We numerically study the level statistics of the Gaussian β ensemble. These statistics generalize Wigner-Dyson level statistics from the discrete set of Dyson indices β = 1, 2, 4 to the continuous range 0 < β < ∞. The Gaussian β ensemble covers Poissonian level statistics for β → 0, and provides a smooth interpolation between Poissonian and Wigner-Dyson level statistics. We establish the physical relevance of the level statistics of the Gaussian β ensemble by showing near-perfect agreement with the level statistics of a paradigmatic model in studies on many-body localization over the entire crossover range from the thermal to the many-body localized phase. In addition, we show similar agreement for a related Hamiltonian with broken time-reversal symmetry.

Nonergodicity in the Anisotropic Dicke ModelBuijsman, W.; Gritsev, V.; Sprik, R.

We study the eigenstates of a paradigmatic model of many-body localization in the Fock basis constructed out of the natural orbitals. By numerically studying the participation ratio, we identify a sharp crossover between different phases at a disorder strength close to the disorder strength at which subdiffusive behaviour sets in, significantly below the many-body localization transition. We repeat the analysis in the conventionally used computational basis, and show that many-body localized eigenstates are much stronger localized in the Fock basis constructed out of the natural orbitals than in the computational basis. arXiv:1712.06892v3 [cond-mat.dis-nn] 30 May 2018 SciPost Physics Submission 3]. Inspired by the seminal work of Basko, Aleiner and Althshuler [4], a large number of investigations has revealed various intriguing properties of the many-body localized phase, among them the persistance up to infinite temperature [5], the separation from the thermal phase by a phase transition [6,7], and the growth of entanglement in the absence of transport [8,9]. The interest for MBL is mainly driven by the notion that many-body localized systems violate the fundamental assumption of statistical mechanics that a non-integrable system can serve as its own heath bath, a phenomenon that has been near-rigorously proven to exist only recently [10].Over the last few years, it has become clear [11] that not only the many-body localized phase, but also the thermal phase in the vicinity of the MBL transition ('critical phase') displays remarkable properties [12], such as subdiffusion [13,14], subthermal entanglement scaling [15], bimodality of the entanglement entropy distribution [16,17], and the violation of the eigenstate thermalization hypothesis [18,19]. The latter can be deduced from the violation of the Berry conjecture [20], roughly stating that the eigenstates of thermal systems are spread out over the full Hilbert space in any local basis. In this work, we study the spreading of eigenstates over the Hilbert space for a paradigmatic model of many-body localization. By numerically studying the participation ratio for a finite-size system, we identify a sharp crossover between different phases at a disorder strength close to the disorder strength at which subdiffusive behaviour [21] and the departure from Poissonian level statistics [7] sets in.We identify the crossover in the Fock basis constructed out of the natural orbitals, and repeat the analysis in the conventionally used computational basis. The natural orbitals and their corresponding occupation numbers resulting from the diagonalization of the oneparticle density matrix [22] recently gained significant attention in the field of MBL [23][24][25][26][27]. It was found [23] that the occupation numbers exhibit qualitatively different statistics in the thermal and the many-body localized phase, allowing them to be used as a probe for the MBL transition [7,28]. Based on these statistics, we argue that the scope can be naturally broadened by studying MBL in the Fock...