Correlations In Momentum Space Research Articles

Recurrence relations for finite-temperature correlators via AdS2/CFT1

This note is aimed at presenting a new algebraic approach to momentum-space correlators in conformal field theory. As an illustration we present a new Lie-algebraic method to compute frequency-space two-point functions for charged scalar operators of CFT$_{1}$ dual to AdS$_{2}$ black hole with constant background electric field. Our method is based on the real-time prescription of AdS/CFT correspondence, Euclideanization of AdS$_{2}$ black hole and projective unitary representations of the Lie algebra $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R})$. We derive novel recurrence relations for Euclidean CFT$_{1}$ two-point functions, which are exactly solvable and completely determine the frequency- and charge-dependences of two-point functions. Wick-rotating back to Lorentzian signature, we obtain retarded and advanced CFT$_{1}$ two-point functions that are consistent with the known results.

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.

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.

New recursion relations and a flat space limit for AdS/CFT correlators

We consider correlation functions of the stress-tensor or a conserved current in ${\mathrm{AdS}}_{d+1}/{\mathrm{CFT}}_{d}$ computed using the Hilbert or the Yang-Mills action in the bulk. We introduce new recursion relations to compute these correlators at tree-level. These relations have an advantage over the Britto-Cachazo-Feng-Witten (BCFW)-like relations described in arXiv:1102.4724 and arXiv:1011.0780 because they can be used in all dimensions including $d=3$. We also introduce a new method of extracting flat-space $S$-matrix elements from AdS/CFT correlators in momentum space. We show that the ($d+1$)-dimensional flat-space amplitude of gravitons or gluons can be obtained as the coefficient of a particular singularity of the $d$-dimensional correlator of the stress-tensor or a conserved current; this technique is valid even at loop-level in the bulk. Finally, we show that our recursion relations automatically generate correlators that are consistent with this observation: they have the expected singularity and the flat-space gluon, or graviton amplitude appears as its coefficient.

The ultraviolet limit and sum rule for the shear correlator in hot Yang-Mills theory

We determine a next-to-leading order result for the correlator of the shear stress operator in high-temperature Yang-Mills theory. The computation is performed via an ultraviolet expansion, valid in the limit of small distances or large momenta, and the result is used for writing operator product expansions for the Euclidean momentum and co-ordinate space correlators as well as for the Minkowskian spectral density. In addition, our results enable us to confirm and refine a shear sum rule originally derived by Romatschke, Son and Meyer.

The rise and fall of the ridge in heavy ion collisions

Recent data from heavy ion collisions at RHIC show unexpectedly large near-angle correlations that broaden longitudinally with increasing centrality. The amplitude of this ridge-like correlation rises rapidly, reaches a maximum, and then falls in the most central collisions. In this Letter we explain how this behavior can be uniquely explained by initial-state coordinate-space anisotropies converted into final-state momentum-space correlations. We propose vn2/εn,part2 as a useful way to study length scales and provide a prediction for the ridge in Pb+Pb collisions at sNN=2.76 TeV.

The rise and fall of the ridge

Recent data from heavy ion collisions at RHIC show unexpectedly large near-angle correlations that broaden longitudinally with centrality. The amplitude of this ridge-like correlation rises rapidly with centrality, reaches a maximum, and then falls in the most central collisions. In this talk we explain how this behavior can be easily understood in a picture where final momentum-space correlations are driven by initial coordinate space density fluctuations. We propose ν n 2 / ε n , part 2 as a useful way to study these effects and explain what it tells us about the collision dynamics.

Probing electron correlation and nuclear dynamics in Momentum Space

Orbital imaging experiments employing Electron Momentum Spectroscopy are subject to many complications, such as distorted wave effects, conformational mobility in the electronic ground state, ultra-fast nuclear dynamics in the final state, or a dispersion of the ionization intensity over electronically excited (shake-up) configurations of the cation. The purpose of the present contribution is to illustrate how a proper treatment of these complications enables us to probe in momentum space the consequences of electron correlation and nuclear dynamics in neutral and cationic states.

Reduced density matrices and coherence of trapped interacting bosons

The first- and second-order correlation functions of trapped interacting Bose-Einstein condensates are investigated numerically on a many-body level from first principles. Correlations in real space and momentum space are treated. The coherence properties are analyzed. The results are obtained by solving the many-body Schr\odinger equation. It is shown in an example how many-body effects can be induced by the trap geometry. A generic fragmentation scenario of a condensate is considered. The correlation functions are discussed along a pathway from a single condensate to a fragmented condensate. It is shown that strong correlations can arise from the geometry of the trap, even at weak interaction strengths. The natural orbitals and natural geminals of the system are obtained and discussed. It is shown how the fragmentation of the condensate can be understood in terms of its natural geminals. The many-body results are compared to those of mean-field theory. The best solution within mean-field theory is obtained. The limits in which mean-field theories are valid are determined. In these limits the behavior of the correlation functions is explained within an analytical model.

