Formula For Correlation Function Research Articles

Correlation Functions Involving Dirac Fields from Homotopy Algebras II: The Interacting Theory

Abstract We extend the formula for correlation functions of free scalar field theories and Dirac field theories in terms of quantum $A_{\infty }$ algebras presented in arXiv:2305.11634 to general scalar-Dirac systems. We obtain the result that the same formula as in the previous paper holds in this case. We show that correlation functions from our formula satisfy the Schwinger–Dyson equations. We therefore confirm that correlation functions from our formula express correlation functions from the ordinary approach of quantum field theory.

Modeling Vibrational Sum Frequency Generation Spectra of Interfacial Water on a Gold Surface: The Role of the Fermi Resonance.

Studying the hydrogen bonding structure of H2O at the metal-water interface is a highly complex yet fascinating endeavor. The intricate interactions and diverse orientations of water molecules on metal surfaces with varying potentials pose a significant challenge in elucidating the coupling between O-H stretching and H-O-H bending modes. In this study, we employed DFT-MD simulation to explore how the orientation of interfacial water molecules changes with the applied potential on the Au(111) surface. Based on the surface-specific velocity-velocity correlation function (ssVVCF) formula, we calculated vibrational sum frequency generation (VSFG) spectra for the O-H stretches. We found that three assigned peaks (∼3300, ∼3450, and 3650 cm-1) shifted toward lower frequencies as the potential moved toward more negative values. Our results align remarkably well with experimental Raman spectroscopy data. Notably, our VSFG analysis revealed a significant change in the VSFG spectra of the hydrogen-bonded O-H groups (∼3300 cm-1), switching from a negative to a positive sign with decreasing potential. This alteration suggests a substantial change in the orientation of these low-frequency O-H groups owing to their increased interactions with the Au surface. In contrast, the orientations of both the high-frequency O-H groups (∼3450 cm-1) and the dangling O-H groups (∼3650 cm-1) remained unaffected by the applied potentials. Furthermore, our analysis of the decomposed vibrational density of states (VDOS) for the H-O-H bending mode uncovered the coupling between the H-O-H bending and O-H stretching vibrations, known as the Fermi resonance. Our work suggests that the H-O-H bending vibration becomes restricted when water molecules transition from the ″one-H-down″ to the ″two-H-down″ conformation, leading to a redshift in the O-H stretching vibration through the Fermi resonance. By constructing the VSFG and decomposed VDOS spectra, we gained valuable insights into the structural changes that Raman spectra alone cannot fully interpret. Specifically, our analysis revealed the critical role of the Fermi resonance effect in shaping the spectroscopic signature of interfacial water molecules on the Au(111) surface.

Memory Tensor for Non-Markovian Dynamics with Random Hamiltonian

In the theory of open quantum systems, the Markovian approximation is very widespread. Usually, it assumes the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation for density matrix dynamics and quantum regression formulae for multi-time correlation functions. Nevertheless, now, quantum non-Markovianity is being actively studied, especially the non-Markovianity of multi-time correlations. In this work, we consider dynamics with a random Hamiltonian, which can lead to GKSL dynamics of the density matrix for some special cases, but correlation functions generally do not satisfy the quantum regression formulae. Despite the fact that random Hamiltonians have been actively studied, dynamics with such Hamiltonians has been little discussed from the viewpoint of multi-time correlations. For specific models with a random Hamiltonian, we provide the formulae for multi-time correlations which occur instead of the usual regression formulae. Moreover, we introduce and calculate the memory tensor, which characterizes multi-time correlations against the Markovian ones. We think that, despite being applied to specific models, the methods developed in this work can be used in a much broader setup.

Integrable crosscap states: from spin chains to 1D Bose gas

The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.

