This chapter focuses on studies of the critical behavior of conductivity in two-dimensional electron systems (2DES) with different amounts and types of disorder as well as different ranges of Coulomb interactions, in order to identify possible universality classes of the 2D metal-insulator transition (MIT). It discusses the manifestations of glassiness in both low- and high-disorder samples. The chapter examines the critical region near the 2D MIT in particular on describing conductivity measurements that have demonstrated dynamical scaling, the main signature of the MIT as a quantum phase transition. It also focuses on experiments that probe the nature of the insulating state, as well as charge dynamics across the MIT, in both high- and low-disorder 2DESs. There is substantial evidence that in many materials near the MIT both strong electronic correlations and disorder play an important role, and thus their competition is expected to lead to glassy behavior of electrons, in analogy with other frustrated systems.
This chapter focuses on studies of the critical behavior of conductivity in two-dimensional electron systems (2DES) with different amounts and types of disorder as well as different ranges of Coulomb interactions, in order to identify possible universality classes of the 2D metal-insulator transition (MIT). It discusses the manifestations of glassiness in both low- and high-disorder samples. The chapter examines the critical region near the 2D MIT in particular on describing conductivity measurements that have demonstrated dynamical scaling, the main signature of the MIT as a quantum phase transition. It also focuses on experiments that probe the nature of the insulating state, as well as charge dynamics across the MIT, in both high- and low-disorder 2DESs. There is substantial evidence that in many materials near the MIT both strong electronic correlations and disorder play an important role, and thus their competition is expected to lead to glassy behavior of electrons, in analogy with other frustrated systems.
This chapter focuses on experimental results obtained in low-disordered strongly interacting two-dimensional (2D) electron systems; in particular in (100)-silicon metal-oxide-semiconductor field-effect transistors (MOSFET). It describes the additional confirmation in favor of zero-temperature zero-field metal-insulator transition is provided by magnetic and thermopower measurements. The chapter argues that the metal-insulator transition in silicon samples with very low level of disorder is driven by interactions. In a low-disordered 2D electron system in silicon MOSFETs, the resistivity at a certain "critical" electron density shows virtually no temperature dependence over a wide range of temperatures. In 2000, it was experimentally found that the ratio between the spin and the cyclotron splittings in silicon MOSFETs strongly increases at low electron densities. The strong enhancement of the spin susceptibility has indicated that at low electron densities, the system behaves well beyond the weakly interacting Fermi liquid.
Conductivity and Charge Stiffness In this chapter, we will consider initially metal–insulator (MI) transitions which occur in pure systems without disorder. In the final section we consider a superfluid–insulator transition induced by some special type of disorder. We will show that the quantities which characterise the phase transitions in both problems obey similar scaling laws and are governed by similar exponents. Differently from the magnetic transitions treated before metal–insulator (MI) transitions have no obvious order parameter to distinguish the metallic from the insulating phase. This precludes a power expansion of the free energy in terms of a small quantity as for magnetic phase transitions. In spite of that, the concepts of phase transitions and critical phenomena turn out to be very useful to describe metal–insulator transitions. In case this problem is approached using the renormalisation group, the localisation transition is associated with an unstable fixed point and the flow of the RG equations to the different attractors is sufficient to distinguish the nature of the phases. Then, in general, we can associate a set of critical exponents with the MI transitions. Besides, these exponents are not independent but obey scaling relations. Since we are concerned with zero temperature instabilities, the quantum hyperscaling law, Eq. 1.18, plays a crucial role in this problem. We will distinguish here two different types of metal–insulator transitions. Those due to a competition between parameters of the relevant Hamiltonian and metal–insulator transitions arising by varying the number of electrons or chemical potential. An example of the former is the Mott transition due to the competition between kinetic energy and the local Coulomb repulsion as described by the half-filled band Hubbard model. The latter will be referred as density-driven metal– insulator transitions in analogy with the superfluid–insulator transition in bosonic systems with varying particle number (Fisher et al ., 1989). These two types of transitions may occur in the same system or Hamiltonian. They in general belong to different universality classes and consequently have different critical exponents.
