We analyze the statistical properties of complex systems with specific conservation laws and symmetry conditions which lead to various constraints and thereby structures in their matrix representation. An increase of constraints leads to a variation of the spectral statistics from Wigner-Dyson to Poisson limits, but the eigenfunction statistics remains weakly multifractal in the bulk. For some constraints, the statistics not only lies between the two limits but is size-independent too, thus indicating a critical point. Our results also reveal an important trend: While the spectral statistics is strongly sensitive to the number of independent matrix elements, the eigenfunction statistics seems to be affected mainly by their relative strengths. This is contrary to a previously held belief of a one-to-one relation between the statistics of the eigenfunctions and eigenvalues (e.g., associating Poisson statistics to the localized eigenfunctions and Wigner-Dyson statistics to delocalized ones). This also indicates the existence of new universality classes based on the matrix constraints (different from the 10 already-known symmetry-based classes).
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