Fractional Regime Research Articles

An Asymptotic-Preserving Scheme for the Kinetic Equation with Anisotropic Scattering: Heavy Tail Equilibrium and Degenerate Collision Frequency

We design an asymptotic preserving (AP) scheme for the linear kinetic equation with anisotropic scattering that leads to a fractional diffusion limit. This limit may be attributed to two reasons: a heavy tail equilibrium or a degenerate collision frequency, both of which are considered in this paper. Our scheme builds on the ideas developed in [L. Wang and B. Yan, J. Comput. Phys., 312 (2016), pp. 157--174] but with two major variations. One is a new splitting of the system that accounts for the anisotropy in the scattering cross section by introducing two extra terms. We then showed, via detailed calculation, that the scheme enjoys a relaxed AP property as opposed to the one step AP for the isotropic scattering. Another contribution is for the degenerate collision frequency case, which brings in additional stiffness. We propose to integrate a “body” term, which appears to be the main component in the diffusion limit. This term is precomputed once with a prescribed accuracy, via a change of variable that alleviates the stiffness. Numerical examples are presented to validate its efficiency in both kinetic and fractional diffusion regimes.

Accounting Meets Economics: Towards an 'Accounting View' of Money

This study lays the foundations of the 'Accounting View' of money. Using international accounting principles, the study argues that state and central bank monies are not debt, and that in fractional reserve regimes only a share of commercial bank money can be regarded as debt. The study determines how the seigniorage associated with the issuance of these monies should be accounted for in the financial statements of the issuing institutions, and examines what this implies for the correct understanding of money. The new view throws light into such issues as the true nature of central bank capital, commercial banks, and digital currencies. Drawing on it, new measurements of seigniorage are derived and applied to the case of the UK (for which recent estimates exist). The results reveal that seigniorage, in particularthat extracted by commercial banks, is a quantitatively relevant phenomenon.

Formation of helical domain walls in the fractional quantum Hall regime as a step toward realization of high-order non-Abelian excitations

We propose an experimentally-feasible system based on spin transitions in the fractional quantum Hall effect regime where parafermions, high-order non-abelian excitations, can be potentially realized. We provide a proof-of-concept experiments showing that in specially designed heterostructures spin transitions at a filling factor 2/3 can be induced electrostatically, allowing local control of polarization and on-demand formation of helical domain walls with fractionalized charge excitations, a pre-requisite ingredient for parafermions formation. We also present exact diagonalization numerical studies of domain walls formed between domains with different spin polarization in the fractional quantum Hall effect regime and show that they indeed possess electronic and magnetic structure needed for parafermion formation when coupled to an s-wave superconductor.

Characterizing Drainage Multiphase Flow in Heterogeneous Sandstones

AbstractIn this work, we analyze the characterization of drainage multiphase flow properties on heterogeneous rock cores using a rich experimental data set and mm‐m scale numerical simulations. Along with routine multiphase flow properties, 3‐D submeter scale capillary pressure heterogeneity is characterized by combining experimental observations and numerical calibration, resulting in a 3‐D numerical model of the rock core. The uniqueness and predictive capability of the numerical models are evaluated by accurately predicting the experimentally measured relative permeability of N2—DI water and CO2—brine systems in two distinct sandstone rock cores across multiple fractional flow regimes and total flow rates. The numerical models are used to derive equivalent relative permeabilities, which are upscaled functions incorporating the effects of submeter scale capillary pressure. The functions are obtained across capillary numbers which span four orders of magnitude, representative of the range of flow regimes that occur in subsurface CO2 injection. Removal of experimental boundary artifacts allows the derivation of equivalent functions which are characteristic of the continuous subsurface. We also demonstrate how heterogeneities can be reorientated and restructured to efficiently estimate flow properties in rock orientations differing from the original core sample. This analysis shows how combined experimental and numerical characterization of rock samples can be used to derive equivalent flow properties from heterogeneous rocks.

Topological model of composite fermions in the cyclotron band generator picture: New insights

Abstract A combinatorial group theory in the braid groups is correlated with the unusual “anyon” statistic of particles in 2D Hall system in the fractional quantum regime well. On this background has been derived cyclotron band generator as a modification and generalization band generator, first established to solve the word and conjugacy problems in the braid group terms. Topological commensurability condition has been embraced by canonical factors – like, based on the concept of parallel descending cycles. Owing to this we can mathematically capture the general hierarchy of correlated states in the lowest Landau level, describing the fractional quantum Hall effect hierarchy, in terms of cyclotron band generators, especially for those being beyond conventional composite fermions model. It has been also shown that cyclotron braid subgroups, developed for interpretation of Laughlin correlations, are a special case of the right-angled Artin groups.

