Power-law Singularity Research Articles

Magnesium Concentration Dependence of Room-Temperature Absorption-Edge Singularity in Alloyed MgZnO Epilayers

Recently, the absorption-edge singularity model accounting for the interaction between excess electrons and photocreated holes has been applied to room-temperature optical spectra in n -type ZnO epilayers [T. Makino et al. : Phys. Rev. B 65 (2002) 121201(R)]. This model predicts that the onset of the spectrum should exhibit a power-law singularity of the form [( E - E 1 ) -α ] with the threshold energy of E 1 located near the fundamental gap. The results of our fitting using this model are in fairly good agreement with available experimental data obtained for MgZnO thin films, which were reported by Kang et al. [Solid State Commun. 115 (2000) 127]. The power exponent α, which is related to the magnitude of screening between photocreated carriers relative to that provided by excess electrons, was enhanced with the incorporation of Mg and had a maximum at the Mg concentration of 8%. The possible reasons for this enhancement are discussed.

Scale-dependent rigidity of polymer-ornamented membranes

We study the fluctuation spectrum of fluid membranes carrying grafted polymers. Contrary to usual descriptions, we find that the modifications induced by the polymers cannot be reduced to the renormalization of the membrane bending rigidity. Instead we show that the ornamented membrane exhibits a scale-dependent elastic modulus that we evaluate. In ornamented lamellar stacks, we further compute the polymer contribution to the Caillé parameter characterizing the power law singularities of the Bragg peaks.

Coulomb singularity effects in the tunneling spectroscopy of individual impurities

Non-equilibrium Coulomb effects in resonant tunnelling processes through deep impurity states are analyzed. It is shown that Coulomb vertex corrections to the tunnelling transfer amplitude lead to a power-law singularity in current- voltage characteristics

Anomalous diffusion in the nonasymptotic regime.

We analyze some properties of the one-dimensional Lévy flights, assuming that the one-step transition rates depend on the flight length x as p(alpha)(x) equivalent to x(-(alpha+2)). For flights on a finite, (2M+1)-site lattice, we can define an effective, size-dependent, diffusion coefficient D(alpha)(M) equivalent to [M(1-alpha) - 1]/(1 - alpha) if alpha < 1, with D1(M) equivalent to ln(M). Using the generalization of statistical mechanics given by Tsallis, we show that for flights on infinite systems, the generalized displacement moments <x(R)> are well defined provided that alpha > R - 3. These moments exhibit a power-law singularity if alpha --> 1(-) and R>2/3. The short- and intermediate-time properties of the generalized mean-square displacement are then studied numerically. This work suggests the conditions under which the asymptotic analytical formulas (obtained in the literature by the use of the generalized central limit theorem) could be applied to finite-time experiments. These formulas should work much better if alpha is close to zero than in the alpha -->1(-) neighborhood.

Oscillatory finite-time singularities in finance, population and rupture

We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between nonlinear trend-followers and nonlinear value investors; (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity; (3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing.

OPTICAL SPECTRA OF SPIN-1/2 FERMI-DEGENERATE SYSTEMS WITH A BOUND STATE: CROSSOVER BETWEEN EXCITONIC PEAKS AND THE POWER-LAW SINGULARITIES

Interband optical absorption spectra of a two-dimensional spin-1/2 Fermi-degenerate system with a bound state are investigated theoretically. We examine how the optical spectra depend on the carrier density to understand the crossover from a normal exciton-peak structure in the low-density (insulator) limit to the power-law singularity in the high-density case. An interplay between the doubly-degenerate bound state and the electronic spin degrees of freedom is of essential importance in determining the threshold behavior of the spectra. Whole spectral shape is calculated with the time-dependent coupled-cluster expansion method, and an exact expression of the critical exponents at these four thresholds is analytically obtained.

Criticality of natural absorbing states.

We study a recently introduced ladder model that undergoes a transition between an active and an infinitely degenerate absorbing phase. In some cases the critical behavior of the model is the same as that of the branching-annihilating random walk with N>/=2 species both with and without hard-core interaction. We show that certain static characteristics of the so-called natural absorbing states develop power-law singularities that signal the approach of the critical point. These results are also explained using random-walk arguments. In addition to that we show that when dynamics of our model is considered as a minimum-finding procedure, it has the best efficiency very close to the critical point.

