Space Correlation Function Research Articles

Universality of modulation length and time exponents

We study systems with a crossover parameter λ, such as the temperature T, which has a threshold value λ(*) across which the correlation function changes from exhibiting fixed wavelength (or time period) modulations to continuously varying modulation lengths (or times). We introduce a hitherto unknown exponent ν(L) characterizing the universal nature of this crossover and compute its value in general instances. This exponent, similar to standard correlation length exponents, is obtained from motion of the poles of the momentum (or frequency) space correlation functions in the complex k-plane (or ω-plane) as the parameter λ is varied. Near the crossover (i.e., for λ→λ(*)), the characteristic modulation wave vector K(R) in the variable modulation length "phase" is related to that in the fixed modulation length "phase" q via |K(R)-q|[proportionality]|T-T(*)|(νL). We find, in general, that ν(L)=1/2. In some special instances, ν(L) may attain other rational values. We extend this result to general problems in which the eigenvalue of an operator or a pole characterizing general response functions may attain a constant real (or imaginary) part beyond a particular threshold value λ(*). We discuss extensions of this result to multiple other arenas. These include the axial next-nearest-neighbor Ising (ANNNI) model. By extending our considerations, we comment on relations pertaining not only to the modulation lengths (or times), but also to the standard correlation lengths (or times). We introduce the notion of a Josephson time scale. We comment on the presence of aperiodic "chaotic" modulations in "soft-spin" and other systems. These relate to glass-type features. We discuss applications to Fermi systems, with particular application to metal to band insulator transitions, change of Fermi surface topology, divergent effective masses, Dirac systems, and topological insulators. Both regular periodic and glassy (and spatially chaotic behavior) may be found in strongly correlated electronic systems.

Peaks in the CMBR power spectrum. I. Mathematical analysis of the associated real space features

The purpose of our study is to understand the mathematical origin in real space of modulated and damped sinusoidal peaks observed in cosmic microwave background radiation anisotropies. We use the theory of the Fourier transform to connect localized features of the two-point correlation function in real space to oscillations in the power spectrum. We also illustrate analytically and by means of Monte Carlo simulations the angular correlation function for distributions of filled disks with fixed or variable radii capable of generating oscillations in the power spectrum. While the power spectrum shows repeated information in the form of multiple peaks and oscillations, the angular correlation function offers a more compact presentation that condenses all the information of the multiple peaks into a localized real space feature. We have seen that oscillations in the power spectrum arise when there is a discontinuity in a given derivative of the angular correlation function at a given angular distance. These kinds of discontinuities do not need to be abrupt in an infinitesimal range of angular distances but may also be smooth, and can be generated by simply distributing excesses of antenna temperature in filled disks of fixed or variable radii on the sky, provided that there is a non-null minimum radius and/or the maximum radius is constrained.

Inflationary perturbation theory is geometrical optics in phase space

A pressing problem in comparing inflationary models with observation is the accurate calculation of correlation functions. One approach is to evolve them using ordinary differential equations (“transport equations”), analogous to the Schwinger-Dyson hierarchy of in-out quantum field theory. We extend this approach to the complete set of momentum space correlation functions. A formal solution can be obtained using raytracing techniques adapted from geometrical optics. We reformulate inflationary perturbation theory in this language, and show that raytracing reproduces the familiar “δN” Taylor expansion. Our method produces ordinary differential equations which allow the Taylor coefficients to be computed efficiently. We use raytracing methods to express the gauge transformation between field fluctuations and the curvature perturbation, ζ, in geometrical terms. Using these results we give a compact expression for the nonlinear gauge-transform part of fNL in terms of the principal curvatures of uniform energy-density hypersurfaces in field space.

Non-perturbative renormalization in coordinate space for [formula omitted] maximally twisted mass fermions with tree-level Symanzik improved gauge action

We present results of a Lattice QCD application of a coordinate space renormalization scheme for the extraction of renormalization constants for flavour non-singlet bilinear quark operators. The method consists in the analysis of the small-distance behaviour of correlation functions in Euclidean space and has several theoretical and practical advantages, in particular: it is gauge invariant, easy to implement and has relatively low computational cost. The values of renormalization constants in the X-space scheme can be converted to the MS¯ scheme via 4-loop continuum perturbative formulae. Our results for Nf=2 maximally twisted mass fermions with tree-level Symanzik improved gauge action are compared to the ones from the RI-MOM scheme and show full agreement with this method.

