Random Dimer Model Research Articles

Dynamical Scaling of Surface Roughness and Entanglement Entropy in Disordered Fermion Models.

Localization is one of the most fundamental interference phenomena caused by randomness, and its universal aspects have been extensively explored from the perspective of one-parameter scaling mainly for static properties. We numerically study dynamics of fermions on disordered one-dimensional potentials exhibiting localization and find dynamical one-parameter scaling for surface roughness, which represents particle-number fluctuations at a given length scale, and for entanglement entropy when the system is in delocalized phases. This dynamical scaling corresponds to the Family-Vicsek scaling originally developed in classical surface growth, and the associated scaling exponents depend on the type of disorder. Notably, we find that partially localized states in the delocalized phase of the random-dimer model lead to anomalous scaling, where destructive interference unique to quantum systems leads to exponents unknown for classical systems and clean systems.

A two-dimensional disordered magnetic metamaterial

We study the effect of a resonant frequency disorder on the eigenstates and the transport of magnetic energy in a two-dimensional (square) array of split-ring resonators (SRRs). In the absence of disorder, we find the dispersion relation of magneto-inductive waves and the mean square displacement (MSD) in closed form, showing that at long times the MSD is ballistic. When disorder is present, we consider two types: the usual Anderson distribution (uncorrelated monomers) and 2×2 units assigned at random to lattice sites (correlated tetramers). This is a direct extension to two dimensions of the one-dimensional random dimer model (RDM). For the uncorrelated case, we see saturation of the MSD for all disorder widths, while for the correlated case we find a disorder window, inside which the MSD does not saturate at long times, with an asymptotic sub-diffusive behavior MSD∼t0.26. Outside this disorder window, the MSD shows the same kind of saturation as in the monomer case. We conjecture that the sub-diffusive behavior is a remanent of a weak resonant transmission of a 2D plane wave across a tetramer unit.

Criticality and conformality in the random dimer model.

In critical systems, the effect of a localized perturbation affects points that are arbitrarily far from the perturbation location. In this paper, we study the effect of localized perturbations on the solution of the random dimer problem in two dimensions. By means of an accurate numerical analysis, we show that a local perturbation of the optimal covering induces an excitation whose size is extensive with finite probability. We compute the fractal dimension of the excitations and scaling exponents. In particular, excitations in random dimer problems on nonbipartite lattices have the same statistical properties of domain walls in spin glass. Excitations produced in bipartite lattices, instead, are compatible with a loop-erased self-avoiding random walk process. In both cases, we find evidence of conformal invariance of the excitations that is compatible with SLE_{κ} with parameter κ depending on the bipartiteness of the underlying lattice only.

Magnetic metamaterials with correlated disorder

We examine the transport of magnetic energy in a simplified model of a magnetic metamaterial, consisting of a one-dimensional array of split-ring resonators (SRR), in the presence of correlated disorder in the resonant frequencies. The computation of the average participation ratio (PR) reveals that on average, the modes for the correlated disorder system are less localized than in the uncorrelated case. The numerical computation of the mean square displacement of an initially localized magnetic excitation shows an asymptotic behavior 〈n2〉∼tα, where α∼0 for uncorrelated disorder, and α>0 for the correlated case, spanning a whole range of behavior ranging from localization to super-diffusive behavior at finite system sizes. The transmission coefficient of a plane wave across a single magnetic dimer reveals the existence of well-defined regions in disorder strength-magnetic coupling space, where unit transmission for some wavevector(s) is possible. This explains the finite degree of mobility for magnetic excitations in the presence of correlated disorder, in the spirit of the random dimer model (Dunlap et al., 1990 [14]). Finally, the transmission of plane waves across an array with correlated disorder shows a power-law decrease with system size, in contrast to the exponential decrease found in the standard Anderson model.

Floquet dynamics of disordered bands with isolated critical energies

We investigate localization properties of driven models which exhibit a subextensive number of extended states in the static setting. We consider instances where the extended modes are or are not protected by topological considerations. To this end, we contrast the strongly driven, disordered, lowest Landau level, which we refer to as the random Landau model (RLM), with the random dimer model (RDM); the RDM also has a subextensive set of delocalized modes in the middle of the spectrum whose origin is not topological. We map the driven models on to a higher-dimensional effective model and numerically compute the localization length as a function of disorder strength, drive amplitude, and frequency using the recursive Green's function method. Our numerical results indicate that, in the presence of a strong drive (low frequency and/or large drive amplitude), the topologically protected RLM continues to exhibit a spectrum with both localized and delocalized (or critical) modes, but the spectral range of delocalized modes is enhanced by the driving. This occurs due to an admixture of the localized modes with extended modes arising due to the topologically protected critical energy in the middle of the spectrum. On the other hand, in the RDM, a weak drive immediately localizes the entire spectrum. This occurs in contrast to the naive expectation from perturbation theory that mixing between localized and delocalized modes generically enhances the delocalization of all modes. Our work highlights the importance of the origin of the delocalized modes in the localization properties of the corresponding Floquet model.

