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arXiv:1008.3256 · Physical Review B

Anderson localization in tight-binding models with flat bands

J. T. Chalker, Thomas Pickles, Pragya Shukla

Published Sep 27, 2010• 141 citations
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Abstract

We consider the effect of weak disorder on eigenstates in a special class of tight-binding models. Models in this class have short-range hopping on periodic lattices; their defining feature is that the clean systems have some energy bands that are dispersionless throughout the Brillouin zone. We show that states derived from these flat bands are generically critical in the presence of weak disorder, being neither Anderson localized nor spatially extended. Further, we establish a mapping between this localization problem and the one of resonances in random impedance networks, which previous work has suggested are also critical. Our conclusions are illustrated using numerical results for a two-dimensional lattice, known as the square lattice with crossings or the planar pyrochlore lattice.

DOI
10.1103/physrevb.82.104209

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We consider the effect of weak disorder on eigenstates in a special class of tight-binding models. Models in this class have short-range hopping on periodic lattices; their defining feature is that the clean systems have some energy bands that are dispersionless throughout the Brillouin zone. We show that states derived from these flat bands are generically critical in the presence of weak disorder, being neither Anderson localized nor spatially extended. Further, we establish a mapping between this localization problem and the one of resonances in random impedance networks, which previous work has suggested are also critical. Our conclusions are illustrated using numerical results for a two-dimensional lattice, known as the square lattice with crossings or the planar pyrochlore lattice.
We present a common mathematical formulation of the level statistics of a disordered tight-binding lattice, with one or many flat bands in the clean limit, in which system-specific details enter through a single parameter. The formulation, applicable to both single- as well as many-particle flat bands, indicates the possibility of two different types of critical statistics: one in weak disorder regime (below a system-specific disorder strength) and insensitive of the disorder strength, another in strong disorder regime and occurs at specific critical disorder strengths. The single-parametric dependence, however, relates the statistics in the two regimes (notwithstanding different scattering conditions therein). This also helps in revealing an underlying universality of the statistics in weakly disordered flat bands, shared by a wide range of other complex systems irrespective of the origin of their complexity.
We investigate localization in two models: a single particle in a two-dimensional square lattice described by the tight binding Hamiltonian, and a two-dimensional square qubit lattice. It is well-known that Anderson localization occurs under suitable conditions in which the system parameters are chosen randomly from some statistical distribution. We propose a situation in which the parameters, specifically the on-site energies, are carefully chosen in such a way that a localization-quantifying parameter is maximized. We demonstrate the optimization procedure with numerical calculations in which the engineered localization significantly exceeds the average localization caused by a random distribution of the on-site energies. We explore the relation between spatial patterns and localization efficiency. Furthermore, we use perturbation theory to gain insight into the localization mechanism and obtain an improved cost function for optimization calculations, leading to enhanced localization in both the single-particle and full Hilbert spaces. Although large-scale simulations for qubit lattices are computationally infeasible, we use small-system simulations to demonstrate that results obtained using the single-particle tight binding model can be adapted to identify optimal settings for qubit lattice systems to achieve maximum decoupling between the qubits, which can be valuable for optimizing the idle-state settings on a quantum processor.
We consider the effect of disorder on the tight-binding Hamiltonians with a flat band and derive a common mathematical formulation of the average density of states and inverse participation ratio applicable for a wide range of them. The system information in the formulation appears through a single parameter which plays an important role in search of the critical points for disorder driven transitions in flat bands [P. Shukla, arXiv:1807.02436]. In a weak disorder regime, the formulation indicates an insensitivity of the statistical measures to disorder strength, thus confirming the numerical results obtained by our as well as previous studies.
We investigate the celebrated problem of the one-dimensional tight-binding model in the presence of disorder leading to Anderson localization from a novel perspective. A binary disorder is assumed to be created by immobile, heavy particles that affect the motion of the lighter, mobile species in the limit of no interaction between mobile particles. Fast, periodic modulations of interspecies interactions allow us to produce an effective model with small diagonal and large off-diagonal disorder previously unexplored in cold-atom experiments. We present an expression for an approximate Anderson localization length and verify the existence of the well-known, extended resonant mode. We also analyze the influence of nonzero next-nearest-neighbor hopping terms. We point out that periodic modulation of interaction allows disorder to work as a tunable bandpass filter for momenta.
Anderson localization is a fundamental phenomenon in disordered quantum systems, where transport is suppressed by wave interference from extensive randomness. Moving beyond traditional multi-impurity scenarios, we investigate impurity-induced localization phenomena in low-dimensional tight-binding systems by focusing on the properties of impurity-generated bound states. By introducing a single on-site impurity into an otherwise extended lattice, we demonstrate that the impurity can host a bound state whose spatial character undergoes a transition from extended to localized as the impurity strength surpasses a critical value. This transition pertains solely to the impurity state, while the bulk states of the host system remain extended. We characterize the localization behavior by analyzing two distinct spatial profiles of the bound states: one with symmetric decay and another with exponential decay from the impurity site. Our results highlight how a local perturbation can induce nontrivial localization behavior at the level of individual eigenstates, without implying a global localization transition of the underlying electronic system.
Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions f(α) defined in Eq. (3) remains positive for α noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization; i.e., the ergodicity can be reached only in the absence of disorder. The one-particle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated.