  We use a recently developed extension of the weak coupling diagrammatic
determinantal quantum Monte Carlo method to investigate the spin, charge and
single particle spectral functions of the one-dimensional quarter-filled
Holstein model with phonon frequency $\omega_0 = 0.1 t$. As a function of the
dimensionless electron-phonon coupling we observe a transition from a Luttinger
to a Luther-Emery liquid with dominant $2k_f$ charge fluctuations. Emphasis is
placed on the temperature dependence of the single particle spectral function.
At high temperatures and in both phases it is well accounted for within a
self-consistent Born approximation. In the low temperature Luttinger liquid
phase we observe features which compare favorably with a bosonization approach
retaining only forward scattering. In the Luther-Emery phase, the spectral
function at low temperatures shows a quasiparticle gap which matches half the
spin gap whereas at temperatures above which this quasiparticle gap closes,
characteristic features of the Luttinger liquid model are apparent. Our results
are based on lattice simulations on chains up to L=20 for two-particle
properties and on CDMFT calculations with clusters up to 12 sites for the
single-particle spectral function.
