Boundary dynamics and phase transitions in the driven-dissipative Ising chain

We study the dynamics of the one-dimensional transverse field Ising model in the presence of a symmetry-breaking boundary field and boundary dephasing. We probe this system by two-time autocorrelation function and polarization of the boundary spin. Previous works by Javed et al. have explored both these effects individually, identifying changes in power-law decays of the autocorrelation function with changes in the edge modes of the model due to symmetry-breaking boundary field [arXiv:2111.11428; PRB 108, L140301] and Zeno physics in the late time behavior of the boundary correlation function with dephasing alone [arXiv:2404.04255; PRB 111, 014316]. Here, we are interested in how the combination of these terms changes the dynamics. Throughout the parameter space, we observe a finite autocorrelation and polarization of the boundary spin at intermediate time scales in the presence of boundary field, even with dephasing. We also note that the bulk magnetic behavior qualitatively determines the value of "steady state" polarization. We further explore the Floquet engineered version of this system as a two-step drive, relevant for Hamiltonian simulation on Noisy Intermediate Scale Quantum(NISQ) devices. This framework allows us to test the robustness of the boundary physics in the presence of time crystallinity.