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Physical review. B, Condensed matter

New approach to localization: Quantum connectivity

Leslie J. Root, J. D. Bauer, James L. Skinner

Published Apr 1, 1988• 67 citations
Publisher

Abstract

We introduce the concept of quantum connectivity, which, along with the phenomenological renormalization method, allows us to study the localization of one-electron eigenstates of various disordered tight-binding Hamiltonians. For the site quantum percolation model in three dimensions we find that the quantum percolation threshold is ${p}_{q}$=0.48\ifmmode\pm\else\textpm\fi{}0.01. For the Anderson model we find that the critical disorder is ${W}_{c}$/J=15.95\ifmmode\pm\else\textpm\fi{}0.25. For a topologically disordered model with exponential interactions of range a, we find that the dimensionless critical density parameter ${\mathit{R}}_{\mathit{c}}$=${\mathrm{\ensuremath{\rho}}}_{\mathit{c}}^{1/3}$a=0.257\ifmmode\pm\else\textpm\fi{}0.014.

DOI
10.1103/physrevb.37.5518

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We introduce the concept of quantum connectivity, which, along with the phenomenological renormalization method, allows us to study the localization of one-electron eigenstates of various disordered tight-binding Hamiltonians. For the site quantum percolation model in three dimensions we find that the quantum percolation threshold is pq{p}_{q}pq​=0.48\ifmmode\pm\else\textpm\fi{}0.01. For the Anderson model we find that the critical disorder is Wc{W}_{c}Wc​/J=15.95\ifmmode\pm\else\textpm\fi{}0.25. For a topologically disordered model with exponential interactions of range a, we find that the dimensionless critical density parameter Rc{\mathit{R}}_{\mathit{c}}Rc​=\ensuremathρc1/3{\mathrm{\ensuremath{\rho}}}_{\mathit{c}}^{1/3}\ensuremathρc1/3​a=0.257\ifmmode\pm\else\textpm\fi{}0.014.
The quantum site and bond percolation problem, which is defined by a disordered tight-binding Hamiltonian with a binary probability distribution, is studied using finite-size-scaling methods. For the simple square lattice, we find that all states are exponentially localized for any amount of disorder, in agreement with the scalng theory of localization and in disagreement with recent claims of a localization transition in two dimensions. The localization length \ensuremath{\lambda} is given by exp{B[p/(1-p)]y{]}^{\mathit{y}}]y} with y very close to 0.5 and p the probability that a site or a bond is present.
Most of the investigations to date on tight-binding, quantum percolation models focused on the quantum percolation threshold, i.e., the analog to the Anderson transition. It appears to occur if roughly 30% of the hopping terms are actually present. Thus, models in the delocalized regime may still be substantially disordered, hence analyzing their transport properties is a nontrivial task which we pursue in the paper at hand. Using a method based on quantum typicality to numerically perform linear response theory we find that conductivity and mean free paths are in good accord with results from very simple heuristic considerations. Furthermore we find that depending on the percentage of actually present hopping terms, the transport properties may or may not be described by a Drude model. An investigation of the Einstein relation is also presented.
A disordered quantum system can be driven, e.g., by increasing disorder, through the Anderson transition between a delocalized and localized phase. Recently, Anderson localization on treelike graphs has attracted much attention in view of its connections with problems of many-body localization. One of the central questions in this context is that of ergodicity of eigenstates. The authors of this paper perform a numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder. The problem can be described as a tight-binding model on a lattice with NNN sites that is locally a tree with constant connectivity. Focusing on the delocalized side of the transition, they show that the data can be interpreted in terms of the finite-size crossover from a small ($N\ensuremath{\ll}{N}_{c}$) to a large ($N\ensuremath{\gg}{N}_{c}$) system, where Nc{N}_{c}Nc​ is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. The results support an analytical prediction that states in the delocalized phase (and at $N\ensuremath{\gg}{N}_{c}$) are ergodic.
We theoretically investigate the quantum percolation problem on Lieb lattices in two and three dimensions. We study the statistics of the energy levels through random matrix theory and determine the level spacing distributions, which, with the aid of finite-size scaling theory, allows us to obtain accurate estimates for site- and bond-percolation thresholds and critical exponents. Our numerical investigation supports a localized-delocalized transition at finite threshold, which decreases as the average coordination number increases. The precise determination of the localization length exponent enables us to claim that quantum site- and bond-percolation problems on Lieb lattices belong to the same universality class, with ν decreasing with lattice dimensionality d, similarly to the classical percolation problem. In addition, we verify that, in three dimensions, quantum percolation on Lieb lattices belongs to the same universality class as the Anderson impurity model.
We perform numerical calculations on a simple cubic lattice for a standard diagonally disordered tight-binding Hamiltonian, whose random site energies are chosen from a Gaussian distribution with variance ∑2. From phenomenological renormalization group studies of the localization length, we determine that the critical disorder is σc≡∑c/J=6.00±0.17, which is in good agreement with previous results (J is the nearest neighbor transfer matrix element). From our calculations we can also determine the mobility edge trajectory, which appears to be analytic at the band center. Defining an order parameter exponent β, which determines how the fraction of extended states vanishes as the critical point is approached from below, this implies that β=1/2, in agreement with a previous study. From a finite-size scaling analysis we find that π2/ν=1.43±0.10, where π2 and ν are the inverse participation ratio and localization length critical exponents, respectively. This ratio of exponents can also be interpreted as the fractal dimension (also called the correlation dimension) D2 of the critical wave functions. Generalizations of the inverse participation ratio lead to a whole set of critical exponents πk, and corresponding generalized fractal dimensions Dk=πk/ν(k−1). From finite-size scaling results we find that D3=1.08±0.10 and D4=0.87±0.09. The inequality of the three dimensions D2, D3, and D4 shows that the critical wave functions have a multifractal structure. Using a generalized phenomenological renormalization technique on the participation ratios, we find that ν=0.99±0.04. This result is in agreement with experiments on compensated or amorphous doped semiconductors.
Anderson localization is a quantum phenomenon in which disorder localizes electronic wave functions. In this work, we propose a new approach to study Anderson localization based on the density matrix formalism. Drawing an analogy to the standard transfer matrix method, we extract the localization length from the modular density matrix in quasi-one-dimensional systems. This approach successfully captures the metal-insulator transition in the three-dimensional Anderson model and in the two-dimensional Anderson model with spin-orbit coupling. It can be also readily extended to multi-orbital systems. We further generalize the formalism to interacting systems, showing that the one-dimensional spinless attractive model exhibits the expected metallic phase, consistent with previous studies. More importantly, we demonstrate the existence of a two-dimensional metallic tendency in the presence of Hubbard interactions and disorder. This method offers a new perspective on Anderson localization and its interplay with interactions.