Wilson loops in large symmetric representations through a double-scaling limit

We derive exact formulas for circular Wilson loops in the mathcal{N} = 4 and mathcal{N} = 2* theories with gauge groups U(N) and SU(N) in the k-fold symmetrized product representation. The formulas apply in the limit of large k and small Yang-Mills coupling g, with fixed effective coupling κ ≡ g2k, and for any finite N. In the SU(2) and U(2) cases, closed analytic formulas are obtained for any k, while the 1/k series expansions are asymptotic. In the N ≫ 1 limit, with N ≪ k, there is an overlapping regime where the formulas can be confronted with results from holography. Simple formulas for correlation functions between the k-symmetric Wilson loops and chiral primary operators are also given.

Influence of Phases of Coherence Functions on the Wind Field Simulation Using Spectral Representation Method

The wave passage effect is a measure of the wave passage delay due to the apparent velocity of waves, which is one of spatially varying properties of multivariate random processes. The phase of coherence function reflects the wave passage effect of wind fields. In the wind field, simulation by the spectral representation method, the classical phase formula, is not rigorous. This may affect the accuracy of simulation results and even cause incorrect simulations. In this study, the influences of the phase on stationary and nonstationary wind field simulations are researched and discussed in detail. Two schemes containing the classical phase formula and the separated phase scheme are compared in four types of wind field simulation. The qualitative analysis based on theoretical correlation function formula is first made to study the influence of the phase. Then, four numerical examples are utilized to quantitatively study the magnitude of the influence on the sample time history and correlation function of the simulated wind field. Results show that the classical phase formula will result in considerable simulation error for all four types of wind fields because it cannot completely represent the phase angle of a complex number.

Waiting time statistics in boundary-driven free fermion chains

We study the waiting-time distributions (WTDs) of quantum chains coupled to two Lindblad baths at each end. Our focus is on free fermion chains, where we derive closed-form expressions in terms of single-particle matrices, allowing one to study arbitrarily large chain sizes. In doing so, we also derive formulas for 2-point correlation functions involving non-Hermitian propagators.

Fourth-neighbour two-point functions of the XXZ chain and the fermionic basis approach

We give a descriptive review of the fermionic basis approach to the theory of correlation functions of the XXZ quantum spin chain. The emphasis is on explicit formulae for short-range correlation functions which will be presented in a way that allows for their direct implementation on a computer. Within the fermionic basis approach a huge class of stationary reduced density matrices, compatible with the integrable structure of the model, assumes a factorized form. This means that all expectation values of local operators and all two-point functions, in particular, can be represented as multivariate polynomials in only two functions ρ and ω and their derivatives with coefficients that are rational in the deformation parameter q of the model. These coefficients are of ‘algebraic origin’. They do not depend on the choice of the density matrix, which only impacts the form of ρ and ω. As an example we work out in detail the case of the grand canonical ensemble at temperature T and magnetic field h for q in the critical regime. We compare our exact results for the fourth-neighbour two-point functions with asymptotic formulae for h, T = 0 and for finite h and T.

Scattering equations in AdS: scalar correlators in arbitrary dimensions

We introduce a bosonic ambitwistor string theory in AdS space. Even though the theory is anomalous at the quantum level, one can nevertheless use it in the classical limit to derive a novel formula for correlation functions of boundary CFT operators in arbitrary space-time dimensions. The resulting construction can be treated as a natural extension of the CHY formalism for the flat-space S-matrix, as it similarly expresses tree-level amplitudes in AdS as integrals over the moduli space of Riemann spheres with punctures. These integrals localize on an operator-valued version of scattering equations, which we derive directly from the ambitwistor string action on a coset manifold. As a testing ground for this formalism we focus on the simplest case of ambitwistor string coupled to two cur- rent algebras, which gives bi-adjoint scalar correlators in AdS. In order to evaluate them directly, we make use of a series of contour deformations on the moduli space of punctured Riemann spheres and check that the result agrees with tree level Witten diagram computations to all multiplicity. We also initiate the study of eigenfunctions of scattering equations in AdS, which interpolate between conformal partial waves in different OPE channels, and point out a connection to an elliptic deformation of the Calogero-Sutherland model.

