Static Correlation Functions Research Articles

Static quark correlators and quarkonium properties at non-zero temperature

We discuss different static quark correlators, including Wilson loops in 2+1 flavor QCD at non-zero temperature and their relation to in-medium quarkonium properties. We present lattice results on static correlation functions obtained with highly improved staggered fermion action and their implications for potential models.

Capillarylike fluctuations of a solid-liquid interface in a noncohesive granular system

One of the most noticeable collective motion of noncohesive granular matter is clustering under certain conditions. In particular, when a quasi-two-dimensional monolayer of monodispersed noncohesive particles is vertically vibrated, a solid-liquid-like transition occurs when the driving amplitude exceeds a critical value. Here the physical mechanism underlying particle clustering relies on the strong interactions mediated by grain collisions, rather than on grain-grain cohesive forces. In average, the solid cluster resembles a drop, with a striking circular shape. We experimentally investigate the coarse-grained solid-liquid interface fluctuations, which are characterized through the static and dynamic correlation functions in the Fourier space. These fluctuations turn out to be well described by the capillary wave theory, which allows us to measure the solid-liquid interface surface tension and mobility once the granular "thermal" kinetic energy is determined. Despite that the system is strongly out of equilibrium and that the granular temperature is not uniform, there is energy equipartition at the solid-liquid interface, for a relatively large range of angular wave numbers. Furthermore, both surface tension and mobility are consistent with a simple order of magnitude estimation considering the characteristic energy, length, and time scales, which is very similar to what can be done for atomic systems.

Effects of size polydispersity on mechanical and geometrical properties of granular system

In this paper, the multi-disperse systems composed of 2048 frictionless particles with different size polydispersities are simulated by using the distinct element method, and effects of size polydispersity on mechanical and geometrical properties of these systems are studied under isotropic confining. The result shows that changing polydispersity can qualitatively affect the coordination number, shear modulus, static structure factor and orientational order correlation function, but it does not influence the cumulative force distribution. At small wave number k, static structure factor of mono-disperse systems shows the power scaling law S(k)∝0.2k-4/3. Moreover, the peak of orientational order correlation function exhibits an exponent scaling law g6 (r)∝ae-r/ξ6, and the scaling exponent ξ6 decreases with polydispersity increasing.

Dynamical Correlations After a Quantum Quench

We consider dynamic (non-equal-time) correlation functions of local observables after a quantum quench. We show that, in the absence of long-range interactions in the final Hamiltonian, the dynamics is determined by the same ensemble that describes static (equal-time) correlations. For many integrable models, static correlation functions of local observables after a quantum quench relax to stationary values, which are described by a generalized Gibbs ensemble. The same generalized Gibbs ensemble then determines dynamic correlation functions, and the basic form of the fluctuation dissipation theorem holds, although the absorption and emission spectra are not simply related as in the thermal case. For quenches in the transverse field Ising chain, we derive explicit expressions for the time evolution of dynamic order parameter correlators after a quench.

Quantum Harmonic Oscillator Systems with Disorder

We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective one-particle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models.

Effect of Electron-Phonon Interaction Range for a Half-Filled Band in One Dimension

We demonstrate that fermion-boson models with nonlocal interactions can be simulated at finite band filling with the continuous-time quantum Monte Carlo method. We apply this method to explore the influence of the electron-phonon interaction range for a half-filled band in one dimension, covering the full range from the Holstein to the Fröhlich regime. The phase diagram contains metallic, Peierls, and phase-separated regions. Nonlocal interactions suppress the Peierls instability, and thereby lead to almost degenerate power-law exponents for charge and pairing correlations.

Numerical renormalization group calculation of impurity internal energy and specific heat of quantum impurity models

We introduce a method to obtain the specific heat of quantum impurity models via a direct calculation of the impurity internal energy requiring only the evaluation of local quantities within a single numerical renormalization group (NRG) calculation for the total system. For the Anderson impurity model, we show that the impurity internal energy can be expressed as a sum of purely local static correlation functions and a term that involves also the impurity Green function. The temperature dependence of the latter can be neglected in many cases, thereby allowing the impurity specific heat, $C_{\rm imp}$, to be calculated accurately from local static correlation functions; specifically via $C_{\rm imp}=\frac{\partial E_{\rm ionic}}{\partial T} + 1/2\frac{\partial E_{\rm hyb}}{\partial T}$, where $E_{\rm ionic}$ and $E_{\rm hyb}$ are the energies of the (embedded) impurity and the hybridization energy, respectively. The term involving the Green function can also be evaluated in cases where its temperature dependence is non-negligible, adding an extra term to $C_{\rm imp}$. For the non-degenerate Anderson impurity model, we show by comparison with exact Bethe ansatz calculations that the results recover accurately both the Kondo induced peak in the specific heat at low temperatures as well as the high temperature peak due to the resonant level. The approach applies to multiorbital and multichannel Anderson impurity models with arbitrary local Coulomb interactions. An application to the Ohmic two state system and the anisotropic Kondo model is also given, with comparisons to Bethe ansatz calculations. The new approach could also be of interest within other impurity solvers, e.g., within quantum Monte Carlo techniques.