Oscillations and damping in the fractional Maxwell materials

This paper examines the oscillatory behaviour of complex viscoelastic systems with power law-like relaxation behaviour. Specifically, we use the fractional Maxwell model, consisting of a spring and fractional dashpot in series, which produces a power-law creep behaviour and a relaxation law following the Mittag-Leffler function. The fractional dashpot is characterised by a parameter beta, continuously moving from the pure viscous behaviour when beta=1 to the purely elastic response when beta=0. In this work, we study the general response function and focus on the oscillatory behaviour of a fractional Maxwell system in four regimes: stress impulse, strain impulse, step stress, and driven oscillations. The solutions are presented in a format analogous to the classical oscillator, showing how the fractional nature of relaxation changes the long-time equilibrium behaviour and the short-time transient solutions. We specifically test the critical damping conditions in the fractional regime, since these have a particular relevance in biomechanics.

The impact of the liquidity coverage ratio on money creation: A stock-flow based dynamic approach

This paper examines money creation process of the banking system when it is complying with the Liquidity Coverage Ratio (LCR). A stock-flow based dynamic model of credit creation process is developed in which the commercial bank supplies loans to the firm. The change of credit is governed by the bank lending and the repayment of the existing loans, where the equilibrium stock of credit could be attained once the lending is exactly equal to the repayment. However, the supply of bank loans is restricted by both the reserve requirement set by the central bank and the LCR prescribed by the banking authority; and, as a result, money creation must be affected by all these regulations. The bank loan supply under the constraint of the required liquidity buffer might have different prescriptions under different economic scenarios, and would eventually result in an equilibrium monetary stock correspondingly. The final formula of money multiplier is derived respectively as the rational response of the bank to the corresponding regulation. When the reserve requirement is tighter than the LCR, the money multiplier has the same expression of that in the prevailing fractional reserve regime. Yet when the situation departs from this regime, the determinants of the money multiplier are found to be associated with the parameters that characterize the behavior of banks subject to the regulation and of the private sector rather than those monetary structural factors. It is noteworthy that there may be a credit contraction and even a significant reduction in money multiplier when the bank is regulated by the LCR. This novel perspective on credit creation of the banking system also offers us an insightful understanding on the impacts of banking regulations on the stability of the banking system and suggests a new guide tool for designing them.

Method of computation of energies in the fractional quantum Hall effect regime

In a previous work, we reported exact results of energies of the ground state in the fractional quantum Hall effect (FQHE) regime for systems with up to $N_{\text{e}} = 6$ electrons at the filling factor $\nu = 1/3$ by using the method of complex polar coordinates. In this work, we display interesting computational details of the previous calculation and extend the calculation to $N_{\text{e}} = 7$ electrons at $\nu = 1/3$. Moreover, similar exact results are derived at the filling $\nu = 1/5$ for systems with up to $N_{\text{e}} = 6$ electrons. The results that we obtained by analytical calculation are in good agreement with their analogues ones derived by the method of Monte Carlo in a precedent work.

Multiscale functions, scale dynamics, and applications to partial differential equations

Modeling phenomena from experimental data always begins with a choice of hypothesis on the observed dynamics such as determinism, randomness, and differentiability. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following: “With a finite set of data concerning a phenomenon, can we recover its underlying nature? From this problem, we introduce in this paper the definition of multi-scale functions, scale calculus, and scale dynamics based on the time scale calculus [see Bohner, M. and Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (Springer Science & Business Media, 2001)] which is used to introduce the notion of scale equations. These definitions will be illustrated on the multi-scale Okamoto’s functions. Scale equations are analysed using scale regimes and the notion of asymptotic model for a scale equation under a particular scale regime. The introduced formalism explains why a single scale equation can produce distinct continuous models even if the equation is scale invariant. Typical examples of such equations are given by the scale Euler-Lagrange equation. We illustrate our results using the scale Newton’s equation which gives rise to a non-linear diffusion equation or a non-linear Schrödinger equation as asymptotic continuous models depending on the particular fractional scale regime which is considered.

