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Motivated by the constrained many-body dynamics, the stability of thelocalization-delocalization properties to the inclusion of the soft constraintsis addressed in random matrix models. These constraints are modeled bycorrelations in long-ranged hopping with Pearson correlation coefficientdifferent from zero or unity. Counterintuitive robustness of delocalizedphases, both ergodic and (multi)fractal, in these models is numericallyobserved and confirmed by the analytical calculations. First, matrix inversiontrick is used to uncover the origin of such robustness. Next, to characterizedelocalized phases a method of eigenstate calculation, sensitive tocorrelations in long-ranged hopping terms, is developed for random matrixmodels and approved by numerical calculations and previous analytical ansatz.The effect of the robustness of states in the bulk of the spectrum theinclusion of to soft constraints is generally discussed for single-particle andmany-body systems.

Robustness of delocalization to the inclusion of soft constraints in long-range random models