Symmetry deduction from spectral fluctuations in complex quantum systems

Abstract

The spectral fluctuations of complex quantum systems, in an appropriate limit, are known to be consistent with those obtained from random matrices. However, this relation between the spectral fluctuations of physical systems and random matrices is valid only if the spectra are desymmetrized. This implies that the fluctuation properties of the spectra are affected by the discrete symmetries of the system. In this Rapid Communication, it is shown that in the chaotic limit the fluctuation characteristics and symmetry structure for any arbitrary sequence of measured or computed levels can be inferred from its higher-order spectral statistics without desymmetrization. In particular, we consider a spectrum composed of k > 0 independent level sequences with each sequence having the same level density. The kth order spacing ratio distribution of such a composite spectrum is identical to its nearest-neighbor counterpart with a modified Dyson index k. This is demonstrated for the spectra obtained from random matrices, quantum billiards, spin chains, and experimentally measured nuclear resonances with disparate symmetry features.

Dynamics of spectral correlations in the entanglement Hamiltonian of the Aubry-André-Harper model

“…11(b). The analytic distributions [58,59] P(r n ) are found to match with our numerically determined distributions.…”

“…We have considered this variant [Eq. ( 20)] here, since benchmark analytic results for the probability distribution of these nth-order spacing ratios have been obtained for Gaussian ensembles [58] and for uncorrelated eigenvalues [59]. For the initial product state (C P ) in the delocalized phase, i.e., λ = 0.5, the distribution for NN spacing ratio n = 1 as well as higher orders n = 2, 3, 4 have been plotted at the initial time, Fig.…”

Exact gap-ratio results for mixed Wigner surmises of up to 4 eigenvalues

“…In recent years, a number of generalizations have been added to the literature on higher-order level spacings [19][20][21][22][23][24][25][26] and non-hermitian Hamiltonians [27]. This short paper adds to the current literature by computing exact r-distribution functions for mixed Wigner surmises, such as two superimposed GUE spectra.…”

Long-range level correlations in quantum systems with finite Hilbert space dimension

“…As mentioned in the Introduction, before analyzing level fluctuations a transformation, called unfolding, is almost always necessary. Although it can be avoided in the study of short-range spectral statistics (as in the notable case of the adjacent level gap ratio [37][38][39][40]), long-range spectral statistics, which are precisely the aim of this paper, always require such a preliminary step. The basic features of this smoothing mechanism are reviewed below.…”