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arXiv:1106.2237 · Physical Review Letters

Localization and Critical Diffusion of Quantum Dipoles in Two Dimensions

I. L. Aleǐner, B. L. Altshuler, K. B. Efetov

Published Aug 9, 2011• 22 citations
arXivHTMLPDF

Abstract

We discuss quantum propagation of dipole excitations in two dimensions. This problem differs from the conventional Anderson localization due to the existence of long-range hops. We find that the critical wave functions of the dipoles always exist which manifest themselves by a scale independent diffusion constant. If the system is T invariant the states are critical for all values of the parameters. Otherwise, there can be a "metal-insulator" transition between this "ordinary" diffusion and the Levy flights (the diffusion constant logarithmically increasing with the scale). These results follow from the two-loop analysis of the modified nonlinear supermatrix σ model.

DOI
10.1103/physrevlett.107.076401

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We discuss quantum propagation of dipole excitations in two dimensions. This problem differs from the conventional Anderson localization due to the existence of long-range hops. We find that the critical wave functions of the dipoles always exist which manifest themselves by a scale independent diffusion constant. If the system is T invariant the states are critical for all values of the parameters. Otherwise, there can be a "metal-insulator" transition between this "ordinary" diffusion and the Levy flights (the diffusion constant logarithmically increasing with the scale). These results follow from the two-loop analysis of the modified nonlinear supermatrix σ model.
We consider a simple model of quantum disorder in two dimensions, characterized by a long-range site-to-site hopping. The system undergoes a metal–insulator transition--its eigenfunctions change from being extended to being localized. We demonstrate that at the point of the transition the nature of the eigenfunctions depends crucially on the magnitude of the hopping amplitude. At small amplitudes they are strongly multifractal. In the opposite limit of large amplitudes, the eigenfunctions do not become fractal. Their density moments do not scale as a power of the system size; instead our result suggests a power of the logarithm of the system size. In this regard, the transition differs from a similar one in the one-dimensional version of the same system, as well as from the conventional Anderson transition in more than two dimensions.
Long-range hoppings in quantum disordered systems are known to yield quantum multifractality, the features of which can go beyond the characteristic properties associated with an Anderson transition. Indeed, critical dynamics of long-range quantum systems can exhibit anomalous dynamical behaviors distinct from those at the Anderson transition in finite dimensions. In this paper, we propose a phenomenological model of wave packet expansion in long-range hopping systems. We consider both their multifractal properties and the algebraic fat tails induced by the long-range hoppings. Using this model, we analytically derive the dynamics of moments and inverse participation ratios of the time-evolving wave packets, in connection with the multifractal dimension of the system. To validate our predictions, we perform numerical simulations of a Floquet model that is analogous to the power law random banded matrix ensemble. Unlike the Anderson transition in finite dimensions, the dynamics of such systems cannot be adequately described by a single parameter scaling law that solely depends on time. Instead, it becomes crucial to establish scaling laws involving both the finite size and the time. Explicit scaling laws for the observables under consideration are presented. Our findings are of considerable interest towards applications in the fields of many-body localization and Anderson localization on random graphs, where long-range effects arise due to the inherent topology of the Hilbert space.
Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $\ensuremath{\nu}$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulator) or nontrivial (a Z2{\mathbb{Z}}_{2}Z2​ insulator supporting a quantum spin Hall effect). This is not true of the boundary multifractal scaling exponents, which we show (numerically) to depend on whether the insulating state is topologically trivial or not.
We present an analytic theory of quantum criticality in quasi one-dimensional topological Anderson insulators. We describe these systems in terms of two parameters (g,χ)(g,χ)(g,χ) representing localization and topological properties, respectively. Certain critical values of χχχ (half-integer for Z\Bbb{Z}Z classes, or zero for Z2\Bbb{Z}_2Z2​ classes) define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow of the integer quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given in terms of the transfer-matrix solution of corresponding supersymmetric non-linear sigma-models. In Z2\Bbb{Z}_2Z2​ classes we uncover a hidden supersymmetry, present at the quantum critical point.
The existence of Anderson localization, characterized by vanishing diffusion due to strong disorder, has been demonstrated in numerous ways. A systematic approach based on the Anderson quantum model of the Fermi gas in random lattices that can describe both diffusive and localized regimes has not yet been fully established. We build on a recent publication [V. Janiš, ] and present a microscopic theory of disordered electrons that covers both the metallic phase with extended Bloch waves and the localized phase, where a propagating particle forms a quantum bound state with the hole left behind at the origin. The general theory provides a framework for constructing controlled approximations to one- and two-particle Green functions that satisfy the necessary conservation laws and causality requirements across the full range of disorder strength. It is used explicitly to derive a local, mean-field-like approximation for the two-particle irreducible vertices, enabling quantitative analysis of the solution's dynamic properties in both metallic and localized phases, including critical behavior at the mobility edge. A new instability line for the dynamical electron-hole correlation function of the metallic phase is introduced.
The phenomenon of localization manifests itself in many quantum mechanical systems ranging from the localization of light in a random medium via Anderson localization of an electronic wave to the motion of atoms in time-dependent laser fields. In all these cases the underlying classical system is chaotic and shows diffusion as a function of time. In contrast, the quantum mechanical counterpart has a localized wave function whose width is governed by the classical diffusion and Planck’s constant. In this paper we show that there exists an additional quantum parameter that controls the localization length. In the system of a two-level ion stored in a Paul trap and interacting with a standing wave it is the detuning between the transition frequency and the laser field. We also discuss the effect of decoherence in form of spontaneous emission. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.