Density wave propagation in a two-dimensional random dimer potential: From a single to a bipartite square lattice

We study the propagation of a density perturbation in a weakly interacting boson gas confined on a lattice and in the presence of square dimerized impurities. Such a two-dimensional random-dimer model (2D-DRDM), previously introduced in [Capuzzi et al., Phys. Rev. A 92, 053622 (2015)], is the disorder transition from a single square lattice, where impurities are absent, to a bipartite square lattice, where the number of impurities is maximum and coincides with half the number of lattice sites. We show that disorder correlations can play a crucial role in the dynamics for a broad range of parameters by allowing density fluctuations to propagate in the 2D-DRDM lattice, even in the limit of strong disorder. In such a regime, the propagation speed depends on the percentage of impurities, interpolating between the speed in a single monoperiodic lattice and that in a bipartite one.

Momentum-space entanglement after a quench in one-dimensional disordered fermionic systems

We numerically investigate the momentum-space entanglement entropy and entanglement spectrum of the random-dimer model and its generalizations, which circumvent Anderson localization, after a quench in the Hamiltonian parameters. The type of dynamics that occurs depends on whether or not the Fermi level of the initial state is near the energy of the delocalized states present in these models. If the Fermi level of the initial state is near the energy of the delocalized states, we observe an interesting slow logarithmic-like growth of the momentum-space entanglement entropy followed by an eventual saturation. Otherwise, the momentum-space entanglement entropy is found to rapidly saturate. We also find that the momentum-space entanglement spectrum reveals the presence of delocalized states in these models for long times after the quench and the many-body entanglement gap decays logarithmically in time when the Fermi level is near the energy of the delocalized states.

How Much Delocalisation is Needed for an Enhanced Area Law of the Entanglement Entropy?

We consider the random dimer model in one space dimension with Bernoulli disorder. For sufficiently small disorder, we show that the entanglement entropy exhibits at least a logarithmically enhanced area law if the Fermi energy coincides with a critical energy of the model where the localisation length diverges.

Violation of the Bell inequality in quantum critical random spin-1/2 chains

We investigate the entanglement and nonlocality properties of two random XX spin-1/2 critical chains, in order to better understand the role of breaking translational invariance to achieve nonlocal states in critical systems. We show that breaking translational invariance is a necessary but not sufficient condition for nonlocality, as the random chains remain in a local ground state up to a small degree of randomness. Furthermore, we demonstrate that the random dimer model does not have the same nonlocality properties of the translationally invariant chain, even though they share the same universality class for a certain range of randomness.

Charge transport properties in a one-dimensional description of polymer polyaniline

The metal-insulator transition taking place in Polyaniline (PAN) is investigated in a one-dimensional configuration when impurities are introduced by doping. The electronic transport is numerically analyzed, representing the system by a Hückel tight-binding Hamiltonian and using the non-equilibrium Green’s function formalism to obtain the conductance of the system under the presence of polaron and bipolaron doping. In a detailed analysis, we highlight the importance of studying extensive N sites chains, evaluating the conductance as a function of the de-coherence parameter η, decreasing it to the limit of 0+ ; this procedure allows us to study the existence of delocalized states in this system. It was possible to verify that although the bipolaron/polaron doping produces a displacement of the Fermi energy into a region of states outside the gap of the pure polymer, the conductance in the limit of zero inelastic scattering is zero, showing the existence of localized states at the Fermi level, due to disorder. This result indicates that the description of PAN as a linear one-dimensional object analyzed in a extensive N sites disordered chain does not manifest correlated disorder as proposed by the Random Dimer Model (RDM), where transport could occur because of the electronic states are extended. The lack of charge diffusion at the Fermi level in this one-dimensional description of the system shows that it is not an adequate model to study the metal-insulator transition when the polymer is doped. The incorporation of the nature of the polymerization process, introducing higher dimensional effects in its early stages, is possibly an essential ingredient to derive an appropriate model to describe the conducting behavior of the PAN system.