TOF-SEMSANS—Time-of-flight spin-echo modulated small-angle neutron scattering

We report on measurements of spatial beam modulation of a polarized neutron beam induced by triangular precession regions in time-of-flight mode and the application of this novel technique spin-echo modulated small-angle neutron scattering (SEMSANS) to small-angle neutron scattering in the very small angle range. It is shown that this method can be implemented straight-forwardly in order to extend the accessible size range of structures to be investigated by SANS towards a microscopic scale by applying a divergent beam and measuring the real space correlation function. The novel approach of SEMSANS enables the application of sophisticated sample environment and measurements of magnetic samples (in contrast to the analogue method SESANS).

Using a grating analyser for SEMSANS investigations in the very small angle range

Spin-echo modulation small-angle neutron scattering (SEMSANS) is based on the detection of spatial beam modulation, which is induced by triangular spin echo precession regions and subsequent spin analyses. In order to detect such signal and exploit it for small angle scattering investigations neutron detection with sub-millimeter spatial resolution is required. Here an approach is reported where instead of a position sensitive detector an absorption grating is used to analyze the beam modulation stepwise. The spin-echo length scan in this case is performed by varying the sample-to-detector distance. The real space correlation functions of reference sample structures in the range 10(2) nm, i.e. giving rise to small-angle scattering in the very small-angle range, are recorded and analyzed successfully. (C) 2012 Elsevier B.V. All rights reserved. (Less)

The clustering of intermediate-redshift quasars as measured by the Baryon Oscillation Spectroscopic Survey

We measure the quasar two-point correlation function over the redshift range 2.2<z<2.8 using data from the Baryon Oscillation Spectroscopic Survey. We use a homogeneous subset of the data consisting of 27,129 quasars with spectroscopic redshifts---by far the largest such sample used for clustering measurements at these redshifts to date. The sample covers 3,600 square degrees, corresponding to a comoving volume of 9.7(Gpc/h)^3 assuming a fiducial LambdaCDM cosmology, and it has a median absolute i-band magnitude of -26, k-corrected to z=2. After accounting for redshift errors we find that the redshift space correlation function is fit well by a power-law of slope -2 and amplitude s_0=(9.7\pm 0.5)Mpc/h over the range 3<s<25Mpc/h. The projected correlation function, which integrates out the effects of peculiar velocities and redshift errors, is fit well by a power-law of slope -1 and r_0=(8.4\pm 0.6)Mpc/h over the range 4<R<16Mpc/h. There is no evidence for strong luminosity or redshift dependence to the clustering amplitude, in part because of the limited dynamic range in our sample. Our results are consistent with, but more precise than, previous measurements at similar redshifts. Our measurement of the quasar clustering amplitude implies a bias factor of b~3.5 for our quasar sample. We compare the data to models to constrain the manner in which quasars occupy dark matter halos at z~2.4 and infer that such quasars inhabit halos with a characteristic mass of <M>~10^{12}Msun/h with a duty cycle for the quasar activity of 1 per cent.

SHORT RANGE ORDERING AND LOCAL DISPLACEMENT OF ALLOYS STUDIED BY CVM

Cluster Variation Method (CVM) is a powerful statistical mechanics means to investigate phase equilibria of an alloy. The advantageous feature of the CVM stems from the fact that wide range of atomic correlations which play an important role at the phase transition is efficiently incorporated into the free energy formula. Hence, configurational fluctuation can be systematically studied through the calculations of correlation functions in the real space and short range order diffuse intensity spectrum in the k-space. However, one of the deficiencies of the conventional CVM is the fact that local lattice distortion (local atomic displacement) is not correctly dealt with. In order to improve such shortcomings, Continuous Displacement Cluster Variation Method (CDCVM) has been developed. In the CDCVM, local lattice distortion is mapped onto the configurational freedom of a multi-component alloy on a rigid (uniformly deformable) lattice. With CDCVM, the applicability of CVM is enlarged and the calculations of diffuse intensity spectrum originating from local lattice distortion can be performed.

