Quenches near Ising quantum criticality as a challenge for artificial neural networks

The near-critical unitary dynamics of quantum Ising spin chains in transversal and longitudinal magnetic fields is studied using an artificial neural network representation of the wave function. A focus is set on strong spatial correlations which build up in the system following a quench into the vicinity of the quantum critical point. We compare correlations obtained by optimizing the parameters of the network states with analytical solutions in integrable cases and time-dependent density matrix renormalization group (tDMRG) simulations, as well as with predictions from a semiclassical discrete truncated Wigner analysis. While the semiclassical approach yields quantitatively correct results only at very short times and near zero transverse fields, the neural-network representation is applicable in a much wider regime. However, for quenches close to the quantum critical point the representation becomes inefficient. For nonintegrable models we show that in regimes where tDMRG is limited to short times due to extensive entanglement growth, also the neural-network parametrization converges only at short times.

Figure 1

(a) Setup of the ANN consisting of one visible and one hidden layer connected via weights $W_{i,j}$ between each pair of variables from different layers and bias $a_i$ (visible) or $b_j$ (hidden) for each variable. (b) Quench protocol in the TFIM for sudden quenches from the deep paramagnetic regime (large $h_{x,i})$ to different distances from the QCP within the paramagnetic and into the ferromagnetic phase, where the ground-state configurations are depicted by the spins at the top.

Figure 2

(a) Correlation length $\xi \left(t\right)$ at a fixed time $t=1$ as a function of the transverse field $h_{x,f}$ after a sudden quench from $\left(h_{x,i}=100,h_{z,i}=0\right)$ with $h_{z,f}=0$ within the paramagnetic and into the ferromagnetic phase. The results of the ANN approach for different $\alpha$ are compared with the dTWA calculations and the exact solutions. (b) Time dependence of the exact correlation function $C_d^{zz}\left(t\right)$ after quenches from the deep paramagnetic phase to several $h_{x,f}$. The correlation length in (a) is calculated by fitting an exponential function to the short-distance decay of the absolute value of $C_d^{zz}$. (c) Time evolution of the correlation function $C_d^{zz}\left(t\right)$ after a sudden quench to $\left(h_{x,f}=1,h_{z,f}=0\right)$ for a spin chain with $N=10$ sites (upper plot) and to $\left(h_{x,f}=2,h_{z,f}=0\right)$ for a spin chain with $N=42$ sites (lower plot).

Figure 3

(a) Exact correlation length ${\xi }_e\left(t\right)$ for $N=10$ spins as a function of time after a sudden quench from $\left(h_{x,i}=100,h_{z,i}=0\right)$ to $h_{z,f}=2$ and different values of $h_{x,f}$. (b), (c), (d) Deviations $\mathrm{\Delta }\xi \left(t\right)=\left|{\xi }_s-{\xi }_e\right|$ of simulation results ${\xi }_s\left(t\right)$ from the exact result. The ANN calculation is not limited by Monte Carlo sampling errors. See also Appendix pp2 for more final values $h_{z,f}$.

Figure 4

(a) Time evolution of the correlation function $C_d^{zz}\left(t\right)$ after a sudden quench from $\left(h_{x,i}=100,h_{z,i}=0\right)$ to $\left(h_{x,f}=0.5,h_{z,f}=1\right)$ in a system with $N=42$ sites. The ANN approach for $\alpha =1$ and $\alpha =2$ is compared to tDMRG calculations with bond dimensions $D=128$ and $D=5$. See also Fig. 8 in Appendix pp5 for quenches described with a larger $\alpha \le 6$. (b) The von Neumann entanglement entropy $S_{\mathrm{vN}}$ as a function of time after the same quench as in (a). The entanglement entropy is calculated using tDMRG with different bond dimensions. The deviations at late times show that in this regime tDMRG breaks down after long times due to the growing entanglement.

Figure 5

Correlation length at a fixed time (a); time evolution of the von Neumann entanglement entropy (b) after a sudden quench from $\left(h_{x,i}=100,h_{z,i}=0\right)$ to different distances $\epsilon$ from the quantum critical point in the TFIM with $h_{z,f}=0$. In the region of large correlation lengths, large $\alpha$ are needed in the ANN approach to capture the exact solution. In the same region also volume-law entanglement is found.

Figure 6

Time evolution of correlation length and von Neumann entropy after sudden quenches from $\left(h_{x,i}=100,h_{z,i}=0\right)$ to different values of $h_{x,f}$ with $h_{z,f}=0$ (a), $h_{z,f}=1$ (b), $h_{z,f}=2$ (c), and $h_{z,f}=3$ (d). The ANN approach with $\alpha =1$ and $\alpha =10$ as well as the dTWA are compared with exact results. Panel (e) shows the time evolution of the von Neumann entanglement entropy extracted from the tDMRG calculations for $h_{z,f}=1,h_{z,f}=2$, and $h_{z,f}=3$. The entanglement entropy is also plotted as contours in the color plot of the exact correlation length in (a), (b), (c), and (d). There one can directly see that large entanglement entropy and volume-law entanglement can be found in the same regimes as large correlation lengths.

Figure 7

Discrete quantum phase space for a spin-$\frac{1}{2}$ system spanned by two variables $a_1,a_2$. The colored lines denote the three sets of parallel lines in the $\left(2\times 2\right)$-dimensional finite mathematical field. With each set a spin operator ${\sigma }^{\alpha },\alpha \in \left\{x,y,z\right\}$, is associated and each line is identified with one eigenvalue of the corresponding operator.

Figure 8

Time evolution of the correlation function $C_d^{zz}\left(t\right)$ after a sudden quench from $\left(h_{x,i}=100,h_{z,i}=0\right)$ to $\left(h_{x,f}=1,h_{z,f}=0\right)$ in a spin chain with $N=12$ (a) and $N=42$ (b) sites. The correlation function is shown for distances $d=1$ (solid lines), $d=2$ (dashed lines), and $d=3$ (dotted lines) between the considered spins. The number of weight parameters in the ANN approach is increased to see that $\alpha =15$ is necessary to capture the exact dynamics for $N=12$, while full convergence cannot be reached for $N=42$ within suitable computation time.