  We develop a linear response theory for materials collectively coupled to a
cavity that is valid in all regimes of light-matter coupling, including
symmetry-broken phases. We present and compare two different approaches. First,
using a coherent path integral formulation for the partition function to obtain
thermal Green functions. This approach relies on a saddle point expansion for
the action, that can be truncated in the thermodynamic limit. Second, by
formulating the equations of motion for the retarded Green functions and
solving them. We use a mean-field decoupling of high-order Green functions in
order to obtain a closed, solvable system of equations. Both approaches yield
identical results in the calculation of response functions for the cavity and
material. These are obtained in terms of the bare cavity and material
responses. In combination, the two techniques clarify the validity of a
mean-field decoupling in correlated light-matter systems and provide
complementary means to compute finite-size corrections to the thermodynamic
limit. The theory is formulated for a general model that encompasses most of
the systems typically considered in the field of cavity QED materials, within a
long-wavelength approximation. Finally, we provide a detailed application of
the theory to the Quantum Hall effect and to a collection of magnetic models.
We validate our predictions against analytical and finite-size
exact-diagonalization results.
