  Non-Fermi liquids in $d=2$ spatial dimensions can arise from coupling a Fermi
surface to a gapless boson. At finite temperature, however, the perturbative
quantum field theory description breaks down due to infrared divergences. These
are caused by virtual static bosonic modes, and afflict both fermionic and
bosonic correlators. We show how these divergences are resolved by
self-consistent boson and fermion self-energies that resum an infinite class of
diagrams and correct the standard Eliashberg equations. Extending a previous
approach in $d=3-\epsilon$ dimensions, we find a new "thermal non-Fermi liquid"
regime that violates the scaling laws of the zero temperature fixed point and
dominates over a wide range of scales. We conclude that basic properties of
quantum phase transitions and quantum-classical crossovers at finite
temperature are modified in crucial ways in systems with soft bosonic
fluctuations, and we begin a study of some of the phenomenological
consequences.
