Power-law Correlation Function Research Articles

Hyperinvariant multiscale entanglement renormalization ansatz: Approximate holographic error correction codes with power-law correlations

We consider a class of holographic tensor networks that are efficiently contractible variational ansatze, manifestly (approximate) quantum error correction codes, and can support power-law correlation functions. In the case when the network consists of a single type of tensor that also acts as an erasure correction code, we show that it cannot be both locally contractible and sustain power-law correlation functions. Motivated by this no-go theorem, and the desirability of local contractibility for an efficient variational ansatz, we provide guidelines for constructing networks consisting of multiple types of tensors that can support power-law correlation. We also provide an explicit construction of one such network, which approximates the holographic HaPPY pentagon code in the limit where variational parameters are taken to be small.

Impact of momentum resolution on factorial moments due to power–law correlations between particles

The effect of momentum resolution on factorial moments due to the power–law correlation function is studied. The study is motivated by searching for the critical point of the strongly interacting matter in heavy-ion collisions using the intermittency method. We observe that factorial moments are significantly affected by the finite momentum resolution. The effect is superficially significant compared to intuitive expectations. The results depend on the power of the correlation function and the number of uncorrelated particles.

Scaling of factorial moments in cumulative variables

A search for power-law fluctuations within the framework of the intermittency method is ongoing to locate the critical point of the strongly interacting matter. In particular, experimental data on proton and pion production in heavy-ion collisions are analyzed in transverse-momentum, pT, space.In this regard, we have studied the dependence of the second scaled factorial moment F2 of particle multiplicity distribution on the number of subdivisions of transverse momentum-interval used in the analysis. The study is performed using a simple model with a power-law two-particle correlation function in pT. We observe that F2 values depend on the size and position of the pT interval. However, when we convert the non-uniform transverse-momentum distribution to uniform one using cumulative transformation, F2 calculated in subdivisions of the cumulative pT becomes independent of the cumulative-pT interval. The scaling behaviour of F2 for the cumulative variable is observed. Moreover, F2 follows a power law with the number of subdivisions of the cumulative-pT interval with the intermittency index close to the correlation function's exponent.

Emergent conformal symmetry in nonunitary random dynamics of free fermions

We present random quantum circuit models for non-unitary quantum dynamics of free fermions in one spatial dimension. Numerical simulations reveal that the dynamics tends towards steady states with logarithmic violations of the entanglement area law and power law correlation functions. Moreover, starting with a short-range entangled many-body state, the dynamical evolution of entanglement and correlations quantitatively agrees with the predictions of two-dimensional conformal field theory with a space-like time direction. We argue that this behavior is generic in non-unitary free quantum dynamics with time-dependent randomness, and show that the emergent conformal dynamics of two-point functions arises out of a simple "nonlinear master equation".

XY model with antinematic interaction.

We consider the XY model with ferromagnetic (FM) and antinematic (AN) nearest-neighbor interactions on a square lattice for a varying interaction strength ratio. Besides the expected FM and AN quasi-long-range order (QLRO) phases we identify at low temperatures another peculiar canted ferromagnetic (CFM) QLRO phase, resulting from the competition between the collinear FM and noncollinear AN ordering tendencies. In the CFM phase neighboring spins that belong to different sublattices are canted by a nonuniversal (dependent on the interaction strength ratio) angle and the ordering is characterized by a fast-decaying power-law intrasublattice correlation function. Compared to the FM phase, in the CFM phase correlations are significantly diminished by the presence of zero-energy domain walls due to the inherent degeneracy caused by the AN interactions. We present the phase diagram as a function of the interaction strength ratio and discuss the character of the respective phases as well as the transitions between them.

Perturbed Kitaev model: Excitation spectrum and long-ranged spin correlations

We developed general approach to the calculation of power-law infrared asymptotics of spin-spin correlation functions in the Kitaev honeycomb model with different types of perturbations. We have shown that in order to find these correlation functions, one can perform averaging of some bilinear forms composed out of free Majorana fermions, and we presented the method for explicit calculation of these fermionic densities. We demonstrated how to derive an effective Hamiltonian for the Majorana fermions, including the effects of perturbations. For specific application of the general theory, we have studied the effect of the Dzyaloshinskii-Moriya anisotropic spin-spin interaction; we demonstrated that it leads, already in the second order over its relative magnitude D/K, to a power-law spin correlation functions, and calculated dynamical spin structure factor of the system. We have shown that an external magnetic field h in presence of the DM interaction, opens a gap in the excitation spectrum of magnitude Δ∝Dh.

