Kibble-Zurek dynamical scaling hypothesis in the Google analog-digital quantum simulator of the $XX$ model

TL;DR

The paper investigates the Kibble-Zurek scaling of quantum quenches near the critical point of the 2D XX model using an analog-digital quantum simulator (Google XX_Google) and complementary tensor-network simulations. It employs an infinite iPEPS with NTU updates to probe the KZ scaling of correlations and a finite 8×8 lattice with TDVP to assess finite-size effects, extracting critical exponents $z=1$, $\nu\approx0.67169$, and $\eta\approx0.03810$. The results on the infinite lattice show a robust data collapse of $\hat{\xi}^{1+\eta}C(t,R)$ and a scaling of $\xi(t_c) \propto (J_r t_r)^{0.41(3)}$, in agreement with KZ predictions, while the 8×8 lattice exhibits a crossover to adiabatic behavior with $\xi(t_c)$ saturating near $1$ for larger ramps and an energy scaling $\Delta E(s_c) \propto (J_r t_r)^{-1.2}$ transitioning to $\Delta E \propto (J_r t_r)^{-2}$. These finite-size effects reconcile with the observation that the experimental data may display nonadiabatic KZ-like power laws, but the simulations indicate a more complex crossover when size and ramp time are limited. The work highlights the importance of lattice size and ramp time in interpreting QKZM dynamics in 2D quantum simulators and provides a benchmark for future experiments and tensor-network methods.

Abstract

State-of-the-art tensor networks are employed to simulate the Hamiltonian ramp in the analog-digital quantum simulation of the quantum phase transition to the quasi-long-range ordered phase of the two-dimensional square-lattice $XX$ model [T.I. Andersen \textit{et al.}, Nature (London) \textbf{638}, 79 (2025)]. We focus on the quantum Kibble-Zurek (KZ) mechanism near the quantum critical point. Using the infinite projected entangled pair state, we simulate an infinite lattice and demonstrate the KZ scaling hypothesis for the $XX$ correlations across a wide range of ramp times. We use the time-dependent variational principle algorithm to simulate a finite $8\times 8$ lattice, similar to the one in the quantum simulation, and find that adiabatic finite-size effects dominate for longer ramp times, where the correlation length's growth with increasing ramp time saturates and the excitation energy's dependence on the ramp time crosses over to a power-law decay characteristic of adiabatic transitions. This finding contradicts the quantum simulation data where the correlation length seems to obey KZ-like power laws, although with modified exponents.

Figures (8)

• Figure 1: The ramp. The values of $G(s)$ and $J(s)$ in the Hamiltonian \ref{['eq:Hsigma']}. The experimental ramp XX_Google is linear $s = (t/t_r)$. In our tensor network simulation, it is slightly modified to $s = (t/t_r)[1-\exp(-40t/t_r)]$ to reduce excitation at the beginning of the ramp Science_Dwave. The left inset shows the initial state of a small $4 \times 4$ system with a staggered field of strength $G(0)=G_{r}$ and nearest-neighbor (NN) coupling $J(0) = 0$. The dots (crosses) correspond to fields going out of (into) the page. The right inset shows the system at the end of the ramp where the staggered field is turned off, $G(1)=0$, and the nearest-neighbor coupling is turned on, $J(1) = J_r$.
• Figure 2: Trotter gate. In (a), an infinite PEPS (iPEPS) tensor network with two sublattice tensors $A$ and $B$ and bond dimension $D$. In (b) left, a two-site Trotter gate is applied to a pair of tensors, increasing the bond dimension from $D$ to $rD$. In (b) right, the dimension is truncated back to $D$. The error of the truncation is the Frobenius norm of the difference between the left and the right. The two tensors on the right, $A'$ and $B'$, are optimized to minimize the error. In (c), the norm squared of (b) right. Here, the balls are double PEPS tensors (contractions of PEPS tensors with their conjugates). The four corner doubles are approximated by their singular value decompositions truncated to one singular value (SVD$_1$). In (d), the optimized $A'$ and $B'$ make the new iPEPS' ready for application of the next Trotter gate.
• Figure 3: Accumulated truncation error. The accumulated truncation error $\delta$ [see \ref{['eq:delta']}] as a function of the ramp parameter $s$ for different ramp times and maximal bond dimension $D=24$. It is a rough upper estimate for errors of local observables inflicted by TN truncation Science_Dwave.
• Figure 4: Correlations at the critical point. The scaled correlation function $\hat{\xi}^{1+\eta}C(t,R)$ as a function of the scaled distance $R/\hat{\xi}$ for several values of the ramp parameter $s=t/t_r$ near the critical point $s_c=0.45(8)$. At the critical $s$, with increasing ramp time $J_rt_r$, the scaled plots are expected to become smoother and collapse to a single scaling function in accordance with the Kibble-Zurek scaling hypothesis. The slower ramps ($J_rt_r\geq4$) are shown in the insets. Their collapse is the best at $s=0.45$, in consistency with the estimate for the critical point $s_c=0.45(8)$. We assume this critical value, $s_c=0.45$, in the following discussion.
• Figure 5: Correlation length near the critical point. The scaled correlation length $\xi(t)/\hat{\xi}$ as a function of the scaled time $(t-t_c)/\hat{t}$ for different ramp times. Here, the KZ timescale $\hat{t}$ is given by $J_r\hat{t} = 0.36 (J_rt_r)^{z\nu/(1+z\nu)}$ with $z\nu=0.67$. The color-shaded areas indicate the error bars of the fit. Within the error bars, the plots collapse to a single scaling function in the KZ regime of the scaled time between $\pm 1$. In particular, the inset shows the correlation length when the ramp crosses the critical point, $\xi(t_c)$, as a function of $J_rt_r$. The best fit $\xi(t_c)\propto (J_rt_r)^{0.41(3)}$ is consistent with the KZ exponent $0.40$ across a wide range of ramp times.