An ab initio analysis is made of the localized (in Mott’s sense) and delocalized (band) states of an electron in a crystal. The criterion of a difference between these states is the behavior of the off-diagonal elements in the one-particle electronic density matrix $$\left\langle {\hat \psi ^ + (r)\hat \psi (r')} \right\rangle$$ for T=0. Localization can be related to its exponential falloff for |r−r′|→∞, and delocalization, to its power-law decay. This corresponds to the analyticity of the density matrix in k space of the Brillouin zone in the first case, and to a singularity (Fermi step) in the second. This analyticity gives rise also to a power-law decay of the correlation functions. In a normal system, localization can be identified with the insulating state, and delocalization, with the conducting state. It is shown that the above localization criterion is applicable to disordered systems as well. Electron localization in superconductors is also discussed. It is pointed out that the above criterion of localization is met also in the BCS superconducting state and in the localized-pair model. The reason for the considerable difference between the properties of the insulating and superconducting states lies in that there are no static fluctuations in the number of electrons, 〈(ΔN)2〉0, in the ground state of an insulator whereas in a superconductor ODLRO results in nonzero fluctuations of N of a quantum nature.
An ab initio analysis is made of the localized (in Mott’s sense) and delocalized (band) states of an electron in a crystal. The criterion of a difference between these states is the behavior of the off-diagonal elements in the one-particle electronic density matrix ⟨ψ^+(r)ψ^(r′)⟩ for T=0. Localization can be related to its exponential falloff for |r−r′|→∞, and delocalization, to its power-law decay. This corresponds to the analyticity of the density matrix in k space of the Brillouin zone in the first case, and to a singularity (Fermi step) in the second. This analyticity gives rise also to a power-law decay of the correlation functions. In a normal system, localization can be identified with the insulating state, and delocalization, with the conducting state. It is shown that the above localization criterion is applicable to disordered systems as well. Electron localization in superconductors is also discussed. It is pointed out that the above criterion of localization is met also in the BCS superconducting state and in the localized-pair model. The reason for the considerable difference between the properties of the insulating and superconducting states lies in that there are no static fluctuations in the number of electrons, 〈(ΔN)2〉0, in the ground state of an insulator whereas in a superconductor ODLRO results in nonzero fluctuations of N of a quantum nature.
It is well known that the metal-insulator transition in two dimensions for non-interacting fermions takes place at infinitesimal disorder. In contrast, the superconductor-to-insulator transition takes place at a finite critical disorder (on the order of Vc ~ 2t), where V is the typical width of the distribution of random site energies and t is the hopping scale. In this article we compare the localization/delocalization properties of one and two particles. Whereas the metal-insulator transition is a consequence of single-particle Anderson localization, the superconductor-insulator transition (SIT) is due to pair localization – or, alternatively, fluctuations of the phase conjugate to pair density. The central question we address is how superconductivity emerges from localized single-particle states. We address this question using inhomogeneous mean field theory and quantum Monte Carlo techniques and make several testable predictions for local spectroscopic probes across the SIT. We show that with increasing disorder, the system forms superconducting blobs on the scale of the coherence length embedded in an insulating matrix. In the superconducting state, the phases on the different blobs are coherent across the system whereas in the insulator long-range phase coherence is disrupted by quantum fluctuations. As a consequence of this emergent granularity, we show that the single-particle energy gap in the density of states survives across the transition, but coherence peaks exist only in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrast to conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the SIT despite a robust single-particle gap.
