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Philosophical Magazine B

Transport around criticalities: Localization-delocalization and paramagnetic-ferromagnetic transitions

Hideo Aoki

Published Sep 1, 2001• 0 citations
Publisher

Abstract

We study two examples of transport properties of electron systems in the vicinity of a critical point. The first example, a one-body problem, is the disorder-driven metal-insulator transition in the integer quantum Hall system. We have numerically examined the multifractality of the critical wavefunction, which is relevant to an anomalous diffusion of electrons. The second example, a many-body problem, is the paramagnetic-ferromagnetic transition in a strongly correlated clean electron system. We show that the proximity to ferromagnetism can give rise to a negative magnetoresistance. These exemplify how a criticality manifests itself in transport properties.

DOI
10.1080/13642810108205778

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•May 30, 2013•0 citations
We study two examples of transport properties of electron systems in the vicinity of a critical point. The first example, a one-body problem, is the disorder-driven metal-insulator transition in the integer quantum Hall system. We have numerically examined the multifractality of the critical wavefunction, which is relevant to an anomalous diffusion of electrons. The second example, a many-body problem, is the paramagnetic-ferromagnetic transition in a strongly correlated clean electron system. We show that the proximity to ferromagnetism can give rise to a negative magnetoresistance. These exemplify how a criticality manifests itself in transport properties.
Two-dimensional ferromagnetic electron gases subject to random scalar potentials and Rashba spin-orbit interactions exhibit a striking quantum criticality. As disorder strength WWW 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{W}_{c,1}Wc,1​. Another transition from the marginal metal to an insulator occurs at a stronger disorder Wc,2{W}_{c,2}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=2D=2D=2 and states in the insulating phase are localized with D=0D=0D=0. The phase diagram for diffusive metals, marginal metals, and the Anderson insulators is plotted in the disorder-magnetic-coupling plane.
In contrast to the first-order correlation-driven Mott metal-insulator transition, continuous disorder-driven transitions are intrinsically quantum critical. Here, we investigate transport quantum criticality in the Falicov-Kimball model, a representative of the latter class in the strong disorder category. Employing cluster-dynamical mean-field theory, we find clear and anomalous quantum critical scaling behavior manifesting as perfect mirror symmetry of scaling curves on both sides of the MIT. Surprisingly, we find that the beta function $\ensuremath{\beta}(g)$ scales as log(g)log(g)log(g) deep into the bad-metallic phase as well, providing a sound unified basis for these findings. We argue that such strong localization quantum criticality may manifest in real three-dimensional systems where disorder effects are more important than electron-electron interactions.
We present a theory of the metal-insulator transition in a disordered two-dimensional electron gas. A quantum critical point, separating the metallic phase, which is stabilized by electronic interactions, from the insulating phase, where disorder prevails over the electronic interactions, has been identified. The existence of the quantum critical point leads to a divergence in the density of states of the underlying collective modes at the transition, causing the thermodynamic properties to behave critically as the transition is approached. We show that the interplay of electron-electron interactions and disorder can explain the observed transport properties and the anomalous enhancement of the spin susceptibility near the metal-insulator transition.
The disordered XXZ model is a prototype model of the many-body localization transition (MBL). Despite numerous studies of this model, the available numerical evidence of multifractality of its eigenstates is not very conclusive due severe finite size effects. Moreover it is not clear if similarly to the case of single-particle physics, multifractal properties of the many-body eigenstates are related to anomalous transport, which is observed in this model. In this work, using a state-of-the-art, massively parallel, numerically exact method, we study systems of up to 24 spins and show that a large fraction of the delocalized phase flows towards ergodicity in the thermodynamic limit, while a region immediately preceding the MBL transition appears to be multifractal in this limit. We discuss the implication of our finding on the mechanism of subdiffusive transport.
Disorder and interactions both play crucial roles in quantum transport. Decades ago, Mott showed that electron-electron interactions can lead to insulating behavior in materials that conventional band theory predicts to be conducting. Soon thereafter, Anderson demonstrated that disorder can localize a quantum particle through the wave interference phenomenon of Anderson localization. Although interactions and disorder both separately induce insulating behavior, the interplay of these two ingredients is subtle and often leads to surprising behavior at the periphery of our current understanding. Modern experiments probe these phenomena in a variety of contexts (e.g. disordered superconductors, cold atoms, photonic waveguides, etc.); thus, theoretical and numerical advancements are urgently needed. In this thesis, we report progress on understanding two contexts in which the interplay of disorder and interactions is especially important. The first is the so-called “dirty” or random boson problem. In the past decade, a strong-disorder renormalization group (SDRG) treatment by Altman, Kafri, Polkovnikov, and Refael has raised the possibility of a new unstable fixed point governing the superfluid-insulator transition in the one-dimensional dirty boson problem. This new critical behavior may take over from the weak-disorder criticality of Giamarchi and Schulz when disorder is sufficiently strong. We analytically determine the scaling of the superfluid susceptibility at the strong-disorder fixed point and connect our analysis to recent Monte Carlo simulations by Hrahsheh and Vojta. We then shift our attention to two dimensions and use a numerical implementation of the SDRG to locate the fixed point governing the superfluid-insulator transition there. We identify several universal properties of this transition, which are fully independent of the microscopic features of the disorder. The second focus of this thesis is the interplay of localization and interactions in systems with high energy density (i.e., far from the usual low energy limit of condensed matter physics). Recent theoretical and numerical work indicates that localization can survive in this regime, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in isolated quantum systems: it separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (“ergodic”) phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present evidence that many-body localization also occurs in quasiperiodic systems that lack true disorder.