Probing Symmetries of Quantum Many-Body Systems through Gap Ratio Statistics

Abstract

The idea of describing properties of complicated physical systems, such as complex atomic nuclei, using random numbers dates back to the 1950s. One application is in quantum mechanics, where random numbers are a tool for making accurate predictions about the statistics of discrete energy levels a system can assume. More precisely, the statistical distribution of the spacings between successive energy levels can be compared with distributions from random matrices, which provides a signature of whether the system behaves in a regular or a chaotic way. Here, we take this toolkit a step further and use it to probe for symmetries, sometimes hidden, in certain physical systems.Because of additional subtleties in the case of interacting particles, the focus in recent years has shifted away from the statistics of the spacings between energy levels and toward the statistics of ratios between successive spacings, which are by now a widely used tool to compare experimental or numerical observations with theoretical predictions. Extra symmetries can modify these statistics to be somewhere between the “regular” and “chaotic” predictions. We show that it is possible to extend the theory of spacing ratio statistics to account for the presence of additional symmetries.Our result provides a tool to not only get a signature of chaos or regularity in systems with symmetries but also uncover these symmetries if they were previously unnoticed.