Full distribution and large deviations of local observables in an exactly solvable current carrying steady state of a strongly driven XXZ chain

Current carrying steady states of interacting spin chains exhibit rich structures generated through an interplay of constraints from the Hamiltonian dynamics and those induced by the current. The $\mathit{\text{XXZ}}$ spin chain when coupled to maximally polarizing Lindblad terms (with opposite signs on either end) admits an exact solution for the steady state in a matrix product state (MPS) form. We use this exact solution to study the correlations and distributions of local spin observables in the nonequilibrium steady state. We present exact expressions for spin correlators, entropy per site and scaled cumulant generating functions (SCGF) for distributions of local observables in the $\mathit{\text{XX}}$ limit (Ising anisotropy $\mathrm{\Delta }=0$). Further, we use the exact MPS solution in the $\mathrm{\Delta }>0$ regime, to calculate numerically exact entropy, correlations, as well as full distributions of spin observables in large systems. In systems where $\mathrm{\Delta }$ is a cosine of rational multiple of $\pi$, we can numerically exactly estimate the large system limit of the SCGF and the large deviation/rate functions of local-$z$ magnetization. For these, we show that the deviations of the SCGF, calculated in finite systems, from the asymptotic large system size limit decay exponentially with system size; however, the decay rate is a discontinuous function of $\mathrm{\Delta }$. The $x$ magnetization density shows a double peak structure at $\mathrm{\Delta }\lesssim 1$, suggesting short-range ferromagnetic ordering in the $x$ direction similar to what was reported for the ground state of the XXZ chain.