General Correlation Functions Research Articles

Inflation correlators with multiple massive exchanges

The most general tree-level boundary correlation functions of quantum fields in inflationary spacetime involve multiple exchanges of massive states in the bulk, which are technically difficult to compute due to the multi-layer nested time integrals in the Schwinger-Keldysh formalism. On the other hand, correlators with multiple massive exchanges are well motivated in cosmological collider physics, with the original quasi-single-field inflation model as a notable example. In this work, with the partial Mellin-Barnes representation, we derive a simple rule, called family-tree decomposition, for directly writing down analytical answers for arbitrary nested time integrals in terms of multi-variable hypergeometric series. We present the derivation of this rule together with many explicit examples. This result allows us to obtain analytical expressions for general tree-level inflation correlators with multiple massive exchanges. As an example, we present the full analytical results for a range of tree correlators with two massive exchanges.

On the testability of the Károlyházy model

Károlyházy’s original proposal, suggesting that space-time fluctuations could be a source of decoherence in space, faced a significant challenge due to an unexpectedly high emission of radiation (13 orders of magnitude more than what was observed in the latest experiment). To address this issue, we reevaluated Károlyházy’s assumption that the stochastic metric fluctuation must adhere to a wave equation. By considering more general correlation functions of space-time fluctuations, we resolve the problem and consequently revive the aforementioned proposal.

Structural Properties of Gibbsian Point Processes in Abstract Spaces

In the language of random counting measures, many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology, are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with cluster-dependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes.

Hybrid quantum-classical algorithm for computing imaginary-time correlation functions

Quantitative descriptions of strongly correlated materials pose a considerable challenge in condensed matter physics and chemistry. A promising approach to address this problem is quantum embedding methods. In particular, the dynamical mean-field theory (DMFT) maps the original system to an effective quantum impurity model comprising correlated orbitals embedded in an electron bath. The biggest bottleneck in DMFT calculations is numerically solving the quantum impurity model, i.e., computing the Green's function. Past studies have proposed theoretical methods to compute the Green's function of a quantum impurity model in polynomial time using a quantum computer. So far, however, efficient methods for computing the imaginary-time Green's functions have not been established despite the advantages of the imaginary-time formulation. We propose a quantum-classical hybrid algorithm for computing imaginary-time Green's functions on quantum devices with limited hardware resources by applying the variational quantum simulation. Using a quantum circuit simulator, we verified this algorithm by computing Green's functions for a dimer model as well as a four-site impurity model obtained by DMFT calculations of the single-band Hubbard model, although our method can be applied to general imaginary-time correlation functions.

Variational methods for contracting projected entangled-pair states

The norms or expectation values of infinite projected entangled-pair states (PEPS) cannot be computed exactly, and approximation algorithms have to be applied. In the last years, many efficient algorithms have been devised -- the corner transfer matrix renormalization group (CTMRG) and variational uniform matrix product state (VUMPS) algorithm are the most common -- but it remains unclear whether they always lead to the same results. In this paper, we identify a subclass of PEPS for which we can reformulate the contraction as a variational problem that is algorithm independent. We use this variational feature to assess and compare the accuracy of CTMRG and VUMPS contractions. Moreover, we devise a new variational contraction scheme, which we can extend to compute general N-point correlation functions.

FZZT branes in JT gravity and topological gravity

We study Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) branes in Witten-Kontsevich topological gravity, which includes Jackiw-Teitelboim (JT) gravity as a special case. Adding FZZT branes to topological gravity corresponds to inserting determinant operators in the dual matrix integral and amounts to a certain shift of the infinitely many couplings of topological gravity. We clarify the perturbative interpretation of adding FZZT branes in the genus expansion of topological gravity in terms of a simple boundary factor and the generalized Weil-Petersson volumes. As a concrete illustration we study JT gravity in the presence of FZZT branes and discuss its relation to the deformations of the dilaton potential that give rise to conical defects. We then construct a non-perturbative formulation of FZZT branes and derive a closed expression for the general correlation function of multiple FZZT branes and multiple macroscopic loops. As an application we study the FZZT-macroscopic loop correlators in the Airy case. We observe numerically a void in the eigenvalue density due to the eigenvalue repulsion induced by FZZT-branes and also the oscillatory behavior of the spectral form factor which is expected from the picture of eigenbranes.

