The insulating state of matter is characterized by the excitation spectrum, but also by qualitative features of the electronic ground state. The insulating ground wavefunction in fact: (i) displays vanishing dc conductivity; (ii) sustains macroscopic polarization; and (iii) is localized. The idea that the insulating state of matter is a consequence of electron localization was first proposed in 1964 by W. Kohn. I discuss here a novel definition of electron localization, rather different from Kohn's, and deeply rooted in the modern theory of polarization. In fact the present approach links the two features (ii) and (iii) above, by means of essentially the same formalism. In the special case of an uncorrelated crystalline solid, the localization of the many-body insulating wavefunction is measured - according to our definition - by the spread of the Wannier orbitals; this spread diverges in the metallic limit. In the correlated case, the novel approach to localization is demonstrated by means of a two-band Hubbard model in one dimension, undergoing a transition from band insulator to Mott insulator.
The insulating state of matter is characterized by the excitation spectrum, but also by qualitative features of the electronic ground state. The insulating ground wavefunction in fact: (i) displays vanishing dc conductivity; (ii) sustains macroscopic polarization; and (iii) is localized. The idea that the insulating state of matter is a consequence of electron localization was first proposed in 1964 by W. Kohn. I discuss here a novel definition of electron localization, rather different from Kohn's, and deeply rooted in the modern theory of polarization. In fact the present approach links the two features (ii) and (iii) above, by means of essentially the same formalism. In the special case of an uncorrelated crystalline solid, the localization of the many-body insulating wavefunction is measured - according to our definition - by the spread of the Wannier orbitals; this spread diverges in the metallic limit. In the correlated case, the novel approach to localization is demonstrated by means of a two-band Hubbard model in one dimension, undergoing a transition from band insulator to Mott insulator.
The insulating state of matter is characterized by the excitation spectrum, but also by qualitative features of the electronic ground state. The insulating ground wave function in fact (i) sustains macroscopic polarization, and (ii) is localized. We give a sharp definition of the latter concept and we show how the two basic features stem from essentially the same formalism. Our approach to localization is exemplified by means of a two-band Hubbard model in one dimension. In the noninteracting limit, the wave function localization is measured by the spread of the Wannier orbitals.
Abstract: The metal–insulator transition (MIT) constitutes one of the most profound and persistent problems in condensed matter physics, sitting at the intersection of electronic structure theory, many-body physics, and emergent quantum phenomena. Conventional band theory, which successfully classifies materials as metals or insulators according to the filling of their electronic bands, fails dramatically when confronted with a class of compounds transition metal oxides chief among them that ought to be metallic by band-counting arguments yet are robustly insulating in the laboratory. The resolution of this paradox required abandoning the independent-electron picture entirely and recognizing that strong Coulomb repulsion between electrons can, by itself, drive localization. This insight, formalized by Mott and later distilled into the Hubbard model, established correlation-driven insulation as a phenomenon distinct from band insulation. The present paper offers a theoretical exploration of the MIT problem, tracing its conceptual development from the Mott criterion through the Hubbard Hamiltonian, the Brinkman–Rice scenario of quasiparticle mass divergence, and the watershed development of dynamical mean-field theory (DMFT), which rendered the Mott transition quantitatively tractable for the first time. We argue that the competition between the electronic bandwidth and the on-site interaction encoded in the ratio U/W provides the unifying organizing principle for the field. Building toward the modern frontier, we examine how the Mott transition connects to quantum phase transitions and quantum criticality, where zero-temperature instabilities governed by quantum rather than thermal fluctuations produce non-Fermi-liquid behavior and anomalous scaling. Case studies of vanadium oxides, rare-earth nickelates, and doped cuprates illustrate the diversity and subtlety of real correlated materials. We conclude that the MIT remains a central and unresolved problem whose continued study is increasingly entangled with the development of quantum materials and quantum technologies. Keywords: Metal–Insulator Transition; Mott Insulator; Strongly Correlated Electrons; Hubbard Model; DMFT; Quantum Criticality; Quantum Materials; Condensed Matter Physics
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.
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.
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.