The past decade has seen a substantial rejuvenation of interest in the study of quantum phase transitions, driven by experiments on cuprate superconductors, heavy fermion materials, organic conductors, and related compounds. Although quantum phase transitions in simple spin systems, like the Ising model in a transverse field, were studied in the early 1970s, much of the subsequent theoretical work examined a particular example: the metal–insulator transition. While this is a subject of considerable experimental importance, the greatest theoretical progress was made for the case of the Anderson transition of non-interacting electrons, which is driven by localization of the electronic states in the presence of a random potential. The critical properties of this transition of noninteracting electrons constituted the primary basis upon which most condensed matter physicists have formed their intuition on the behavior of the systems near a quantum phase transition. However, it is clear that strong electronic interactions play a crucial role in the systems of current interest noted earlier, and simple paradigms for the behavior of such systems near quantum critical points are not widely known. It is the purpose of this book to move interactions to center stage by describing and classifying the physical properties of the simplest interacting systems undergoing a quantum phase transition. The effects of disorder will be neglected for the most part but will be considered in the concluding chapters. Our focus will be on the dynamical properties of such systems at nonzero temperature, and it will become apparent that these differ substantially from the noninteracting case.
The metal–insulator transition (MIT) observed in a two-dimensional dilute electron liquid raises the question about the applicability of the scaling theory of disordered electrons, the approach pioneered by Phil Anderson and his collaborators,8 for the description of this transition. In this context, we review here the scaling theory of disordered electrons with electron–electron interactions. We start with the disordered Fermi liquid, and show how to adjust the microscopic Fermi-liquid theory to the presence of disorder. Then we describe the non-linear sigma model (NLSM) with interactions. This model has a direct relation with the disordered Fermi liquid, but can be more generally applicable, since it is a minimal model for disordered interacting electrons. The discussion is mostly about the general structure of the theory emphasizing the connection of the scaling parameters entering the NLSM with conservation laws. Next, we show that the MIT, as described by the NLSM with interactions, is a quantum phase transition and identify the parameters needed for the description of the kinetics and thermodynamics of the interacting liquid in the critical region of the transition. Finally, we discuss the MIT observed in Si -MOSFETs. We consider it as an example of the Anderson transition in the presence of the electron interactions. We demonstrate that the two-parameter RG equations, which treat disorder in the one-loop approximation but incorporate the full dependence on the interaction amplitudes, describe accurately the experimental data in Si -MOSFETs including the observed non-monotonic behavior of the resistance and its strong drop at low temperatures. The fact that this drop can be reproduced theoretically, together with the argument that Anderson localization should occur at strong disorder, justified the existence of the MIT within the scaling theory.
Two-dimensional ferromagnetic electron gases subject to random scalar potentials and Rashba spin-orbit interactions exhibit a striking quantum criticality. As disorder strength W increases, such a system undergoes two transitions. It first changes from a normal diffusive metal consisting of extended states to a marginal metal consisting of critical states at a critical disorder Wc,1. Another transition from the marginal metal to an insulator occurs at a stronger disorder Wc,2. Through highly accurate numerical procedures based on the recursive Green's function and the exact diagonalization methods, we elucidate the nature of the quantum criticality and the properties of the pertinent states. The conductance is described by an unorthodox one-parameter scaling law: Conductance of various system sizes and disorders collapse into two branches of a scaling curve corresponding to diffusive metal and insulating phases with an exponentially diverging correlation length, $\ensuremath{\xi}\ensuremath{\propto}exp[\ensuremath{\alpha}/\sqrt{|W\ensuremath{-}{W}_{c}|}]$, near transition points. Finite-size scaling analysis of the inverse participation ratio reveals that the states of the marginal metal are fractals of a universal fractal dimension $D=1.90\ifmmode\pm\else\textpm\fi{}0.02$, while those of the diffusive metal spread over the whole system with D=2 and states in the insulating phase are localized with D=0. The phase diagram for diffusive metals, marginal metals, and the Anderson insulators is plotted in the disorder-magnetic-coupling plane.
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