Precise matching of the correlated hopping model to monitored Majorana effective Hamiltonians

Determine the precise parameter correspondence and model-level correspondence between the effective Hamiltonians generated by monitored Majorana circuits and the constructed random power-law hopping Hamiltonian with correlated hopping magnitudes.

Background

The paper constructs a random Majorana Hamiltonian with algebraically decaying hoppings and tunable correlations among hopping magnitudes. This model reproduces qualitative features of the effective Hamiltonians arising in monitored Majorana circuits, including negative correlations between nearest-neighbor and long-range hopping magnitudes and positive correlations among long-range hopping magnitudes.

Numerical results indicate that the constructed model reproduces the unconventional entanglement scaling observed in the monitored dynamics: the ground-state entanglement scales logarithmically when hopping correlations are absent and appears to scale as [ln(L)]² when correlations are present. Nevertheless, the construction is phenomenological, and the precise relation between its parameters or microscopic structure and the effective Hamiltonians generated by the monitored circuits remains unresolved.

References

However, precise matching between the effective Hamiltonian for monitored Majorana circuits and the model parameters or model itself used in this analysis is left for future work.

— Effective Hamiltonian description on monitored Majorana chains: correlated power-law hoppings and unconventional entanglement scaling  (2609.04091 - Mochizuki et al., 3 Sep 2026) in Section 3.2, “Scaling of gap and entanglement” (Subsection 3.2)