Single-hole spectral function and spin-charge separation in thet−Jmodel

Worm algorithm quantum Monte Carlo simulations of the hole Green function with subsequent spectral analysis were performed for J/t 0.1, 0.2, 0.4 on lattices with up to LxL=32x32 sites at temperatures as low as T=J/40, and present, apparently, the hole spectral function in the thermodynamic limit. Spectral analysis reveals a delta-function-sharp quasiparticle peak at the lower edge of the spectrum which is incompatible with the power-law singularity and thus rules out the possibility of spin-charge separation in this parameter range. Spectral continuum features two peaks separated by a gap ~4t.

Dynamically induced multichannel Kondo effect

We study how the multichannel Kondo effect is dynamically induced to affect the photoemission and the inverse photoemission spectrum when an electron is emitted from (or added to) the completely screened Kondo impurity with spin $S&gt;1/2.$ The spectrum thereby shows a power-law edge singularity characteristic of the multichannel Kondo model. We discuss this anomalous behavior by using the exact solution of the multichannel Kondo model and boundary conformal field theory. The idea is further applied to the photoemission for quantum spin systems, in which the edge singularity is controlled by the dynamically induced overscreening effect with a mobile Kondo impurity.

First-order phase transition with a logarithmic singularity in a model with absorbing states.

Recently, Lipowski [Phys. Rev. E 62, 4401 (2000)] investigated a stochastic lattice model which exhibits a discontinuous transition from an active phase into infinitely many absorbing states. Since the transition is accompanied by an apparent power-law singularity, it was conjectured that the model may combine features of first- and second-order phase transitions. In the present work it is shown that this singularity emerges as an artifact of the definition of the model in terms of products. Instead of a power law, we find a logarithmic singularity at the transition. Moreover, we generalize the model in such a way that the second-order phase transition becomes accessible. As expected, this transition belongs to the universality class of directed percolation.

First-order transition with power-law singularity in models with absorbing states

We study one- and two-dimensional models which undergo a transition between active and absorbing phases. The transition point in these models is of a novel type: jump of the order parameter coincides with its power-law singularity. Some arguments supported by Monte Carlo simulations prompted us to predict the exact location of the transition point. Both models possess gaugelike symmetry.

Overscreening Kondo effect induced by photoemission

We study how the overscreening Kondo effect is dynamically induced to affect the photoemission spectrum when a core electron in the Kondo impurity with higher spin is emitted by the photoemission. The photoemission spectrum thereby shows an anomalous power-law edge singularity characteristic of the overscreening model. We discuss this anomalous behavior by using the exact solution and boundary conformal field theory. We also apply the idea to a quantum spin system, for which the edge singularity is controlled by the dynamically induced overscreening effect with a mobile Kondo impurity.

A Model with a Singular Electronic Density of States and Some Properties of High-Temperature Superconductors

The influence of singularities in the electronic density of states (DOS) on high-temperature superconductors is studied in the framework of the conventional BCS theory. By using a DOS with singularities at two different points we are able to explain the asymmetry in the isotope exponent α with respect to the point where Tc is maximum. A DOS with a power law singularity reproduces the vanishingly small values of α and large values of R2 = ΔC(Tc)/Cn(Tc), the ratio of the jump in the electronic specific heat to the normal specific heat at T = Tc, observed in certain high-Tc materials. However, the value of the ratio R1 = 2Δ0/kBTc does not deviate much from its canonical BCS value of 3.53, in agreement with earlier results.

Exact Dynamics of theSU(K)Haldane-Shastry Model

The dynamical structure factor $S(q,\omega)$ of the SU(K) (K=2,3,4) Haldane-Shastry model is derived exactly at zero temperature for arbitrary size of the system. The result is interpreted in terms of free quasi-particles which are generalization of spinons in the SU(2) case; the excited states relevant to $S(q,\omega)$ consist of K quasi-particles each of which is characterized by a set of K-1 quantum numbers. Near the boundaries of the region where $S(q,\omega)$ is nonzero, $S(q,\omega)$ shows the power-law singularity. It is found that the divergent singularity occurs only in the lowest edges starting from $(q,\omega) = (0,0)$ toward positive and negative q. The analytic result is checked numerically for finite systems via exact diagonalization and recursion methods.

