Dynamics of entanglement entropy and entanglement spectrum crossing a quantum phase transition

We study the time evolution of entanglement entropy and entanglement spectrum in a finite-size system which crosses a quantum phase transition at different speeds. We focus on the transverse-field Ising model with a time-dependent magnetic field, which is linearly tuned on a time scale $\tau$. The time evolution of the entanglement entropy displays different regimes depending on the value of $\tau$, showing also oscillations which depend on the instantaneous energy spectrum. The entanglement spectrum is characterized by a rich dynamics where multiple crossings take place with a gap-dependent frequency. Moreover, we investigate the Kibble-Zurek scaling of the entanglement entropy and Schmidt gap.

Figure 1

Dynamics of the entanglement entropy for $L=50$, $h_i=1.4$, $h_f=0.4$. Main panel: $S\left(t\right)$ for different values of $\tau$ (dashed vertical line: location of the critical point). Inset: $\tilde{S}\left(t\right)$ as defined in Eq. (14) for the same values of $\tau$ as in the main panel.

Figure 2

Sum of the first four eigenvalues of the reduced density matrix $W_4$ [see Eq. (13)] for $L=50$, $h_i=1.4$, $h_f=0.4$, and different values of $\tau$.

Figure 3

Dynamics of the entanglement spectrum for $L=50$, $h_i=1.4$, and $h_f=0.4$. Black dashed, red dot-dot-dashed, green solid, and blue dash-dotted lines correspond to the dynamical first, second, third, and fourth eigenvalue of the reduced density matrix of the half chain, respectively. Different panels refer to (a): $\tau =500$; (b): $\tau =0.1$; (c): $\tau =1$; (d): $\tau =8$; (e): $\tau =30$; (f): $\tau =100$. In panels (b), (c), and (d) the red and green lines are overlaid. The cyan lines in panel (a) show the ground-state values of the first four eigenvalues. The inset in panel (d) is a zoom of the crossing point.

Figure 4

Panel (a): evolution of the entanglement entropy according to the protocol (16) for $\tau =10$. Panel (b): oscillation period as a function of the energy gap ($h_i=1.4$, $h_f\in [0.31,0.4]$); the fit is performed by means of the formula $y=a_0+a_1/x$. Panel (c): time at which the eigenvalues of the reduced density matrix cross as a function of $\tau$, for $\tau \in [8,17]$. Black dots: numerical data; red line: fitting formula $y=a_0+a_1/x^{a_2}$, giving $a_0=0.428193$ (critical point: 0.4). Panel (d): same as in (b), but with $h_i=1.5$, $h_f\in [0.41,0.5]$, and $\tau =30$.

Figure 5

Main panel: time evolution of the expectation value of the density of left spins on the first four eigenstates of the reduced density matrix [cf. Eq. (17)] for a ramping of the magnetic field from $h_i=6$ to $h_f=-1$ with $\tau =1$ and $L=18$ sites. Panel (a): zoom of the main panel around the crossing point, with the usual color code. Panel (b): corresponding dynamics of the entanglement spectrum in the same time interval of panel (a).

Figure 6

Panel (a): dynamics of the entanglement spectrum for a ramping from $h_i=6.0$ to $h_f=0$, with $\tau =10$ and $L=18$. Panel (b): expectation value of the density of left spins.

Figure 7

Main panel: entanglement entropy at the final instant of the evolution for $\tau \in [1,200]$ at different system sizes ($L=10\text{--}100$, from bottom to top). Dashed maroon line: $y=\frac{1}{12}{log}_{2}x+\text{const}$. Inset: entanglement entropy at the final instant of the evolution for $\tau =100$ as a function of the system size $L$.

Figure 8

Main panel: Schmidt gap at the final instant of the evolution for $\tau \in [15,150]$ at different system sizes ($L=80\text{--}150$, from bottom to top). Black line: $y\approx x^{-1/2}$. Inset: Schmidt gap at the final instant of the evolution for $\tau =100$ as a function of the system size $L$.

Figure 9

Main panel: dynamics of the entanglement entropy $S\left(t\right)$ for $L=50$, $h_i=0.5$, $h_f=1.5$, and different values of $\tau$ (dashed vertical line: location of the critical point). Inset, black dots: period of oscillation of the entropy as a function of the gap at $h=h_f$ ($h_i=0.5$, $h_f=1.41,1.42,\cdots ,1.5$) with $\tau =50$. Red line: fit with $y=a_0+a_1/x$.

Figure 10

Dynamics of the entanglement spectrum with $L=50$, $h_i=0.5$, and $h_f=1.5$. Red/blue/green/brown line: dynamical first/second/third/fourth eigenvalue of the reduced density matrix of the half chain. Panels (a)–(f): $\tau =500$, $0.1$, 5, 30, 40, 100. The cyan lines in panel (a) show the ground-state values of the first four eigenvalues. In panels (b) and (c) black and red, green and blue data are almost coincident.

Figure 11

Schmidt gap at the final instant of the evolution for $\tau \in [1,200]$ at different system sizes for a ramping from $h_i=1.5$ to $h_f=0.5$.