Degeneracy in excited-state quantum phase transitions of two-level bosonic models and its influence on system dynamics (2024)

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  Figure 1

  (a)Excitation energy scaled by the system size $\varepsilon =(E-E_0)/N$ as a function of the control parameter $\xi$ for the LMG model with the model Hamiltonian (4) and system size $N=50$. Blue solid (red dashed) lines mark even- (odd-)parity levels. (b)Excitation energy scaled by the system size $\varepsilon =(E-E_0)/N$ as a function of the control parameter $\xi$ for the 2DVM Hamiltonian (7) with a system size $N=50$. Blue solid (red dashed) lines mark levels with angular momentum $\ell =0$ (1). In both panels the color-filled area marks the energy difference between selected states with different (a)parity or (b)angular momentum. (c)Energy difference between selected pairs of states of the LMG model Hamiltonian (4) having different parity as a function of the control parameter $\xi$. (d)Energy difference between selected states of the Hamiltonian (7) with angular momentum $\ell =0$ and 1 as a function of the control parameter $\xi$. In both cases, the labels of the selected pairs of levels are provided in the legends and we use for each pair of states the same color used to fill the corresponding area in (a)and (b).

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  Figure 2

  (a)Difference of energy between even and odd energy levels in the LMG model, $\mathrm{\Delta }{E}_i=E_i^{-}-E_i^{+}$ with $i=0,1,2,3$, as a function of the system size $N$ using lin-log axes. Black lines are the result of the fit of the depicted data to an exponential law. (b)Difference of energy between $\ell =0$ and $\ell =1$ energy levels in the 2DVM, $\mathrm{\Delta }{E}_{\nu }=E_{\nu }^{\ell =1}-E_{\nu }^{\ell =0}$ with $\nu =0,1,2,3$, as a function of the system size $N$ using log-log axes. Black lines are the result of the fit of the depicted data to a power law. In both cases, the control parameter value is $\xi =0.5$.

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  Figure 3

  Excitation energy values as a function of the control parameter $\xi$ for the 2DVM Hamiltonian (5) with a system size $N=50$ and angular momenta (a) $\ell =0,1$, (b) $\ell =14,15$, and (c) $\ell =30,31$. (d)Shown in logarithmic scale is the excitation energy for minimum-energy states of angular momenta $\ell =1$ (aqua solid line), $\ell =14$ (orange dashed line), and $\ell =30$ (violet dotted line) with respect to the ground state, divided by the angular momentum value $\ell$. These differences have been highlighted in (a)–(c)using the same color code.

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  Figure 4

  (a)Time-averaged MOTOC $\overline{{\mathcal{F}}_{VW}^{\left(j\right)}}\left(T\right)$ for the LMG model with $\widehat{V}=\widehat{W}={\widehat{J}}_x$ as a function of the system's excitation energy scaled by the system size $E/N$ for even-parity eigenstates of the Hamiltonian (1) with a system size $N=300$. (b)Time-averaged MOTOC $\overline{{\mathcal{F}}_{VW}^{\left(j\right)}}\left(T\right)$ for $\ell =0$ eigenstates of the 2DVM Hamiltonian in Eq.(7) with $N=300$ with $\widehat{V}={\widehat{D}}_{-}$ and $\widehat{W}={\widehat{D}}_{+}$. Results are plotted as a function of the system's excitation energy scaled by the system size $(E-E_{gs})/N$. In both panels the control parameter value is $\xi =0.6$ and the red solid line is the stationary value obtained with Eq.(11). Dashed lines are the result of averaging for different time interval values (see the legends). The inset in (a)is a close-up of the vicinity of the ESQPT critical energy. A vertical pink dash-dotted line marks the ESQPT critical energy in the mean-field limit.

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  Figure 5

  (a)Time-averaged value of the MOTOC $\overline{{\mathcal{F}}_{VW}^{\left(j\right)}}\left(T\right)$ for the LMG model with $\widehat{V}=\widehat{W}={\widehat{J}}_x$ as a function of the system's excitation energy scaled by the system size $(E-E_{gs})/N$ for even-parity eigenstates of the Hamiltonian (1) for various system-size values (see the legend). (b)Time-averaged value of the MOTOC $\overline{{\mathcal{F}}_{VW}^{\left(j\right)}}\left(T\right)$ for the 2DVM with $\widehat{V}={\widehat{D}}_{-}$ and $\widehat{W}={\widehat{D}}_{+}$ as a function of the system's excitation energy scaled by the system size $(E-E_{gs})/N$ for $\ell =0$ for eigenstates of the 2DVM Hamiltonian (7) for various system-size values $N$ (see the legend). In both panels calculations are carried out for a control parameter value $\xi =0.6$ and the time average is performed over a time interval $T=1000$. Dashed lines are the results for different system-size values. A vertical pink dash-dotted line marks the ESQPT critical energy in the mean-field limit.

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  Figure 6

  (a)Excitation energy scaled by the system size $\varepsilon =(E-E_0)/N$ as a function of the control parameter $\xi$ for the VM with model Hamiltonian (A3) and system size $N=50$. Blue solid (red dashed) lines mark levels with angular momentum $J=0$ (1). (b)Excitation energy scaled by the system size $\varepsilon =(E-E_0)/N$ as a function of the control parameter $\xi$ for the IBM Hamiltonian Eq.(A6) with a system size $N=50$. Blue solid (red dashed) lines mark levels with seniority ${\nu }_s={\nu }_d=0$ (${\nu }_s={\nu }_d=1$). In both panels the color-filled area marks the energy difference between selected states with different (a)angular momentum or (b)seniority. (c)Energy difference between selected pairs of states of the VM Hamiltonian (A3) having different angular momentum as a function of the control parameter $\xi$. (d)Energy difference between selected states of the Hamiltonian (A6) with different seniority as a function of the control parameter $\xi$. In both cases, the labels of the selected pairs of levels are provided in the legends of (c)and (d)and the color used for each pair of states is the same color used to fill the corresponding area in (a)and (b).

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