''DOUBLE-HUMPED EFFECT'' IN THE TURBULENT COLLISION MAGNETIZED PLASMA

Statistical moments of the spatial power spectrum of multiple scattered ordinary and extraordinary waves in the turbulent collision magnetized plasma with aligned anisotropic electron density irregularities are investigated using modify smooth perturbation method taking into account diffraction effects. Correlation function and variances of the phase fluctuations are obtained for arbitrary correlation function of the electron density fluctuations. Double-humped Effect is investigated analytically and numerically using the anisotropic Gaussian spectral function of electron density irregularities for the polar ionospheric F-region applying the experimental data. Investigation of the anisotropy of the F layer irregularities is of great interest in ionospheric physics. It is well known that the irregularities are elongated in the direction of the geomagnetic field of lines, but especially at high latitudes they are also found to be anisotropic in the plane perpendicular to the external magnetic field vector. The anisotropy is usually investigated by correlation analysis of radio signals from satellites, observed by space receiver on the ground. The fluctuations in amplitude and phase (scintillations) of radio waves passing through the ionosphere are caused by spatial irregularities in the electron density. The spatial fluctuations in phase cause fluctuations of the angle of arrival at the observation points. Broadening of the spatial power spectrum (SPS) of scattered electromagnetic (EM) waves in the turbulent collision magnetized plasma for both power-law and anisotropic Gaussian correlation functions of electron density fluctuations was analyzed in (1) using the complex ray (-optics) approximation. Double-humped Effect, the features of the SPS, spectral width and shift of its maximum of multiple scattered EM waves in the turbulent collisionless magnetized plasma were considered in (2, 3). The influence of the collision frequency between plasma particles on the statistical characteristics of scattered EM waves in the turbulent magnetized plasma with both electron density and external magnetic field fluctuations was investigated in (4). The present paper reports the results of an analysis of the SPS, which is related to fluctuations of the radio refractive index in the F region. Analytical expressions of the correlation function of phase fluctuations of scattered both ordinary and extraordinary waves are obtained for the arbitrary correlation functions of electron density fluctuation using modify smooth perturbation method taking into account the diffraction effects. The Double-humped Effect in the collision magnetized plasma with elongated electron density irregularities is considered for the first time. Formation of a gap in the SPS of scattered radiation is analyzed numerically for the polar ionospheric F -region using the anisotropic Gaussian spectral function of electron density fluctuations. This spectrum contains angle of aligned plasma irregularities to the direction of an external magnetic field and anisotropy factor. The numerical calculation is computed based on experimental data. Finally, paper concludes the obtained results and their significance.

Field equation of the correlation function of mass-density fluctuations for self-gravitating systems

We study the mass density distribution of Newtonian self-gravitating systems. Modeling the system as a fluid in hydrostatical equilibrium, we obtain from first principle the field equation and its solution of correlation function $\xi(r)$ of the mass density fluctuation itself. We apply thid to studies of the large-scale structure of the Universe within a small redshift range. The equation tells that $\xi(r)$ depends on the point mass $m$ and the Jeans wavelength scale $\lambda_{0}$, which are different for galaxies and clusters. It explains several longstanding, prominent features of the observed clustering : that the profile of $\xi_{cc}(r)$ of clusters is similar to $\xi_{gg}(r)$ of galaxies but with a higher amplitude and a longer correlation length, and that the correlation length increases with the mean separation between clusters as a universal scaling $r_0\simeq 0.4d$. Our solution $\xi(r)$ also yields the observed power-law correlation function of galaxies $\xi_{gg}(r)\simeq (r_0/r)^{1.7}$ valid only in a range $1<r<10 h^{-1}$Mpc. At larger scales the solution $\xi(r)$ breaks below the power law and goes to zero around $\sim 50h^{-1}$Mpc, just as the observational data have demonstrated. With a set of fixed model parameters, the solutions $\xi_{gg}(r)$ for galaxies, the corresponding power spectrum, and $\xi_{cc}(r)$ for clusters, simultaneously, agree with the observational data from the major surveys of galaxies, and of clusters.