Structure of quantum correlations in momentum space and off-diagonal long-range order inηpairing and BCS states

The quantum states built with the $\mathrm{\ensuremath{\eta}}$ pairing mechanism, i.e., $\mathrm{\ensuremath{\eta}}$ pairing states, were first introduced in the context of high-temperature superconductivity where they were recognized as important example of states allowing for off-diagonal long-range order (ODLRO). In this paper we describe the structure of the correlations present in these states when considered in their momentum representation and we explore the relations between the quantum bipartite and multipartite correlations exhibited in $k$ space and the direct lattice superconducting correlations. In particular, we show how the negativity between paired momentum modes is directly related to the ODLRO. Moreover, we investigate the dependence of the block entanglement on the choice of the modes forming the block and on the ODLRO; consequently we determine the multipartite content of the entanglement through the evaluation of the generalized Meyer-Wallach measure in the direct and reciprocal lattice. The determination of the persistency of entanglement shows how the network of correlations depicted exhibits a self-similar structure which is robust with respect to ``local'' measurements. Finally, we recognize how a relation between the momentum-space quantum correlations and the ODLRO can be established even in the case of BCS states.

Screening correlators with chiral fermions

We study screening correlators of quark-antiquark composites at $T=2{T}_{c}$, where ${T}_{c}$ is the QCD phase transition temperature, using overlap quarks in the quenched approximation of lattice QCD. As the lattice spacing is changed from $1/4T$ to $a=1/6T$ and $1/8T$, we find that screening correlators change little, in contrast with the situation for other types of lattice fermions. All correlators are close to the ideal gas prediction at small separations. The long distance falloff is clearly exponential, showing that a parametrization by a single screening length is possible at distances $z\ensuremath{\ge}1/T$. The correlator corresponding to the thermal vector is close to the ideal gas value at all distances, whereas that for the thermal scalar deviates at large distances. This is examined through the screening lengths and momentum space correlators. There is strong evidence that the screening transfer matrix does not have reflection positivity.

Dynamical dimer correlations at bipartite and non-bipartite Rokhsar–Kivelsonpoints

We determine the dynamical dimer correlation functions of quantum dimer models at theRokhsar–Kivelson point on the bipartite square and cubic lattices and the non-bipartitetriangular lattice. On the basis of an algorithmic idea by Henley, we simulate astochastic process of classical dimer configurations in continuous time and perform astochastic analytical continuation to obtain the dynamical correlations in momentumspace and the frequency domain. This approach allows us to observe directlythe dispersion relations and the evolution of the spectral intensity within theBrillouin zone beyond the single-mode approximation. On the square lattice,we confirm analytical predictions related to soft modes close to the wavevectors(π,π) and (π,0) and further reveal the existence of shadow bands close to the wavevector(0,0). On the cubic lattice the spectrum is also gapless but here only a single soft mode at(π,π,π) is found, as predicted by the single-mode approximation. The soft mode has a quadraticdispersion at very long wavelength, but crosses over to a linear behavior very rapidly. Webelieve this to be the remnant of the linearly dispersing ‘photon’ of the Coulomb phase.Finally the triangular lattice is in a fully gapped liquid phase where the bottom of thedimer spectrum exhibits a rich structure. At the M point the gap is minimal and thespectral response is dominated by a sharp quasiparticle peak. On the other hand, at the Xpoint the spectral function is much broader. We sketch a possible explanation basedon the crossing of the coherent dimer excitations into the two-vison continuum.

Spatial noise correlations of a chain of ultracold fermions: A numerical study

We present a numerical study of noise correlations, i.e., density-density correlations in momentum space, in the extended fermionic Hubbard model in one dimension. In experiments with ultracold atoms, these noise correlations can be extracted from time-of-flight images of the expanding cloud. Using the density-matrix renormalization group method to investigate the Hubbard model at various fillings and interactions, we confirm that the noise correlations contain full information on the most important fluctuations present in the system. We point out the importance of the sum rules fulfilled by the noise correlations and show that they yield nonsingular structures beyond the predictions of bosonization approaches. Noise correlations can thus serve as a universal probe of order and can be used to characterize the many-body states of cold atoms in optical lattices.

Shannon entropies of atomic structure factors, off-diagonal order and radial correlation

Shannon entropies of one- and two-electron atomic structure factors in the position and momentum representations are used to examine the behaviour of the off-diagonal elements of density matrices with respect to the uncertainty principle and to analyse the effects of radial correlation on off-diagonal order. We show that radial correlation induces off-diagonal order in position space which is characterized by larger entropic values. Radial correlation in momentum space is characterized by smaller entropic values as information is forced into regions closer to the diagonal. Related off-diagonal correlation functions are also discussed.