The Ising correlation C(M, N) for ν = −k

We present Painlevé VI sigma form equations for the general Ising low and high temperature two-point correlation functions C(M, N) with M ⩽ N in the special case ν = −k where ν = sinh 2E h /k B T/sinh 2E v /k B T. More specifically four different non-linear ODEs depending explicitly on the two integers M and N emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with M + N even or odd. These four different non-linear ODEs are also valid for M ⩾ N when ν = −1/k. For the low-temperature row correlation functions C(0, N) with N odd, we exhibit again for this selected ν = −k condition, a remarkable phenomenon of a Painlevé VI sigma function being the sum of four Painlevé VI sigma functions having the same Okamoto parameters. We show in this ν = −k case for T < T c and also T > T c, that C(M, N) with M ⩽ N is given as an N × N Toeplitz determinant.

Power System Mode Metering in the Presence of Non-Gaussian Noise

This letter presents a robust generalized correlation approach for monitoring of power system oscillatory modes. Due to the non-Gaussian nature of measurement noise, and presence of impulsive outliers, the wide-sense stationarity assumption of power system measurement data is violated even under ambient conditions thus limiting the effectiveness of conventional autocorrelation based approaches in mode metering. The fundamental definition of correlation function for power system measurements is extended to a generalized correlation function in order to incorporate higher order statistics into existing modal estimation technique. Simulations on synthetic and real-life test signals have been conducted to demonstrate the robustness of the proposed method.

Dual formulations of Polyakov loop lattice models

Dual representations are constructed for non-abelian lattice spin models with U(N) and SU(N) symmetry groups, for all N and in any dimension. These models are usually related to the effective models describing the interaction between Polyakov loops in the strong coupled QCD. The original spin degrees of freedom are explicitly integrated out and a dual theory appears to be a local theory for the dual integer-valued variables. The construction is performed for the partition function and for the most general correlation function. The latter include the two-point function corresponding to quark-anti-quark free energy and the N-point function related to the free energy of a baryon. We consider both pure gauge models and models with static fermion determinant for both the staggered and Wilson fermions with an arbitrary number of flavours. While the Boltzmann weights of such models are complex in the presence of non-zero chemical potential the dual Boltzmann weights appear to be strictly positive on admissible configurations. An essential part of this work with respect to previous studies is an extension of the dual representation to the case of 1) an arbitrary value of the temporal coupling constant in the Wilson action and 2) an arbitrary number of flavours of static quark determinants. The applications and extensions of the results are discussed in detail. In particular, we outline a possible approach to Monte-Carlo simulations of the dual theory, to the large N expansion and to the development of a tensor renormalization group.

Toverline{T} deformed CFT as a non-critical string

We present a new exact treatment of Toverline{T} deformed 2D CFT in terms of the worldsheet theory of a non-critical string. The transverse dimensions of the non-critical string are represented by the undeformed CFT, while the two longitudinal light-cone di- rections are described by two scalar fields X+ and X− with free field OPE’s but with a modified stress tensor, arranged so that the total central charge adds up to 26. The relation between our X± field variables and 2D dilaton gravity is indicated. We compute the physical spectrum and the partition function and find a match with known results. We describe how to compute general correlation functions and present an integral expression for the three point function, which can be viewed as an exact formula for the OPE coefficients of the Toverline{T} deformed theory. We comment on the relationship with other proposed definitions of local operators.