Fluctuations and Criticality of a Granular Solid-Liquid-Like Phase Transition

We present an experimental study of density and order fluctuations in the vicinity of the solid-liquid-like transition that occurs in a vibrated quasi-two-dimensional granular system. The two-dimensional projected static and dynamic correlation functions are studied. We show that density fluctuations, characterized through the structure factor, increase in size and intensity as the transition is approached, but they do not change significantly at the transition itself. The dense, metastable clusters, which present square symmetry, also increase their local order in the vicinity of the transition. This is characterized through the bond-orientational order parameter Q4, which in Fourier space shows an Ornstein-Zernike-like behavior. Depending on the filling density and vertical height, the transition can be of first- or second-order type. In the latter case, the associated correlation length ξ4, the relaxation time τ4, the zero k limit of Q4 fluctuations (static susceptibility), the pair correlation function of Q4, and the amplitude of the order parameter obey critical power laws, with saturations due to finite size effects. Their respective critical exponents are ν(perpendicular))=1, ν(parallel)=2, γ=1, η=0.67, and β=1/2, whereas the dynamical critical exponent z=ν(parallel)/ν(perpendicular)=2. These results are consistent with model C of dynamical critical phenomena, valid for a nonconserved critical order parameter (bond-orientation order) coupled to a conserved field (density).

Quantum Renormalization Effect in One-Dimensional Heisenberg Antiferromagnets

The dispersion relations of the lowest magnetic excitations at low temperatures and the temperature dependence of the inverse magnetic correlation length were determined using inelastic pulsed neutron scattering experiments on the following one-dimensional Heisenberg antiferromagnets: CsVCl 3 ( S = 3/2), CsVBr 3 ( S = 3/2), CsCrCl 3 ( S = 2), and CsNiCl 3 ( S = 1). From studying the exchange constant, J , it has been found that the observed energies of the magnetic excitations require a quantum renormalization factor that is dependent on the spin quantum number, S . The values of J were determined from the temperature dependence of the inverse magnetic correlation length, which describes the static correlation function that can be deduced from the energy-integrated dynamical structure factor. We demonstrated that the S -dependent quantum renormalization effect was detected by only inelastic pulsed neutron scattering experiments, and that the effect was consistent with quantum Monte Carlo calculations.

Finite temperature correlation functions from discrete functional equations

We present a new approach to the static finite temperature correlation functions of the Heisenberg chain based on functional equations. An inhomogeneous generalization of the n-site density operator is considered. The lattice path integral formulation with a finite but arbitrary Trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters. We show that these equations yield a unique characterization of the density operator. Our functional equations are a discrete version of the reduced q-Knizhnik–Zamolodchikov equations which played a central role in the study of the zero temperature case. As a natural result, and independent of the arguments given by Jimbo et al (2009 J. Phys. A: Math. Theor. 42 304018), we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature: they are a sum of products of nearest-neighbour correlators.

Peculiarities of the correlation functions, X-ray and neutron scattering in BaTiO3

Based on molecular-dynamics (MD) simulations, we have calculated the static and dynamic correlation functions in a BaTiO3 crystal. The static correlation functions have been used to study the peculiarities of diffuse scattering in barium titanate showing the experimentally observed anomalous planes. Based on time-dependent pair correlation functions, we have calculated the phonon spectra of BaTiO3 and studied the central peak of inelastic scattering. The phonon frequencies calculated by the MD method agree well with those obtained previously in the quasi-harmonic approximation. We show that the central peak of inelastic scattering is associated mainly with the soft optic mode and has the same symmetry. The large anisotropy in the displacements of atoms in the soft mode allows the presence of peculiarities in both X-ray scattering and EXAFS spectroscopy to be explained. The characteristic shape of the EXAFS spectra is shown to be explained by the quasi-one-dimensional motion of the oxygen ions in the cubic lattice of BaTiO3. Our calculation of triple correlation functions shows that the titanium atom in the described model oscillates around the cubic cell center. Explaining the experimental data that have caused disagreement about the nature of the phase transition in BaTiO3 using the developed model gives grounds to treat the phase transition in barium titanate as a displacive one.

