One-parameter Scaling Research Articles

Renormalization group analysis of the Anderson model on random regular graphs

We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.

Numerical comparison of nonlinear thermal radiation and chemically reactive bio-convection flow of Casson-Carreau nano-liquid with gyro-tactic microorganisms: Lie group theoretic approach

The current research presents a mathematical model to study the flow of a non-Newtonian magnetohydrodynamics (MHD) Casson-Carreau nanofluid (CCNF) over a stretching porous surface, considering mass and heat transport rates with Stefan blowing, non-linear thermal radiation, heat source-sink, chemical reaction, thermophoretic and Brownian motions, convective heating, Joule heating, motile microorganisms, and bio-convection. The presence of microorganisms is utilized to control the suspension of nanomaterials within the nanofluid. The current flow model has been rendered by the boundary layer approximation and we get the highly nonlinear partial differential equations (PDEs). These nonlinear PDEs are simplified by the novel Lie group theoretic method. The one-parameter Lie scaling method simplified the PDEs and convert it into the ordinary differential equations (ODEs). Numerical solutions for these ODEs are obtained using the bvp4c scheme built-in function in MATLAB, ensuring reliable outcomes for temperature, velocity, concentration, and motile microorganism density profiles. The numerical results are presented through graphs and compared with available data, showing good agreement. These numerical outcomes reveal several important flow characteristics. Rates of change for Nr are 0.0007 and 0.0005, and for Ω, they are −0.0754 and −0.0536, respectively. Similarly, the rate of change for Rb in both models is −0.002 and −0.0002. Analysis shows a positive impact of the bioconvection Rayleigh number in both models, notably higher for the Casson fluid compared to the Carreau fluid model. Buoyancy ratio parameter exhibits consistent rates of change, while the reduction in impact is more pronounced for the Casson fluid model in the case of the mixed bioconvection parameter. The mixed bio-convection parameter reduces momentum velocity for both Casson and Carreau fluids, whereas the Darcy parameter boosts fluid velocity. As the Newtonian heating parameter increases, the temperature velocity distribution of both fluids also increases. The concentration profile of both Casson-Carreau fluid phases declines as the heat source-sink parameter and Schmidt number increase. Microbial velocity shows a decrease with increasing R values, whereas the opposite trend is observed for the Peclet number Pe.

Vertical nanoscale strain-induced electronic localization in epitaxial La2/3Sr1/3MnO3 films with ZrO2 nanopillar inclusions

Unusual electrical transport properties associated with weak or strong localization are sometimes found in disordered electronic materials. Here, we report experimental observation of a crossover of electronic behavior from weak localization to enhanced weak localization due to the spatial influence of disorder induced by ZrO2 nanopillars in (La2/3Sr1/3MnO3)1−x:(ZrO2)x (x = 0, 0.2, and 0.3) nanocomposite films. The spatial strain regions, identified by scanning transmission electron microscopy and high-resolution x-ray diffraction, induce a coexistence of two-dimentional (2D) and three-dimentional (3D) localization and switches to typical 2D localization with increasing density of ZrO2 pillars due to length scale confinement, which interestingly accords with enhancing vertically interfacial strain. Based on the excellent agreement of our experimental results with one-parameter scaling theory of localization, the enhanced weak localization exists in metal range close to the fixed point. These films provide a tunable experimental model for studying localization in particular the transition regime by appropriate choice of the second epitaxial phase.Graphical

Theory of localization-hindered thermalization in nonlinear multimode photonics

Our society’s appetite for ultra-high bandwidth communication networks and high-power optical sources, together with recent breakthroughs in mode multiplexing/demultiplexing schemes, forced the photonics community to reconsider the deployment of nonlinear multimode systems. These developments pose fundamental challenges stemming from the complexity of nonlinear mode-mode mixing by which they exchange energy in the process towards an equilibrium Rayleigh-Jeans (RJ) distribution. Here we develop a universal one-parameter scaling theory for the relaxation rates of out-of-equilibrium excitations towards their RJ thermal state. The theory predicts an exponential suppression of the rates with increasing disorder due to the formation of stable localization clusters resisting the nonlinear mode-mode interactions that tend to separate them. For low optical temperatures, the rates experience a crossover from linear to nonlinear temperature dependence which reflects a disorder-induced reorganization of the low frequency eigenmodes. Our theory will guide the design of nonlinear multimode photonic networks with tailored relaxation-scales.