Integral Formulas and Antisymmetrization Relations for the Six-Vertex Model

We study the relationship between various integral formulas for nonlocal correlation functions of the six-vertex model with domain wall boundary conditions. Specifically, we show how the known representation for the emptiness formation probability can be derived from that for the so-called row configuration probability. A crucial ingredient in the proof is a relation expressing the result of antisymmetrization of some given function with respect to permutations in two sets of its variables in terms of the Izergin-Korepin partition function. This relation generalizes another one obtained by Tracy and Widom in the context of the asymmetric simple exclusion process.

Twin prime correlations from the pair correlation of Riemann zeros

We establish, via a heuristic Fourier inversion calculation, that the Hardy–Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height E on the critical line. Previously it was known that the Hardy–Littlewood conjecture implies the pair correlation formula, and we show that the reverse implication also holds. An averaged form of the Hardy–Littlewood conjecture is obtained by inverting the limit of the two-point correlation function and the precise form of the conjecture is found by including asymptotically lower order terms in the two-point correlation function formula.

Conformal field theory on top of a breathing one-dimensional gas of hard core bosons

The recent results of [J. Dubail, J.-M. Stéphan, J. Viti, P. Calabrese, Scipost Phys. 2, 002 (2017)], which aim at providing access to large scale correlation functions of inhomogeneous critical one-dimensional quantum systems —e.g. a gas of hard core bosons in a trapping potential— are extended to a dynamical situation: a breathing gas in a time-dependent harmonic trap. Hard core bosons in a time-dependent harmonic potential are well known to be exactly solvable, and can thus be used as a benchmark for the approach. An extensive discussion of the approach and of its relation with classical and quantum hydrodynamics in one dimension is given, and new formulas for correlation functions, not easily obtainable by other methods, are derived. In particular, a remarkable formula for the large scale asymptotics of the bosonic \boldsymbol{n}𝐧-particle function \boldsymbol{\left< \Psi^\dagger (x_1,t_1) \dots \Psi^\dagger (x_n,t_n) \Psi(x_1',t_1') \dots \Psi(x_n',t_n') \right>} is obtained. Numerical checks of the approach are carried out for the fermionic two-point function —easier to access numerically in the microscopic model than the bosonic one— with perfect agreement.

Product Matrix Processes as Limits of Random Plane Partitions

AbstractWe consider a random process with discrete time formed by squared singular values of products of truncations of Haar-distributed unitary matrices. We show that this process can be understood as a scaling limit of the Schur process, which gives determinantal formulas for (dynamical) correlation functions and a contour integral representation for the correlation kernel. The relation with the Schur processes implies that the continuous limit of marginals for q-distributed plane partitions coincides with the joint law of squared singular values for products of truncations of Haar-distributed random unitary matrices. We provide structural reasons for this coincidence that may also extend to other classes of random matrices.

Exact signal correlators in continuous quantum measurements

This article provides an exact formula for the signal n-point correlation functions of detectors continuously measuring an arbitrary quantum system, in the presence of detection imperfections. The derivation uses only continuous stochastic calculus techniques, but the final result is easily understood from a discrete picture of repeated interactions with qubits or from a parallel with continuous matrix product states. This result provides a crude yet efficient a way to estimate system parameters directly from experimental data, without requiring non-linear state reconstruction.

