Fate of Measurement-Induced Phase Transition in Long-Range Interactions

We consider quantum many-body dynamics under quantum measurements, where the measurement-induced phase transitions (MIPs) occur when changing the frequency of the measurement. In this work, we consider the robustness of the MIP for long-range interaction that decays as $r^{-\alpha }$ with distance $r$. The effects of long-range interactions are classified into two regimes: (i) the MIP is observed $(\alpha >{\alpha }_c)$, and (ii) the MIP is absent even for arbitrarily strong measurements $(\alpha <{\alpha }_c)$. Using fermion models, we demonstrate both regimes in integrable and nonintegrable cases. We identify the underlying mechanism and propose sufficient conditions to observe the MIP, that is, $\alpha >d/2+1$ for general bilinear systems and $\alpha >d+1$ for general nonintegrable systems ($d$: spatial dimension). Numerical calculation indicates that these conditions are optimal.

Figure 1

Schematic of findings. We consider fermion systems with long-range interactions, where each site is constantly measured by the amplitude (frequency) $\gamma$. The function ${\gamma }_c(\alpha )$ is the critical measurement amplitude [see Fig. 3 for the free fermion case]. In the regime $\alpha <{\alpha }_c$, the MIP does not exist. See Statement 1 for sufficient conditions of $\alpha$ for the existence of the MIP.

Figure 2

(a) and (b) ${\overline{S}}_{\ell }$ as a function of $\sin (\pi \ell /L)$ ($L=200$) for $\alpha =1.8$ in (a) and for $\alpha =0.8$ in (b), where the $x$ axis is log scale. The CFT behavior can be observed around $\ell \sim L/2$ as indicated in (a), where the solid lines are guides for the eyes. (c) and (d) The mutual information as a function of $\gamma$.

Figure 3

(a) Finite-size scaling for the mutual information with the BKT scenario $g(L)\gamma \overline{I}(\gamma ,\alpha )$ versus $\log \{L\xi [\gamma ,{\gamma }_c(\alpha )]\}=\log L-\nu /\sqrt{\gamma -{\gamma }_c(\alpha )}$, where $\nu =6.0$, 5.5, 4.5, and 3.0 for $\alpha =1.6$, 1.8, 2.3, and $\alpha =\infty$, respectively and $g(L)=[1+1/(2\log L-4){]}^{-1}$. (b) Critical amplitude ${\gamma }_c(\alpha )$ as a function of $\alpha$. No critical amplitude exists for $\alpha <1.5$.

Figure 4

(a) and (b) The mutual information $\overline{I}(\gamma ,\alpha )$ between the farthest two sites for $\alpha =3.0$ in (a) and for $\alpha =0.5$ in (b). The values ${\gamma }_p$ indicated by arrows are amplitudes that give the peaks. (c) The system-size dependence of ${\gamma }_p$. This indicates that ${\gamma }_p$ robustly remains finite for $\alpha >2$ in the thermodynamic limit. (d) The finite-size scaling for $\alpha =3.0$ with the ansatz $\overline{I}(\gamma ,\alpha )=L^{-\beta }f[(\gamma -{\gamma }_p)L^{1/\nu }]$ with the exponents $\beta =2.59\pm 0.02$ and $\nu =1.4\pm 0.1$.