  An exact description of integrable spin chains at finite temperature is
provided using an elementary algebraic approach in the complete Hilbert space
of the system. We focus on spin chain models that admit a description in terms
of free fermions, including paradigmatic examples such as the one-dimensional
transverse-field quantum Ising and XY models. The exact partition function is
derived and compared with the ubiquitous approximation in which only the
positive parity sector of the energy spectrum is considered. Errors stemming
from this approximation are identified in the neighborhood of the critical
point at low temperatures. We further provide the full counting statistics of
an arbitrary observable at thermal equilibrium and characterize in detail the
thermal distribution of the kink number and transverse magnetization in the
transverse-field quantum Ising chain.