We developed an analytic theory of inhomogeneous superconducting pairing in strongly disordered materials, which are moderately close to superconducting-insulator transition. Single-electron eigenstates are assumed to be Anderson localized, with a large localization volume. Superconductivity develops due to coherent delocalization of originally localized preformed Cooper pairs. The key assumption of the theory is that each such pair is coupled to a large number $Z\ensuremath{\gg}1$ of similar neighboring pairs. We derived integral equations for the probability distribution $P(\mathrm{\ensuremath{\Delta}})$ of local superconducting order parameter $\mathrm{\ensuremath{\Delta}}(\mathbit{r})$ and analyzed their solutions in the limit of small dimensionless Cooper coupling constant $\ensuremath{\lambda}\ensuremath{\ll}1$. The shape of the order-parameter distribution is found to depend crucially upon the effective number of nearest neighbors ${Z}_{\text{eff}}=2{\ensuremath{\nu}}_{0}{\mathrm{\ensuremath{\Delta}}}_{0}Z$, with ${\ensuremath{\nu}}_{0}$ being the single-particle density of states at the Fermi level. The solution we provide is valid both at large and small Zeff; the latter case is nontrivial as the function $P(\mathrm{\ensuremath{\Delta}})$ is heavily non-Gaussian. One of our key findings is the discovery of a broad range of parameters where the distribution function $P(\mathrm{\ensuremath{\Delta}})$ is non-Gaussian but also noncritical (in the sense of superconductor-insulator transition criticality). The analytic results are supplemented by numerical data and good agreement between them is observed.
While the modern theory of the insulating state shows that the conducting or insulating properties of a system can be extracted solely from the ground state properties via the so-called localization tensor (LT), no chemical reading of this important quantity has ever been offered. Here, a remarkable link between the LT and the bond orders as described by the delocalization indices (DIs) of chemical bonding theory is reported. This is achieved through a real space partition of the LT into intra- and interatomic contributions. We show that the convergence or divergence of the LT in the thermodynamic limit, which signals the insulating or conducting nature of an extended system, respectively, can be nailed down to DIs. This allows for the exploitation of traditional chemical intuition to identify essential and spectator atomic groups in determining electrical conductivity. The thermodynamic limit of the LT is controlled by the spatial decay rate of the interatomic DIs, exponential in insulators and power-law in conductors. Computational data of a few selected toy systems corroborate our results.
We discuss transport and localization properties on the insulating side of the disorder dominated superconductor-insulator transition, described in terms of the dirty boson model. Analyzing the spectral properties of the interacting bosons in the absence of phonons, we argue that the Bose glass phase admits three distinct regimes. For strongest disorder the boson system is a fully localized, perfect insulator at any temperature. At smaller disorder, only the low temperature phase exhibits perfect insulation while delocalization takes place above a finite temperature. We argue that a third phase must intervene between these perfect insulators and the superconductor. This conducting Bose glass phase is characterized by a mobility edge in the many body spectrum, located at finite energy above the ground state. In this insulating regime purely electronically activated transport occurs, with a conductivity following an Arrhenius law at asymptotically low temperatures, while a tendency to superactivation is predicted at higher T. These predictions are in good agreement with recent transport experiments in highly disordered films of superconducting materials.
