Figure 1

Plot of the first few energy levels (ordered by increasing energy at any given $\rho$) for ${\rho }^{*}=20$. Solid lines represent states which persist in the $\beta \to 0$ limit. Dashed lines refer to states which exist only for $\beta \ne 0$, disappearing altogether when one takes the limit $\beta \to 0$ (their threshold ${\rho }_{\mathcal{N}}={\rho }^{*}/\mathcal{N},\mathcal{N}=1,3,\dots$ becomes increasingly large since ${\rho }^{*}\to \infty$). Each energy level is represented by a broken line which is alternatively solid and dashed: e.g. the ground state $(k=1)$ starts with a dashed line both for $\rho \ge {\rho }^{*}$ and $\rho \le -{\rho }^{*}$ [in the online version each color follows alternatively a solid and dashed line and represents a given energy level: $k=1$, ground state (red); $k=2$, first excited level (light orange); $k=3$ (yellow); $k=4$ (green), …]. Each energy level is thus characterized by a broken line with infinitely many discontinuities and changes of slope. Such points of discontinuity and changes of slope in the energy levels signal a quantum phase transition. Thus each energy level crosses the critical point at $\rho =0$ undergoing infinitely many quantum phase transitions.