Excited-state quantum phase transitions in systems with two degrees of freedom. III. Interacting boson systems

The series of articles [Ann. Phys. 345, 73 (2014) and Ann. Phys. 356, 57 (2015)] devoted to excited-state quantum phase transitions (ESQPTs) in systems with $f=2$ degrees of freedom is continued by studying the interacting boson model of nuclear collective dynamics as an example of a truly many-body system. The intrinsic Hamiltonian formalism with angular momentum fixed to $L=0$ is used to produce a generic first-order ground-state quantum phase transition with an adjustable energy barrier between the competing equilibrium configurations. The associated ESQPTs are shown to result from various classical stationary points of the model Hamiltonian, whose analysis is more complex than in previous cases because of (i) a nontrivial decomposition to kinetic and potential energy terms and (ii) the boundedness of the associated classical phase space. Finite-size effects resulting from a partial separability of both degrees of freedom are analyzed. The features studied here are inherent in a great majority of interacting boson systems.

Figure 1

Excited-state quantum phase diagram and classical stationary points of the IBM Hamiltonian (30) with ${\beta }_0^{\prime }=\sqrt{2}$. The top panel shows density plots of the first derivative $\frac{\partial }{\partial E}{\overline{\rho }}^{\lambda }\left(E\right)$ of the smooth level density as a function of control parameter $\lambda$ and energy $E$ (the darker shades represent larger absolute values). The blue and red areas indicate positive and negative derivatives, respectively, white areas correspond to zero derivatives. The middle panel displays the ESQPT critical borderlines, as described in the text [see items (i)–(vi)]. Vertical lines (a)–(d) demarcate the cuts shown in Fig. 3. The bottom panel depicts the coordinate $x=\beta cos\gamma$ and momentum $p_x$ demarcating $\lambda$-dependent positions of stationary points in the classical phase space. The line types correspond to those indicating the respective ESQPT borderlines in the middle panel and in Fig. 3. The thick black line at $p_x=0$ in the bottom panel contains all remaining lines. All axes are dimensionless.

Figure 2

The same as in Fig. 1, but for ${\beta }_0^{\prime }=1.7$, which yields a richer phase structure. Vertical lines (a)–(d) demarcate the cuts of the level density described in Fig. 4.

Figure 3

The potential energy function ${\mathcal{V}}^{\lambda }(x,y=0)$ (left column) and the level density derivative $\frac{\partial }{\partial E}{\overline{\rho }}^{\lambda }\left(E\right)$ (right column) for four values of parameter $\lambda$ in Hamiltonian (30) with ${\beta }_0^{\prime }=\sqrt{2}$. Rows (a)–(d) correspond to the respective cuts at $\lambda =0.2,1.0,1.6$, and 2.5 in Fig. 1. Recall: $({\lambda }^{*},{\lambda }_c,{\lambda }^{**})=(0.707,1,1.333)$. Energies of ESQPTs are indicated by the same types of horizontal and vertical lines in the left and right columns, respectively, also in correspondence to Fig. 1. All axes are dimensionless.

Figure 4

The same as in Fig. 3 but for ${\beta }_0^{\prime }=1.7$. Rows (a)–(d) correspond to the respective cuts at $\lambda =0.12,0.65,1.45$ and 2.90 in Fig. 2. Recall: $({\lambda }^{*},{\lambda }_c,{\lambda }^{**})=(0.460,1,1.257)$.

Figure 5

The spectrum of Hamiltonian (30) with $N=50$ and ${\beta }_0^{\prime }=\sqrt{2}$ as a function of parameter $\lambda$ (upper panel) and the corresponding oscillatory component ${\tilde{\rho }}^{\lambda }\left(E\right)$ of the level density (lower panel). The breaks of all levels (except the $E_0^{\lambda }=0$ ground state) in the upper panel at $\lambda ={\lambda }_c=1$ result from the piecewise form of the Hamiltonian. The size of ${\tilde{\rho }}^{\lambda }\left(E\right)$ in the lower panel is expressed by the shades of gray (darker shades mark larger values). Colored curves depict energies of various stationary points of the excited potential energy surfaces (see text); colors encode the numbers $N_{\gamma }$ of $\gamma$-excitation quanta consistently with Fig. 6. Vertical lines (a), (b), and (c) demarcate position of the spinodal, critical, and antispinodal points, respectively, where the potentials of Fig. 6 are drawn. The axes are dimensionless.

Figure 6

Excited potential energy surfaces (24) for three parameter values of Hamiltonian (30) with ${\beta }_0^{\prime }=\sqrt{2}$. The condensate boson follows from Eq. (A3), the excited mode is associated with the $\gamma$ boson (A4). We show the $y=0$ cuts of surfaces depending on $(x,y)$. Individual curves in each panel depict potentials with the given numbers of the $\gamma$ boson. The values of $\lambda$ correspond to the spinodal (a), critical (b), and antispinodal (c) points of the ground-state potential (cf. Fig. 5). For a more detailed explanation see Appendix. All axes are dimensionless.