Improving the condition number of estimated covariance matrices

High dimensional error covariance matrices and their inverses are used to weight the contribution of observation and background information in data assimilation procedures. As observation error covariance matrices are often obtained by sampling methods, estimates are often degenerate or ill-conditioned, making it impossible to invert an observation error covariance matrix without the use of techniques to reduce its condition number. In this paper, we present new theory for two existing methods that can be used to ‘recondition’ any covariance matrix: ridge regression and the minimum eigenvalue method. We compare these methods with multiplicative variance inflation, which cannot alter the condition number of a matrix, but is often used to account for neglected correlation information. We investigate the impact of reconditioning on variances and correlations of a general covariance matrix in both a theoretical and practical setting. Improved theoretical understanding provides guidance to users regarding method selection, and choice of target condition number. The new theory shows that, for the same target condition number, both methods increase variances compared to the original matrix, with larger increases for ridge regression than the minimum eigenvalue method. We prove that the ridge regression method strictly decreases the absolute value of off-diagonal correlations. Theoretical comparison of the impact of reconditioning and multiplicative variance inflation on the data assimilation objective function shows that variance inflation alters information across all scales uniformly, whereas reconditioning has a larger effect on scales corresponding to smaller eigenvalues. We then consider two examples: a general correlation function, and an observation error covariance matrix arising from interchannel correlations. The minimum eigenvalue method results in smaller overall changes to the correlation matrix than ridge regression but can increase off-diagonal correlations. Data assimilation experiments reveal that reconditioning corrects spurious noise in the analysis but underestimates the true signal compared to multiplicative variance inflation.

Multi-time formulation of Matsubara dynamics.

Matsubara dynamics has recently emerged as the most general form of a quantum-Boltzmann-conserving classical dynamics theory for the calculation of single-time correlation functions. Here, we present a generalization of Matsubara dynamics for the evaluation of multitime correlation functions. We first show that the Matsubara approximation can also be used to approximate the two-time symmetrized double Kubo transformed correlation function. By a straightforward extension of these ideas to the multitime realm, a multitime Matsubara dynamics approximation can be obtained for the multitime fully symmetrized Kubo transformed correlation function. Although not a practical method, due to the presence of a phase-term, this multitime formulation of Matsubara dynamics represents a benchmark theory for future development of Boltzmann preserving semiclassical approximations to general higher order multitime correlation functions. It also reveals a connection between imaginary time-ordering in the path integral and the classical dynamics of multitime correlation functions.

Graph Database Solution for Higher-order Spatial Statistics in the Era of Big Data

Abstract We present an algorithm for the fast computation of the general N-point spatial correlation functions of any discrete point set embedded within an Euclidean space of . Utilizing the concepts of kd-trees and graph databases, we describe how to count all possible N-tuples in binned configurations within a given length scale, e.g., all pairs of points or all triplets of points with side lengths &lt; r MAX. Through benchmarking, we show the computational advantage of our new graph-based algorithm over more traditional methods. We show measurements of the three-point correlation function up to scales of ∼200 Mpc (beyond the baryon acoustic oscillation scale in physical units) using current Sloan Digital Sky Survey (SDSS) data. Finally, we present a preliminary exploration of the small-scale four-point correlation function of 568,776 SDSS Constant (stellar) Mass (CMASS) galaxies in the northern Galactic cap over the redshift range of 0.43 &lt; z &lt; 0.7. We present the publicly available code GRAMSCI (GRAph Made Statistics for Cosmological Information; bitbucket.org/csabiu/gramsci), under a Gnu is Not Unix (GNU) General Public License.

Generalised NMR-moisture correlation function of building materials based on a capillary bundle model

Quantitative determination of moisture with nuclear magnetic resonance is usually a simple task, since the initial amplitude is proportional to the moisture content. However, situations can arise where the combination of short relaxation times, low signal-to-noise ratio and the ill-posed nature of the inverse Laplace transformation lead to a significant bias of the initial amplitude impeding it being an accurate proxy for the moisture content. This is often met when applying single-sided NMR to building materials with small pores, like concrete. We present a method to derive a generalised correlation function linking not only the initial amplitude, but relaxation attenuated signal amplitudes with moisture content. Therefore, the effect of de-saturation on the T2 distribution of the respective material near saturation, which is the input to the model, is simulated using a capillary bundle model. Predicted and measured moisture coincide, and the uncertainty is significantly smaller compared to the conventional approach.

