Power-law Singularity Research Articles

Probing Anomalous Relaxation by Coherent Multidimensional Optical Spectroscopy

We propose to study the origin of algebraic decay of two-point correlation functions observed in glasses, proteins, and quantum dots by their nonlinear response to sequences of ultrafast laser pulses. Power-law spectral singularities and temporal relaxation in two-dimensional correlation spectroscopy signals are predicted for a continuous time random walk model of stochastic spectral jumps in a two-level system with a power-law distribution of waiting times psi(t) approximately t;{-alpha-1}. Spectroscopic signatures of stationary ensembles for 1<alpha<2 and aging effects in nonstationary ensembles with 0<alpha<1 are identified.

Scalar field probes of power-law space-time singularities

We analyse the effective potential of the scalar wave equation near generic space-time singularities of power-law type (Szekeres-Iyer metrics) and show that the effective potential exhibits a universal and scale invariant leading x^{-2} inverse square behaviour in the ``tortoise coordinate'' x provided that the metrics satisfy the strict Dominant Energy Condition (DEC). This result parallels that obtained in hep-th/0403252 for probes consisting of families of massless particles (null geodesic deviation, a.k.a. the Penrose Limit). The detailed properties of the scalar wave operator depend sensitively on the numerical coefficient of the x^{-2}-term, and as one application we show that timelike singularities satisfying the DEC are quantum mechanically singular in the sense of the Horowitz-Marolf (essential self-adjointness) criterion. We also comment on some related issues like the near-singularity behaviour of the scalar fields permitted by the Friedrichs extension.

Dynamic Structure Factor of the Calogero-Sutherland Model

We evaluate the dynamic structure factor S(q, omega) of a one-dimensional quantum Hamiltonian with the inverse-square interaction (Calogero-Sutherland model). For a fixed small q, the structure factor differs from zero in a finite interval of frequencies of the width deltaomega proportional to q2/m. At the borders of this interval S(q, omega) exhibits power-law singularities with exponents depending on the interaction strength. The singularities are similar in origin to the well-known Fermi-edge singularity in the x-ray absorption spectra of metals.

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

We have obtained exact results for the Ising model on a hierarchical lattice incorporating three key features characterizing many real-world networks--a scale-free degree distribution, a high clustering coefficient, and the small-world effect. By varying the probability p of long-range bonds, the entire spectrum from an unclustered, non-small-world network to a highly clustered, small-world system is studied. Using the self-similar structure of the network, we obtain analytic expressions for the degree distribution P(k) and clustering coefficient C for all p, as well as the average path length l for p = 0 and 1. The ferromagnetic Ising model on this network is studied through an exact renormalization-group transformation of the quenched bond probability distribution, using up to 562,500 renormalized probability bins to represent the distribution. For p < 0.494, we find power-law critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with p, and exponential decay of correlations away from Tc. For p > or = 0.494, in fact where the network exhibits small-world character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between a low-temperature phase with nonzero magnetization and finite correlation length and a high-temperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching Tc from below, the magnetization and the susceptibility, respectively, exhibit the singularities of exp(-C/square root of Tc - T) and exp(D/square root of Tc - T), with C and D positive constants. With long-range bond strengths decaying with distance, we see a phase transition with power-law critical singularities for all p, and evaluate an unusually narrow critical region and important corrections to power-law behavior that depend on the exponent characterizing the decay of long-range interactions.

Dynamic Response of One-Dimensional Interacting Fermions

We evaluate the dynamic structure factor S(q, omega) of interacting one-dimensional spinless fermions with a nonlinear dispersion relation. The combined effect of the nonlinear dispersion and of the interactions leads to new universal features of S(q, omega). The sharp peak S(q, omega) approximately q(delta(omega -uq), characteristic for the Tomonaga-Luttinger model, broadens up; for a fixed becomes finite at arbitrarily large . The main spectral weight, however, is confined to a narrow frequency interval of the width deltaomega approximately q(2)/m. At the boundaries of this interval the structure factor exhibits power-law singularities with exponents depending on the interaction strength and on the wave number q.

