport and statistics in open quantum systems. Such control is enabled by a
first-order-type dynamic phase transition in current statistics and the
associated coexistence of different transport channels (or nonequilibrium
steady states) classified by symmetry. Microreversibility then ensues, via the
Gallavotti-Cohen fluctuation theorem, a twin dynamic phase transition for rare
current fluctuations. Interestingly, the symmetry present in the initial state
is spontaneously broken at the fluctuating level, where the quantum system
selects the symmetry sector that maximally facilitates a given fluctuation. We
illustrate these results in a qubit network model motivated by the problem of
coherent energy harvesting in photosynthetic complexes, and introduce the
concept of a symmetry-controlled quantum thermal switch, suggesting
symmetry-based design strategies for quantum devices with controllable
transport properties.