GALILEAN CONFORMAL ALGEBRA IN SEMI-INFINITE SPACE

In the present paper, we considered Galilean conformal algebras (GCAs), which arises as a contraction relativistic conformal algebras (xi→ϵxi, t→t, ϵ→0). We can use the Galilean conformal (GC) symmetry to constrain two-point and three-point functions. Correlation functions in space–time without boundary condition were found [A. Bagchi and I. Mandal, Phys. Lett. B675, 393 (2009).]. In real situations, there are boundary conditions in space–time, so we have calculated correlation functions for GC invariant fields in semi-infinite space with boundary condition in r = 0. We have calculated two-point and three-point functions with boundary condition in fixed time.

Gauge Theories on ALE Space and Super Liouville Correlation Functions

We present a relation between \({\mathcal{N}=2}\) quiver gauge theories on the ALE space \({\mathcal{O}_{\mathbb{P}^1}(-2)}\) and correlators of \({\mathcal{N}=1}\) super Liouville conformal field theory, providing checks in the case of punctured spheres and tori. We derive a blow-up formula for the full Nekrasov partition function and show that, up to a U(1) factor, the \({\mathcal{N}=2^*}\) instanton partition function is given by the product of the character of \({\widehat{SU}(2)_2}\) times the super Virasoro conformal block on the torus with one puncture. Moreover, we match the perturbative gauge theory contribution with super Liouville three-point functions.

2D PIV measurements in the near field of grid turbulence using stitched fields from multiple cameras

We present measurements of grid turbulence using 2D particle image velocimetry taken immediately downstream from the grid at a Reynolds number of ReM = 16500 where M is the rod spacing. A long field of view of 14M × 4M in the down- and cross-stream directions was achieved by stitching multiple cameras together. Two uniform biplanar grids were selected to have the same M and pressure drop but different rod diameter D and cross-section. A large data set (104 vector fields) was obtained to ensure good convergence of second-order statistics. Estimations of the dissipation rate \(\varepsilon\) of turbulent kinetic energy (TKE) were found to be sensitive to the number of mean-squared velocity gradient terms included and not whether the turbulence was assumed to adhere to isotropy or axisymmetry. The resolution dependency of different turbulence statistics was assessed with a procedure that does not rely on the dissipation scale η. The streamwise evolution of the TKE components and \(\varepsilon\) was found to collapse across grids when the rod diameter was included in the normalisation. We argue that this should be the case between all regular grids when the other relevant dimensionless quantities are matched and the flow has become homogeneous across the stream. Two-point space correlation functions at x/M = 1 show evidence of complex wake interactions which exhibit a strong Reynolds number dependence. However, these changes in initial conditions disappear indicating rapid cross-stream homogenisation. On the other hand, isotropy was, as expected, not found to be established by x/M = 12 for any case studied.

IR divergences and kinetic equation in de Sitter space. (Poincare patch; principal series)

We explicitly show that the one loop IR correction to the two--point function in de Sitter space scalar QFT does not reduce just to the mass renormalization. The proper interpretation of the loop corrections is via particle creation revealing itself through the generation of the quantum averages $<a^+_p a_p>$, $<a_p a_{-p}>$ and $<a^+_p a^+_{-p}>$, which slowly change in time. We show that this observation in particular means that loop corrections to correlation functions in de Sitter space can not be obtained via analytical continuation of those calculated on the sphere. We find harmonics for which the particle number $<a^+_p a_p>$ dominates over the anomalous expectation values $<a_p a_{-p}>$ and $<a^+_p a^+_{-p}>$. For these harmonics the Dyson--Schwinger equation reduces in the IR limit to the kinetic equation. We solve the latter equation, which allows us to sum up all loop leading IR contributions to the Whiteman function. We perform the calculation for the principle series real scalar fields both in expanding and contracting Poincare patches.