First-principles quantum simulations of dissociation of molecular condensates: Atom correlations in momentum space

We investigate the quantum many-body dynamics of dissociation of a Bose-Einstein condensate of molecular dimers into pairs of constituent bosonic atoms and analyze the resulting atom-atom correlations. The quantum fields of both the molecules and atoms are simulated from first principles in three dimensions using the positive-P representation method. This allows us to provide an exact treatment of the molecular field depletion and s-wave scattering interactions between the particles, as well as to extend the analysis to nonuniform systems. In the simplest uniform case, we find that the major source of atom-atom decorrelation is atom-atom recombination which produces molecules outside the initially occupied condensate mode. The unwanted molecules are formed from dissociated atom pairs with non-opposite momenta. The net effect of this process -- which becomes increasingly significant for dissociation durations corresponding to more than about 40% conversion -- is to reduce the atom-atom correlations. In addition, for nonuniform systems we find that mode-mixing due to inhomogeneity can result in further degradation of the correlation signal. We characterize the correlation strength via the degree of squeezing of particle number-difference fluctuations in a certain momentum-space volume and show that the correlation strength can be increased if the signals are binned into larger counting volumes.

Mutual information and electron correlation in momentum space

Mutual information and information entropies in momentum space are proposed as measures of the nonlocal aspects of information. Singlet and triplet state members of the helium isoelectronic series are employed to examine Coulomb and Fermi correlations, and their manifestations, in both the position and momentum space mutual information measures. The triplet state measures exemplify that the magnitude of the spatial correlations relative to the momentum correlations depends on and may be controlled by the strength of the electronic correlation. The examination of one- and two-electron Shannon entropies in the triplet state series yields a crossover point, which is characterized by a localized momentum density. The mutual information density in momentum space illustrates that this localization is accompanied by strong correlation at small values of p.

Probing Pair-Correlated Fermionic Atoms through Correlations in Atom Shot Noise

Pair-correlated fermionic atoms are created through dissociation of weakly bound molecules near a magnetic-field Feshbach resonance. We show that correlations between atoms in different spin states can be detected using the atom shot noise in absorption images. Furthermore, using time-of-flight imaging we have observed atom pair correlations in momentum space.

Coherence and correlation properties of a one-dimensional attractive Fermi gas

A recently developed quantum Monte Carlo algorithm based on the stochastic evolution of Hartree–Fock states has been applied to compute the static correlation functions of a one-dimensional model of attractively interacting two component fermions. The numerical results have been extensively compared to existing approximate approaches. The crossover to a condensate of pairs can be identified as the first-order pair coherence extending throughout the whole size of the system. The possibility of revealing the onset of the transition with other observables such as the density–density correlations or the second-order momentum space correlations is discussed.

Hydrodynamic theory for spatially inhomogeneous semiconductor lasers. I. A microscopic approach

Starting from the microscopic semiconductor Bloch equations including the Boltzmann transport terms in the distribution function equations for electrons and holes, we derived a closed set of diffusion equations for carrier densities and temperatures with self-consistent coupling to Maxwell's equation and to an effective optical polarization equation. The coherent many-body effects are included within the screened Hartree-Fock approximation, while scatterings are treated within the second Born approximation including both the in- and out-scatterings. Microscopic expressions for electron-hole $(e$-$h)$ and carrier--LO-phonon $(c$-LO) scatterings are directly used to derive the momentum and energy relaxation rates. These rates, expressed as functions of temperatures and densities, lead to microscopic expressions for self- and mutual-diffusion coefficients in the coupled density-temperature diffusion equations. Approximations for reducing the general two-component description of the electron-hole plasma to a single-component one are discussed. In particular, we show that a special single-component reduction is possible when e-$h$ scattering dominates over $c$--LO-phonon scattering. The ambipolar diffusion approximation is also discussed and we show that the ambipolar diffusion coefficients are independent of e-$h$ scattering, even though the diffusion coefficients of individual components depend sensitively on the e-$h$ scattering rates. Our discussions lead to deeper insights into the roles played in the single-component reduction by the electron-hole correlation in momentum space induced by scatterings and the electron-hole correlation in real space via internal static electrical field. Finally, the theory is completed by coupling the diffusion equations to the lattice temperature equation and to the effective optical polarization, which in turn couples to the laser field.

Equilibrium correlations in interfaces between two immiscible liquids

Static density and concentration correlations in momentum space are determined quantitatively in molecular dynamics computer experiments on the interface between two liquid phases of a binary mixture. The interaction between spherical particles of equal sizes is modeled by 6–12 Lennard-Jones potentials. The resulting four density-density correlations are inverted to four direct correlation functions. Transformation to density-concentration variables inspired by the Bhatia–Thornton transformation and the hitherto unknown characteristic shapes of their projections, including the generalization of the Yvon–Zwanzig–Triezenberg projection, are discussed in detail; it is demonstrated that the q4 coefficient again is negative. This precludes its interpretation as a rigidity coefficient. The interfacial contributions decay rather quickly with the momentum q, becoming totally submerged by short-range bulk and bulklike fluctuations at q-values of the order of 1/5 of the q-value of the nearest-neighbor (first) peak in the scattering factor.