David Logan

Abstract:

We study classical percolation models in Fock space as proxies for the quantum many-body localization (MBL) transition. Percolation rules are defined for two models of disordered quantum spin chains using their microscopic quantum Hamiltonians and the topologies of the associated Fock-space graphs. The percolation transition is revealed by the statistics of Fock-space cluster sizes, obtained by exact enumeration for finite-sized systems. As a function of disorder strength, the typical cluster size shows a transition from a volume law in Fock space to subvolume law, directly analogous to the behavior of eigenstate participation entropies across the MBL transition. Finite-size scaling analyses for several diagnostics of cluster size statistics yield mutually consistent critical properties. We show further that local observables averaged over Fock-space clusters also carry signatures of the transition, with their behavior across it in direct analogy to that of corresponding eigenstate expectation values across the MBL transition. The Fock-space clusters can be explored under a mapping to kinetically constrained models. Dynamics within this framework likewise show the ergodicity-breaking transition via Monte Carlo averaged local observables and yield critical properties consistent with those obtained from both exact cluster enumeration and analytic results derived in our recent work [arXiv:1812.05115]. This mapping allows access to system sizes two orders of magnitude larger than those accessible in exact enumerations. Simple physical pictures based on freezing of local real-space segments of spins are also presented and shown to give values for the critical disorder strength and correlation length exponent ν consistent with numerical studies.

Abstract:

A canonical model for many-body localization (MBL) is studied, of interacting spinless fermions on a lattice with uncorrelated quenched site disorder. The model maps onto a tight-binding model on a ‘Fock-space (FS) lattice’ of many-body states, with an extensive local connectivity. We seek to understand some aspects of MBL from this perspective, via local propagators for the FS lattice and their self-energies (SE's), focusing on the SE probability distributions, over disorder and FS sites. A probabilistic mean-field theory (MFT) is first developed, centered on self-consistent determination of the geometric mean of the distribution. Despite its simplicity this captures some key features of the problem, including recovery of an MBL transition, and predictions for the forms of the SE distributions. The problem is then studied numerically in 1d by exact diagonalization, free from MFT assumptions. The geometric mean indeed appears to act as a suitable order parameter for the transition. Throughout the MBL phase the appropriate SE distribution is confirmed to have a universal form, with long-tailed Lévy behavior as predicted by MFT. In the delocalized phase for weak disorder, SE distributions are clearly log-normal, while on approaching the transition they acquire an intermediate Lévy-tail regime, indicative of the incipient MBL phase.

Abstract:

We consider some aspects of a standard model employed in studies of many-body localization: interacting spinless fermions with quenched disorder, for non-zero flling fraction, here on d-dimensional hypercubic lattices. The model may be recast as an equivalent tight-binding model on a `Fock-space (FS) lattice' with an extensive local connectivity. In the thermodynamic limit exact results are obtained for the distributions of local FS coordination numbers, FS site-energies, and the density of many-body states. All such distributions are well captured by exact diagonalisation on the modest system sizes amenable to numerics. Care is however required in choosing the appropriate variance for the eigenvalue distribution, which has implications for reliable identification of mobility edges.

Abstract:

The periodic Anderson model (PAM) is studied within the framework of dynamical mean-field theory, with particular emphasis on the interaction-driven Mott transition it contains, and on resultant Mott insulators of both Mott-Hubbard and charge-transfer type. The form of the PAM phase diagram is first deduced on general grounds using two exact results, over the full range of model parameters and including metallic, Mott, Kondo and band insulator phases.The effective low-energy model which describes the PAM in the vicinity of a Mott transition is then shown to be a one-band Hubbard model, with effective hoppings that are not in general solely nearest neighbour, but decay exponentially with distance. This mapping is shown to have a range of implications for the physics of the problem, from phase boundaries to single-particle dynamics; all of which are confirmed and supplemented by NRG calculations. Finally we consider the locally degenerate, non-Fermi liquid Mott insulator, to describe which requires a two-self-energy description. This is shown to yield a number of exact results for the associated local moment, charge, and interaction-renormalised levels, together with a generalisation of Luttinger's theorem to the Mott insulator.