Estimating the Many-Body Localization Transition Critical Disorder from Consecutive-Gap-Ratio Statistics

The consecutive-gap-ratio distribution is a powerful tool that reveals exciting properties of many-body complex systems. Here, we use a recently proposed two-parameter surmise ratio distribution [1] that describes the crossover between the Gaussian orthogonal ensemble (GOE) and Poisson statistics, which is believed to occur in systems undergoing the many-body localization (MBL) transition. Several statistical quantities related to the moments of the ratio distribution, such as the $\braket{r}$ together with the sample-to-sample variance $(\Delta_s r)^2$ [2], have been used to estimate the critical field at which the MBL transition occurs. In this study, we analyze the ensemble averaged consecutive-gap-ratio distribution of the zero-magnetization sector of the isotropic Heisenberg spin-$1/2$ chain with disordered local fields, considering both open and periodic boundary conditions. We determine the best-fit parameters of the surmise distribution for varying the maximal local field strength $h$ across several chain sizes $L$. Using finite-size scaling analysis of our data, we determine the critical field $h_{\rm c}$ . Finally, we compare our estimated values for both boundary conditions with those reported in the literature.