Microscopic analysis of order parameters in nuclear quantum phase transitions

Microscopic signatures of nuclear ground-state shape phase transitions in Nd isotopes are studied using excitation spectra and collective wave functions obtained by diagonalization of a five-dimensional Hamiltonian for quadrupole vibrational and rotational degrees of freedom, with parameters determined by constrained self-consistent relativistic mean-field calculations for triaxial shapes. As a function of the physical control parameter, the number of nucleons, energy gaps between the ground state and the excited vibrational states with zero angular momentum, isomer shifts, and monopole transition strengths exhibit sharp discontinuities at neutron number $N=90$, which is characteristic of a first-order quantum phase transition.

Figure 1

(a) Self-consistent $\mathrm{RMF}+\mathrm{BCS}$ triaxial quadrupole binding energy map of ${}^{150}\mathrm{Nd}$ in the $\beta –\gamma$ plane ($0⩽\gamma ⩽{60}^{°}$). The contours join points on the surface with the same energy (in MeV). (b) Projection of the binding energy map on the $\gamma =0^{°}$ axis. (c) The dependence of the binding energy on the deformation parameter $\gamma$, for two values of the axial deformation $\beta =0.2$ and 0.25.

Figure 2

Calculated differences between squares of ground-state charge radii: $\langle r_c^2\rangle {}_{0_1^{+}}(A+2)-\langle r_c^2\rangle {}_{0_1^{+}}(A)$ (a) and isomer shifts $\langle r_c^2\rangle {}_{2_1^{+}}-\langle r_c^2\rangle {}_{0_1^{+}}$ (b) as functions of neutron number in Nd isotopes.

Figure 3

Evolution of the first and second excited $0^{+}$ states (a) and the isomer shifts $\langle r_c^2\rangle {}_{0_2^{+}}-\langle r_c^2\rangle {}_{0_1^{+}}$ (b) with the neutron number in Nd isotopes. Microscopic values calculated with the PC-F1 energy-density functional are compared with data [22].

Figure 4

The calculated monopole transition strength ${\rho }^2(E0;0_2^{+}\to 0_1^{+})$ as a function of neutron number $N$ in Nd isotopes.