Local measures of dynamical quantum phase transitions

In recent years, dynamical quantum phase transitions (DQPTs) have emerged as a useful theoretical concept to characterize nonequilibrium states of quantum matter. DQPTs are marked by singular behavior in an effective free energy $\lambda \left(t\right)$, which, however, is a global measure, making its experimental or theoretical detection challenging in general. We introduce two local measures for the detection of DQPTs with the advantage of requiring fewer resources than the full effective free energy. The first, called the real-local effective free energy ${\lambda }_M\left(t\right)$, is defined in real space and is therefore suitable for systems where locally resolved measurements are directly accessible such as in quantum-simulator experiments involving Rydberg atoms or trapped ions. We test ${\lambda }_M\left(t\right)$ in Ising chains with nearest-neighbor and power-law interactions, and find that this measure allows extraction of the universal critical behavior of DQPTs. The second measure we introduce is the momentum-local effective free energy ${\lambda }_k\left(t\right)$, which is targeted at systems where momentum-resolved quantities are more naturally accessible, such as through time-of-flight measurements in ultracold atoms. We benchmark ${\lambda }_k\left(t\right)$ for the Kitaev chain, a paradigmatic system for topological quantum matter, in the presence of weak interactions. Our introduced local measures for effective free energies can further facilitate the detection of DQPTs in modern quantum-simulator experiments.

Figure 1

(a) Effective free energy $\lambda \left(t\right)$ (black crossed line) and real-local effective free energy ${\lambda }_M\left(t\right)$ (colored dotted lines) as a function of time $t$ for different values of $M=2,4,8,16,32,64,128$ sites. Recall that ${\lambda }_{M=N}\left(t\right)=\lambda \left(t\right)$. We find that even when $M\ll N\to \infty , {\lambda }_M\left(t\right)$ reliably detects the DQPT, even yielding a practically exact estimate of the critical time $t_c=\pi /4$. (b) Zoom-in of ${\lambda }_M\left(t\right)$ as a function of $t-t_c$ around the critical time. (c) Time derivative of real-local free energy, $d{\lambda }_M\left(t\right)/dt$, as a function of $t-t_c$. As $M$ increases, the discontinuity jump at the critical time $t_c$ becomes sharper. (d) $|{\lambda }_M\left(t\right)-{\lambda }_M\left(t_c\right)|M$ as a function of $|t-t_c|M$. The plot shows good collapse of the curves for different $M$, particularly when $t$ is close to $t_c$, and reveals a critical exponent $\alpha =1$.

Figure 2

DQPTs arising in the long-range quantum Ising chain given by the Hamiltonian (17) at $\mu =2$ in the wake of a quench from the paramagnetic [$h_i>h_c^e(\mu =2)$] to the ferromagnetic phase [$h_f=0.25<h_c^e(\mu =2)\approx 2.5$]. The initial value of the transverse-field strength is $h_i=5$ in (a) and (c) and $h_i\to \infty$ for (b) and (d). In (a) and (b) we show the real-local effective free energy ${\lambda }_M\left(t\right)$ as a function of evolution time $t$ (colored dotted lines), while the black crossed line represents the effective free energy $\lambda \left(t\right)$. The insets contain a zoom-in of the sharp feature occurring at the critical time, showing ${\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)$ as a function of $t-t_{c,M}$, where $t_{c,M}$ is the estimated critical time from ${\lambda }_M\left(t\right)$, taken as the time of the corresponding peak in the latter. Note how the estimate $t_{c,M}$ is off in the case of $h_i=5$, while it is very accurate in the case of $h_i\to \infty$, and this is because in the latter case the configuration projected on in ${\lambda }_M\left(t\right)$ has the same spin alignment as the ground state of Hamiltonian (17) at $h_i\to \infty$. In (c) and (d) we present the scaling analysis of ${\lambda }_M\left(t\right)$ shown in (a) and (b), respectively. The collapse is obtained by plotting $|{\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)|M$ as a function of $|t-t_{c,M}|M$. Despite having the same critical exponent $\alpha =1$, the scaling function here is linear, unlike the case of the classical Ising model where it is quadratic; cf. Fig. 1.