Dynamical spin correlations in the Heisenberg ladder under a magnetic field and correlation functions in the SO(5) ladder

The zero-temperature dynamical spin-spin correlation functions are calculated for the spin-$\frac{1}{2}$ two-leg antiferromagnetic Heisenberg ladder in a magnetic field above the lower critical field ${H}_{c1}$. The dynamical structure factors are calculated, which exhibit both massless modes and massive excitations. These modes appear in different sectors characterized by the parity in the rung direction and by the momentum in the direction of the chains. The structure factors have power-law singularities at the lower edges of their support. The results are also applicable to the spin-1 Heisenberg chain. The implications are briefly discussed for the various correlation functions and the $\ensuremath{\pi}$ resonance in the SO(5) symmetric ladder model.

Solution of Hilbert’s problem for an annulus in a specific case and its application to an explosion problem

A method for solving Hilbert's boundary value problem with discontinuous coefficients is studied for a function single-valued and analytic in an annulus in the case in which the solution may have power-law singularities at finitely many points on the boundary of the annulus. To illustrate the results obtained, we consider an explosion problem for two pinching charges in a homogenous medium with a circular cylinder lying in the flow caused by the explosion.

Universal Finite-Size Scaling Functions in the 3D Ising Spin Glass

We study the three-dimensional Edwards-Anderson model with binary interactions by Monte Carlo simulations. Direct evidence of finite-size scaling is provided, and the universal finite-size scaling functions are determined. Monte Carlo data are extrapolated to infinite volume with an iterative procedure up to correlation lengths xi \approx 140. The infinite volume data are consistent with a conventional power law singularity at finite temperature Tc. Taking into account corrections to scaling, we find Tc = 1.156 +/- 0.015, nu = 1.8 +/- 0.2 and eta = -0.26 +/- 0.04. The data are also consistent with an exponential singularity at finite Tc, but not with an exponential singularity at zero temperature.

Griffiths-McCoy singularities in the random transverse-field Ising spin chain

We consider the paramagnetic phase of the random transverse-field Ising spin chain and study the dynamical properties by numerical methods and scaling considerations. We extend our previous work [Phys. Rev. B 57, 11404 (1998)] to new quantities, such as the non-linear susceptibility, higher excitations and the energy-density autocorrelation function. We show that in the Griffiths phase all the above quantities exhibit power-law singularities and the corresponding critical exponents, which vary with the distance from the critical point, can be related to the dynamical exponent z, the latter being the positive root of [(J/h)^{1/z}]_av=1. Particularly, whereas the average spin autocorrelation function in imaginary time decays as [G]_av(t)~t^{-1/z}, the average energy-density autocorrelations decay with another exponent as [G^e]_av(t)~t^{-2-1/z}.

Spectral Functions of One-Dimensional Electron Systems with Ising-Like Spin Gap

The effect of a spin gap on the photoemission spectral functions is studied in the one-dimensional t - J X X Z model. We find that spinon bands show up in the spectral functions but holon bands are obscured in the spin-gap cases. The reason for this is studied using a simple interpretation of the spectral functions, which is a convolution between the spectral function of spinless fermions and a spin excitation weight Z ( Q ). The spin gap suppresses the singularity of Z ( Q ), and as a result, only the holon bands are obscured. It is also shown that the spinon peaks lie along a cosine band and extend over the region | k |> k F . Finally, the momentum-dependent critical exponents of the power-law singularity at the spinon bands are determined.

Stress of an anisotropic plate with angular inclusions

The plane problem of anisotropic elastic bodies with angular inclusions has been reduced to a Fredholm system of integral equations. The principal terms in the potentials have been determined in explicit form, taking account of power-law singularities at the corner points. Applications of the results have been considered.