We prove the occurrence of Anderson localisation for a system of infinitely many particles interacting with a short range potential, within the ground state Hartree–Fock approximation. We assume that the particles hop on a discrete lattice and that they are submitted to an external periodic potential which creates a gap in the non-interacting one particle Hamiltonian. We also assume that the interaction is weak enough to preserve a gap. We prove that the mean-field operator has exponentially localised eigenvectors, either on its whole spectrumor at the edges of its bands, depending on the strength of the disorder.
We prove the occurrence of Anderson localisation for a system of infinitely many particles interacting with a short range potential, within the ground state Hartree–Fock approximation. We assume that the particles hop on a discrete lattice and that they are submitted to an external periodic potential which creates a gap in the non-interacting one particle Hamiltonian. We also assume that the interaction is weak enough to preserve a gap. We prove that the mean-field operator has exponentially localised eigenvectors, either on its whole spectrumor at the edges of its bands, depending on the strength of the disorder.
In the presence of strong enough disorder one-dimensional systems of interacting spinless fermions at nonzero filling factor are known to be in a many-body localized phase. When represented in `Fock space,' the Hamiltonian of such a system looks like that of a single `particle' hopping on a Fock lattice in the presence of a random disordered potential. The coordination number of the Fock lattice increases linearly with the system size L in one dimension. Thus in the thermodynamic limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the disordered interacting problem in one dimension maps on to an Anderson model with infinite coordination number. Despite this, this system displays localization which appears counterintuitive. A close observation of the on-site disorder potentials on the Fock lattice reveals a large degree of correlation among them as they are derived from an exponentially smaller number of on-site disorder potentials in real space. This indicates that the correlations between the on-site disorder potentials on a Fock lattice has a strong effect on the localization properties of the corresponding many-body system. This intuition is also consistent with studies of quantum random energy model where the typical mid-spectrum states are ergodic and the on-site potentials in Fock space are completely uncorrelated. In this work we perform a systematic quantitative exploration of the nature of correlations of the Fock space potential required for localization. We study different functional variations of the disorder correlation in Fock lattice by analyzing the eigenspectrum obtained through exact diagonalization. Without changing the typical strength of the on-site disorder potential in Fock lattice we show that changing the correlation strength can induce thermalization or localization in systems. From among the various forms of correlations we study, we find that only the linear variation of correlations with Hamming distance in Fock space is able to drive a thermal-MBL phase transition where the transition is driven by the correlation strength. Systems with the other forms of correlations we study are found to be ergodic.
Interplay of electron correlation and randomness is studied by using the Anderson-Hubbard model within the Hartree-Fock approximation. Under the coexistence of short-range interaction and diagonal disorder, we obtain the ground-state phase diagram in three dimensions, which includes an antiferromagnetic insulator, an antiferromagnetic metal, a paramagnetic insulator (Anderson-localized insulator) and a paramagnetic metal. Although only the short-range interaction is present in this model, we find unconventional soft gaps in the insulating phases irrespective of electron filling, spatial dimensions and long-range order, where the single-particle density of states (DOS) vanishes with a power-law scaling in one dimension (1D) or even faster in two dimensions (2D) and three dimensions (3D) toward the Fermi energy. We call it soft Hubbard gap. Moreover, exact-diagonalization results in 1D support the formation of the soft Hubbard gap beyond the mean-field level. The formation of the soft Hubbard gap cannot be attributed to a conventional theory by Efros and Shklovskii (ES) owing the emergence of soft gaps to the long-range Coulomb interaction. Indeed, based on a picture of multivalley energy landscape, we propose a phenomenological scaling theory, which predicts a scaling of the DOS in perfect agreement with the numerical results. We further discuss a correction of the scaling of the DOS by the long-range part of the Coulomb interaction, which modifies the scaling of Efros and Shklovskii. Furthermore, explicit formulae for the temperature dependence of the DC resistivity via variable-range hopping under the influence of the soft gaps are derived. Finally, we compare the present theory with experimental results of SrRu_{1-x}Ti_xO_3.
Non-interacting particles with long-range hopping are known to be delocalized in disordered systems of infinite size. It is thus natural to assume that such particles can traverse any finite-size lattice. Here, we demonstrate that this is not generally true. The delocalization mechanism is induced by resonances between distant lattice sites. The number of such resonances diverges with the system size. For a finite-size lattice the number of resonances is finite and may not be sufficient to result in delocalization. We consider particles with long-range (dipolar) hopping in three-dimensional lattices with diagonal disorder and random dilution. We compute the wavepacket dynamics of particles placed in an individual lattice site, the inverse participation ratios and the fluctuations of the wavefunctions with disorder. We show that, for certain finite-size disordered lattices, particles remain localized within a finite volume much smaller than the lattice size, and that the wavefunctions exhibit the log-normal fluctuations characteristic of Anderson localization. We characterize the localization properties as functions of dilution and diagonal disorder. We combine our results with scaling theory to obtain the size dependence of the localization--diffusion crossover. Our results indicate that particles with long-range hopping undergo exponential localization in lattices of finite size, even macroscopically finite. Our phase diagrams illustrate a rather unusual phenomenon: quantum particles can diffuse through a lattice of size 10A, but not through a lattice of size A.
