Entanglement dynamics of monitored non-interacting fermions on Graphic-Processing-Units

Abstract

The description of the entanglement dynamics of monitored non-interacting fermions, including the existence of measurement-induced phase transitions (MIPT), is a challenging problem with conflicting results in the literature. The mapping of the problem onto a non-linear sigma model (NLSM) indicates that relatively large lattice sizes are required to determine the nature of the entanglement entropy (EE) in the thermodynamics limit. Here we address this problem numerically for monitored non-interacting fermions with $U(1)$ symmetry. The use of Graphic-Processing-Unit (GPU) techniques, even with outdated hardware, makes possible to reach much larger lattice sizes ($L = 16384$ and $160\times160$ in one (1d) and two (2d) dimensions respectively) than in previous studies which enables us to characterize quantitatively the entanglement dynamics. In 1d, we show that in order to confirm the absence of a MIPT, for both projective and homodyne measurements, predicted by the NLSM it is necessary to reach $L \sim 10000$. In 2d, also as predicted by the NLSM, we observe for both protocols a MIPT at finite monitoring rate characterized by a scale invariant mutual information. The critical monitoring strength depends on the protocol while the critical exponent $ν\approx 1.3$ governing the approach to the MIPT is similar in both cases. These features are not correctly predicted by the NLSM. Our results paves the way for a fully quantitative description of the entanglement dynamics of monitoring quantum systems.

Figures (6)

• Figure 1: Left: $C(\tilde{r})$ Eq. (\ref{['eq:Cr']}) using the PM protocol, see Appendix \ref{['app:protocol']} for details, with $L = 8192$, $34$ trajectories and an additional time average over four equidistant points for $t \ge L/2$. The dashed curves stand for the fittings to: ballistic ($\log^2$ decay), diffusive ($1/r^2$ decay) and exponential decay. The monitoring strength is $\gamma = 0.5$ and the fitting parameters are $l_0 \approx 63\pm 3$, $p \approx 2.20\pm0.0025$, $l_{\rm cor} \approx 1450 \pm 2$. Right: The correlation length $l_{\rm corr}$ from the fitting of $C(r)$ as a function of $\gamma$. Depending on $\gamma$ and $L$ we average over both $30-100$ trajectories and $4-8$ equally spaced time points $t \in [L/2,L]$.
• Figure 2: Left: $C(r)$ Eq. (\ref{['eq:Cr']}) for the QSD protocol, different monitoring rates $\gamma$, $L = 16384$ for $\gamma=0.4$, $L=12000$ for $\gamma=0.45$, and $L=8192$ otherwise. The time step is $dt=0.05$. The width of each curve is the error bar. For each $\gamma$, we average over both at least $10$ trajectories and $33$ equally spaced time points $t\in [L/2,L]$ after saturation. For $\gamma\leq 0.5$, $C(r)\sim r^{-p}$ with, $p \sim 2$ ($p = 2.1$ for $\gamma = 0.3$). We need to reach $L = 12000, 16384$, much larger than in previous studies, to observe an exponential decay $\sim \exp(-r/l_{\rm cor})$ for $\gamma = 0.45, 0.4$ respectively. Right: Correlation length $l_{\rm cor}$ from the fitting of $C(r)$ for the QSD protocol for different sizes $L$ compared with the analytic prediction Eq. (\ref{['eq:RG']}). For instance, $l_{cor}=1620 \pm 165$ for $\gamma = 0.4$. Shifting $\gamma\rightarrow \gamma-\gamma_c$ and treating $\gamma_c$ as an additional fitting parameter yields $\gamma_c = 0.00 \pm 0.10$, which confirms the absence of a MIPT.
• Figure 3: Particle number covariance $\overline{G_{AB}}$ Eq. (\ref{['eq:GAB']}) as a function of $\gamma$ for different sizes $L$. The lines stand for least‐squares polynomial fittings, see Ohtsuki2014 and main text for details. Left: QSD protocol. A sharp crossing occurs at $\gamma_c \approx 4.77 \pm 0.01$ indicating the existence of a MIPT characterized by a scale invariant $\overline{G_{AB}}$. The inset depicts the optimal data collapse achieved by rescaling $\overline{G_{AB}}$ with $\lvert \gamma/\gamma_c - 1\rvert\,L^{1/\nu}$ around $\gamma_c$, where $\nu \approx 1.28 \pm 0.03$ is the critical exponent governing the divergence of the correlation length. Right: PM protocol. Similarly, the crossing is at $\gamma_c \approx 5.72 \pm 0.02$, (note that the most of the disagreement in $\gamma_c$ with respect to Ref. poboiko2023a is due to a different definition of the waiting times needed to compare both protocols) the optimal data collapse (inset) is at $\gamma_c = 5.72 \pm 0.02$ and $\nu = 1.31 \pm 0.08$. The value of $\nu$ agrees with the analogue for the 3d non‑Hermitian Anderson transition Shindou2021. Note that poboiko2023a$\mathcal{I}_2 \simeq {2\pi^2\over 3}\overline{G_{AB}}$ for $\gamma \lesssim \gamma_c$.
• Figure 4: Mutual information $\mathcal{I}_2$ and the particle‐number covariance $\overline{G_{AB}}$ Eq. (\ref{['eq:GAB']}) versus system size $L$. Left: PM protocol: $\gamma = 3$, $\mathcal{I}_2 \simeq {2\pi^2\over 3}\overline{G_{AB}} \propto L$ so the system is in the volume-law phase. Inset: $\gamma = 6.4$, both $\mathcal{I}_2$ and $\overline{G_{AB}}$ decays exponentially so the system is in the area-law phase. Right: $\mathcal{I}_2$ and $\overline{G_{AB}}$ at $\gamma_c$ do not depend on $L$ for both PM and QSD protocols. Moreover, its value at the MIPT does not depend much on the protocol.
• Figure 5: The correlation length $l_{\rm cor}$ resulting from the fitting of the exponential decay $C(r)$ as a function of $\gamma$. To extract $l_{\rm cor}$ from $C(r)$ when $\gamma \ge 0.6$, we only fit the region when $C(r)$ decays exponentially, but with $r \ll L/2$ to get rid of the boundary effects. For $\gamma \le 0.55$, we rescaled the distance following the details introduced in the main text. We perform the fitting with both $l_{\rm cor} \sim {1\over |\gamma - \gamma_c|} \exp({a\over |\gamma - \gamma_c|})$ Eq. (\ref{['eq:RG']}) poboiko2023 and the BKT prediction Eq. (\ref{['eq:BKT']}) alberton2021a$l_{\rm cor} \sim \exp({b\over \sqrt{|\gamma - \gamma_c|}})$. For size $L = 4096$, we get a critical monitoring strength $\gamma_c = 0.1\pm 0.1$ for Eq. (\ref{['eq:RG']}) and $\gamma_c = 0.22\pm 0.07$ for the BKT prediction. Similar results, suggesting a non-zero $\gamma_c$, occur for the BKT fitting function and smaller $N$. However, for size $L = 8192$, we obtain a critical monitoring strength $\gamma_c = 0.0 \pm 0.1$ using Eq. (\ref{['eq:RG']}) and $\gamma_c = 0.06\pm 0.1$ for the BKT prediction Eq. (\ref{['eq:BKT']}).