Viscoplastic solution to field at steadily propagating crack tip in linear-hardening materials

An elastic-viscoplastic constitutive model was adopted to analyze asymptotically the tip-field of moving crack in linear-hardening materials under plane strain condition. Under the assumption that the artificial viscosity coefficient was in inverse proportion to power law of the rate of effective plastic strain, it is obtained that stress and strain both possess power law singularity and the singularity exponent is uniquely determined by the power law exponent of the rate of effective plastic strain. Variations of zoning structure according to each material parameter were discussed by means of numerical computation for the tip-field of mode II dynamic propagating crack, which show that the structure of crack tip field is dominated by hardening coefficient rather than viscosity coefficient. The secondary plastic zone can be ignored for weak hardening materials while the secondary plastic zone and the secondary elastic zone both have important influence on crack tip field for strong hardening materials. The dynamic solution approaches to the corresponding quasi-static solution when the crack moving speed goes to zero, and further approaches to the HR (Hui-Riedel) solution when the hardening coefficient is equal to zero.

High frequency quasi-normal modes for black holes with generic singularities: II. Asymptotically non-flat spacetimes

The possibility that the asymptotic quasi-normal mode (QNM) frequencies can be used to obtain the Bekenstein–Hawking entropy for the Schwarzschild black hole—commonly referred to as Hod's conjecture—has received considerable attention. To test this conjecture, using the monodromy technique, attempts have been made to analytically compute the asymptotic frequencies for a large class of black hole spacetimes. In an earlier work, two of the current authors computed the high frequency QNMs for scalar perturbations of (D + 2)-dimensional spherically symmetric, asymptotically flat, single horizon spacetimes with generic power-law singularities. In this work, we extend these results to asymptotically non-flat spacetimes. Unlike in the earlier analyses, we treat asymptotically flat and de Sitter spacetimes in a unified manner, while the asymptotic anti-de Sitter spacetimes are considered separately. We obtain master equations for the asymptotic QNM frequency for all three cases. We show that for all three cases, the real part of the asymptotic QNM frequency—in general—is not proportional to ln(3), thus indicating that Hod's conjecture may be restrictive.

On the Hagedorn behaviour of singular scale-invariant plane waves

As a step towards understanding the properties of string theory in time-dependent and singular spacetimes, we study the divergence of density operators for string ensembles in singular scale-invariant plane waves, i.e. those plane waves that arise as the Penrose limits of generic power-law spacetime singularities. We show that the scale invariance implies that the Hagedorn behaviour of bosonic and supersymmetric strings in these backgrounds, even with the inclusion of RR or NS fields, is the same as that of strings in flat space. This is in marked contrast to the behaviour of strings in the BFHP plane wave which exhibit quantitatively and qualitatively different thermodynamic properties.

Quantum Griffiths effects in itinerant Heisenberg magnets

We study the influence of quenched disorder on quantum phase transitions in itinerant magnets with Heisenberg spin symmetry, paying particular attention to rare disorder fluctuations. In contrast to the Ising case where the overdamping suppresses the tunneling of the rare regions, the Heisenberg system displays strong power-law quantum Griffiths singularities in the vicinity of the quantum critical point. We discuss these phenomena based on general scaling arguments, and we illustrate them by an explicit calculation for O(N) spin symmetry in the large-N limit. We also discuss broad implications for the classification of quantum phase transitions in the presence of quenched disorder.

Dielectric critical behaviour at a metal–insulator transition under lattice compression

We study dielectric critical behaviour around a continuous metal–insulator transition in crystalline Cesium Iodide induced by changing the lattice parameter. The ab initio calculations of band structure and various quantities related to the dielectric response are performed in the transition region, within the local density approximation of the density functional theory. These calculations allow us to establish the power-law singularities of various quantities on two sides of the transition. The exponents obtained here are mean-field like due to the approximation in which interactions and disorder are treated. The critical behaviour is discussed by applying the scaling principle to the wavevector and frequency dependent dielectric function. We further investigate the effect of dielectric anomalies on optical properties by calculating the reflectance around transition region taking the ionic contribution to the dielectric function also into account. We find that the reflectance as a function of frequency shows very different kind of behaviour on both sides of the metal–insulator transition.