Numerical analysis of longitudinal ultrasonic attenuation in sintered materials using a simplified two-phase model

Techniques of quantitative nondestructive evaluation using attenuation of ultrasonic waves have been proposed as a potential tool for monitoring sintering processes because of the direct connection between the changes of wave propagation characteristics and microstructure properties. However, the influence of these changes during sintering on sound propagation remains unclear. In addition to theoretical investigations, numerical models can be utilized to provide key information for interpreting experimental data quantitatively. In this article, a simplified two-phase model using Voronoi polycrystals is applied to study wave propagation through sintered materials. Finite element simulations are developed with various material and geometric parameters of the two-phase model. Example longitudinal attenuation results are obtained and compared with the scattering theory for different input wave frequencies. The comparison of the numerical results with the theory shows the dependence of the attenuation on the parameters of the correlation function and the two-phase geometry. The results also validate the correlation function formula used in the theory. The influence of the input wave frequency and material properties on the correlation lengths is also discussed. Such numerical models can be used to verify theoretical models efficiently and to design further experimental methods for characterization of microstructures.

Cluster algebras based on vertex operator algebras

Starting from Zhu recursion formulas for correlation functions for vertex operator algebras with formal parameters associated to local coordinates around marked points on a Riemann surfaces, we introduce a cluster algebra structure over a noncommutative set of variables. Cluster elements and mutation rules are explicitly defined. In particular, we propose an elliptic version of vertex operator cluster algebras arising from correlation functions and Zhu reduction procedure for vertex operators on the torus.

Two-point Correlation Function and Feynman-Kac Formula for the Stochastic Heat Equation

In this paper, we obtain an explicit formula for the two-point correlation function for the solutions to the stochastic heat equation on \(\mathbb {R}\). The bounds for p-th moments proved in Chen and Dalang (Ann. Probab. 2015) are simplified. We validate the Feynman-Kac formula for the p-point correlation function of the solutions to this equation with measure-valued initial data.

Quantum-critical scaling of fidelity in 2D pairing models

The laws of quantum-critical scaling theory of quantum fidelity, dependent on the underlying system dimensionality D, have so far been verified in exactly solvable 1D models, belonging to or equivalent to interacting, quadratic (quasifree), spinless or spinfull, lattice-fermion models. The obtained results are so appealing that in quest for correlation lengths and associated universal critical indices ν, which characterize the divergence of correlation lengths on approaching critical points, one might be inclined to substitute the hard task of determining an asymptotic behavior at large distances of a two-point correlation function by an easier one, of determining the quantum-critical scaling of the quantum fidelity. However, the role of system's dimensionality has been left as an open problem. Our aim in this paper is to fill up this gap, at least partially, by verifying the laws of quantum-critical scaling theory of quantum fidelity in a 2D case. To this end, we study correlation functions and quantum fidelity of 2D exactly solvable models, which are interacting, quasifree, spinfull, lattice-fermion models. The considered 2D models exhibit new, as compared with 1D ones, features: at a given quantum-critical point there exists a multitude of correlation lengths and multiple universal critical indices ν, since these quantities depend on spatial directions, moreover, the indices ν may assume larger values. These facts follow from the obtained by us analytical asymptotic formulae for two-point correlation functions. In such new circumstances we discuss the behavior of quantum fidelity from the perspective of quantum-critical scaling theory. In particular, we are interested in finding out to what extent the quantum fidelity approach may be an alternative to the correlation-function approach in studies of quantum-critical points beyond 1D.

Correlation functions in resonance fluorescence with spectral resolution: Signal-processing approach

In the framework of the signal processing approach to single-atom resonance fluorescence with spectral resolution, we diagrammatically derive an analytical formula for arbitrary-order spectral correlation functions of the scattered fields that pass through Fabry-Perot interferometers. Our general expression is then applied to study correlation signals in the limit of well separated spectral lines of the resonance fluorescence spectrum. In particular, we study the normalized second-order temporal intensity correlation functions in the case of the interferometers tuned to the components of the spectrum and obtain interferential corrections to the approximate results derived in the secular limit. In addition, we explore purely spectral correlations and show that they can fully be understood in terms of the two-photon cascades down the dressed state ladder.

Non-Markovian correlation functions for open quantum systems

Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. We approach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non-Markovian characteristics are demonstrated.