Deformation induced structural evolution in bulk metallic glasses

The structural behavior of binary Cu50Zr50 and ternary Cu50Zr45Ti5 bulk metallic glasses (BMGs) under applied stress was investigated by means of in-situ high energy X-ray synchrotron diffraction. The components of the strain tensors were determined from the shifts of the maxima of the atomic pair correlation functions (PDF) in real space. The anisotropic atomic reorientation in the first-nearest-neighbor shell versus stress suggests structural rearrangements in short-range order. Within the plastic deformation range the overall strain of the metallic glass is equal to the yield strain. After unloading, the atomic structure returns to the stress-free state, and the short-range order is identical to that of the undeformed state. Plastic deformation, however, leads to localized shear bands whose contribution to the volume averaged diffraction pattern is too weak to be detected. A concordant region evidenced by the anisotropic component is activated to counterbalance the stress change due to the atomic bond reorientation in the first-nearest-neighbor shell. The size of the concordant region is an important factor dominating the yield strength and the plastic strain ability of the BMGs.

An efficient method for accurate evaluation of the site–site direct correlation functions of molecular fluids

The direct correlation function (DCF) plays an important role in liquid integral-equation theories and non-mean-field applications of the classical density functional theory (DFT). While for a simple fluid the DCF can easily be calculated from the radial distribution function via the Fourier transform and/or, for special cases, can be derived from analytical solutions of the Ornstein–Zernike equation, computation of the site–site DCFs of a molecular fluid is more challenging because of numerical issues associated with solving the matrix integral equations. This paper describes a new theoretical method for accurate evaluation of the site–site DCFs of molecular fluids by combination of molecular simulation and analytical asymptotic analysis. The computational procedure entails four steps: (1) molecular simulation is used to calculate the site–site total correlation functions (TCFs) in real space; (2) the reference-interaction-site model (RISM) is used to calculate the site–site DCFs in Fourier space at large wavenumbers; (3) asymptotic expressions are derived for the TCFs and DCFs in the limit of small wavenumbers; and (4) site–site DCFs over the entire range are obtained by interpolation of the asymptotic results. The numerical procedure has been illustrated by application to bulk SPC/E water. Accurate evaluation of the site–site DCFs for water lays a foundation for future applications of the DFT to aqueous systems with atomic details.

On graviton non-gaussianities during inflation

We consider the most general three point function for gravitational waves produced during a period of exactly de Sitter expansion. The de Sitter isometries constrain the possible shapes to only three: two preserving parity and one violating parity. These isometries imply that these correlation functions should be conformal invariant. One of the shapes is produced by the ordinary gravity action. The other shape is produced by a higher derivative correction and could be as large as the gravity contribution. The parity violating shape does not contribute to the bispectrum [1106.3228, 1108.0175], even though it is present in the wavefunction. We also introduce a spinor helicity formalism to describe de Sitter gravitational waves with circular polarization. These results also apply to correlation functions in Anti-de Sitter space. They also describe the general form of stress tensor correlation functions, in momentum space, in a three dimensional conformal field theory. Here all three shapes can arise, including the parity violating one.

The Hagedorn spectrum and large N c QCD in 2 + 1 and 3 + 1 dimensions

We show that a Hagedorn spectrum (i.e., spectrum where the number of hadrons grows exponentially with the mass) emerges automatically in large $N_c$ QCD in 2+1 and 3+1 dimensions. The approach is based on the study of Euclidean space correlation functions for composite operators constructed from quark and gluon fields and exploits the fact that the short time behavior of the correlators is known in QCD. The demonstration relies on one critical assumption: that perturbation theory accurately describes the trace of the logarithm of a matrix of point-to-point correlation functions in the regime where the perturbative corrections to the asymptotically free value are small.

Asymptotic correlation functions and FFLO signature for the one-dimensional attractive spin-1/2 Fermi gas

We investigate the long distance asymptotics of various correlation functions for the one-dimensional spin-1/2 Fermi gas with attractive interactions using the dressed charge formalism. In the spin polarized phase, these correlation functions exhibit spatial oscillations with a power-law decay whereby their critical exponents are found through conformal field theory. We show that spatial oscillations of the leading terms in the pair correlation function and the spin correlation function solely depend on Δ k F and 2 Δ k F , respectively. Here Δ k F = π ( n ↑ − n ↓ ) denotes the mismatch between the Fermi surfaces of spin-up and spin-down fermions. Such spatial modulations are characteristics of a Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. Our key observation is that backscattering among the Fermi points of bound pairs and unpaired fermions results in a one-dimensional analog of the FFLO state and displays a microscopic origin of the FFLO nature. Furthermore, we show that the pair correlation function in momentum space has a peak at the point of mismatch between both Fermi surfaces k = Δ k F , which has recently been observed in numerous numerical studies.