Fermi-Edge Resonance and Tunneling in Nonequilibrium Electron Gas

Fermi-edge singularity changes in a nonequilibrium system, acquiring features that reflect the structure of energy distribution. In particular, it splits into several components if the energy distribution exhibits multiple steps. While conventional approaches, such as bosonization, fail to describe the nonequilibrium problem, an exact solution for a generic energy distribution can be obtained with the help of the method of functional determinants. In the case of a split Fermi distribution, the "open loop" part of the Greens function possesses power law singularities. At the same time, the resulting tunneling density of states exhibits broadened peaks centered at Fermi sublevels.

The excited random walk in one dimension

We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q=1-p. If the walk hops onto an empty site, there is no bias. For the 1-excited walk on the half-line (one cookie initially at each site), the probability of first returning to the starting point at time t scales as t^{-(2-p)}. Although the average return time to the origin is infinite for all p, the walk eats, on average, only a finite number of cookies until this first return when p<1/2. For the infinite line, the probability distribution for the 1-excited walk has an unusual anomaly at the origin. The positions of the leftmost and rightmost uneaten cookies can be accurately estimated by probabilistic arguments and their corresponding distributions have power-law singularities near the origin. The 2-excited walk on the infinite line exhibits peculiar features in the regime p>3/4, where the walk is transient, including a mean displacement that grows as t^{nu}, with nu>1/2 dependent on p, and a breakdown of scaling for the probability distribution of the walk.

Response of a catalytic reaction to periodic variation of the CO pressure: IncreasedCO2production and dynamic phase transition

We present a kinetic Monte Carlo study of the dynamical response of a Ziff-Gulari-Barshad model for CO oxidation with CO desorption to periodic variation of the CO pressure. We use a square-wave periodic pressure variation with parameters that can be tuned to enhance the catalytic activity. We produce evidence that, below a critical value of the desorption rate, the driven system undergoes a dynamic phase transition between a CO2 productive phase and a nonproductive one at a critical value of the period and waveform of the pressure oscillation. At the dynamic phase transition the period-averaged CO2 production rate is significantly increased and can be used as a dynamic order parameter. We perform a finite-size scaling analysis that indicates the existence of power-law singularities for the order parameter and its fluctuations, yielding estimated critical exponent ratios beta/nu approximately 0.12 and gamma/nu approximately 1.77. These exponent ratios, together with theoretical symmetry arguments and numerical data for the fourth-order cumulant associated with the transition, give reasonable support for the hypothesis that the observed nonequilibrium dynamic phase transition is in the same universality class as the two-dimensional equilibrium Ising model.

Analysis of Stress Singularity Field in Three-Dimensional Joints Using Three-Dimensional Boundary Element Method

Stress singularity at a vertex in three-dimensional joints is closely related to the strength of joints. It is very important to perform an analysis for evaluating the reliability of a real three-dimensional joint. However, the stress singularity fields near the vertex and along stress singularity lines are still unclear. In the present study, we investigated the characteristics of stress singularity field at a vertex and a point located on the stress singularity line in three-dimensional joints using a three-dimensional BEM and an eigen analysis based on FEM. It was shown that the order of stress singularity at the vertex is larger than that at the point on the stress singularity line, and the order of logarithmic singularity along the singularity line is larger than that at vertex. In the present analysis, a triple root and a quintple root of eigen value, p=1, occur at the vertex and a point on the singularity line, respectively. The order of stress singularity determined from the eigen analysis was agreed with that from BEM. The value of coefficient in a power-law singularity increased as approaching to the vertex along the singularity line. The stress field along the singularity line is more influenced on the logarithmic singularity than that at the vertex. In particular, coefficients of terms in logarithmic singularity vary in the same way with the angle from an interface.

Power-law singularities in string theory and M-theory

We extend the definition of the Szekeres–Iyer power-law singularities to supergravity, string and M-theory backgrounds, and find that are characterized by Kasner-type exponents. The near singularity geometries of brane and some intersecting brane backgrounds are investigated and the exponents are computed. The Penrose limits of some of these power-law singularities have profiles A ∼ u−γ for γ ⩾ 2. We find the range of the exponents for which γ = 2 and the frequency squares are bounded by 1/4. We propose some qualitative tests for deciding whether a null or timelike spacetime singularity can be resolved within string theory and M-theory based on the near singularity geometry and its Penrose limits.