Probabilistic properties of generalized stochastic processes in algebras of generalized functions

Stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. The notion of point values of Colombeau stochastic processes in compactly supported generalized points is established, which uniquely characterize the process, and relying on this result we prove the measurability of the corresponding random variables with values in the Colombeau algebra of compactly supported generalized constants endowed with the topology generated by sharp open balls. The generalized characteristic function and the generalized correlation function of Colombeau stochastic processes are introduced and their properties are investigated. It is shown that the characteristic function of classical stochastic processes can be embedded into the space of generalized characteristic functions. The generalized expectation and the generalized correlation function can be retrieved from the generalized characteristic function. The structural representation of the correlation function which is supported on the diagonal is given. Examples of generalized characteristic functions related to Gaussian Colombeau stochastic processes are given.

The shear and bulk relaxation times from the general correlation functions

In this paper we present two quantum field theoretical analyses on the shear and bulk relaxation times. First, we discuss how to find Kubo formulas for the shear and the bulk relaxation times. Next, we provide results on the shear viscosity relaxation time obtained within the diagrammatic approach for the massless λϕ4 theory.

Thermalization after holographic bilocal quench

We study thermalization in the holographic (1 + 1)-dimensional CFT after simultaneous generation of two high-energy excitations in the antipodal points on the circle. The holographic picture of such quantum quench is the creation of BTZ black hole from a collision of two massless particles. We perform holographic computation of entanglement entropy and mutual information in the boundary theory and analyze their evolution with time. We show that equilibration of the entanglement in the regions which contained one of the initial excitations is generally similar to that in other holographic quench models, but with some important distinctions. We observe that entanglement propagates along a sharp effective light cone from the points of initial excitations on the boundary. The characteristics of entanglement propagation in the global quench models such as entanglement velocity and the light cone velocity also have a meaning in the bilocal quench scenario. We also observe the loss of memory about the initial state during the equilibration process. We find that the memory loss reflects on the time behavior of the entanglement similarly to the global quench case, and it is related to the universal linear growth of entanglement, which comes from the interior of the forming black hole. We also analyze general two-point correlation functions in the framework of the geodesic approximation, focusing on the study of the late time behavior.

Generalized TMDs and the exclusive double Drell–Yan process

Generalized transverse momentum dependent parton distributions (GTMDs) are the most general parton correlation functions of hadrons. By considering the exclusive double Drell–Yan process it is shown for the first time how quark GTMDs can be measured. Specific GTMDs can be addressed by means of polarization observables.

On form factors and correlation functions in twistor space

In this paper, we continue our study of form factors and correlation functions of gauge-invariant local composite operators in the twistor-space formulation of mathcal{N}=4 super Yang-Mills theory. Using the vertices for these operators obtained in our recent papers [1, 2], we show how to calculate the twistor-space diagrams for general NkMHV form factors via the inverse soft limit, in analogy to the amplitude case. For general operators without α indices, we then reexpress the NMHV form factors from the position-twistor calculation in terms of momentum twistors, deriving and expanding on a relation between the two twistor formalisms previously observed in the case of amplitudes. Furthermore, we discuss the calculation of generalized form factors and correlation functions as well as the extension to loop level, in particular providing an argument promised in [3].

Hexagonalization of correlation functions

We propose a nonperturbative framework to study general correlation functions of single-trace operators in mathcal{N} = 4 supersymmetric Yang-Mills theory at large N . The basic strategy is to decompose them into fundamental building blocks called the hexagon form factors, which were introduced earlier to study structure constants using integrability. The decomposition is akin to a triangulation of a Riemann surface, and we thus call it hexagonalization. We propose a set of rules to glue the hexagons together based on symmetry, which naturally incorporate the dependence on the conformal and the R-symmetry cross ratios. Our method is conceptually different from the conventional operator product expansion and automatically takes into account multi-trace operators exchanged in OPE channels. To illustrate the idea in simple set-ups, we compute four-point functions of BPS operators of arbitrary lengths and correlation functions of one Konishi operator and three short BPS operators, all at one loop. In all cases, the results are in perfect agreement with the perturbative data. We also suggest that our method can be a useful tool to study conformal integrals, and show it explicitly for the case of ladder integrals.