We introduce the basic concepts of catastrophe theory needed to derive analytically the phase diagram of the proton-neutron interacting boson model (IBM-2). Previous studies [1,2,3] were based on numerical solutions. We here explain the whole IBM-2 p
hase diagram including the precise order of the phase transitions in terms of the cusp catastrophe.
We study the quantum phase diagram and excitation spectrum of the frustrated $J_1$-$J_2$ spin-1/2 Heisenberg Hamiltonian. A hierarchical mean-field approach, at the heart of which lies the idea of identifying {it relevant} degrees of freedom, is deve
loped. Thus, by performing educated, manifestly symmetry preserving mean-field approximations, we unveil fundamental properties of the system. We then compare various coverings of the square lattice with plaquettes, dimers and other degrees of freedom, and show that only the {it symmetric plaquette} covering, which reproduces the original Bravais lattice, leads to the known phase diagram. The intermediate quantum paramagnetic phase is shown to be a (singlet) {it plaquette crystal}, connected with the neighboring Neel phase by a continuous phase transition. We also introduce fluctuations around the hierarchical mean-field solutions, and demonstrate that in the paramagnetic phase the ground and first excited states are separated by a finite gap, which closes in the Neel and columnar phases. Our results suggest that the quantum phase transition between Neel and paramagnetic phases can be properly described within the Ginzburg-Landau-Wilson paradigm.
The evolution pattern of exceptional points is studied in a non-integrable limit of the complex-extended 3-level Richardson-Gaudin model. The appearance of a pseudo-diabolic point from the fusion of two exceptional points is demonstrated in the anti-
hermitian limit of the model and studied in some details.
