  An atomistic effective Hamiltonian is used to investigate electrocaloric (EC)
effects of Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$ (PMN) relaxor ferroelectrics in its
ergodic regime, and subject to electric fields applied along the pseudocubic
[111] direction. Such Hamiltonian qualitatively reproduces (i) the electric
field-versus-temperature phase diagram, including the existence of a critical
point where first-order and second-order transitions meet each other; and (ii)
a giant EC response near such critical point. It also reveals that such giant
response around this critical point is microscopically induced by field-induced
percolation of polar nanoregions. Moreover, it is also found that, for any
temperature above the critical point, the EC coefficient-versus-electric field
curve adopts a maximum (and thus larger electrocaloric response too), that can
be well described by the general Landau-like model proposed in [Jiang et al,
Phys. Rev. B 96, 014114 (2017)] and that is further correlated with specific
microscopic features related to dipoles lying along different rhombohedral
directions. Furthermore, for temperatures being at least 40 K higher than the
critical temperature, the (electric field, temperature) line associated with
this maximal EC coefficient is below both the Widom line and the line
representing percolation of polar nanoregions.