Determining accurate measurements of the growth rate from the galaxy correlation function in simulations

We use high-resolution N-body simulations to develop a new, flexible, empirical approach for measuring the growth rate from redshift-space distortions (RSD) in the 2-point galaxy correlation function. We quantify the systematic error in measuring the growth rate in a $1 \, h^{-3}$ Gpc$^3$ volume over a range of redshifts, from the dark matter particle distribution and a range of halo-mass catalogues with a number density comparable to the latest large-volume galaxy surveys such as the WiggleZ Dark Energy Survey and the Baryon Oscillation Spectroscopic Survey (BOSS). Our simulations allow us to span halo masses with bias factors ranging from unity (probed by emission-line galaxies) to more massive haloes hosting Luminous Red Galaxies. We show that the measured growth rate is sensitive to the model adopted for the small-scale real-space correlation function, and in particular that the "standard" assumption of a power-law correlation function can result in a significant systematic error in the growth rate determination. We introduce a new, empirical fitting function that produces results with a lower (5-10%) amplitude of systematic error. We also introduce a new technique which permits the galaxy pairwise velocity distribution, the quantity which drives the non-linear growth of structure, to be measured as a non-parametric stepwise function. Our (model-independent) results agree well with an exponential pairwise velocity distribution, expected from theoretical considerations, and are consistent with direct measurements of halo velocity differences from the parent catalogues. In a companion paper we present the application of our new methodology to the WiggleZ Survey dataset.

Distribution of first-return times in correlated stationary signals

We present an analytical expression for the first return time (FRT) probability density function of a stationary correlated signal. Precisely, we start by considering a stationary discrete-time Ornstein-Uhlenbeck (OU) process with exponential decaying correlation function. The first return time distribution for this process is derived by adopting a well-known formalism typically used in the study of the FRT statistics for nonstationary diffusive processes. Then, by a subordination approach, we treat the case of a stationary process with power-law tail correlation function and diverging correlation time. We numerically test our findings, obtaining in both cases a good agreement with the analytical results. We notice that neither in the standard OU nor in the subordinated case a simple form of waiting time statistics, like stretched-exponential or similar, can be obtained while it is apparent that long time transient may shadow the final asymptotic behavior.

Beams of gravitationally bound ultracold neutrons in rough waveguides

We investigate the propagation of ultracold neutrons through a rough waveguide in conjunction with recent experiments in which the ultracold neutrons were beamed between a perfect mirror and a rough scatterer and absorber. The main goal is to find a way to resolve the lowest gravitationally quantized discrete states in the peV range. We compare the neutron count for various types of mirrors with Gaussian, power-law, and exponential correlation functions of surface inhomogeneities. The main conclusion is that all the information about inhomogeneities, including their amplitude, correlation radius, and the rate of decay of the correlation function, enter the exit neutron count via just a single constant $\ensuremath{\Phi}$, which effectively renormalizes the amplitude of roughness. To observe well-defined quantum steps, one should have an experimental setup with $\ensuremath{\Phi}g40$. For a wide variety of correlation functions, the constant $\ensuremath{\Phi}$ is proportional to the square of the amplitude of the surface roughness and is inversely proportional to the square root of the correlation radius. The strong dependence of $\ensuremath{\Phi}$ on roughness parameters and the shape of the correlation function opens a novel way for improving the resolution of gravitationally bound states by optimizing the roughness pattern without reverting to an undesirable strong roughness. We discuss how to optimize the scatterer and absorber by first generating numerically the desired roughness profile and then transferring it to the mirror. We also study the effect of beam preparation on the initial occupancies of gravitational states before the beam enters the waveguide. It turns out that there are simple ways to manipulate the beam in front of the waveguide that can help to resolve the gravitationally bound quantum states. Our results are in good agreement with available experimental data.

Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems

In this paper we suggest a consistent approach to derivation of generalized Fokker-Planck equation (GFPE) for Gaussian non-Markovian processes with stationary increments. This approach allows us to construct the probability density function (PDF) without a need to solve the GFPE. We employ our method to obtain the GFPE and PDFs for free generalized Brownian motion and the one in harmonic potential for the case of power-law correlation function of the noise. We prove the fact that the considered systems may be described with Einstein-Smoluchowski equation at high viscosity levels and long times. We also compare the results with those obtained by other authors. At last, we calculate PDF of thermodynamical work in the stochastic system which consists of a particle embedded in a harmonic potential moving with constant velocity, and check the work fluctuation theorem for such a system.