Metal-insulator crossover in monolayer MoS2

We report on transport measurements in monolayer MoS2 devices, close to the bottom of the conduction band edge. These devices were annealed in situ before electrical measurements. This allows us to obtain good ohmic contacts at low temperatures, and to measure precisely the conductivity and mobility via four-probe measurements. The measured effective mobility up to μ eff = 180 cm2 V−1 s−1 is among the largest obtained in CVD-grown MoS2 monolayer devices. These measurements show that electronic transport is of the insulating type for σ ≤ 1.4e 2/h and n ≤ 1.7 × 1012 cm−2, and a crossover to a metallic regime is observed above those values. In the insulating regime, thermally activated transport dominates at high temperature (T > 120 K). At lower temperatures, conductivity is driven by Efros–Schklovkii variable range hopping in all measured devices, with a universal and constant hopping prefactor, that is a clear indication that hopping is not phonon-mediated. At higher carrier density, and high temperature, the conductivity is well modeled by the Boltzmann equation for a non-interacting Fermi gas, taking into account both phonon and impurity scatterings. Finally, even if this apparent metal-insulator transition can be explained by phonon-related phenomena at high temperature, the possibility of a genuine 2D MIT cannot be ruled out, as we can observe a clear power-law diverging localization length close to the transition, and a one-parameter scaling can be realized.

A One-Parameter Memoryless DFP Algorithm for Solving System of Monotone Nonlinear Equations with Application in Image Processing

In matrix analysis, the scaling technique reduces the chances of an ill-conditioning of the matrix. This article proposes a one-parameter scaling memoryless Davidon–Fletcher–Powell (DFP) algorithm for solving a system of monotone nonlinear equations with convex constraints. The measure function that involves all the eigenvalues of the memoryless DFP matrix is minimized to obtain the scaling parameter’s optimal value. The resulting algorithm is matrix and derivative-free with low memory requirements and is globally convergent under some mild conditions. A numerical comparison showed that the algorithm is efficient in terms of the number of iterations, function evaluations, and CPU time. The performance of the algorithm is further illustrated by solving problems arising from image restoration.

Dynamical many-body delocalization transition of a Tonks gas in a quasi-periodic driving potential

The quantum kicked rotor is well-known for displaying dynamical (Anderson) localization. It has recently been shown that a periodically kicked Tonks gas will always localize and converge to a finite energy steady-state. This steady-state has been described as being effectively thermal with an effective temperature that depends on the parameters of the kick. Here we study a generalization to a quasi-periodic driving with three frequencies which, without interactions, has a metal-insulator Anderson transition. We show that a quasi-periodically kicked Tonks gas goes through a dynamical many-body delocalization transition when the kick strength is increased. The localized phase is still described by a low effective temperature, while the delocalized phase corresponds to an infinite-temperature phase, with the temperature increasing linearly in time. At the critical point, the momentum distribution of the Tonks gas displays different scaling at small and large momenta (contrary to the non-interacting case), signaling a breakdown of the one-parameter scaling theory of localization.

Regional Analysis of Poverty in Ecuador: Sensitivity to the Choice of Equivalence Scales.

Economies of scale and equivalent consumption units, which are present in households, must be considered in the measurement of monetary poverty, in order to obtain indicators that approximate the reality of each household. Therefore, in this research, monetary poverty in Ecuador is measured and analyzed at the provincial level for the period 2009–2016. It works with data from the National Survey of Employment, Unemployment and Underemployment (ENEMDU) and the INEC methodology is used to measure poverty, but per-capita income is replaced by equivalent income, generated by applying a one-parameter scale (Buhmann et al., 1988) and two two-parameter scales (OECD, 1984; Cutler and Katz, 1992). The main results show that monetary poverty rates are significantly lower when equivalent income is applied, that there is high poverty sensitivity depending on the equivalence scale used and that the provinces with the highest levels of poverty are located in the Amazon region.

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.

Nonunitary Scaling Theory of Non-Hermitian Localization.

Non-Hermiticity can destroy Anderson localization and lead to delocalization even in one dimension. However, a unified understanding of non-Hermitian delocalization has yet to be established. Here, we develop a scaling theory of localization in non-Hermitian systems. We reveal that non-Hermiticity introduces a new scale and breaks down the one-parameter scaling, which is the central assumption of the conventional scaling theory of localization. Instead, we identify the origin of unconventional non-Hermitian delocalization as the two-parameter scaling. Furthermore, we establish the threefold universality of non-Hermitian localization based on reciprocity; reciprocity forbids delocalization without internal degrees of freedom, whereas symplectic reciprocity results in a new type of symmetry-protected delocalization.

Presence and absence of delocalization-localization transition in coherently perturbed disordered lattices.

