In this paper a new and more comprehensive characterization of the insulating state of matter is developed. This characterization includes the conventional insulators with energy gap as well as systems discussed by Mott which, in band theory, would be metals. The essential property is this: Every low-lying wave function $\ensuremath{\Phi}$ of an insulating ring breaks up into a sum of functions, $\ensuremath{\Phi}={{\ensuremath{\Sigma}}_{\ensuremath{-}\ensuremath{\infty}}}^{\ensuremath{\infty}}{\ensuremath{\Phi}}_{M}$, which are localized in disconnected regions of the many-particle configuration space and have essentially vanishing overlap. This property is the analog of localization for a single particle and leads directly to the electrical properties characteristic of insulators. An Appendix deals with a soluble model exhibiting a transition between an insulating and a conducting state.
In this paper a new and more comprehensive characterization of the insulating state of matter is developed. This characterization includes the conventional insulators with energy gap as well as systems discussed by Mott which, in band theory, would be metals. The essential property is this: Every low-lying wave function \ensuremathΦ of an insulating ring breaks up into a sum of functions, \ensuremathΦ=\ensuremathΣ\ensuremath−\ensuremath∞\ensuremath∞\ensuremathΦM, which are localized in disconnected regions of the many-particle configuration space and have essentially vanishing overlap. This property is the analog of localization for a single particle and leads directly to the electrical properties characteristic of insulators. An Appendix deals with a soluble model exhibiting a transition between an insulating and a conducting state.
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 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 study three cases of insulating states of matter in terms of the renormalization procedure where the conventional renormalization group scheme is not simply applicable. The first subject is the spectral weight structure of hole-doped Mott insulators. As the mixing between two separate Hubbard bands is dynamically generated, additional charge degrees of freedom is required to give a proper description to the relevant low-energy physics. On this account, we first discuss how the low-energy Hubbard band should be partitioned to account for the extra degrees of freedom. Following the exact integration procedure of the upper Hubbard band, we explicitly demonstrate that the conserved charge cannot be exhausted by counting quasiparticles. In addition, we argue that it is the existence of dynamically generated charge degrees of freedom that gives rise to the coexistence of poles and zeroes in the single-particle Green function. In comparison to the Fermi arc structure, which is intrinsic to cuperate phenomenology, we suggest that the suppression of the spectral weight at the back side of the arc is a consequence of composite excitations, arising from dynamical mixing. The second topic we study is the nature of the transition between two insulating states of matter in a weakly disordered bosonic system. In particular, we investigate the instabilities of the Mott-insulating phase within a renormalization group analysis of the replica field theory obtained by a strong-coupling expansion around the atomic limit. To this end, we identify a new order parameter and associated correlation length scale that are capable of capturing the transition from a state with zero compressibility, the Mott insulator, to another insulating state with finite compressibility, the Bose glass. The order parameter is the relative variance of the disorder-induced mass distribution. From its distinctive behavior on each phase, we find that the divergence of the relative variance in the Bose glass signals the breakdown of self-averaging. Lastly, we also emphasize that the transition at commensurate filling is governed by a different fixed point in the renormalization group flow. The last topic discussed is the thermoelectric properties of the correlated Kondo lattice system at low temperature. Specifically, we explore the periodic Anderson impurity model with spin-orbit interactions, using the slave-particle method. As the thermopower is related to the entropy per carrier, particle-hole asymmetry and a large density of states at the chemical potential make Kondo insulator an ideal platform for thermoelectrics. In this respect, we further explore the possibility to improve the thermoelectric performance by tuning the crystal electric field splitting, the spin-orbit interaction strength, and orbital degeneracy.
What theoretical approaches are appropriate for describing topological quantum matter? The answer depends on the type of topological phase we want to study, e.g., noninteracting or interacting, and on the type of information we are interested in, e.g., generic properties or system-specific properties that are relevant to a particular experiment or type of experiment. Noninteracting topological phases can be described within topological band theory using the basic tools of the band theory of solids. However, the complexity of the model, i.e., how much detail has to be included, depends on whether we are only interested in the topological properties of the system or, in addition, we need to consider certain non-topological features. For example, predicting whether or not a specific material is a topological insulator requires detailed knowledge of its band structure, which can be acquired using density functional theory (DFT) methods. Similar treatments may be required to account for certain experimental features involving both topological and nontopological contributions. On the other hand, to understand the classification of topological insulators, it is sufficient to consider “bare bone” effective models, such as continuum Dirac models or simple tight-binding models. Topologically, these descriptions with extremely different levels of complexity are equivalent whenever one can smoothly connect the corresponding Hamiltonians (up to the addition/subtraction of trivial bands) without closing the bulk gap. However, the effective models provide only a crude description of experimentally relevant properties such as energy gaps, band dispersion, and real space properties of boundary states. In some sense, the situation is analogous to the characterization of a given mathematical object within topology, which only deals with properties that are robust against deformations, and geometry, which accounts for properties described by distances, angles, and shapes.
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.
W HAT THEORETICAL APPROACHES ARE APPROPRIATE for describing topological quantum matter? The answer depends on the type of topological phase we want to study, e.g., non-interacting or interacting, and on the type of information we are interested in, e.g., system-specific properties in the case of a theory designed to provide support for a particular experiment or type of experiment. Non-interacting topological phases can be described within topological band theory using the basic tools of the band theory of solids. However, the complexity of the model, i.e., how much detail has to be included, depends on whether we are only interested in the topological properties of the system or, in addition, we need to consider certain nontopological features. For example, predicting whether or not a specific material is a topological insulator requires detailed knowledge of its band structure, which can be acquired using density functional theory (DFT) methods. Similar treatments may be required to account for certain experimental features involving both topological and non-topological contributions. On the other hand, to understand the classification of topological insulators, it is sufficient to consider “bare bone” effective models, such as continuum Dirac models or simple tight-binding models. Topologically, these descriptions with extremely different levels of complexity are equivalent whenever one can smoothly connect the corresponding Hamiltonians (up to the addition/subtraction of trivial bands) without closing the insulating gap. However, the effective models provide only a crude description of experimentally relevant properties such as energy gaps, band dispersion, and real space properties of boundary states. In some sense, the situation is analogous to the characterization of a given mathematical object within topology, which only deals with properties that are robust against deformations, and geometry, which accounts for properties described by distances, angles, and shapes.