Figure 3

Quench in the transverse-field strength of the long-range quantum Ising chain of Eq. (17) for $\mu =1.8$ from $h_i=0$ to $h_f=5>h_c^d(\mu =1.8,h_i=0)\approx 2.05$, which results in a regular DQPT at $t_c\sim 0.371$. (a) Real-local effective free energy ${\lambda }_M\left(t\right)$ (colored dotted lines) as a function of time $t$ for three values of the configuration size $M=128,256,512$ sites, along with the exact effective free energy $\lambda \left(t\right)$ (black crossed line). The DQPT can be detected in ${\lambda }_M\left(t\right)$, where the corresponding peak sharpens with increasing $M$. The approximate critical time $t_{c,M}$ estimated from ${\lambda }_M\left(t\right)$ also becomes more accurate at larger $M$. The inset shows a zoom-in of ${\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)$ as a function of $t-t_{c,M}$, where the shifted real-local and exact effective free energies overlap nicely. (b) $|{\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)|M$ as a function of $|t-t_{c,M}{|}^{\alpha }M$. The value of the critical exponent $\alpha$ should be such that the curves exhibited in (a) collapse onto each other. The best overlap achieved occurs for $\alpha =1$.

Figure 4

Quench in the transverse-field strength of the long-range quantum Ising chain of Eq. (17) for $\mu =1.6$ from $h_i=0$ to $h_f=6>h_c^d(\mu =1.6,h_i=0)\approx 2.35$, which results in a regular DQPT. (a) Real-local effective free energy ${\lambda }_M\left(t\right)$ (colored dotted lines) as a function of evolution time $t$ for three values of configuration size $M=128,256,512$ sites, in addition to the effective free energy $\lambda \left(t\right)$ (black crossed line). The DQPT can be detected in ${\lambda }_M\left(t\right)$ through the sharpening of the associated peak occurring at the estimated critical time $t_{c,M}$, which comes closer to the exact critical time $t_c$ with larger $M$. The inset shows a critical time $t_c\sim 0.3$, where we plot and zoom in on ${\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)$ as a function of $t-t_{c,M}$. The real-local and exact effective free energies fall nicely on top of each other, indicating good robustness of ${\lambda }_M\left(t\right)$ to capture this DQPT. (b) $|{\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)|M$ as a function of $|t-t_{c,M}{|}^{\alpha }M$. The scaling analysis reveals the best collapse at a critical exponent $\alpha =1.05$.

Figure 5

Quench in the transverse-field strength of the long-range quantum Ising chain described by the Hamiltonian (17) for $\mu =2$ from $h_i=0$ to $h_f=1.25<h_c^d(\mu =2)\approx 1.85$, which results in an anomalous DQPT. (a) Real-local effective free energy ${\lambda }_M\left(t\right)$ (colored dotted lines) as function of time $t$ for three values of configuration size $M=128,256,512$ sites, along with the exact effective free energy $\lambda \left(t\right)$ (black crossed line). Much the same way as in the case of regular DQPTs, here with increasing $M$ the real-local effective free energy ${\lambda }_M\left(t\right)$ exhibits a sharper peak and its estimated critical time $t_{c,M}$ approaches its exact counterpart $t_c\sim 1.84$ at which $\lambda \left(t\right)$ shows the anomalous DQPT. The inset displays a zoom-in of ${\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)$ as a function of $t-t_{c,M}$, indeed showing a sharpening peak with increasing $M$. (b) $|{\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)|M$ as a function of $|t-t_{c,M}{|}^{\alpha }M$. The value of the critical exponent $\alpha$ should be such that the curves exhibited in (a) collapse onto each other. The best overlap achieved occurs for $\alpha =1.3$, although the result is conclusive only at relatively small $|t-t_{c,M}|$.

