Exact thermal properties of free-fermionic spin chains

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
a wide class of observables at thermal equilibrium and characterize in detail the thermal
distribution of the kink number and transverse magnetization in the transverse-field
quantum Ising chain.