  Microscopic signatures of nuclear ground-state shape phase transitions in
odd-mass Eu isotopes are explored starting from excitation spectra and
collective wave functions obtained by diagonalization of a core-quasiparticle
coupling Hamiltonian based on energy density functionals. As functions of the
physical control parameter -- the number of nucleons -- theoretical low-energy
spectra, two-neutron separation energies, charge isotope shifts, spectroscopic
quadrupole moments, and $E2$ reduced transition matrix elements accurately
reproduce available data, and exhibit more pronounced discontinuities at
neutron number $N=90$, compared to the adjacent even-even Sm and Gd isotopes.
The enhancement of the first-order quantum phase transition in odd-mass systems
can be attributed to a shape polarization effect of the unpaired proton which,
at the critical neutron number, starts predominantly coupling to Gd core nuclei
that are characterized by larger quadrupole deformation and weaker proton
pairing correlations compared to the corresponding Sm isotopes.