The clear distinction between metals and insulators is a well established and appartently simple concept which can be clarified by studying the behavior of the electrical conductivity. However, a theoretical description of the processes leading to one of the two extremes is no trivial task. The widely accepted band structure theory relies on an one-electron picture to characterize the metallic/insulating behavior of a material. However, in spite of the fact that it can be used to make successful predictions, this description does not capture one of the essential elements characterizing the nature of an electronic quantum system, i.e. the electron correlation. The key to the different behaviors is clearly contained in the correlated many- electron wave function, but the analysis of such a high-dimensional object in configuration space can be quite cumbersome. Therefore the development of powerful tools to simplify this task is necessary. The total position-spread (TPS) tensor, which was derived by Walter Kohn’s theory of the insulating state, plays a key role in the field of metal-insulator transitions (MITs). This tool provides information about the electron localization, which is strongly connected to the electrical conductivity. Therefore, in this work particular attention was paid to the calculation of the TPS tensor of small and extended systems. Moreover, part of this thesis focuses on a new formalism, which allows the study of spin mobility and spin entanglement and provides therefore deeper insight into the effects of electron correlation. In the framework of quantum information theory (QIT), the quantum entanglement can be quantified by means of measures such as the von Neumann entropy. Also, the analysis of the elements of the two-orbital reduced density matrix provides an overview of the complexity of the many-electron wave function. These insights were used in this work for the analysis of the metallic/insulating character of a system. However, in order to perform such investigastions, the problem of achieving an accurate description of the many- electron wave function has to be dealt with. Despite the tremedous progress in the efficiency of quantum-chemical methods, the treatement of extended systems using standard approaches is still a difficult (if not unfeasible) task. Therefore the development and testing of sophisticated, low-scaling methods is currently the subject of many investigations. In this work, two approaches were employed: 1) the ab-initio density matrix renormalization group (DMRG) approach which, based on a tensor product ansatz, provides very precise results and exploits QIT to improve its performance; 2) the method of increments (MoI) which, using a many-body expansion in terms of localized orbitals, yields accurate correlation energies. By using both methods on strongly correlated one-dimesional model systems, we worked on a new formalism of the method of increments which can be used to describe the whole dissociation curve without inconvenient discontinuities. Finally, aiming to calculate the total position-spread tensor of extended systems, we worked on the application of the MoI for quantities other than the correlation energy.
In this thesis we consider a variational wave function approach as a possible route to describe the competition between disorder and strong electron-electron interaction in two dimensions. In particular we aim to obtain a transparent and physically intuitive understanding of the competition between these two localizing forces within the simplest model where they both are active, namely the disordered Hubbard model at half filling and in a square lattice. Our approach is based on an approximate form of the ground-state wave function, which we believe contains the physically relevant ingredients for a correct description of both the Mott and the Anderson insulators, where electrons are localized by the Coulomb repulsion and by disorder, respectively. For strongly interacting fermionic systems, a standard variational wave function is constructed by a correlation term acting on a Slater determinant, the latter being an uncorrelated metallic state. Previous variational calculations showed that a long-range density-density correlation factor, so called Jastrow factor, is needed to correctly describe the Mott insulator [9]. This term, which is collective by definition, correlates spatially charge uctuations, thus preventing their free motion that would otherwise imply metallic conductance. For this reason, our variational wave function does include such a term. Anderson localization is instead mostly a matter of single-particle wave functions, hence it pertains to the uncorrelated Slater determinant which the Jastrow factor acts onto. We consider both the case in which we enforce paramagnetism in the wave function and the case in which we allow for magnetic ordering. Summarizing briefly our results, we find that, when the variational wave function is forced to be paramagnetic, the Anderson insulator to Mott insulator transition is continuous. This transition can be captured by studying several quantities. In particular, a novel one that we have identified and that is easily accessible variationally is the disconnected density-density fluctuation at long wavelength, defined by lim where ^nq is the Fourier transform of the charge density at momentum q, (...) denotes quantum average at fixed disorder and the overbar represents the average over disorder configurations. We find that Ndisc q!0 is everywhere finite in the Anderson insulator and vanishes critically at the Mott transition, staying zero in the Mott insulator. When magnetism is allowed and the hopping only connects nearest neighbor sites, upon increasing interaction the paramagnetic Anderson insulator first turns antiferromagnetic and finally the magnetic and compressible Anderson insulator gives way to an incompressible antiferromagnetic Mott insulator. The optimized uncorrelated Slater determinant is always found to be the eigenstate of a disordered non-interacting effective Hamiltonian, which suggests that the model is never metallic. Finally, when magnetism is frustrated by a next to nearest neighbor hopping, the overall sequence of phases does not change. However, the paramagnetic to magnetic transition within the Anderson insulator basin of stability turns first order. Indeed, within the magnetically ordered phase, we find many almost degenerate paramagnetic states with well defined local moments. This is suggestive of an emerging glassy behavior when the competition between disorder and strong correlation is maximum.