Role of Fock-space correlations in many-body localization

Models of many-body localization (MBL) can be represented as tight-binding models in the many-body Hilbert space (Fock space). We explore the role of correlations between matrix elements of the effective Fock-space Hamiltonians in the scaling of MBL critical disorder Wc(n) with the size n of the system. For this purpose, we consider five models, which all have the same distributions of diagonal (energy) and off-diagonal ("hopping") Fock-space matrix elements but different Fock-space correlations. These include quantum-dot (QD) and one-dimensional (1D) MBL models, their modifications (uQD and u1D models) with removed correlations of off-diagonal matrix elements, as well a quantum random energy model (QREM) with no correlations at all. Our numerical results are in full consistency with analytical arguments predicting n3/4 (ln n)-1/4 Wc n ln n for the scaling of Wc(n) in the QD model (we find Wc ~ n numerically), Wc(n) ~ const. for the 1D model, Wc ~ n ln n for the uQD and u1D models without off-diagonal correlations, and Wc ~ n1/2 ln n for QREM. The key difference between the QD and 1D models is in the structure of correlations of many-body energies. Removing off-diagonal Fock-space correlations makes both these models "maximally chaotic". Our findings demonstrate that the scaling of Wc(n) for MBL transitions is governed by a combined effect of Fock-space correlations of diagonal and off-diagonal matrix elements.