  We investigate the normal mode spectrum of a trapped ion chain at the
symmetry-breaking linear to zigzag transition and at finite temperatures. For
this purpose we modulate the amplitude of the Doppler cooling laser in order to
excite and measure mode oscillations. The expected mode softening at the
critical point, a signature of the second-order transition, is not observed.
Numerical simulations show that this is mainly due to the finite temperature of
the chain. Inspection of the trajectories suggest that the thermal shifts of
the normal-mode spectrum can be understood by the ions collectively jumping
between the two ground state configurations of the symmetry broken phase. We
develop an effective analytical model, which allows us to reproduce the
low-frequency spectrum as a function of the temperature and close to the
transition point. In this model the frequency shift of the soft mode is due to
the anharmonic coupling with the high frequency modes of the spectrum, acting
as an averaged effective thermal environment. Our study could prove important
for implementing ground-state laser cooling close to the critical point.
