Excited-state quantum phase transitions in a two-fluid Lipkin model

Background: Composed systems have become of great interest in the framework of ground-state quantum phase transitions (QPTs) and many of their properties have been studied in detail. However, in these systems, the study of the so-called excited-state quantum phase transitions (ESQPTs) has not received so much attention.

Purpose: A quantum analysis of the ESQPTs in the two-fluid Lipkin model is presented in this work. The study is performed through the Hamiltonian diagonalization for selected values of the control parameters in order to cover the most interesting regions of the system phase diagram.

Method: A Hamiltonian that resembles the consistent-Q Hamiltonian of the interacting boson model (IBM) is diagonalized for selected values of the parameters. Properties such as the density of states, the Peres lattices, the nearest-neighbor spacing distribution, and the participation ratio are analyzed.

Results: An overview of the spectrum of the two-fluid Lipkin model for selected positions in the phase diagram has been obtained. The location of the excited-state quantum phase transition can be easily singled out with the Peres lattice, with the nearest-neighbor spacing distribution, with Poincaré sections, or with the participation ratio.

Conclusions: This study completes the analysis of QPTs for the two-fluid Lipkin model, extending the previous study to excited states. The ESQPT signatures in composed systems behave in the same way as in single ones, although the evidences of their presence can be sometimes blurred. The Peres lattice turns out to be a convenient tool to look into the position of the ESQPT and to define the concept of phase in the excited states realm.

Figure 1

Phase diagram of the consistent-$Q$-like two-fluid Lipkin model. In the diagram, the different phases are represented: spherical and deformed, the first-order QPT surfaces, and the second-order QPT line. Moreover, the relevant control parameters ($x, y=(y_1+y_2)/2$, and $y^{\prime }=(y_1-y_2)/2$) and dynamical symmetries also are shown. The marked points correspond to the cases studied in Sec. 4 (see text).

Figure 2

Density of states for a one-fluid Lipkin Hamiltonian $H=x{n}_t-\frac{1-x}{N}{Q}^{\left(y\right)}{Q}^{\left(y\right)}$ ($n_t$ and $Q^{\left(y\right)}$ defined in Eqs. (5) and (7) with $x=1/2, y=0$, and $N=1000$ (a) and with $x=1/2, y=1/2$, and $N=1000$ (b); the insets show schematically the corresponding potential energy surface in both cases.

Figure 3

Density of states as a function of the excitation energy for a Hamiltonian with parameters $y=0, y^{\prime }=0$, and $x=1/2$ and a number of bosons $N_1=N_2=70$. (a) For selected values of $j$; (b) for all the states without distinction in $j$.

Figure 4

Peres lattice for $\frac{\langle n_{t_1}\rangle }{N_1}$ vs the excitation energy for the same Hamiltonian and parameters as Fig. 3. (a) For selected values of $j$; (b) for all the states without distinction in $j$. Poincaré sections for energies $E=-10, E=0, E=20$, and $E=50$, in panels (c), (d), (e), and (f), respectively [energies marked in panel (b)].

Figure 5

Normalized participation ratio as a function of the excitation energy for the same Hamiltonian and parameters as in Fig. 3. (a) For selected values of $j$; (b) for all the states without distinction in $j$.

Figure 6

Density of states as a function of the excitation energy for a Hamiltonian with parameters $y=1, y^{\prime }=0$, and $x=1/2$ and a number of bosons $N_1=N_2=70$. (a) For selected values of $j$; (b) for all the states without distinction in $j$.

Figure 7

Peres lattice for $\frac{\langle n_{t_1}\rangle }{N_1}$ vs the excitation energy for the same Hamiltonian and parameters shown in Fig. 6. (a) For selected values of $j$; (b) for all the states without distinction in $j$. Poincaré sections for energies $E=-10, E=0, E=20$, and $E=50$, in panels (c), (d), (e), and (f), respectively [energies marked in panel (b)].

Figure 8

Participation ratio as a function of the excitation energy for the same Hamiltonian and parameters as in Fig. 6. (a) For selected $j$ values; (b) for all the states without distinction in $j$.

Figure 9

Different observables as a function of the excitation energy for a Hamiltonian with parameters $y=1/4, y^{\prime }=3/4$, and $x=1/2$ and a number of bosons $N_1=N_2=70$: (a) Density of states, (b) Peres lattice (black points) for $\frac{\langle n_{t_1}\rangle }{N_1}$ and $\eta$ value (red solid curve), (c) participation ratio, and (d)–(g) Poincaré sections for energies $E=-20, E=0, E=30$, and $E=50$, respectively [energies marked in panel (b)].

Figure 10

Same as in Fig. 9 but for $y=0, y^{\prime }=1/2$, and $x=1/2$. Panels (d), (e), (f), and (g) are Poincaré sections for energies $E=-10, E=0, E=20$, and $E=50$.

Figure 11

Same as in Fig. 9 but for $y=0, y^{\prime }=1$, and $x=1/2$. Panels (d), (e), (f), and (g) are Poincaré sections for energies $E=-10, E=0, E=30$, and $E=50$.

Figure 12

Same as Fig. 9 but for $y=0, y^{\prime }=3/2$, and $x=1/2$. Panels (d), (e), (f), and (g) are Poincaré sections for energies $E=-20, E=0, E=30$, and $E=50$.

Figure 13

$\langle n_{t_1}\rangle /N_1$ vs $\langle n_{t_2}\rangle /N_2$ for the Hamiltonians studied in Fig. 10 with parameters $x=1/2, y=0$, and $y^{\prime }=1/2$ (a), in Fig. 11 with parameters $x=1/2, y=0$, and $y^{\prime }=1$ (b), and in Fig. 12 with parameters $x=1/2, y=0$, and $y^{\prime }=3/2$ (c). Black points correspond to states with negative energy, while the red ones correspond to states with positive energy. In panels (d)–(f), we depict $\langle n_{t_2}\rangle /N_2-\langle n_{t_1}\rangle /N_1$ as a function of the energy for the same parameters as in panels (a)–(c), respectively.