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb–Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in |x|-v|t| is replaced by exponential decay in |x|-v|t|^\alpha with 0<\alpha<1 . In fact, we can characterize the values of \alpha for which such a bound holds as those exceeding \alpha_u^+ , the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of [14], we relate Lieb–Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan–Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in [8, 9, 2, 3] to our purposes. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. Along the way, we prove a new result about the one-body Fibonacci Hamiltonian: the upper transport exponent agrees with the time-averaged upper transport exponent, see Corollary 2.9. We also explain why our method does not extend to yield anomalous Lieb–Robinson bounds of power-law type for the random dimer model.

Tight-binding chains with off-diagonal disorder: Bands of extended electronic states induced by minimal quasi–one-dimensionality

It is shown that an entire class of off-diagonally disordered linear lattices composed of two basic building blocks and described within a tight-binding model can be tailored to generate absolutely continuous energy bands. It can be achieved if linear atomic clusters of an appropriate size are side-coupled to a suitable subset of sites in the backbone, and if the nearest-neighbor hopping integrals, in the backbone and in the side-coupled cluster, bear a certain ratio. We work out the precise relationship between the number of atoms in one of the building blocks in the backbone and that in the side attachment. In addition, we also evaluate the definite correlation between the numerical values of the hopping integrals at different subsections of the chain, that can convert an otherwise point spectrum (or a singular continuous one for deterministically disordered lattices) with exponentially (or power law) localized eigenfunctions to an absolutely continuous spectrum comprising one or more bands (subbands) populated by extended, totally transparent eigenstates. The results, which are analytically exact, put forward a non-trivial variation of the Anderson localization (Anderson P. W., Phys. Rev., 109 (1958) 1492), pointing towards its unusual sensitivity to the numerical values of the system parameters and, go well beyond the other related models such as the Random Dimer Model (RDM) (Dunlap D. H. et al., Phys. Rev. Lett., 65 (1990) 88).

Transport of localized and extended excitations in chains embedded with randomly distributed linear and nonlinear n-mers.

We examine the transport of extended and localized excitations in one-dimensional linear chains populated by linear and nonlinear symmetric identical n-mers (with n=3, 4, 5, and 6), randomly distributed. First, we examine the transmission of plane waves across a single linear n-mer, paying attention to its resonances, and looking for parameters that allow resonances to merge. Within this parameter regime we examine the transmission of plane waves through a disordered and nonlinear segment composed by n-mers randomly placed inside a linear chain. It is observed that nonlinearity tends to inhibit the transmission, which decays as a power law at long segment lengths. This behavior still holds when the n-mer parameters do not obey the resonance condition. On the other hand, the mean square displacement exponent of an initially localized excitation does not depend on nonlinearity at long propagation distances z, and shows a superdiffusive behavior ∼z(1.8) for all n-mers, when parameters obey the resonance merging condition; otherwise the exponent reverts back to the random dimer model value ∼z(1.5).

Enhancing quantum coherence with short-range correlated disorder

We introduce a two-dimensional short-range correlated disorder that is the natural generalization of the well-known one-dimensional dual random dimer model [Phys. Rev. Lett 65, 88 (1990)]. We demonstrate that, as in one dimension, this model induces a localization-delocalization transition in the single-particle spectrum. Moreover we show that the effect of such a disorder on a weakly-interacting boson gas is to enhance the condensate spatial homogeneity and delocalisation, and to increase the condensate fraction around an effective resonance of the two-dimensional dual dimers. This study proves that short-range correlations of a disordered potential can enhance the quantum coherence of a weakly-interacting many-body system.

Role of Correlations and Off-Diagonal Terms in Binary Disordered One-Dimensional Systems

We investigate one dimensional tight binding model in the presence of a correlated binary disorder. The disorder is due to the interaction of particles with heavy immobile other species. Off-diagonal disorder is created by means of a fast periodic modulation of interspecies interaction. The method based on transfer matrix techniques allows us to calculate the energies of extended modes in the correlated binary disorder. We focus on $N$-mer correlations and regain known results for the case of purely diagonal disorder. For off-diagonal disorder we find resonant energies. We discuss ambiguous properties of those states and compare analytical results with numerical calculations. Separately we describe a special case of the dual random dimer model.

Measurement-induced disturbance near Anderson localization in one-dimensional systems

We study the localization transition in several typical one-dimensional single-particle systems by means of measurement-induced disturbance (MID). The results show that the MID presents a rapid drop around the boundary between the localized state and extended ones, and the corresponding first-order derivative exhibits a behavior of divergence around the critical point for deterministic on-site potential systems (e.g. the quasi-periodic model). These characteristics can capture a phase diagram as well as the traditional method. For the non-deterministic on-site systems (e.g. the random dimer model), the states around the resonant energies possess relatively large values for MID, which means that they are extended. In addition, as the random potential ϵb exceeds the critical value, the states possessing a large MID vanish completely. These results show that MID can be useful in detecting localization transition in these typical one-dimensional systems.

