Abstract
The level statistics in the transition between delocalized and localized phases of many body interacting systems is considered. We recall the joint probability distribution for eigenvalues resulting from the statistical mechanics for energy level dynamics as introduced by Pechukas and Yukawa. The resulting single parameter analytic distribution is probed numerically via Monte Carlo method. The resulting higher order spacing ratios are compared with data coming from different quantum many body systems. It is found that this Pechukas–Yukawa distribution compares favorably with β–Gaussian ensemble—a single parameter model of level statistics proposed recently in the context of disordered many-body systems. Moreover, the Pechukas–Yukawa distribution is also only slightly inferior to the two-parameter β–h ansatz shown earlier to reproduce level statistics of physical systems remarkably well.
Highlights
Fritz Haake made fundamental contributions first to quantum optics to the fast developing in the eighties of the last century area of Quantum Chaos with his seminal monograph on the subject [3] and with many original works from introducing a celebrated kicked top model [4] to providing a link between Gutzwiller’s periodic orbit theory [5] and random matrix statistics [6,7,8].Those late works provided a highlight of Haake’s fascination of the link between spectral properties of physical models and random matrix theory
We recall the joint probability distribution for eigenvalues resulting from the statistical mechanics for energy level dynamics as introduced by Pechukas and Yukawa
It is found that this Pechukas-Yukawa distribution compares favorably with β– Gaussian ensemble – a single parameter model of level statistics proposed recently in the context of disordered many-body systems
Summary
Fritz Haake made fundamental contributions first to quantum optics (see e.g. [1, 2]) to the fast developing in the eighties of the last century area of Quantum Chaos with his seminal monograph on the subject [3] and with many original works from introducing a celebrated kicked top model [4] to providing a link between Gutzwiller’s periodic orbit theory [5] and random matrix statistics [6,7,8]. Haake devoted particular interest to periodically driven (Floquet) systems in this context, in particular since a transition between level clustering for integrable systems to level repulsion (for quantally chaotic system) may be viewed as a relaxation toward equilibrium [11] For this transition, following the original Wigner 2 × 2 matrices approach, Lenz and Haake found an interpolating spacing distribution [12] which compared well with random matrix simulations adding a significant contribution to the topic which originated with the early work of Rosenzweig and Porter [13]. The Rosenzweig-Porter distribution was an important twist on random matrices As it is well known [3, 14, 15] for generalized time reversal invariant systems (the case we shall solely concentrate on) the Gaussian orthogonal ensemble of random matri-.
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