  Driven-dissipative many-body systems are difficult to analyze analytically
due to their non-equilibrium dynamics, dissipation and many-body interactions.
In this paper, we consider a driven-dissipative infinite-range Ising model with
local spontaneous emission, which naturally emerges from the open Dicke model
in the large-detuning limit. Utilizing an adaptation of the Suzuki-Trotter
quantum-to-classical mapping, we develop an exact field-theoretical analysis
and a diagrammatic representation of the spin model that can be understood from
a simple scattering picture. With this representation, we are able to analyze
critical behavior, finite-size scaling and the effective temperature near the
respective phase transition. Our formalism further allows a detailed study of
the ordered phase where we find a "heating" region within which the effective
temperature becomes negative, thereby exhibiting a truly non-equilibrium
behavior. At the phase transition, we find two distinct critical behaviors with
overdamped and underdamped critical dynamics at generic and weakly-dissipative
critical points, respectively. We further show that the underdamped critical
behavior is robust against short-range perturbations and is not an artifact of
the mean-field nature of the model. To treat such perturbations, we extend our
diagrammatic representation to include the coupling to spin waves due to the
short-range interactions. The field-theoretical approach and the diagrammatics
developed in this work should prove useful in applications to generic
short-range driven-dissipative spin systems.