Prospect for characterizing interacting soft colloidal structures using spin-echo small angle neutron scattering

Spin-echo small angle neutron scattering (SESANS) provides a new experimental tool for structural investigation. Due to the action of spin-echo encoding, SESANS measures a spatial correlation function in real space, as opposed to the structure factor S(Q), I(Q), in momentum (Q) space measured by conventional small angle neutron scattering. To establish the usefulness of SESANS in structural characterization, particularly for interacting colloidal suspensions, we have previously conducted a theoretical study of the SESANS correlation functions for model systems consisting of particles with uniform density profiles [X. Li, C.-Y. Shew, Y. Liu, R. Pynn, E. Liu, K. W. Herwig, G. S. Smith, J. L. Robertson, and W.-R. Chen J. Chem. Phys. 132, 174509 (2010)]. Within the same framework, we explore in the present paper the prospect of using SESANS to investigate the structural characteristics of colloidal systems consisting of particles with nonuniform intraparticle mass distribution. As an example, a Gaussian model of interacting soft colloids is used to investigate the manifestation of structural softness in a SESANS measurement. The exploration shows a characteristically different SESANS correlation function for interacting soft colloids, in comparison to that of a uniform hard sphere system. The difference arises from the Abel transform imbedded in the mathematical formalism bridging the SESANS spectra and the spatial autocorrelation function.

Neutron studies of an inorganic plastic glass

A neutron diffraction investigation has been performed of the structure of four chalcogenide glasses, using the D4c diffractometer on the high-flux reactor at the Institut Laue-Langevin (ILL). Vitreous Ge 3As 52S 45 is shown to have a structure based on As 4S 3 molecules, whereas that of its selenide analogue, vitreous Ge 3As 52Se 45, involves far fewer As 4Se 3 molecules. Two As 2X 3 reference glasses (X = S or Se) have purely network structures. As–X, and As–As bond lengths have been extracted, together with their associated co-ordination numbers, from peak fits to the real space correlation functions, T( r), and suggest that all four glasses are chemically ordered; i.e. that they contain the maximum possible number of As–X (and Ge–X) bonds. The molecular nature of vitreous Ge 3As 52S 45 is evident from the enhanced first diffraction peak relative to the As 2S 3 reference glass, and has been further investigated by comparing its interference function, Qi( Q), with that expected for an isolated As 4S 3 molecule. On the other hand, the second diffraction peak for vitreous Ge 3As 52S 45 is not reproduced by a random orientation molecular model, indicating that there is orientational correlation between adjacent molecules, and this is discussed with respect to the structure of the corresponding As 4S 3 crystalline phases. The neutron-weighted vibrational density of states (VDOS) for vitreous Ge 3As 52S 45, obtained with the ISIS MARI spectrometer, unambiguously confirms the existence of As 4S 3 molecules, as revealed by a peak at ~ 34 meV arising from the triangle of As atoms that form the base of the molecule. MARI data recorded at ambient temperature also reveal a strong quasi-elastic component in the scattered intensity, which has been further investigated using the IN5 quasi-elastic scattering spectrometer (ILL) and indicates the presence of rotational diffusion of the As 4S 3 molecules; i.e. that, at ambient temperature, vitreous Ge 3As 52S 45 behaves as a plastic glass. A possible structural model for vitreous Ge 3As 52S 45 comprises a two-dimensional tangled GeS 2 net with As 4S 3 molecules trapped in its folds whereas, for vitreous Ge 3As 52Se 45, a clathrate model appears more appropriate.

A new formulation of two-time correlation functions of Markov chains applied to gene networks

We developed a new formulation and derived a set of equations for two-time correlation functions in discrete state space for general Markov chains. Numerical solutions of these equations lead to exact results of correlation functions, which is more efficient and accurate for complex systems than normal Gillespie simulations. The formalism is applied to two simple systems as examples: a two state system and self-regulators with results compared with Monte Carlo simulations and mean-field theory. For self-regulators, from non-adiabatic to adiabatic regimes, we observed both monotonic and turnover kinetic behavior of response time, which can be can tested by experiments.