A new type of delocalization induced by coherent harmonic perturbations in one-dimensional Anderson-localized disordered systems is investigated. With only a few M frequencies a normal diffusion is realized, but the transition to a localized state always occurs as the perturbation strength is weakened below a critical value. The nature of the transition qualitatively follows the Anderson transition (AT) if the number of degrees of freedom M+1 is regarded as the spatial dimension d. However, the critical dimension is found to be d=M+1=3 and is not d=M+1=2, which should naturally be expected by the one-parameter scaling hypothesis.

RETRACTED ARTICLE: One-parameter scaling transformations of Maxwell nanofluid with Ludwig–Soret and pedesis motion passed over stretching–shrinking surfaces

RETRACTED ARTICLE: One-parameter scaling transformations of Maxwell nanofluid with Ludwig–Soret and pedesis motion passed over stretching–shrinking surfaces

Surface Conductivity of the Ytterbium-Doped Topological Kondo-Insulator SmB6

The temperature dependence of the conductivity of the topological Kondo-insulator (TKI) Sm1 – xYbxB6 is investigated in the temperature range 2 < T < 300 K for compositions with x ≤ 0.024. It is found that the ytterbium impurity most strongly affects the low-temperature (T < 20 K) electrical conductivity: when changing from x = 0 to x = 0.024, the activation energy of bulk conductivity decreases by 1.8 times from 4 to 2.2 meV, and the 2D surface conductivity increases by six times. The separation made for the contributions to the conductivity enables us to establish that the one-parameter scaling model can be used for describing the conductivity of 2D surface states in the Sm1 – xYbxB6 TKI in which both the interaction with phonons and the electron–electron scattering effects are taken into account.

Remarkable Accuracy of an O(N6) Perturbative Correction to Opposite-Spin CCSD: Are Triples Necessary for Chemical Accuracy in Coupled Cluster?

The focus of this work is OS-CCSD-SPT(2), which is a second-order similarity transformed perturbation theory correction to opposite spin coupled cluster singles doubles, where in the latter the same-spin amplitudes are removed and the opposite-spin ones are solved self-consistently. OS-CCSD-SPT(2) is free of empirical parameters, has an instrinsic scaling of O(N6), and makes no use of triples. We demonstrate that, for non-multireference molecules, OS-CCSD-SPT(2) produces relative energies whose accuracy is significantly higher than what is generally expected of a triples-free model. For example, using PBE0 orbitals in the reference, OS-CCSD-SPT(2) exhibits a mean absolute deviation (MAD) of 1.13 kcal/mol with respect to CCSD(2F) benchmark values for the non-multireference subset of W4-08 atomization energies (cf. a MAD > 6.5 kcal/mol for CCSD) and a MAD of 0.68 kcal/mol for the energies of reactions generated from the W4-08 molecules. These MADs are reduced to 0.61 and 0.63 kcal/mol, respectively, by a simple one-parameter spin-component scaling of the OS-CCSD-SPT(2) same-spin correlation energy. OS-CCSD is also naturally amenable to higher order corrections: the associated third-order correction, OS-CCSD-SPT(3), which does involve connected triples and quadruples, exhibits a MAD of 0.44 kcal/mol for the same atomization-energy benchmark.

Critical phenomena of dynamical delocalization in quantum maps: Standard map and Anderson map.

Following the paper exploring the Anderson localization of monochromatically perturbed kicked quantum maps [Phys. Rev. E 97, 012210 (2018)2470-004510.1103/PhysRevE.97.012210], the delocalization-localization transition phenomena in polychromatically perturbed quantum maps (QM) is investigated focusing particularly on the dependency of critical phenomena on the number M of the harmonic perturbations, where M+1=d corresponds to the spatial dimension of the ordinary disordered lattice. The standard map and the Anderson map are treated and compared. As the basis of analysis, we apply the self-consistent theory (SCT) of the localization for our systems, taking a plausible hypothesis on the mean-free-path parameter which worked successfully in the analyses of the monochromatically perturbed QMs. We compare in detail the numerical results with the predictions of the SCT by largely increasing M. The numerically obtained index of critical subdiffusion t^{α} (t:time) agrees well with the prediction of one-parameter scaling theory α=2/(M+1), but the numerically obtained critical exponent of localization length significantly deviates from the SCT prediction. Deviation from the SCT prediction is drastic for the critical perturbation strength of the transition: If M is fixed, then the SCT presents plausible prediction for the parameter dependence of the critical value, but its value is 1/(M-1) times smaller than the SCT prediction, which implies existence of a strong cooperativity of the harmonic perturbations with the main mode.

Evidence of Nonrigidity Effects in the Description of Low-Energy Anisotropic Molecular Collisions of Hydrogen Molecules with Excited Metastable Helium Atoms.

