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

Abstract

We investigate the structure of effective Hamiltonians governing monitored dynamics of a one-dimensional Majorana chain using Lyapunov spectral analysis. We focus on the gapless phase of the Lyapunov spectrum, where the effective Hamiltonians behave differently from conventional critical Hamiltonians in isolated systems; the spectral gap closes faster than $1/L$ but slower than $1/L^2$ and the entanglement entropy grows as $[\ln(L)]^2$ where $L$ is the system size. We find that the 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 $[\ln(L)]^2$ scaling over the system sizes studied. These results suggest that correlations among long-range hopping magnitudes are responsible for the entanglement scaling rarely encountered in the ground states of conventional isolated quantum systems.