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 ${n}^{3/4}{(lnn)}^{\ensuremath{-}1/4}\ensuremath{\lesssim}{W}_{c}\ensuremath{\lesssim}nlnn$ for the scaling of
Wc(n) in the QD model (we find ${W}_{c}\ensuremath{\sim}n$ numerically), ${W}_{c}(n)\ensuremath{\sim}\text{const}$ for the 1D model, ${W}_{c}\ensuremath{\sim}nlnn$ for the uQD and u1D models without off-diagonal correlations, and ${W}_{c}\ensuremath{\sim}{n}^{1/2}lnn$ 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.
A particle-number conserving approach is presented to solve the nuclear mean-field plus pairing Hamiltonian problem with a realistic deformed Woods-Saxon single-particle potential. The method is designed for the state-dependent monopole pairing Hamiltonian $H{\^{}}_{\mathrm{pair}}={\ensuremath{\sum}}_{\ensuremath{\alpha}\ensuremath{\beta}}{G}_{\ensuremath{\alpha}\ensuremath{\beta}}{c}_{\ensuremath{\alpha}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}\ifmmode\bar\else\textasciimacron\fi{}}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}\ifmmode\bar\else\textasciimacron\fi{}}{c}_{\ensuremath{\beta}}$ with an arbitrary set of matrix elements ${G}_{\ensuremath{\alpha}\ensuremath{\beta}}.$ Symmetries of the Hamiltonians on the many-body level are discussed using the language of
P symmetry introduced earlier in the literature and are employed to diagonalize the problem; the only essential approximation used is a many-body (Fock-space) basis cutoff. An optimal basis construction is discussed and the stability of the final result with respect to the basis cutoff is illustrated in details. Extensions of the concept of
P symmetry are introduced and their consequences for an optimal many-body basis cutoff construction are exploited. An algorithm is constructed allowing to solve the pairing problems in the many-body spaces corresponding to $p\ensuremath{\sim}40$ particles on $n\ensuremath{\sim}80$ levels and for several dozens of lowest lying states with precision $\ensuremath{\sim}$(1--2) % within seconds of the CPU time on a CRAY computer. Among applications, the presence of the low-lying seniority
s=0 solutions, that are usually poorly described in terms of the standard approximations (BCS, HFB), is discussed and demonstrated to play a role in the interpretation of the spectra of rotating nuclei.
We construct and solve a classical percolation model with a phase transition that we argue acts as a proxy for the quantum many-body localization transition. The classical model is defined on a graph in the Fock space of a disordered, interacting quantum spin chain, using a convenient choice of basis. Edges of the graph represent matrix elements of the spin Hamiltonian between pairs of basis states that are expected to hybridize strongly. At weak disorder, all nodes are connected, forming a single cluster. Many separate clusters appear above a critical disorder strength, each typically having a size that is exponentially large in the number of spins but a vanishing fraction of the Fock-space dimension. We formulate a transfer matrix approach that yields an exact value $\ensuremath{\nu}=2$ for the localization length exponent, and also use complete enumeration of clusters to study the transition numerically in finite-sized systems.