The “Coulomb Phase” in Frustrated Systems

The “Coulomb phase” is an emergent state for lattice models (particularly highly frustrated antiferromagnets), which have local constraints that can be mapped to a divergence-free “flux.” The coarse-grained versions of this flux or polarization behave analogously to electric or magnetic fields; in particular, defects at which the local constraint is violated behave as effective charges with Coulomb interactions. I survey the derivation of the characteristic power-law correlation functions and the pinch points in reciprocal space plots of diffuse scattering, as well as applications to magnetic relaxation, quantum-mechanical generalizations, phase transitions to long-range-ordered states, and the effects of disorder.

Anomalous diffusion in a generalized Langevin equation

We analyze the motion of a particle governed by a generalized Langevin equation with the colored noise described by a combination of power-law and generalized Mittag–Leffler function. This colored noise generalizes the power-law correlation function and an exponential one. We obtain exact results for the relaxation function. Further, we obtain the first moments and variances of the displacement and velocity. The long-time behaviors of these quantities are also investigated. We show that normal diffusion processes can be generated by a class of these colored noises.

Propagation and localization of acoustic and elastic waves in heterogeneous materials: renormalization group analysis and numerical simulations

We describe and discuss the recent progress in the study of propagation and localization of acoustic and elastic waves in heterogeneous media. The heterogeneity is represented by a spatial distribution of the local elastic moduli. Both randomly distributed elastic moduli as well as those with long-range correlations with a nondecaying power-law correlation function, are considered. The motivation for the study is twofold. One is that recent analysis of experimental data for the spatial distribution of the elastic moduli of rock indicated that the distribution is characterized by the type of long-range correlations that we consider in this study. The second motivation for the problem is to understand whether localization of electrons (which, in quantum mechanics, are described by wave functions) has any analogy in the propagation of classical waves in disordered media. The problem is studied by two approaches. One of them is based on developing a dynamic renormalization group (RG) approach to analytical analysis of the governing equations for wave propagation. The RG analysis indicates that, depending on the type of the disorder (correlated vs. uncorrelated), one may have a transition between localized and extended regimes in any spatial dimension. The second approach utilizes numerical simulations of the governing equations in two- and three-dimensional media. The results obtained by the two approaches are in agreement with each other. Using numerical simulations, we also describe how the characteristics of a propagating wave may be used for probing the differences between heterogeneous media with short- and long-range correlations. To do so, we study the evolution of several distinct characteristics of the waves, such as the amplitude of the coherent wave front, its width, the spectral densities, the scalogram (wavelet transformation of the waves’ amplitudes at different scales and times), and the dispersion relation. It is demonstrated that such properties have completely different characteristics in uncorrelated and correlated media. Finally, it is shown how wave propagation may be used for establishing a link between the static and dynamical properties of heterogeneous media.

Power-law scaling and fractal nature of medium-range order in metallic glasses

The atomic structure of metallic glasses has been a long-standing scientific problem. Unlike crystalline metals, where long-range ordering is established by periodic stacking of fundamental building blocks known as unit cells, a metallic glass has no long-range translational or orientational order, although some degrees of short- and medium-range order do exist. Previous studies have identified solute- (minority atom)-centred clusters as the fundamental building blocks or short-range order in metallic glasses. Idealized cluster packing schemes, such as efficient cluster packing on a cubic lattice and icosahedral packing as in a quasicrystal, have been proposed and provided first insights on the medium-range order in metallic glasses. However, these packing schemes break down beyond a length scale of a few clusters. Here, on the basis of neutron and X-ray diffraction experiments, we propose a new packing scheme-self-similar packing of atomic clusters. We show that the medium-range order has the characteristics of a fractal network with a dimension of 2.31, and is described by a power-law correlation function over the medium-range length scale. Our finding provides a new perspective of order in disordered materials and has broad implications for understanding their structure-property relationship, particularly those involving a change in length scales.