Cold collisions serve as a sensitive probe of the interaction potential. In the recent study of Klein et al. (Nature Phys.2017, 13, 35–38), the one-parameter scaling of the interaction potential was necessary to obtain agreement between theoretical and observed patterns of the orbiting resonances for excited metastable helium atoms colliding with hydrogen molecules. Here, we show that the effect of nonrigidity of the H2 molecule on the resonant structure, absent in the previous study, is critical to predict the correct positions of the resonances in that case. We have complemented the theoretical description of the interaction potential and revised reaction rate coefficients by proper inclusion of the flexibility of the molecule. The calculated reaction rate coefficients are in remarkable agreement with the experimental data without empirical adjustment of the interaction potential. We have shown that even state-of-the-art calculations of the interaction energy cannot ensure agreement with the experiment if such an important physical effect as flexibility of the interacting molecule is neglected. Our findings about the significance of the nonrigidity effects can be especially crucial in cold chemistry, where the quantum nature of molecules is pronounced.

Topological phase transitions in glassy quantum matter

Amorphous systems have rapidly gained promise as novel platforms for topological matter. In this work we establish a scaling theory of amorphous topological phase transitions driven by the density of lattice points in two dimensions. By carrying out a finite-size scaling analysis of topological invariants averaged over discrete and continuum random geometries, we discover unique critical properties of Chern and $\mathbb{Z}_2$ glass transitions. Even for short-range hopping models the Chern glass phase may persist down to the fundamental lower bound given by the classical percolation threshold. While the topological indices accurately satisfy the postulated one-parameter scaling, they do not generally flow to the closest integer value in the thermodynamic limit. Furthermore, the value of the critical exponent describing the diverging localization length varies continuously along the phase boundary and is not fixed by the symmetry class of the Hamiltonian. We conclude that the critical behaviour of amorphous topological systems exhibit characteristic features not observed in disordered systems, motivating a wealth of new research directions.

Low-high voltage duality in tunneling spectroscopy of the Sachdev-Ye-Kitaev model

The Sachdev-Ye-Kitaev (SYK) model describes a strongly correlated metal with all-to-all random interactions (average strength $J$) between $N$ fermions (complex Dirac fermions or real Majorana fermions). In the large-$N$ limit a conformal symmetry emerges that renders the model exactly soluble. Here we study how the non-Fermi liquid behavior of the closed system in equilibrium manifests itself in an open system out of equilibrium. We calculate the current-voltage characteristic of a quantum dot, described by the complex-valued SYK model, coupled to a voltage source via a single-channel metallic lead (coupling strength $\Gamma$). A one-parameter scaling law appears in the large-$N$ conformal regime, where the differential conductance $G=dI/dV$ depends on the applied voltage only through the dimensionless combination $\xi=eVJ/\Gamma^2$. Low and high voltages are related by the duality $G(\xi)=G(\pi/\xi)$. This provides for an unambiguous signature of the conformal symmetry in tunneling spectroscopy.

Controlling symmetry and localization with an artificial gauge field in a disordered quantum system

Anderson localization, the absence of diffusion in disordered media, draws its origins from the destructive interference between multiple scattering paths. The localization properties of disordered systems are expected to be dramatically sensitive to their symmetries. So far, this question has been little explored experimentally. Here we investigate the realization of an artificial gauge field in a synthetic (temporal) dimension of a disordered, periodically driven quantum system. Tuning the strength of this gauge field allows us to control the parity–time symmetry properties of the system, which we probe through the experimental observation of three symmetry-sensitive signatures of localization. The first two are the coherent backscattering, marker of weak localization, and the recently predicted coherent forward scattering, genuine interferential signature of Anderson localization. The third is the direct measurement of the β(g) scaling function in two different symmetry classes, allowing to demonstrate its universality and the one-parameter scaling hypothesis.

Localization in one-dimensional tight-binding model with chaotic binary sequences

We have numerically investigated localization properties in the one-dimensional tight-binding model with chaotic binary on-site energy sequences generated by a modified Bernoulli map with the stationary-nonstationary chaotic transition (SNCT). The energy sequences in question might be characterized by their correlation parameter B and the potential strength W. The quantum states resulting from such sequences have been characterized in the two ways: Lyapunov exponent at band centre and the dynamics of the initially localized wavepacket. Specifically, the B−dependence of the relevant Lyapunov exponent’s decay is changing from linear to exponential one around the SNCT (B ≃ 2). Moreover, here we show that even in the nonstationary regime, mean square displacement (MSD) of the wavepacket is noticeably suppressed in the long-time limit (dynamical localization). The B−dependence of the dynamical localization lengths determined by the MSD exhibits a clear change in the functional behaviour around SNCT, and its rapid increase gets much more moderate one for B ≥ 2. Moreover we show that the localization dynamics for B > 3/2 deviates from the one-parameter scaling of the localization in the transient region.