Power-law Singularity Research Articles

NUMERICAL EVIDENCE OF A FIRST-ORDER TRANSITION IN A BRANCHED POLYMER: FINITE SIZE EFFECT AT THE TRANSITION

Within the ε-expansion scheme, we argue that the upper critical dimension for the continuous transition (if it exists) in the dilute limit of branched polymers should be du=4. However, Monte-Carlo simulations in d=2, 3 and 4 for a single branched polymer on a finite lattice show evidence of a first-order transition at K=K0 and not a continuous transition which we observe for a linear polymer. Here, K represents bond activity. Our extensive simulation in d=2 is compared with previously known results. The existence of a first-order transition at K0 is not inconsistent with power-law singularities as [Formula: see text], which have been observed by various workers. However, for K<K0, the self-similarity which is a prerequisite of a fractal (i.e., a critical) object is only approximate. At K=K0 the branched polymer is a regular, compact object and not a fractal. In constrast, a linear polymer is a fractal at the transition, as is expected and as our simulation also suggests. Thus, our work shows that the order of the thermodynamic limit N→∞ and K→K0 is important, provided the transition remains first-order in the thermodynamic limit. In previous works, one considers [Formula: see text] after the limit N→∞ has already been taken, whereas we are interested in K→K0, followed by N→∞.

Momentum distribution near k=3kF in the Tomonaga-Luttinger model.

In contrast to normal Fermi liquids, where the momentum distribution n(k) has a discontinuous jump at k=${\mathit{k}}_{\mathit{F}}$, in Luttinger liquids a power-law singularity appears in n(k). It was argued that this singularity shows up not only at ${\mathit{k}}_{\mathit{F}}$ but at 3${\mathit{k}}_{\mathit{F}}$,5${\mathit{k}}_{\mathit{F}}$, . . . as well, and the anomaly exponent has been detemined for the one-dimensional Hubbard model using numerical methods or assuming conformal invariance. We present an exact calculation of the anomaly exponent for the Tomonaga-Luttinger model and compare it with the results for the Hubbard model.

Magnetohydrodynamic equilibria in the vicinity of an X-type neutral line specified by footpoint shear

Results pertaining to two-dimensional (∂/∂z=0) magnetohydrodynamic equilibria in the presence of an X-type neutral line are presented. Naive analyses indicate that there may be tangential discontinuities in B, specifically discontinuities in Bz across the separatrix connected to the X line. However, such analyses indicate an infinite z component of footpoint displacement (or safety factor q in the toroidal case) at the separatrix. The solutions presented here allow the specification of footpoint displacement (or safety factor q) that is finite as the separatrix is approached. These solutions are scale-invariant, or similarity, solutions. They are appropriate near the X line on length scales intermediate between the boundary layer width because of resistivity (or other nonideal effects) and the macroscopic length scale. Force balance across the separatrix implies identical radial dependence in all four quadrants and continuity of B2z across the separatrix. The latter shows that there are two classes of solutions: those with Bz continuous across the separatrix and those with ‖Bz‖ continuous but with a sign change in Bz. The former class has fractional power-law singularities at the separatrix. The latter class has, in addition, a sheet current along the separatrix in the x-y plane associated with the jump in Bz. Detailed properties of these solutions are explored. In particular, sheetlike one-dimensional solutions are found to be limiting cases of the general solutions. Except for one special case, these sheet solutions cannot have finite footpoint displacement if they are force free, but can in the presence of pressure gradient. The general solutions can have arbitrary separatrix angle and ratio of flux between the quadrants. Discussion is presented regarding the matching of these solutions to the physical conditions on boundaries where the field lines leave the system, e.g., the photospheric surface in the solar coronal context.

Cancellation of infrared and collinear singularities in relativistic thermal field theories

Self-energy insertions lead to severe infrared divergences in relativistic thermal field theories. These singularities are studied at two-loop order in a renormalizable scalar field theory: gϕ 3 in dimension D = 6. Using a dimensional regularization D = 6 + 2 ϵ of infrared and collinear singularities, we show that the leading (power law) singularities cancel out in the calculation of the rate for a physical process. We show that the surviving singularities are of the form ϵ − n , n ⩽ 3, and we are able to prove the cancellation of ϵ −3 and ϵ −2 singularities. This result makes plausible the validity of the Kinoshita-Lee-Nauenberg theorem at finite temperature.