Dynamic renormalization group analysis of propagation of elastic waves in two-dimensional heterogeneous media

We study localization of elastic waves in two-dimensional heterogeneous solids with randomly distributed Lam\'e coefficients, as well as those with long-range correlations with a power-law correlation function. The Matin-Siggia-Rose method is used, and the one-loop renormalization group (RG) equations for the the coupling constants are derived in the limit of long wavelengths. The various phases of the coupling constants space, which depend on the value $\rho$, the exponent that characterizes the power-law correlation function, are determined and described. Qualitatively different behaviors emerge for $\rho<1$ and $\rho>1$. The Gaussian fixed point (FP) is stable (unstable) for $\rho<1$ ($\rho>1$). For $\rho<1$ there is a region of the coupling constants space in which the RG flows are toward the Gaussian FP, implying that the disorder is irrelevant and the waves are delocalized. In the rest of the disorder space the elastic waves are localized. We compare the results with those obtained previously for acoustic wave propagation in the same type of heterogeneous media, and describe the similarities and differences between the two phenomena.

Quasars Probing Quasars. II. The Anisotropic Clustering of Optically Thick Absorbers around Quasars

With close pairs of quasars at different redshifts, a background quasar sightline can be used to study a foreground quasar's environment in absorption. We used a sample of 17 Lyman limit systems with column density N_HI > 10^19 cm^-2 selected from 149 projected quasar pair sightlines, to investigate the clustering pattern of optically thick absorbers around luminous quasars at z ~ 2.5. Specifically, we measured the quasar-absorber correlation function in the transverse direction, and found a comoving correlation length of r_0=9.2_{+1.5}_{-1.7} Mpc/h (comoving) assuming a power law correlation function, with gamma=1.6. Applying this transverse clustering strength to the line-of-sight, would predict that ~ 15-50% of all quasars should show a N_HI > 10^19 cm^-2 absorber within a velocity window of v < 3000 km/s. This overpredicts the number of absorbers along the line-of-sight by a large factor, providing compelling evidence that the clustering pattern of optically thick absorbers around quasars is highly anisotropic. The most plausible explanationfor the anisotropy is that the transverse direction is less likely to be illuminated by ionizing photons than the line-of-sight, and that absorbers along the line-of-sight are being photoevaporated. A simple model for the photoevaporation of absorbers subject to the ionizing flux of a quasar is presented, and it is shown that absorbers with volume densities n_H < 0.1 cm^-3 will be photoevaporated if they lie within ~ 1 Mpc (proper) of a luminous quasar. Using this simple model, we illustrate how comparisons of the transverse and line-of-sight clustering around quasars can ultimately be used to constrain the distribution of gas in optically thick absorption line systems.

Renormalization group analysis and numerical simulation of propagation and localization of acoustic waves in heterogeneous media

Propagation of acoustic waves in strongly heterogeneous elastic media is studied using renormalization group analysis and extensive numerical simulations. The heterogeneities are characterized by a broad distribution of the local elastic constants. We consider both Gaussian-white distributed elastic constants, as well as those with long-range correlations with a nondecaying power-law correlation function. The study is motivated in part by recent analysis of experimental data for the spatial distribution of the elastic moduli of rock at large length scales, which indicated that the distribution contains the same type of long-range correlations as what we consider in the present paper. The problem that we formulate and the results are, however, applicable to acoustic wave propagation in any disordered elastic material that contains the types of heterogeneities that we consider in the present paper. Using the Martin-Siggia-Rose method, we analyze the problem analytically and find that, depending on the type of disorder, the renormalization group (RG) flows exhibit a transition to a localized or extended regime in any dimension. We also carry out extensive numerical simulations of acoustic wave propagation in one-, two-, and three-dimensional systems. Both isotropic and anisotropic media (with anisotropy being due to stratified) are considered. The results for the isotropic media are consistent with the RG predictions. While the RG analysis, in its present form, does not make any prediction for the anisotropic media, the results of our numerical simulations indicate the possibility of the existence of a regime of superlocalization in which the waves' amplitudes decay as $\mathrm{exp}[\ensuremath{-}{(\ensuremath{\mid}\mathbf{x}\ensuremath{\mid}∕\ensuremath{\xi})}^{\ensuremath{\gamma}}]$, with $\ensuremath{\gamma}&gt;1$, where $\ensuremath{\xi}$ is the localization length. However, further investigations may be necessary in order to establish the possible existence of such a localization regime.

Generalized Langevin equation with fractional derivative and long-time correlation function

We investigate the motion of a particle governed by a generalized Langevin equation with fractional derivative, nonlocal dissipative force, and long-time correlation function. We derive general expressions for the variances with a linear external force. We also analyze their asymptotic behaviors for the power-law correlation function without external force.