Disordered Lattice Model Research Articles

Delocalization and re-entrant localization of flat-band states in non-Hermitian disordered lattice models with flat bands

Abstract We present a numerical study of Anderson localization in disordered non-Hermitian lattice models with flat bands. Specifically, we consider 1D stub and 2D kagome lattices that have a random scalar potential and a uniform imaginary vector potential and calculate the spectra of the complex energy, the participation ratio, and the winding number as a function of the strength of the imaginary vector potential, h. The flat-band states are found to show a double transition from localized to delocalized and back to localized states with h, in contrast to the dispersive-band states going through a single delocalization transition. When h is sufficiently small, all flat-band states are localized. As h increases above a certain critical value h1, some pairs of flat-band states become delocalized. The participation ratio associated with them increases substantially and their winding numbers become nonzero. As h increases further, more and more flat-band states get delocalized until the fraction of the delocalized states reaches a maximum. For larger h values, a re-entrant localization takes place and, at another critical value h2, all flat-band states return to compact localized states with very small participation ratios and zero winding numbers. This re-entrant localization transition, which is due to the interplay among disorder, non-hermiticity, and the flat band, is a phenomenon occurring in many models having an imaginary vector potential and a flat band simultaneously. We explore the spatial characteristics of the flat-band states by calculating the local density distribution.

Numerical linked-cluster expansions for disordered lattice models

Imperfections in correlated materials can alter their ground state as well as finite-temperature properties in significant ways. Here, we develop a method based on numerical linked-cluster expansions for calculating exact finite-temperature properties of disordered lattice models directly in the thermodynamic limit. We show that a continuous distribution for disordered parameters can be achieved using a set of carefully chosen discrete modes in the distribution, which allows for the averaging of properties over all disorder realizations. We benchmark our results for thermodynamic properties of the square lattice Ising and quantum Heisenberg models with bond disorder against Monte Carlo simulations and study them as the strength of disorder changes. We also apply the method to the disordered Heisenberg model on the frustrated checkerboard lattice, which is closely connected to Sr$_2$Cu(Te$_{0.5}$W$_{0.5}$)O$_6$. Our method can be used to study finite-temperature properties of other disordered quantum lattice models including those for interacting lattice fermions.

Fidelity susceptibility in one-dimensional disordered lattice models

We investigate quantum phase transitions in one-dimensional quantum disordered lattice models, the Anderson model and the Aubry-Andr\'{e} model, from the fidelity susceptibility approach. First, we find that the fidelity susceptibility and the generalized adiabatic susceptibility are maximum at the quantum critical points of the disordered models, through which one can locate the quantum critical point in disordered lattice models. Second, finite-size scaling analysis of the fidelity susceptibility and of the generalized adiabatic susceptibility show that the correlation length critical exponent and the dynamical critical exponent at the quantum critical point of the one-dimensional Anderson model are respectively 2/3 and 2 and of the Aubry-Andr\'{e} model are respectively 1 and 2.375. Thus the quantum phase transitions in the Anderson model and in the Aubry-Andr\'{e} model are of different universality classes. Because the fidelity susceptibility and the generalized adiabatic susceptibility are directly connected to the dynamical structure factor which are experimentally accessible in the linear response regime, the fidelity susceptibility in quantum disordered systems may be observed experimentally in near future.

Entanglement entropy of disordered quantum wire junctions

We consider different disordered lattice models composed of M linear chains glued together in a star-like manner, and study the scaling of the entanglement between one arm and the rest of the system using a numerical strong-disorder renormalization group method. For all studied models, the random transverse-field Ising model (RTIM), the random XX spin model, and the free-fermion model with random nearest-neighbor hopping terms, the average entanglement entropy is found to increase with the length L of the arms according to the form . For the RTIM and the XX model, the effective central charge is universal with respect to the details of junction, and only depends on the number M of arms. Interestingly, for the RTIM, decreases with M, whereas for the XX model it increases. For the free-fermion model, depends also on the details of the junction, which is related to the sublattice symmetry of the model. In this case, both increasing and decreasing tendency with M can be realized with appropriate junction geometries.

Alignment and nonlinear elasticity in biopolymer gels.

We present a Landau-type theory for the nonlinear elasticity of biopolymer gels with a part of the order parameter describing induced nematic order of fibers in the gel. We attribute the nonlinear elastic behavior of these materials to fiber alignment induced by strain. We suggest an application to contact guidance of cell motility in tissue. We compare our theory to simulation of a disordered lattice model for biopolymers. We treat homogeneous deformations such as simple shear, hydrostatic expansion, and simple extension, and obtain good agreement between theory and simulation. We also consider a localized perturbation which is a simple model for a contracting cell in a medium.

Percolation in insect nest networks: Evidence for optimal wiring

Optimization has been shown to be a driving force for the evolution of some biological structures, such as neural maps in the brain or transport networks. Here we show that insect networks also display characteristic traits of optimality. By using a graph representation of the chamber organization of termite nests and a disordered lattice model, it is found that these spatial nests are close to a percolation threshold. This suggests that termites build efficient systems of galleries spanning most of the nest volume at low cost. The evolutionary consequences are outlined.