Dynamical properties of the Anderson impurity model within a diagrammatic pseudoparticle approach

The Anderson model of a twofold spin degenerate impurity level in the limit of infinite Coulomb repulsion, $U\ensuremath{\rightarrow}\ensuremath{\infty}$, coupled to one and two degenerate conduction bands or channels, is considered in pseudo-particle representation. We extend the conserving T-matrix approximation (CTMA), a general diagrammatic approximation scheme based on a fully renormalized computation of two-particle vertex functions in the spin and in the charge channel, to the calculation of thermodynamic and spectral properties. In the single-channel case, the CTMA yields in the Kondo regime a temperature independent Pauli spin susceptibility for temperatures below the Kondo temperature ${T}_{K}$ and down to the lowest temperatures considered, reproducing the exact spin screening in the Fermi liquid state. The impurity spectral density appears to remain non-singular down to the lowest temperatures, in agreement with Fermi liquid behavior. However, the unitarity sum rule, which is crucial for an impurity solver like the CTMA to be applicable within dynamical mean field theories for strongly correlated lattice models, is overestimated at the lowest temperatures. We argue that this shortcoming may be due to numerical imprecision and discuss an appropriate scheme for its correction. In the two-channel case, the spectral density calculated within CTMA exhibits qualitatively the correct non-Fermi liquid behavior at low temperatures, i.e., a power law singularity.

Elastic-viscoplastic fields near the tip of a propagating crack under anti-plane shear

The elastic-viscoplastic model proposed by Bingham was used to analyse the stress and strain surrounding the tip of a propagating crack under antiplane shear. The proper displacement pattern was given; the asymptotic equations were derived and solved numerically. The analysis and calculation show that for smaller viscosity the crack-tip possesses logarthmic singularity, and for larger viscosity it possesses power-law singularity. In critical case, the two kinds of singularity are consistent with each other. The result revealed the important role of viscosity for crack-tip field.

The Universality of Penrose Limits near Space-Time Singularities

We prove that Penrose limits of metrics with arbitrary singularities of power-law type show a universal leading u^{-2}-behaviour near the singularity provided that the dominant energy condition is satisfied and not saturated. For generic power-law singularities of this type the oscillator frequencies of the resulting homogeneous singular plane wave turn out to lie in a range which is known to allow for an analytic extension of string modes through the singularity. The discussion is phrased in terms of the recently obtained covariant characterisation of the Penrose limit; the relation with null geodesic deviation is explained in detail.

Evidence for a percolation-driven transition to coherent surface superconductivity.

Above the upper critical field we have investigated the field dependences of the surface conductance, G'-iG" and the critical current J(c) of an electropolished pure niobium cylinder. The low frequency limits of G', G", and J(c) display power-law singularities, defining a transition to coherent surface superconductivity at H(c)(c3). The critical exponents as well as the dynamical scaling of G'-iG" are consistent with predictions for a two-dimensional percolation transition. Relating H(c)(c3) to the conventional onset field, we find H(c)(c3)/H(c3)=0.81, and, surprisingly, this ratio turns out to be independent of significant variations of H(c3) due to differently structured surfaces.

Finite-time singularity signature of hyperinflation

We present a novel analysis extending the recent work of Mizuno et al. (Physica A 308 (2002) 411) on the hyperinflations of Germany (1920/1/1–1923/11/1), Hungary (1945/4/30–1946/7/15), Brazil (1969–1994), Israel (1969–1985), Nicaragua (1969–1991), Peru (1969–1990) and Bolivia (1969–1985). On the basis of a generalization of Cagan's model of inflation based on the mechanism of “inflationary expectation” of positive feedbacks between realized growth rate and people's expected growth rate, we find that hyperinflations can be characterized by a power law singularity culminating at a critical time t c . Mizuno et al.'s double-exponential function can be seen as a discrete time-step approximation of our more general non-linear ODE formulation of the price dynamics which exhibits a finite-time singular behavior. This extension of Cagan's model, which makes natural the appearance of a critical time t c , has the advantage of providing a well-defined end of the clearly unsustainable hyperinflation regime. We find an excellent and reliable agreement between theory and data for Germany, Hungary, Peru and Bolivia. For Brazil, Israel and Nicaragua, the super-exponential growth seems to be already contaminated significantly by the existence of a cross-over to a stationary regime.