THE HUBBARD MODEL: FROM SMALL TO LARGE U

The Hubbard model is investigated starting from both the small and large U limits. This allows one to derive an interpolation formula for the double occupancy at half-filling for dimensionalities d = 1, 2, 3. It shows a smooth behavior as a function of U and tends to zero only for U → ∞. A quantity that probes more sensitively the nature of the ground state is the momentum distribution function n(k). At half filling n(k) is smooth at k F both for d = 1 and d = 2, at least for not too small values of U. In one dimension for all other band fillings the slope of n(k) has a power-law singularity at k F with an exponent α increasing steadily from zero at U = 0 to 1/8 for U → ∞; the system is a "marginal Fermi liquid". A similar behavior may occur close to half-filling for d = 2, but for small densities one expects the usual step function of a normal Fermi liquid.

Dielectric constant and the electric conductivity near the consolute point of the critical binary liquid mixture nitroethane-3-methylpentane.

Experimental data are presented for the dielectric constant \ensuremath{\epsilon} and the electric conductivity \ensuremath{\sigma} near the consolute point of a binary liquid mixture nitroethane\char21{}3-methylpentane. Fitting the data for \ensuremath{\epsilon} and \ensuremath{\sigma} as a function of temperature, we find for both a 1-\ensuremath{\alpha} power-law singularity with an \ensuremath{\alpha} value consistent with the Ising value \ensuremath{\alpha}=0.11. The temperature dependence of the amplitude of the observed low-frequency Maxwell-Wagner critical dispersion has been explained on the basis of a droplet model for the critical fluctuations.

Analysis of series with stochastic coefficients.

For functions with power-law singularities we consider series expansions whose coefficients have been determined by Monte Carlo simulation. In many practical problems the relative noise of the coefficients is constant or slowly increasing with the order. We modeled real Monte Carlo expansions by test series with known singularity structure where noise with different strength and form is imposed on the coefficients. The efficiency of different standard methods of series analysis (ratio method, Pad\'e approximants, differential approximants) has been tested together with smoothing methods based on repeated partial summation of the series. We found the Pad\'e method to give reasonable estimates and its accuracy is independent of the smoothing, while the estimates of the ratio and the differential approximant methods are greatly improved when smoothing is applied. Indeed, we found the ratio method with optimally selected smoothing to give the most reliable results.

Nonmaximality and phase transitions

Detailed analysis of an infinite class of mean-field, Ginzburg-Landau models of continuous phase transitions which violate the twenty-year old Ascher-Michel maximality conjecture is presented. In all previous counterexamples to the conjecture, fluctuations were found to drive the transitions discontinuous, thus restoring the validity of the conjecture. Our renormalization-group ε-expansion calculations for this infinite class of models uncover the first counterexamples also to the restored conjecture. In contrast to the mean-field behavior, direction of the order parameter near the transition is determined by a stable fixed point and depends only on thermodynamic variables perpendicular to the critical surface, showing a characteristic power-law singularity. Analogous models for the Higgs problem are not known.

The distribution of the reflection phase of disordered conductors

The reflection coefficients of one-dimensional disordered conductors is asymptotically a pure phase, when the sample length is much larger than the localisation length. The authors study the stationary distribution of this reflection phase for tight-binding chains with diagonal disorder, i.e. random site potentials. They derive the Dyson integral equation obeyed by this distribution and relate it to the Lyapunov exponent and the integrated density of states. This general approach allows them to consider then more specifically two types of disorder. In the case of a binary alloy, where the potentials take two values, the distribution of the reflection phase is shown to have generically an infinity of power-law singularities related to the occurrence of periodic patterns of potentials. This phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy-dependent threshold. They also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials. This non-trivial solvable model allows them to examine analytically various properties of the phase distribution, such as the anomalies which occur at weak disorder.

