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S. is overwhelming for small . M. Zhao et al. 0, The {GJ≠16} are TBRE ×ε. Here j=17/2, n=4. 0, The {GJ≠0 } are TBRE ×ε. Here j=17/2, n=4. 0 Fig. 12. s. s. probabilities obtained by ﬁxing G0 = ±1 and all other GJ being the TBRE Hamiltonian multiplied by . In this ﬁgure, j = 17 2 and n = 4. s. probability is also sizable because of the contributions from J = 6 and J = 12 (refer to Table 6). The above numerical experiments are not trivial. s. dominance. s. s. (J = 0, 6, 8, 12, and 22, refer to the last row of Table 6).

Physics Reports 400 (2004) 1 – 66 for GJ = − JJ : J (j ) (EI )2 = J (EI,i )2 /(DI ) i (46) and 2 (EI ) 2 = J i J (j ) EI,i (DI )2 . (47) J (j ) Although it is oversimpliﬁed to assume that all non-zero eigenvalues EI,i ’s are equal, it is very instructive to estimate the I ’s by using this assumption. One can then obtain 2 I ∼ j+ 1 2 J (j ) i (EI,i )2 DI − 36 j+ 1 2 . By using the analytical expressions of the number of states with angular momentum I for four fermions in a single-j shell [48], one ﬁnally obtains12 for I = 0: for I = 2: for I = 4: 2 I 2 I 2 I ∼ 12 − 36/ j + ∼ 8 − 36/ j + ∼ 7 − 36/ j + 1 2 1 2 , 1 2 , .

S. dominance results from an interplay between the diagonal and off-diagonal matrix elements. In Ref. [77], Kaplan and Papenbrock studied the structure of eigenstates for many-body fermion systems in the presence of the TBRE Hamiltonian. They found that near the edge of the spectrum, wave function intensities of the TBRE Hamiltonian exhibit ﬂuctuations which deviate signiﬁcantly from the expectations of the random matrix theory. A simple formula was given which relates these ﬂuctuations to the ﬂuctuations for the TBRE Hamiltonian.