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Quantum phase transitions between competing ground-state shapes of atomicnuclei with an odd number of protons or neutrons are investigated in amicroscopic framework based on nuclear energy density functional theory and theparticle-plus-boson-core coupling scheme. The boson-core Hamiltonian, as wellas the single-particle energies and occupation probabilities of the unpairednucleon, are completely determined by constrained self-consistent mean-fieldcalculations for a specific choice of the energy density functional and paringinteraction, and only the strength parameters of the particle-core coupling areadjusted to reproduce selected spectroscopic properties of the odd-mass system.We apply this method to odd-A Eu and Sm isotopes with neutron number $N \approx90$, and explore the influence of the single unpaired fermion on the occurrenceof a shape phase transition. Collective wave functions of low-energy states areused to compute quantities that can be related to quantum order parameters:deformations, excitation energies, E2 transition rates and separation energies,and their evolution with the control parameter (neutron number) is analysed.

Signatures of shape phase transitions in odd-mass nuclei