1/f noise and spectral singularities in strongly disordered electronic systems

Discusses the common origin of 1/f noise in the power spectrum of random walks in a random environment and the 1/E-like spectral anomaly for tight-binding electronic systems with off-diagonal disorder. In one dimension the existence of the Dyson singularity, ( rho (E)) varies as mod ln-3 mod E mod mod / mod E mod , for the density of states of the random chain as mod E mod to 0 is inevitably linked to Sinai ultraslow diffusion with the law (x2(t)) varies as ln4t as t to 0 and the presence of S(f) varies as ln4f/f power spectral density for small frequencies f. Different forms of power-law singularities are expected instead for a correlated disorder model appropriate to describe random symmetric diffusion and magnon or phonon excitations. The two problems are discussed in terms of the averaged moments of the electronic wavefunction. The 1/E behaviour is shown to rely on the underlying very general validity of the log-normal distribution for strongly disordered electronic systems. Analytical arguments are given and numerical evidence reported for the asymptotic presence of these singularities in the dimensions d=2, 3 when the disorder is sufficiently strong.

Upper and lower bounds on the Renyi dimensions and the uniformity of multifractals

We prove that for arbitrary probability measures the generalized Renyi dimensions D( q) satisfy the inequality D( q)( q − 1)/ q ≤ D( q')( q' − 1)/ q' ≤ D( q)( q' − 1)/ q'; q' > q, q, q' ∉ [0,1], sign q' = sign q. I n particular, 1 2 D(2) ≤ D(∞) ≤ D(2) and 1 2 D(-∞)≤D(-1)≤D(-∞) . The upper bound corresponds to uniform measures whereas the lower bound describes non-uniform measures such as measures with a power law singularity. In order to characterize intermediate cases we introduce a quantity that measures the uniformity of multifractals. We discuss the relation to the ⨍(α) spectrum and apply the formalism to several examples such as power law measures, the Feigenbaum attractor and the attractor and mode locking structure of the critical circle map.

Refractive index near the critical point and pressure dependence of the critical temperature of a water-sodium di-2-ethylhexylsulfosuccinate-decane microemulsion.

Experimental data are presented for the refractive-index behavior in the homogeneous phase below the cloud-point temperature of a critical microemulsion composed of sodium di-2-ethylhexylsulfosuccinate (AOT), water, and decane. The pressure dependence of the critical temperature (${\mathrm{dT}}_{c}$/dP) is also reported. Fitting the refractive-index data as a function of temperature, we find a (1-\ensuremath{\alpha}) power-law singularity with an \ensuremath{\alpha} value consistent with the Ising value ${\ensuremath{\alpha}}_{I}$=0.11. It is shown that the refractive-index anomaly observed in this microemulsion is an intrinsic effect, most likely related to the electric field dependence of the critical temperature (${\mathrm{dT}}_{c}$/${\mathrm{dE}}^{2}$). Further, using two-scale-factor universality and experimental values for ${\mathrm{dT}}_{c}$/dP and the correlation-length critical amplitude ${\ensuremath{\xi}}_{0}$, we also found that the density contribution to the refractive-index anomaly is negligibly small.

REGULAR AND CHAOTIC TIME EVOLUTION IN SPIN CLUSTERS

We calculate spin-autocorrelation functions (as time averages over chaotic trajectories) and their intensity spectra for clusters of two classical spins, interacting via a nonintegrable Hamiltonian. The long-time behaviour observed includes both power-law decay and persistent oscillatory components, resulting in an intensity spectrum with power-law singularities and discrete lines.

Equimagnetization lines in the hybrid-order phase diagram of the d=3 random-field Ising model (invited)

The three-dimensional random-field Ising model has been approximately solved by pursuing the global renormalization-group trajectories of the full, coupled probability distributions of local fields and bonds. This study, which in effect is the global analysis of the renormalization-group flows of 220 physical quantities (based on a generalization of the Migdal–Kadanoff approximation), yields several results that were previously unsuspected. The boundary between the ferromagnetic and paramagnetic phases is hybrid order, in the sense that the magnetization is discontinuous (as in a first-order transition) and the specific heat has a power-law singularity (as in a second-order transition). The magnetic properties of the d=3 random-field Ising model are presented in the form of equimagnetization lines. The study has been repeated for a variety of dimensions d. The lower critical dimension dl=2 is correctly obtained. Moreover, this study indicates the existence of an intermediate-critical dimension dt (between the lower- and upper-critical dimensions) where a singularity occurs in the critical properties as a function of dimension.

