Work statistics and symmetry breaking in an excited-state quantum phase transition

We examine how the presence of an excited-state quantum phase transition manifests in the dynamics of a many-body system subject to a sudden quench. Focusing on the Lipkin-Meshkov-Glick model initialized in the ground state of the ferromagnetic phase, we demonstrate that the work probability distribution displays non-Gaussian behavior for quenches in the vicinity of the excited-state critical point. Furthermore, we show that the entropy of the diagonal ensemble is highly susceptible to critical regions, making it a robust and practical indicator of the associated spectral characteristics. We assess the role that symmetry breaking has on the ensuing dynamics, highlighting that its effect is only present for quenches beyond the critical point. Finally, we show that similar features persist when the system is initialized in an excited state and briefly explore the behavior for initial states in the paramagnetic phase.

Figure 1

Spectrum of the LMG model (4), with respect to the magnetic field $h$. Main panel: energy spectrum for $N=100$, showing the crossing between the critical line of the ESQPT $E_c=0$ and the lifting of the degeneracy. Only $\frac{1}{5}$ of the total eigenstates are shown for clarity. Solid and dashed lines refer to eigenstates with opposite parity. Lower right inset: zoom around the critical energy for $N=500$ spins. Upper right and lower left insets: sketch of the effective potential above and below the ESQPT, respectively.

Figure 2

Symmetric ground state. (a), (b) The survival probability (7) and the work probability distribution (9), respectively, for a system size $N=2000$ and initial magnetic field $h_i=0.5$, when the quench is performed to below ($h_f=0.6$), above ($h_f=0.9$), and at the ESQPT critical point ($h_f=0.75$). (c) The Shannon entropy (11) with respect to the magnetic field $h_f$ for various system sizes $N=100\left[\text{bottom,}\text{blue}\right]\to 1000\left[\text{top,}\text{cyan}\right]$, the inset show the scaling of the maximum of $S_W$ with respect to ${log}_2\left(N\right)$.

Figure 3

Fully symmetry-broken ground state. (a), (b) The survival probability (7) and the work probability distribution (9), respectively, for a system size $N=2000$ and initial ground state at $h_i=0.5$, when the quench is performed to below ($h_f=0.6$), above ($h_f=0.9$), and at the ESQPT critical point ($h_f=0.75$). (c) The Shannon entropy (11) with respect to the magnetic field $h_f$ for various system sizes $N=100\left[\text{bottom,}\text{blue}\right]\to 1000\left[\text{top,}\text{yellow}\right]$.

Figure 4

(a) The work probability distribution $P_W$ [Eq. (9)] and (b) the Shannon entropy $S_W$ [Eq. (11)] with respect to $h_f$ in the LMG model (13). In both panels we quench from $h_i=0.25$ in a system of size $N=1000$, initialized in the symmetric ground state (S), fully symmetry-broken (FSB) ground state, and in superposition between the two ground states. Note the peak at $h_f\approx 0.625$ which corresponds the critical value $h_f^c=(1+h_i)/2$ in this case [cf. Eq. (12)].

Figure 5

Symmetric excited state. (a) The survival probability (7) for a system size $N=2000$, when the quench is performed from $h_i=0.5$ to below ($h_f=0.6$), above ($h_f=0.9$), and to the ESQPT critical point ($h_f=0.75$). (b) The work probability distribution (9) for a system of size $N=2000$ quenched from $h_i=0.5$ to the ESQPT critical point ($h_f=0.75$).

Figure 6

Paramagnetic ground state. (a), (b) The survival probability (7) and the work probability distribution (9), respectively, for a system size $N=2000$, when the quench is performed to below ($h_f=1.2$), above ($h_f=0.5$), and the second-order QPT critical point ($h_f=1.0$). (c) The Shannon entropy (11) with respect to the magnetic field $h_f$ for various system sizes $N=100\left[\text{bottom,}\text{blue}\right]\to 1000\left[\text{top,}\text{cyan}\right]$.