Analytical results for quasiparticle excitations in the Fractional Quantum Hall Effect regime

In this work, quasiparticle energies for systems with N=3,4,5,6 and 7 electrons are calculated analytically in both Laughlin and composite fermions (CF) theories by considering the electron–electron interaction potential. The exact results we have obtained for the first and the second excited states agree with previous numerical results. This study shows that at this level the CF-wave function has lower energy in comparison with Laughlin wave function energy.

Dephasing of an electronic two-path interferometer

This Rapid Communication was motivated by the quest for observing interference of fractionally charged quasiparticles. Here, we study the behavior of an electronic Mach-Zehnder interferometer at the integer quantum Hall effect regime at filling factors greater than 1. Both the visibility and the velocity were measured and found to be highly correlated as a function of the filling factor. As the filling factor approached unity, the visibility quenched, not to recover for filling factors smaller than unity. Alternatively, the velocity saturated around a minimal value at the unity filling factor. We highlight the significant role interactions between the interfering edge and the bulk play as well as that of the defining potential at the edge. Shot-noise measurements suggest that phase averaging (due to phase randomization), rather than single-particle decoherence, is likely to be the cause of the dephasing in the fractional regime.

An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit

We present a new asymptotic-preserving scheme for the linear Boltzmann equation which, under appropriate scaling, leads to a fractional diffusion limit. Our scheme rests on novel micro-macro decomposition to the distribution function, which splits the original kinetic equation following a reshuffled Hilbert expansion. As opposed to classical diffusion limit, a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter. Our idea is, while solving the macro-micro part in a truncated velocity domain (truncation only depends on numerical accuracy), to incorporate an integrated tail over the velocity space that is beyond the truncation, and its major component can be precomputed once with any accuracy. Such an addition is essential to drive the solution to the correct asymptotic limit. Numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes.

Fractional charge and spin states in topological insulator constrictions

We investigate theoretically properties of two-dimensional topological insulator constrictions both in the integer and fractional regimes. In the presence of a perpedicular magnetic field, the constriction functions as a spin filter with near-perfect efficiency and can be switched by electric fields only. Domain walls between different topological phases can be created in the constriction as an interface between tunneling, magnetic fields, charge density wave, or electron-electron interactions dominated regions. These domain walls host non-Abelian bound states with fractional charge and spin and result in degenerate ground states with parafermions. If a proximity gap is induced bound states give rise to an exotic Josephson current with 8$\pi$-peridiodicity.

Metal–insulator transition in the quasi-one-dimensional transport of fractional quantum hall states

We investigate edge state transmission in quantum point contacts (QPCs) in the fractional quantum-Hall regime, finding behavior reminiscent of a metal–insulator transition. The transition is suggested by an unusual behavior of the differential conductance in the fractional-quantum-Hall regime, and by the presence of a fixed point and universal scaling in the temperature dependence of the linear conductance. Noting that the 0.7 feature evolves continuously into a last fractional plateau at high magnetic fields, we suggest that this still unresolved feature may itself be viewed as a manifestation of a local, microscopic, metal–insulator transition.

Integer and fractional quantum anomalous Hall effect in a strip of stripes model

We study the quantum anomalous Hall effect in a strip of stripes model coupled to a magnetic texture with zero total magnetization and in the presence of strong electron-electron interactions. A helical magnetization along the stripes and a spin-selective coupling between the stripes gives rise to a bulk gap and chiral edge modes. Depending on the ratio between the period of the magnetic structure and the Fermi wavelength, the system can exhibit the integer or fractional quantum anomalous Hall effect. In the fractional regime, the quasiparticles have fractional charges and non-trivial Abelian braid statistics.

Surface Acoustic Waves Probe of the Spin Phase Transition at v=2/3 in n-GaAs/AlGaAs structure

High frequency (ac) conductivity in the single quantum well AlGaAs/GaAs/AlGaAs with high mobility was investigated by contactless acoustic methods in the fractional quantum Hall effect regime in perpendicular and tilted magnetic fields. We studied the dependence of ac conductivity σac=σ1-iσ2 on both the temperature and magnetic field tilt angle. Tilting the magnetic field relative to the sample surface enabled us to change the position of the conductivity oscillation minimum at v= 2/3. We measured the temperature dependence of ac conductivity for each tilt angle and for the 2/3 state we calculated the activation energy ΔE which was derived by constructing the Arrhenius plot ln σ1 against 1/T. Analyzing behavior of the activation energy in total magnetic field for the filling factor 2/3 we observed a distinct minimum which can be interpreted as the spin unpolarized-polarized phase transition.

