See, e.g

(unpublished)

The sense in which the eigenvalues are Poisson distributed requires two qualifications. First, the average level spacing varies systematically and substantially within each spectrum

it is exponentially small in volume near the middle and at best power law small in the tails. Second, since the number of random values of the potential is very small compared to the number of (many-body) levels generated, there are bound to be significant correlations in the latter, even deep inside the insulating phase. However, to the best of our knowledge, these correlations are undetectable provided the thermodynamic limit is taken first before defining and computing statistical measures. Loosely speaking, for a system of L sites, any statistical measure involving a finite number G of consecutive gaps will converge to its Poisson value in the insulating phase as L→∞

GOE data was obtained from 1000 random matrices of size 3432, which is the number of states in a half-filled chain of L=14

The simplest one-parameter finite-size scaling would have the spectral statistics at the mobility edge intermediate between Poisson and GOE, as appears to be the case for single-particle localization (e. g., Ref. [14]). In this case, drifting crossing points would be due to slow decay with increasing L of effects from operators that are irrelevant (in the RG sense) at the critical fixed point. It should be possible then to eliminate or even change the sign of these effects by either changing the observable being studied or the Hamiltonian itself, to cross the phase boundary where the effects due to the irrelevant operators are either quantitatively small or of the opposite sign. However, it is possible that the sign of the irrelevant perturbation is constrained by physical considerations, e.g., in scalar ϕ4 theory the quartic term is irrelevant above four dimensions but must be positive for the low-temperature phase to exist (the vortex fugacity at the Kosterlitz-Thouless transition in another example, but there it is the high-T phase that needs it). We discuss (in main text) how something along these lines that would invalidate the simplest one-parameter scaling analysis might be responsible for the drift of our crossings in Fig. 2, and for the reason that we have been unable to eliminate or reverse this drift by varying the model or the quantities we measure

arXiv: