Spontaneous symmetry breaking in non-relativistic quantum systems has previously been addressed in the framework of effective field theory. Low-lying excitations are constructed from Nambu-Goldstone modes using symmetry arguments only. We extend that
approach to finite systems. The approach is very general. To be specific, however, we consider atomic nuclei with intrinsically deformed ground states. The emergent symmetry breaking in such systems requires the introduction of additional degrees of freedom on top of the Nambu-Goldstone modes. Symmetry arguments suffice to construct the low-lying states of the system. In deformed nuclei these are vibrational modes each of which serves as band head of a rotational band.
We present a model-independent approach to electric quadrupole transitions of deformed nuclei. Based on an effective theory for axially symmetric systems, the leading interactions with electromagnetic fields enter as minimal couplings to gauge potent
ials, while subleading corrections employ gauge-invariant non-minimal couplings. This approach yields transition operators that are consistent with the Hamiltonian, and the power counting of the effective theory provides us with theoretical uncertainty estimates. We successfully test the effective theory in homonuclear molecules that exhibit a large separation of scales. For ground-state band transitions of rotational nuclei, the effective theory describes data well within theoretical uncertainties at leading order. In order to probe the theory at subleading order, data with higher precision would be valuable. For transitional nuclei, next-to-leading order calculations and the high-precision data are consistent within the theoretical uncertainty estimates. We also study the faint inter-band transitions within the effective theory and focus on the $E2$ transitions from the $0^+_2$ band (the $beta$ band) to the ground-state band. Here, the predictions from the effective theory are consistent with data for several nuclei, thereby proposing a solution to a long-standing challenge.
The coupled-cluster wave function factorizes to a very good approximation into a product of an intrinsic wave function and a Gaussian for the center-of-mass coordinate. The width of the Gaussian is in general not identical to the oscillator length of
the underlying single-particle basis. The quality of the separation can be verified by a simple procedure.
We compute the binding energies, radii, and densities for selected medium-mass nuclei within coupled-cluster theory and employ the bare chiral nucleon-nucleon interaction at order N3LO. We find rather well-converged results in model spaces consisting
of 15 oscillator shells, and the doubly magic nuclei 40Ca, 48Ca, and the exotic 48Ni are underbound by about 1 MeV per nucleon within the CCSD approximation. The binding-energy difference between the mirror nuclei 48Ca and 48Ni is close to theoretical mass table evaluations. Our computation of the one-body density matrices and the corresponding natural orbitals and occupation numbers provides a first step to a microscopic foundation of the nuclear shell model.
