Detecting non-equilibrium quantum phase transitions by violations of Leggett-Garg inequalities

Non-equilibrium quantum phase transitions in correlated systems have been the object of recent intense research, and new characterization methods are valuable for their analysis. In this work, we discuss how single-site two-time correlation (STC) and Leggett-Garg inequalities (LGIs) identify the critical points of such transitions. We exemplify this on a testbed out-equilibrium configuration, namely a one-dimensional spin-1/2 lattice connected to Markovian reservoirs of different magnetization at its boundaries. When described by an XXZ Hamiltonian, the steady state features a transition from ballistic to diffusive spin transport at the isotropic point. By performing large-scale matrix product simulations we obtain STC for dichotomic operators, and demonstrate that the transition is determined by a change of direction of the maximal violation of LGIs. In addition we show that LGIs are sensible to transport changes induced by dephasing. In particular we find that for strong interactions and large driving, where dephasing leads to an insulating-diffusive transition, a large increase of their violations occurs. This corresponds to a new observation of quantum behavior enhancement by decoherence.