Ballistic transport in one-dimensional random dimer photonic crystals

Using the transfer-matrix technique and the Kronig Penney model, we numerically and analytically investigate the effect of short-range correlated disorder in Random Dimer Model (RDM) on transmission properties of the light in one dimensional photonic crystals made of three different materials. Such systems consist of two different structures randomly distributed along the growth direction, with the additional constraint that one kind of these layers always appear in pairs. It is shown that the one dimensional random dimer photonic crystals support two types of extended modes. By shifting of the dimer resonance toward the host fundamental stationary resonance state, we demonstrate the existence of the ballistic response in these systems.

Quantum discord near Anderson metal–insulator transition with correlated disorder

Localization transitions in the one-dimensional Anderson model with long range correlated disorder are studied by means of quantum discord. The results show that the quantum discord exhibits a sudden transition around the boundary between the localized state and extended ones, and does not depend on the system size L in the region of extended states. This feature can depict an identical phase diagraph as well as the traditional method. As an another example, we also consider this quantum correlation in the random dimer model. It is then shown that the quantum discord can be a good quantity to detect the localization transition in these one-dimensional systems with correlated disorder.

Magnetic responses of randomly depleted spin ladders

The magnetic responses of a spin-1/2 ladder doped with non-magnetic impurities are studied using various methods and including the regime where frustration induces incommensurability. Several improvements are made on the results of the seminal work of Sigrist and Furusaki [J. Phys. Soc. Jpn. 65, 2385 (1996)]. Deviations from the Brillouin magnetic curve due to interactions are also analyzed. First, the magnetic profile around a single impurity and effective interactions between impurities are analyzed within the bond-operator mean-field theory and compared to density-matrix renormalization group calculations. Then, the temperature behavior of the Curie constant is studied in details. At zero-temperature, we give doping-dependent corrections to the results of Sigrist and Furusaki on general bipartite lattice and compute exactly the distribution of ladder cluster due to chain breaking effects. Using exact diagonalization and quantum Monte-Carlo methods on the effective model, the temperature dependence of the Curie constant is compared to a random dimer model and a real-space renormalization group scenario. Next, the low-part of the magnetic curve corresponding to the contribution of impurities is computed using exact diagonalization. The random dimer model is shown to capture the bulk of the curve, accounting for the deviation from the Brillouin response. At zero-temperature, the effective model prediction agrees relatively well with density-matrix renormalization group calculations. Finite-temperature effects are displayed within the effective model and for large depleted ladder models using quantum Monte-Carlo simulations. In all, the effect of incommensurability does not display a strong qualitative effect on both the magnetic susceptibility and the magnetic curve. Consequences for experiments on the BiCu2PO6 compound and other spin-gapped materials are briefly discussed.

Quenching of the transmittivity of a one-dimensional binary random dimer model through side-attached atoms

By means of a tight-binding model we study the electronic and transport properties of a one-dimensional system with short-range spatially correlated disorder composed by a periodic chain functionalized by a random distribution of side-attached atoms with enegies ${E}_{A}$ or ${E}_{B}$ assigned in pairs as in the random dimer model. The attached atoms are spatially distributed so as to make the original periodic chain a random binary dimer model alloy with energy-dependent on-site parameters. We first consider the case of a single atom and of a couple of atoms, with the same site energy, side-attached to consecutive chain sites. In the case of a couple of attached atoms we evidence significative differences in the local density of states with respect to the case of a single side-attached atom; moreover we find the presence of two transmitting states instead of a single one as it happens in the case of a single substitutional dimer in the chain. We then address the case in which each chain site is decorated by an attached atom as specified above. After a decimation-renormalization procedure we are led to a random binary dimer system with energy-dependent site parameters. We study the localization properties of this system by evaluating the Lyapunov coefficient; we find for it the existence of four delocalized states in the spectrum instead of the two found for embedded random dimers. The Lyapunov coefficient as a function of the electron Fermi energy shows a quite complex behavior with zeros, relative minima, and maxima; their energy positions and heights depend on the site parameters and the coupling interaction of the attached atoms. The analysis of the average transmittivity of finite samples allows us also to show the effects of the competition between the strong localization induced by the presence of the attached atoms and the delocalization of the states due to the short-range order in the distribution of their energy. We show that it is possible to individuate specific parameters of the system to quench the transmittivity of the binary random dimer model.