Abstract
We study the level-spacing statistics for noninteracting Hamiltonians defined on the two-dimensional quasiperiodic Ammann-Beenker (AB) tiling. When applying the numerical procedure of ``unfolding,'' these spectral properties in each irreducible sector are known to be well described by the universal Gaussian orthogonal random matrix ensemble. However, the validity and numerical stability of the unfolding procedure has occasionally been questioned due to the fractal self-similarity in the density of states for such quasiperiodic systems. Here, using the so-called $r$-value statistics for random matrices, $P(r)$, for which no unfolding is needed, we show that the Gaussian orthogonal ensemble again emerges as the most convincing level statistics for each irreducible sector. The results are extended to random-AB tilings where random flips of vertex connections lead to the irreducibility.
Highlights
The statistical description of energy levels in complex systems has a long and distinguished history [1,2]
In the context of many-body localization (MBL), r-value statistics has proven its worth by allowing the numerical determination of the transition from MBL to the so-called ergodic phase at weak disorders without the need to unfold spectra [10,12,13]
In this Letter, we employ the r-value statistics to QP Hamiltonians defined on the Ammann-Beenker (AB) octagonal tiling [14] and to randomized versions of the tiling
Summary
The statistical description of energy levels in complex systems has a long and distinguished history [1,2]. The P(s) was shown to follow PGOE(s) better than the celebrated Wigner expression PWigner(s) = π /2 exp (−π s2/4) of GOE which is based on a (2×2)-matrix surmise [2]. We shall use a surmise based on a (5×5) matrix (see Appendix) which improves upon (2) such that the deviation to r GOE is reduced from 1% to < 0.4%.
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