  We present exact numerical results for the effects of thermal fluctuations on
the experimentally relevant thermodynamic and spectral properties of Peierls
chains. To this end, a combination of classical Monte Carlo sampling and exact
diagonalization is used to study adiabatic half-filled Holstein and
Su-Schrieffer-Heeger models. The classical nature of the lattice displacements
in combination with parallel tempering permit simulations on large system sizes
and a direct calculation of spectral functions in the frequency domain. Most
notably, the long-range order and the associated Peierls gap give rise to a
distinct low-temperature peak in the specific heat. The closing of the gap and
suppression of order by thermal fluctuations involves in-gap excitations in the
form of soliton-antisoliton pairs, and is also reflected in the dynamic density
and bond structure factors as well as in the optical conductivity. We compare
our data to the widely used mean-field approximation, and highlight relations
to symmetry-protected topological phases and disorder problems.