Quantum relaxation and finite-size effects in the XY chain in a transverse field after global quenches

We consider global quenches in the quantum XY chain in a transverse field and study the nonequilibrium relaxation of the magnetization and the correlation function as well as the entanglement entropy in finite systems. For quenches in the ordered phase the exact results are well described by a semiclassical theory (SCT) in terms of ballistically moving quasi-particle pairs. For finite systems quasi-periodic behaviour of the dynamical evolution of the local order parameter and the correlation functions is predicted correctly including the period length, an exponential relaxation, a quasi-stationary regime and an exponential recurrence in one period. In the thermodynamic limit the SCT is exact for the entanglement entropy and its modified version following the method of Calabrese P. et al., J. Stat. Mech. (2012) P07016, is exact for the magnetization and the correlation function, too. The stationary correlation function is shown to be described by a generalized Gibbs ensemble.

Spin nematic phase in one-dimensional and quasi-one-dimensional frustrated magnets in a strong magnetic field

We discuss spin-1/2 one-dimensional (1D) and quasi-1D magnets with competing ferromagnetic nearest-neighbor $J_1$ and antiferromagnetic next-nearest-neighbor $J$ exchange interactions in strong magnetic field $H$. It is well known that due to attraction between magnons quantum phase transitions (QPTs) take place at $H=H_s$ from the fully polarized phase to nematic ones if $J>|J_1|/4$. Such a transition at $J>0.368|J_1|$ is characterized by softening of the two-magnon bound-state spectrum. Using a bond operator formalism we propose a bosonic representation of the spin Hamiltonian containing, aside from bosons describing one-magnon spin-1 excitations, a boson describing spin-2 excitations which spectrum coincides at $H\ge H_s$ with the two-magnon bound-state spectrum obtained before. The presence of the bosonic mode in the theory that softens at $H=H_s$ makes substantially standard the QPT consideration. In 1D case at $H<H_s$, we find an expression for the magnetization which describes well existing numerical data. Expressions for spin correlators are obtained which coincide with those derived before either in the limiting case of $J\gg|J_1|$ or using a phenomenological theory. In quasi-1D magnets, we find that the boundary in the $H$--$T$ plane between the fully polarized and the nematic phases is given by $H_s(0)-H_s(T)\propto T^{3/2}$. Simple expressions are obtained in the nematic phase for static spin correlators, spectra of magnons and the soft mode, magnetization and the nematic order parameter. All static two-spin correlation functions are short ranged with the correlation length proportional to $1/\ln(1+|J_1|/J)$. Dynamical spin susceptibilities are discussed and it is shown that the soft mode can be observed experimentally in the longitudinal channel.

Structural correlations and cooperative dynamics in supercooled liquids

The relationships between diffusivity and the excess, pair and residual multiparticle contributions to the entropy are examined for Lennard-Jones liquids and binary glassformers, in the context of approximate inverse power law mappings of simple liquids. In the dense liquid where diffusivities are controlled by collisions and cage relaxations, Rosenfeld-type excess entropy scaling of diffusivities is found to hold for both crystallizing as well as vitrifying liquids. The crucial differences between the two categories of liquids emerge only when local cooperative effects in the dynamics result in significant caging effects in the time-dependent behaviour of the single-particle mean square displacement. In the case of glassformers, onset of such local cooperativity coincides with onset of deviations from Rosenfeld-type excess entropy scaling of diffusivities and increasing spatiotemporal heterogeneity. In contrast, for two- and three-dimensional liquids with a propensity to crystallise, the onset of local cooperative dynamics is sufficient to trigger crystallization provided that the liquid is sufficiently supercooled that the free energy barrier to nucleation of the solid phase is negligible. The state points corresponding to onset of transient caging effects can be associated with typical values, within reasonable bounds, of the excess, pair, and residual multiparticle entropy as a consequence of the isomorph-invariant character of the excess entropy, diffusivity and related static and dynamic correlation functions.

Non-ergodicity transition and multiple glasses in binary mixtures: on the accuracy of the input static structure in the mode coupling theory

We examine the question of the accuracy of the static correlation functions used as input in the mode coupling theory (MCT) of non-ergodic states in binary mixtures. We first consider hard-sphere mixtures and compute the static pair structure from the Ornstein–Zernike equations with the Percus–Yevick closure and more accurate ones that use bridge functions deduced from Rosenfeld’s fundamental measures functional. The corresponding MCT predictions for the non-ergodicity lines and the transitions between multiple glassy states are determined from the long-time limit of the density autocorrelation functions. We find that while the non-ergodicity transition line is not very sensitive to the input static structure, up to diameter ratios D2/D1 = 10, quantitative differences exist for the transitions between different glasses. The discrepancies with the more accurate closures become even qualitative for sufficiently asymmetric mixtures. They are correlated with the incorrect behavior of the PY structure at high size asymmetry. From the example of ultra-soft potential it is argued that this issue is of general relevance beyond the hard-sphere model.