Quantum spin Hall effect in strip of stripes model

We consider the quantum spin Hall effect in an anisotropic strip of stripes and address both integer and fractional filling factors. The first model is based on a gradient of spin-orbit interaction in the direction perpendicular to the stripes. The second model is based on two weakly coupled strips with reversed dispersion relations. We demonstrate that these systems host helical modes, i.e., modes in which opposite spins propagate in opposite directions. In the integer regime, the modes carry an elementary electron charge whereas in the fractional regime they carry fractional charges, and their excitations possess anyonic braiding statistics. These simple quasi-one-dimensional models can serve as a platform for understanding effects arising due to electron-electron correlations in topological insulators.

Proliferation of neutral modes in fractional quantum Hall states.

The fractional quantum Hall effect is a canonical example of topological phases. While electric currents flow downstream in edge modes, neutral edge modes, observed only in hole-conjugate states and in ν=5/2, flow upstream. It is believed that the latter transport results from multiple counter-propagating channels-mixed by disorder that is accompanied by Coulomb interaction. Here we report on sensitive shot noise measurements that reveal unexpected presence of neutral modes in non-hole-conjugate fractional states; however, not in the integer states. Furthermore, the incompressible bulk is also found to allow energy transport. While density reconstructions along the edge may account for the energy carrying edge modes, the origin of the bulk energy modes is unidentified. The proliferation of neutral modes changes drastically the accepted transport picture of the fractional quantum Hall effects. Their apparent ubiquitous presence may explain the lack of interference of fractional quasiparticles-preventing observation of fractional statistics.

Front propagation in reaction-diffusion systems with anomalous diffusion

A numerical study of the role of anomalous diffusion in front propagation in reaction-diffusion systems is presented. Three models of anomalous diffusion are considered: fractional diffusion, tempered fractional diffusion, and a model that combines fractional diffusion and regular diffusion. The reaction kinetics corresponds to a Fisher-Kolmogorov nonlinearity. The numerical method is based on a finite-difference operator splitting algorithm with an explicit Euler step for the time advance of the reaction kinetics, and a Crank-Nicholson semi-implicit time step for the transport operator. The anomalous diffusion operators are discretized using an upwind, flux-conserving, Grunwald-Letnikov finite-difference scheme applied to the regularized fractional derivatives. With fractional diffusion of order $\alpha$, fronts exhibit exponential acceleration, $a_L(t) \sim e^{\gamma t/\alpha}$, and develop algebraic decaying tails, $\phi \sim 1/x^{\alpha}$. In the case of tempered fractional diffusion, this phenomenology prevails in the intermediate asymptotic regime $\left(\chi t \right)^{1/\alpha} \ll x \ll 1/\lambda$, where $1/\lambda$ is the scale of the tempering. Outside this regime, i.e. for $x > 1/\lambda$, the tail exhibits the tempered decay $\phi \sim e^{-\lambda x}/x^{\alpha+1}$, and the front velocity approaches the terminal speed $v_*= \left(\gamma-\lambda^\alpha \chi\right)/ \lambda$. Of particular interest is the study of the interplay of regular and fractional diffusion. It is shown that the main role of regular diffusion is to delay the onset of front acceleration. In particular, the crossover time, $t_c$, to transition to the accelerated fractional regime exhibits a logarithmic scaling of the form $t_c \sim \log \left(\chi_d/\chi_f\right)$ where $\chi_d$ and $\chi_f$ are the regular and fractional diffusivities.

Finite clusters of fast-rotating spinless bosons in a harmonic trap

Rapidly rotating two-dimensional ultracold Bose–Einstein condensates of spinless bosons in a harmonic trap have attracted considerable interest during the recent years. It is expected that, in the fast-rotation limit, the system of bosons will exhibit collective behavior similar to that of two-dimensional electrons in the fractional quantum Hall effect regime. It is predicted that the most robust correlated bosonic state in this regime will be the Bose Laughlin state at a half filling factor. An exact treatment of such a state is generally a formidable task due to the inherent many-particle nature of the wave function. We report in this work that a transformation to Jacobi coordinates allows one to obtain much desirable exact analytic closed-form expressions for various quantities of interest corresponding to a Bose Laughlin wave function for various finite systems of particles.