Fidelity of the diagonal ensemble signals the many-body localization transition

In this work, we use exact matrix diagonalization to explore the many-body localization (MBL) transition in a random-field Heisenberg chain. We demonstrate that the fidelity and fidelity susceptibility can be utilized to characterize the interaction-driven many-body localization transition in this closed spin system which is in agreement with previous analytical and numerical results [S. Garnerone, N. T. Jacobson, S. Haas, and P. Zanardi, Phys. Rev. Lett. 102, 057205 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.057205; P. Zanardi and N. Paunkovic, Phys. Rev. E 74, 031123 (2006)PLEEE81539-375510.1103/PhysRevE.74.031123]. In particular, instead of ground-state fidelity, we test the fidelity between two diagonal ensembles related by a small parameter perturbation δh, it is special that here the parameter perturbation δh_{i} for each site are random variables like h_{i}. It shows that fidelity of the diagonal ensemble develop a pronounced drop at the transition. We utilize fidelity to estimate the critical disorder strength h_{c} for different system size, we get h_{c}∈ [2.5,3.9] and get a power-law decay with an exponent of roughly -1.49(2) for system size N, and can extrapolate h_{c}^{inf} of the infinite system is about 2.07 which all agree with a recent work by Huse and Pal, in which the MBL transition in the same model was predicted to be hc [2,4]. We also estimate the scaling of maximum of averaged fidelity susceptibility as a function of system size N, it shows a power law increase with an exponent of about 5.05(1).