Fracture size effects from disordered lattice models

We study size effects in the fracture strength of notched quasi-brittle materials using numerical simulations of lattice models for fracture. In particular, we consider the random fuse model, the random spring model and the random beam model, which all give similar results. These allow us to establish and understand the crossover between a regime controlled by disorder-induced statistical effects and a stress-concentration controlled regime ruled by fracture mechanics. The crossover is described by a scaling law that accounts for the presence of fracture process zone which we quantify by averaging over several disordered configurations of the model. The models can be used to study the development of the fracture process zone as the load is increased and to express this in terms of crack resistance (R-curve).

Disorder-induced zero-energy spectral singularity for random matrices with correlations

A zero-energy mid-band singularity has been found in the energy spectrum of random matrices with correlations between diagonal and off-diagonal elements typical of vibrational problems. Two representative classes of matrices, characterizing the instantaneous configurations in liquids and mechanically unstable lattices (which mimic the former) have been analysed. At least for disordered lattice models, the singularity is universal and its origin can be explained within the mean-field treatment.

Monte Carlo calculations of hole size distributions: Simulation of positron annihilation spectroscopy

AbstractConsequences are explored of a hole size distribution in an amorphous polymer for the ortho‐positronium (o‐Ps) lifetime (τ3) and intensity (I3), determined by positron annihilation lifetime spectroscopy. The disordered lattice model, with a vacancy fraction h as a central quantity, is used to represent the equation‐of‐state behavior of the polymer. By means of Monte Carlo simulations, we obtain the cluster size distribution as a function of h and hence temperature. The predicted average cluster size and the cluster concentration are compared to τ3 and I3 data, respectively, for bisphenol‐a polycarbonate. Furthermore, the influence of an o‐Ps lifetime distribution on the experimental mean τ3 is investigated. By mimicking the computational methods used in experimental analysis, agreement between experiment and theory in respect to τ3 and to I3 in the melt ensues. In the glass, however, the experimental I3 becomes increasingly smaller with decreasing temperature than is computed. These deviations may result from a distortion of the equilibrium free volume. © 1992 John Wiley & Sons, Inc.

Multifractal amplitude fluctuations in the transfer-matrix approach to the statistics of disordered lattice models

It is shown that the amplitude fluctuations of vector iterates in the transfer-matrix approach to the statistics of the one-dimensional random Ising chain correspond to an exact strange set characterised by a multifractal spectrum. The temperature and disorder dependence of the spectrum are also investigated.

Neutron Diffraction from Clay-Water Systems

AbstractThe use of neutron diffraction to determine some of the structural properties of montmorillonite-water systems at low water concentrations is described. The samples were prepared by compression or suction to give clay samples with between one and three molecular layers of water between the plates.About 10% of the platelets in the clay are randomly oriented. The remainder are partially oriented in the plane of the sample, with an angular spread of 40° about the mean orientation. It is suggested that these oriented domains are formed from the larger platelets present in the system. The Bragg diffraction pattern is better explained by a disordered lattice model rather than by a mixture model with small particles having a well-defined lattice spacing. We have fitted both the intensities of (00l) reflections and the shape of the (001) reflection quantitatively to a model which allows for a Gaussian spread of platelet spacing about a mean value. The half width of the spread is about 10% of the lattice spacing.No significant structural differences are found between Li, Na, K, and Cs montmorillonites. The method of preparation has no effect on the structural properties of the large platelet particles but does affect the randomly oriented fraction. The lattice spacing of the latter appears to be better defined for samples prepared by compression.Experiments on the variation of lattice spacing with humidity indicate that the structural model we have used is adequate except at humidities where the system is changing over from one to two, or two to three water layers.

Electric Conductivity for Lloyd's Disordered Lattice Model

Static conductivity is investigated for a three-dimensional disordered lattice model proposed by Lloyd by using the diagrammatical expansion technique of the Green function method. The ensemble-averaged transport time τtr is shown to go to zero not slower than (ℏV2/Γ3) when (V/Γ) tends to zero, where V is the typical transfer energy and Γ the fluctuation breadth of the random site energies while the life time τl remains to be of the order of (ℏ/Γ). This reduction arises from the vertex corrections to the ensemble-averaged two-particle Green function.

Existence of localization in Lloyd's disordered lattice model

Electrical conductivity is calculated for a disordered lattice model proposed by Lloyd. It is shown rigorously in the narrow band limit that the ensemble-averaged conductivity is zero and that Anderson's criterion for localization holds for this model.

Electrostatic interactions in electrolyte solutions

A simple equation for the electrostatic terms of activity and osmotic coefficients and of the Gibbs function is developed from the statistically derived equations of Kirkwood. This follows Kirkwood’s equations accurately up to ka = 0·8 and then turns into a cube root function appropriate to a disordered lattice model at higher concentrations. The use of these equations removes many anomalies which occur when the Debye-Hückel equation is used at concentrations for which it was not intended and gives a consistent picture up to values of ka = 20 for all types of aqueous electrolyte solutions.