Effective Hamiltonian description on monitored Majorana chains: correlated power-law hoppings and unconventional entanglement scaling

Abstract: We investigate the structures of effective Hamiltonians governing monitored dynamics of a one-dimensional Majorana chain through the Lyapunov spectral analysis. We focus on a gapless phase characterized by finite-size scalings different from those in conventional critical and/or frustration-free systems; the spectral gap closing faster than $1/L$ but slower than $1/L^2$ and the entanglement entropy growing as $[\ln(L)]<sup>2$ with $L$ being the system size. We find that the corresponding effective Hamiltonians have random long-range power-law hoppings with nontrivial magnitude correlations, rather than being independently and identically distributed. To elucidate the role of these non-Gaussian correlations, we construct random power-law hopping models that capture the essential features of the effective Hamiltonians. The spectral gaps of the constructed models decay faster than $1/L$ but slower than $1/L^2$. We find that, in the absence of hopping correlations, the ground-state entanglement exhibits $\ln(L)$ scaling. In the presence of correlations, by contrast, the entanglement entropy is enhanced and its system-size dependence is consistent with $[\ln(L)]<sup>2$ scaling over the system sizes studied. These results suggest that correlations among long-range hopping magnitudes are responsible for the entanglement scaling that seldom appears in ground states of conventional isolated quantum systems.