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arXiv:1411.0412 · Physical Review E

Transport in tight-binding bond percolation models

Daniel Schmidtke, Abdellah Khodja, Jochen Gemmer B

Published Sep 19, 2014• 5 citations
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Abstract

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.

DOI
10.1103/physreve.90.032127

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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.
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​=${\mathrm{\ensuremath{\rho}}}_{\mathit{c}}^{1/3}$a=0.257\ifmmode\pm\else\textpm\fi{}0.014.
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.
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.
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We investigate the transport properties of the Holstein model using the numerically exact quantum typicality (QT) approach. Roughly speaking, QT exploits the fact that even a single, randomly chosen pure state can effectively represent the full statistical ensemble in a high-dimensional Hilbert space. This allows us to compute frequency-dependent mobilities, representative of the thermodynamic limit, that are well converged with respect to all numerical parameters. Our results are compared against other numerically exact methods, and used to analyze the contribution of vertex corrections to frequency-dependent mobility. The promising accuracy and efficiency of the QT approach suggest its applicability to a broader class of Hamiltonians.
We investigate numerically the nature of energy eigenstates in one-dimensional bond-disordered Anderson models with hopping amplitudes decreasing as ${H}_{\mathrm{ij}}\ensuremath{\propto}1/|i\ensuremath{-}j{|}^{\ensuremath{\alpha}}.$ The eigenstates become delocalized whenever the hopping amplitudes decay slower than 1/r.1/r.1/r. By performing an exact diagonalization scheme on finite chains, we compute the participation ratio of all energy eigenstates. Employing a finite-size scaling analysis, we report on the relevant scaling exponents characterizing this delocalization transition as well as the level-spacing distribution at the critical point $\ensuremath{\alpha}=1.$ The random hopping amplitudes are taken from both uniform and random sign distributions. We show that these models display similar critical behavior in the vicinity of $\ensuremath{\alpha}=1.$ However, the random sign model exhibits an asymptotic delocalization in the limit of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ and the universal scaling behavior in this regime is also reported.