Exchange and correlation effects on density excitation spectra of metallic quantum wires at finite temperature

We have studied the effect of exchange and correlations on the density excitation spectra of metallic quantum wires at finite temperature. The correlations are treated by incorporating the first-order self and exchange contributions into the random-phase approximation (RPA). Numerical results are presented for the spectra of the density response function and the plasmon dispersion for the gold wire on Si(557) substrate—a system studied recently by Nagao et al. (2006 Phys. Rev. Lett. 97 116802) for plasmons using electron energy loss spectroscopy. Our results for plasmons are found to agree with the experimental data. Though the first-order correction is small at currently accessible wire parameters, it becomes significant with increasing coupling parameter rs. The effect of temperature on plasmons is found to be small for the wire system investigated experimentally. However, temperature has a significant effect on the spectra of the response function. We have also calculated the static structure factor, the pair-correlation function and the correlation energy at zero temperature in the first-order theory to check its applicability in dealing with correlations. Results are compared directly with the available Monte Carlo simulation data. It is found that the static correlation functions improve significantly over the RPA with the increase of rs. On the other hand, the correlation energy shows very good agreement for rs ≤ 5 and wire widths b ≥ a0. For smaller b, the agreement is good up to relatively smaller rs.

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

We consider an overdamped Brownian particle moving in a confining asymptotically logarithmic potential, which supports a normalized Boltzmann equilibrium density. We derive analytical expressions for the two-time correlation function and the fluctuations of the time-averaged position of the particle for large but finite times. We characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential, and we support our predictions with extensive Langevin simulations. While the Boltzmann measure is used to obtain stationary correlation functions, we show how the non-normalizable infinite covariant density is related to the superaging behavior.

Exact Solution for the Stationary Kardar-Parisi-Zhang Equation

We obtain the first exact solution for the stationary one-dimensional Kardar-Parisi-Zhang equation. A formula for the distribution of the height is given in terms of a Fredholm determinant, which is valid for any finite time t. The expression is explicit and compact enough so that it can be evaluated numerically. Furthermore, by extending the same scheme, we find an exact formula for the stationary two-point correlation function.

Magnetic Field Dependence of Critical Attenuation in Ferromagnet

Many aspects of sound attenuation above and below critical point have been discussed in the literature [1, 2]. However, little attention has been paid so far to critical attenuation in the presence of the magnetic eld. A notable exception are the meaneld theory results [3 5]. The aim of this paper is to provide a dynamical matching formalism of the renormalization group which allows us to calculate the ultrasonic attenuation coe cient α(t, ω,H) by means of the e expansion not only in the asymptotic region H = 0 but also in the whole magneticeld range. We implement the method used in the calculation of the dynamic susceptibility in the ordered phase [6] for the evaluation of the sound attenuation coe cient in an external magnetic eld but use a modi ed matching condition suitable for H = 0. One of the best method for computation of the scaling functions is the method introduced by Nelson [7] for computation of the static correlation function above the critical temperature. It was a generalization of the renormalization group technique developed by Nelson and Rudnick [8] used to obtain the equation of state to rst order in e = 4 − d as well as the scaling functions for the susceptibility, speci c heat and etc. both in the ordered and disordered phase. Later this method was generalized by Achiam and Kosterlitz [9] to calculate the static momentum-dependent correlation function for arbitrary temperature and magnetic eld. Dengler et al. [10] were rst who generalized this method into dynamic correlation function in the disordered phase. In a recent paper Pawlak and Erdem [6] extended these results by obtaining (to rst order in e) the expressions for the dynamic susceptibility and correlation function for nonconserved Ising order parameter both above and below the critical temperature in zero magnetic eld. Generalization of this method to nonzero elds will be used for calculation of the ultrasonic attenuation.

Using point-to-set correlations to probe unjamming of frictionless grains

We present a detailed analysis of the unjamming transition in 2D frictionless disk packingsusing a static correlation function that has been widely used to study disordered systems.We show that this point-to-set (PTS) correlation function exhibits a dominantlength scale that diverges as the unjamming transition is approached throughdecompression. In addition, we identify deviations from mean-field predictions, andpresent detailed analysis of the origin of non-mean-field behavior. A mean-fieldbulk-surface argument is reviewed. Corrections to this argument are identified,which lead to a change in the functional form of the critical PTS boundary sizeR0. An entropic description of the origin of the correlations is presented, andsimple rigidity assumptions are shown to predict the functional form ofR0 as a function ofthe pressure P.