The continuous melting transition of a three-dimensional crystal at a planar elastic instability

The structure and scattering cross section are analysed for a crystal undergoing an elastic instability in which the elastic constant decreases to zero over a plane or planes of propagation directions. It is shown that, at this phase transition, long-range order is destroyed and that the crystal undergoes a novel form of melting. In the scattering cross section, the Bragg peaks disappear and are replaced by power-law singularities which are analogous to those occurring in the scattering from a smectic liquid crystal or a two-dimensional crystal. Expressions are obtained to describe the behaviour as the phase transition is approached.

Thermodynamic properties of the classical Heisenberg chain with nearest- and next-nearest-neighbor interactions

Exact results for the thermodynamic quantities of the one-dimensional classical Heisenberg model with nearest- and next-nearest-neighbor exchange interactions are obtained by means of the numerical transfer matrix method. For a wide range of exchange constants, the system exhibits helical short-range order on which we focus our attention. We find that the Fourier-transformed spin correlation function shows a maximum with asymmetric shape at the characteristic wave-number ±qm (≠0, ±π). The correlation length defined as the inverse of the width atq=qm obeys a simple scaling law and shows a power-law singularity at zero temperature. Results for the heat capacity and the susceptibility are also presented and discussed in connection with the helical short-range order.

Scaling properties of a structure intermediate between quasiperiodic and random

We consider a one-dimensional structure obtained by stringing two types of “beads” (short and long bonds) on a line according to a quasiperiodic rule. This model exhibits a new kind of order, intermediate between quasiperiodic and random, with a singular continuous Fourier transform (i.e., neither Dirac peaks nor a smooth structure factor). By means of an exact renormalization transformation acting on the two-parameter family of circle maps that defines the model, we study in a quantitative way the local scaling properties of its Fourier spectrum. We show that it exhibits power-law singularities around a dense set of wavevectorsq, with a local exponentγ(q) varying continuously with the ratio of both bond lengths. Our construction also sheds some new light on the interplay between three characteristic properties of deterministic structures, namely: (1) a bounded fluctuation of the atomic positions with respect to their average lattice; (2) a quasiperiodic Fourier transform, i.e., made of Dirac peaks; and (3) for sequences generated by a substitution, the number-theoretic properties of the eigenvalue spectrum of the substitution.

Diffraction of an electromagnetic wave at a sphere. Summation of Mie series

The authors suggest a new method for the summation of functional series which diverge absolutely at the edge of the convergence circle. The essence of the method is based on the multiplicative weakening of the logarithmic and power law singularities of functions which are represented by the series. By way of example we sum the Mie series which determine the diffraction of an electromagnetic wave at a sphere when the source and observation points are positioned on the surface of the sphere. It is shown that a double transformation of the original series is, with a good reserve, sufficient for practical purposes.

Power law singularities in the scale covariant theory

Power asymptote singularities are discussed in the scale covariant theory of gravitation. Some general results are derived. Special attention is paid to the Friedmann [A. Friedmann, Z. Phys. 10, 377 (1922)] and Kasner [E. Kasner, Am. J. Math. 43, 217 (1921)] models. A wider class of behavior is exhibited and it is shown that the results obtained constitute a generalization of the corresponding general relativistic results.

Dynamics of concentration fluctuations for a micellar solution.

The decay rate $\ensuremath{\Gamma}$ for a micellar solution pentaethylene glycol mono $n$-dodecyl ether + water has been examined near the critical mixing point by light scattering. We find that the dynamic scaling behavior of ${\ensuremath{\Gamma}}^{*}[\frac{6\ensuremath{\pi}\ensuremath{\eta}(T)}{{k}_{\mathrm{B}}T}](\frac{\ensuremath{\Gamma}}{{k}^{3}})$ as a function of $k\ensuremath{\xi}$ over the range of $0.14\ensuremath{\lesssim}k\ensuremath{\xi}\ensuremath{\lesssim}6.2$ has an obvious kinship with that for a fluid mixture near the critical mixing point, where $\ensuremath{\eta}(T)$ is the shear viscosity and $\ensuremath{\xi}$ the correlation length. It is also found that the shear viscosity exhibits a critical-like power-law singularity and depends explicitly on shear gradients.