Figure 6

Dynamics of the interacting Kitaev chain given by Eq. (18) for $J=\mathrm{\Delta }=1$, and upon quenching $h$ from $h_i\to \infty$ to $h_f=0.2$. (a) Effective free energy $\lambda \left(t\right)$ as a function of time $t$ for different values of $U=0,0.05,0.1,0.2,0.4$. For each of these values $\lambda \left(t\right)$ shows a cusp around $t_c\sim 0.8$. The inset shows a zoom-in of $\lambda \left(t\right)-\lambda \left(t_c\right)$ as a function of $t-t_c$ around $t-t_c\approx 0$. (b) Momentum-local effective free energy ${\lambda }_k\left(t\right)$ as a function of evolution time $t$ for the same values of the interaction strength $U$ considered in (a). For $U=0$, we have the exact limit ${\lambda }_k\left(t\right)=\lambda \left(t\right)$. The momentum-local effective free energy reliably detects the underlying DQPT exhibiting a sharp feature around ${\tilde{t}}_c\sim 0.8$, where we see that the approximate critical time ${\tilde{t}}_c$ extracted from ${\lambda }_k\left(t\right)$ approaches $t_c$ with decreasing $U$. The inset shows a zoom-in of ${\lambda }_k\left(t\right)-{\lambda }_k\left({\tilde{t}}_c\right)$ as a function of $t-{\tilde{t}}_c$ while zooming in around $t-{\tilde{t}}_c\approx 0$.

Figure 7

Dynamics in the long-range quantum Ising chain given by Eq. (17) with $\mu =2$, in the wake of a quench in the transverse-field strength from $h_i=0$ to $h_f=3>h_c^d(\mu =2,h_i=0)\approx 1.85$, leading to a regular DQPT. (a) Effective free energy (black crossed line) $\lambda \left(t\right)$ and its real-local counterpart (colored dotted lines) ${\lambda }_M\left(t\right)$ for configuration size $M=128,256,512$ sites as function of time $t$. Whereas $\lambda \left(t\right)$ shows a clear DQPT, ${\lambda }_M\left(t\right)$ is smooth, although we see a sharpening of the associated peak with increasing $M$, as well as better accuracy in estimating the approximate critical time $t_{c,M}$ in the associated peak of ${\lambda }_M\left(t\right)$. The inset shows a zoom-in of ${\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)$ as a function of $t-t_{c,M}$. (b) Scaling analysis where we show $|{\lambda }_M\left(t\right)-{\lambda }_M\left(t_{c,M}\right)|M$ as a function of $|t-t_{c,M}{|}^{\alpha }M$. The best collapse occurs for $\alpha =1.1$, and seems rather conclusive.

Figure 8

Same as Fig. 7 but for $\mu =2.2$, the dynamical critical point of which is $h_c^d(\mu =2.2,h_i=0)\approx 1.7$. Unlike Fig. 7, the critical exponent that gives the best collapse in the scaling analysis is $\alpha =1.05$ rather than 1.1. Nevertheless, within the precision of our scaling analysis, one cannot conclude that the associated DQPTs are of different universality.

Figure 9

Same as Figs. 7 and 8 but for $\mu =2.4$, the dynamical critical point of which is $h_c^d(\mu =2.4,h_i=0)\approx 1.51$. The scaling analysis yields the best collapse for $\alpha =1.05$. The precision of our scaling analysis cannot definitively ascertain whether this regular DQPT is of a different universality than its counterparts in Figs. 3 and 7.

Figure 10

Convergence with MPS maximal bond dimension $D$ in the case of the anomalous DQPT discussed in Sec. 3b3 and shown in Fig. 5, which is one of the most computationally demanding of our calculations. (a) The real-local effective free energy at configuration size $M=512$ sites shows good convergence already at maximal bond dimension $D=400$. (b) The exact effective free energy shows good convergence already at maximal bond dimension $D=450$.