The insulating state of matter can be probed by means of a ground state geometrical marker, which is closely related to the modern theory of polarization (based on a Berry phase). In the present work we show that this marker can be applied to determine the metal-insulator transition in disordered systems. In particular, for noninteracting systems the geometrical marker can be obtained from the configurational average of the norm-squared one-body density matrix, which can be calculated within open as well as periodic boundary conditions. This is in sharp contrast to a classification based on the static conductivity, which is only sensible within periodic boundary conditions. We exemplify the method by considering a simple lattice model, known to have a metal-insulator transition as a function of the disorder strength, and demonstrate that the transition point can be obtained accurately from the one-body density matrix. The approach has a general ab initio formulation and could in principle be applied to realistic disordered materials by standard electronic structure methods.
An insulator differs from a metal because of a different organization of the electrons in their ground state. In recent years this feature has been probed by means of a geometrical property, the quantum metric tensor, which addresses the system as a whole, and is therefore limited to macroscopically homogenous samples. Here we show that an analogous approach leads to a localization marker, which can detect the metallic versus insulating character of a given sample region using as the sole ingredient the ground state electron distribution, even in the Anderson case (where the spectrum is gapless). When applied to an insulator with a nonzero Chern invariant, our marker is capable of discriminating the insulating nature of the bulk from the conducting nature of the boundary. Simulations (both model Hamiltonian and first principles) on several test cases validate our theory.
Periodic boundary conditions are not always used in the study of disordered systems, but it can be advantageous to apply them to mimick thermodynamically large systems. In this case, polarization and its cumulants can not be obtained directly, but through the tools of the modern theory of polarization. This theory casts the polarization in crystalline systems as a geometric phase, rather than an operator expectation value. We develop disorder averaging techniques within the context of this theory which can calculate the variance of the polarization, its higher order moments, and the excess kurtosis (or Binder cumulant). We also derive an indicator of delocalization based on the degeneracy as a function of boundary conditions. We apply the computational techniques to two model systems. To test localization, we use a one-dimensional disordered model which is fully Anderson localized. Our calculations verify this. We also apply our techniques to the one dimensional de Moura-Lyra model, developed to study power law correlated (controlled by a parameter,
α) disorder. While this model is a pathological one, our method is validated. We also point out the significance of pairwise degeneracies found in the parameter range,
α>2 and near the band center (or near half filling), where the model was conjectured to exhibit a mobility edge.
Understanding the metal-insulator transition in disordered many-fermion systems, both with and without interactions, is one of the most challenging and consequential problems in condensed matter physics. In this paper, we address this issue from the perspective of the modern theory of the insulating state (MTIS), which has already proven to be effective for band and Mott insulators in clean systems. First, we consider noninteracting systems with different types of aperiodic external potentials: uncorrelated disorder (one-dimensional Anderson model), deterministic disorder (Aubry-Andr\'e Hamiltonian and its modification including next-nearest-neighbor hopping), and disorder with long-range correlations (self-affine potential). We show how the many-body localization tensor defined within the MTIS may be used as a powerful probe to discriminate the insulating and the metallic phases, and to locate the transition point. Then, we investigate the effect of weak repulsive interactions in the Aubry-Andr\'e Hamiltonian, a model which describes a recent cold-atoms experiment. By treating the weak interactions within a mean-field approximation we obtain a linear shift of the transition point towards stronger disorder, providing evidence for delocalization induced by interactions.
We study the (Anderson) metal-insulator transition (MIT) in tight binding models (TBM) of disordered systems using the scaling behavior of the typical density of states (GDOS) as localization criterion. The GDOS is obtained as the geometrical mean value of the local density of states (LDOS) averaged over many different lattice sites and disorder realizations. The LDOS can efficiently be obtained within the kernel polynomial method (KPM). To check the validity and accuracy of the method, we apply it here to the standard Anderson model of disordered systems, for which the results (for instance for the critical disorder strength of the Anderson transition) are well known from other methods.
The ground-state phase diagrams of the Falicov-Kimball model with local disorder are derived within the dynamical mean-field theory and using the geometrically averaged (``typical'') local density of states. Correlated metal, Mott insulator, and Anderson insulator phases are identified. The metal-insulator transitions are found to be continuous. The interaction and disorder compete with each other stabilizing the metallic phase against the occurring one of the insulators. The Mott and Anderson insulators are found to be continuously connected.
We have used the density-matrix renormalization group method to study the ground-state properties of the symmetric periodic Anderson model in one dimension. We have considered lattices with up to
Ns=50 sites, and electron densities ranging from quarter to half filling. Through the calculation of energies, correlation functions, and their structure factors, together with careful extrapolations (toward ${N}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$), we were able to map out a phase diagram
Uvsn, where
U is the electronic repulsion on
f orbitals, and
n is the electronic density, for a fixed value of the hybridization. At quarter filling,
n=1, our data is consistent with a transition at ${U}_{{c}_{1}}\ensuremath{\simeq}2$, between a paramagnetic (PM) metal and a spin-density-wave (SDW) insulator; overall, the region $U\ensuremath{\lesssim}2$ corresponds to a PM metal for all
n<2. For $1<n\ensuremath{\lesssim}1.5$ a ferromagnetic phase is present within a range of
U, while for $1.5\ensuremath{\lesssim}n<2$, we find an incommensurate SDW phase; above a certain
Uc(n), the system displays a Ruderman-Kittel-Kasuya-Yosida behavior, in which the magnetic wave vector is determined by the occupation of the conduction band. At half filling, the system is an insulating spin liquid, but with a crossover between weak and strong magnetic correlations.