We investigate the inter-play of f-electron band dispersion and inter-f electron interactions on the gap in the one electron density of states (DOS) of the periodic Anderson model. The calculations are based on dynamical mean field theory and the numerical renormalization group method for the model in infinite dimensions. The local f-electron DOS at low energies for the electron-hole symmetric model with interactions is found to have a complete-gap, a pseudo-gap or to be gap-less, whenever these structures occur for the non-interacting model but the parameters that characterize these features are strongly renormalized. As the degree of f-electron dispersion is increased, when there is a complete gap, the many-body renormalization remains almost constant, but in the pseudo-gap case the renormalization increases. We calculate the sizes of gaps both in the local DOS and in the dynamic local susceptibilities. When the gap width is defined by the edges in the spectrum for the complete-gap model, the widths of the indirect gap and the spin gap are equal, independent of the degree of electron correlation. However, if the width is defined as the excitation energy between the peaks in the spectrum, then the ratio of the indirect gap to the spin gap increases and eventually saturates as the correlation increases. The value of this ratio depends on the f-electron band dispersion. We have also investigated the dependence of the shape of the interacting DOS on the f-band filling for different forms of f-band dispersion. When the f-band dispersion is small, the gap structure in the symmetric model vanishes as the filling factor decreases. When the model, including the dispersion, has no gap structure, the filling dependence on the DOS in low energies is similar to that for a single impurity Anderson model. The energy dispersion of the quasi-particle in the range, ε f < ε < ε f + U , is approximately explained by the renormalized band model in the cases of both large and small f-band dispersion.
We study the interplay of disorder localization and strong local interactions within the Anderson-Hubbard model. Taking into account local Mott-Hubbard physics and static screening of the disorder potential, the system is mapped onto an effective single-particle Anderson model, which is studied within the self-consistent theory of electron localization. For fermions, we find rich nonmonotonic behavior of the localization length ξ, particularly in two-dimensional systems, including an interaction-induced exponential enhancement of ξ for small and intermediate disorders and a strong reduction of ξ due to hopping suppression by strong interactions. In three dimensions, we identify for half filling a Mott-Hubbard-assisted Anderson localized phase existing between the metallic and the Mott-Hubbardgapped phases. For small U there is re-entrant behavior from the Anderson localized phase to the metallic phase. For bosons, the unrestricted particle occupation number per lattice site yields a monotonic enhancement of ξ as a function of decreasing interaction, which we assume to persist until the superfluid Bose- Einstein condensate phase is entered. Besides, we study cold atomic gases expanding, by a diffusion process, in a weak random potential. We show that the density-density correlation function of the expanding gas is strongly affected by disorder and we estimate the typical size of a speckle spot, i.e., a region of enhanced or depleted density. Both a Fermi gas and a Bose-Einstein condensate (in a mean-field approach) are considered.
We study Anderson localisation on high-dimensional graphs with spatial structure induced by long-ranged but distance-dependent hopping. To this end, we introduce a class of models that interpolate between the short-range Anderson model on a random regular graph and fully connected models with statistically uniform hopping, by embedding a random regular graph into a complete graph and allowing hopping amplitudes to decay exponentially with graph distance. The competition between the exponentially growing number of neighbours with graph distance and the exponentially decaying hopping amplitude positions our models effectively as power-law hopping generalisation of the Anderson model on random regular graphs. Using a combination of numerical exact diagonalisation and analytical renormalised perturbation theory, we establish the resulting localisation phase diagram emerging from the interplay of the lengthscale associated to the hopping range and the onsite disorder strength. We find that increasing the hopping range shifts the localisation transition to stronger disorder, and that beyond a critical range the localised phase ceases to exist even at arbitrarily strong disorder. Our results indicate a direct Anderson transition between delocalised and localised phases, with no evidence for an intervening multifractal phase, for both deterministic and random hopping models. A scaling analysis based on inverse participation ratios reveals behaviour consistent with a Kosterlitz-Thouless-like transition with two-parameter scaling, in line with Anderson transitions on high-dimensional graphs. We also observe distinct critical behaviour in average and typical correlation functions, reflecting the different scaling properties of generalised inverse participation ratios.