Observed Correlation Function Research Articles

Simulating two-dimensional correlation spectroscopies with third-order infrared and fifth-order infrared-Raman processes of liquid water.

To investigate the possibility of measuring the intermolecular and intramolecular anharmonic coupling of balk water, we calculate third-order two-dimensional (2D) infrared spectra and fifth-order 2D IR-IR-Raman-Raman spectra expressed in terms of four-body correlation functions of optical observables. For this purpose, a multimode Brownian oscillator model of four interacting anharmonic oscillators strongly coupled to their respective heat baths is employed. The nonlinearity of system-bath interactions is considered to describe thermal relaxation and vibrational dephasing. The linear and nonlinear spectra are then computed in a non-Markovian and nonperturbative regime in a rigorous manner using discretized hierarchical equations of motion in mixed Liouville-Wigner space. The calculated 2D spectra for stretching-bending, bending-librational, stretching-librational, and stretching-translational modes consist of various positive and negative peaks exhibiting essential details of intermolecular and intramolecular mode-mode interactions under thermal relaxation and dephasing at finite temperature.

Dynamics of a lattice 2-group gauge theory model

We study a simple lattice model with local symmetry, whose construction is based on a crossed module of finite groups. Its dynamical degrees of freedom are associated both to links and faces of a four-dimensional lattice. In special limits the discussed model reduces to certain known topological quantum field theories. In this work we focus on its dynamics, which we study both analytically and using Monte Carlo simulations. We prove a factorization theorem which reduces computation of correlation functions of local observables to known, simpler models. This, combined with standard Krammers-Wannier type dualities, allows us to propose a detailed phase diagram, which form is then confirmed in numerical simulations. We describe also topological charges present in the model, its symmetries and symmetry breaking patterns. The corresponding order parameters are the Polyakov loop and its generalization, which we call a Polyakov surface. The latter is particularly interesting, as it is beyond the scope of the factorization theorem. As shown by the numerical results, expectation value of Polyakov surface may serve to detects all phase transitions and is sensitive to a value of the topological charge.

Many-body quantum chaos and dual-unitarity round-a-face.

We propose a new type of locally interacting quantum circuits-quantum cellular automata-that are generated by unitary interactions round-a-face (IRF). Specifically, we discuss a set (or manifold) of dual-unitary IRFs with local Hilbert space dimension d [DUIRF (d)], which generate unitary evolutions both in space and time directions of an extended 1+1 dimensional lattice. We show how arbitrary dynamical correlation functions of local observables can be evaluated in terms of finite-dimensional completely positive trace preserving unital maps in complete analogy to recently studied circuits made of dual-unitary brick gates (DUBGs). The simplest non-vanishing local correlation functions in dual-unitary IRF circuits are shown to involve observables non-trivially supported on two neighboring sites. We completely characterize the ten-dimensional manifold of DUIRF (2) for qubits ( d=2) and provide, for d=3,4,…,7, empirical estimates of its dimensionality based on numerically determined dimensions of tangent spaces at an ensemble of random instances of dual-unitary IRF gates. In parallel, we apply the same algorithm to determine dimDUBG(d) and show that they are of similar order though systematically larger than dimDUIRF(d) for d=2,3,…,7. It is remarkable that both sets have a rather complex topology for d≥3 in the sense that the dimension of the tangent space varies among different randomly generated points of the set. Finally, we provide additional data on dimensionality of the chiral extension of DUBG circuits with distinct local Hilbert spaces of dimensions d≠d' residing at even/odd lattice sites.

Wilson loops and area laws in lattice gauge theory tensor networks

Tensor network states have been a very prominent tool for the study of quantum many-body physics, thanks to their physically relevant entanglement properties and their ability to encode symmetries. In the last few years, the formalism has been extended and applied to theories with local symmetries to - lattice gauge theories. In the contraction of tensor network states as well as correlation functions of physical observables with respect to them, one uses the so-called transfer operator, whose local properties dictate the long-range behaviour of the state. In this work we study transfer operators of tensor network states (in particular, PEPS - projected entangled pair states) in the context of lattice gauge theories, and consider the implications of the local symmetry on their structure and properties. We focus on the Wilson loop - a nonlocal, gauge-invariant observable which is central to pure gauge theories, whose long range decay behaviour probes the confinement or deconfinement of static charges. Using the symmetry, we show how to handle its contraction, and formulate conditions relating local properties to its decay fashion.

Angular clustering and host halo properties of [O ii] emitters at z > 1 in the Subaru HSC survey

Abstract We study the angular correlation function of star-forming galaxies and properties of their host dark matter halos at z > 1 using the Hyper-Suprime Cam (HSC) Subaru Stragetic Program (SSP) survey. We use [O ii] emitters identified using two narrow-band (NB) filters, NB816 and NB921, in the Deep/UltraDeep layers, which respectively cover large angular areas of 16.3 deg2 and 16.9 deg2. Our sample contains 8302 and 9578 [O ii] emitters at z = 1.19 (NB816) and z = 1.47 (NB921), respectively. We detect a strong clustering signal over a wide angular range, $0{_{.}^{\circ}} 001$ < θ < 1 °, with bias $b=1.61^{+0.13}_{-0.11}$ (z = 1.19) and $b=2.09^{+0.17}_{-0.15}$ (z = 1.47). We also find a clear deviation of the correlation from a simple power-law form. To interpret the measured clustering signal, we adopt a halo occupation distribution (HOD) model that is constructed to explain the spatial distribution of galaxies selected by star formation rate. The observed correlation function and number density are simultaneously explained by the best-fitting HOD model. From the constrained HOD model, the average mass of halos hosting the [O ii] emitters is derived to be $\log {M_{\rm eff}/(h^{-1}\, {M}_{\odot })}=12.70^{+0.09}_{-0.07}$ and $12.61^{+0.09}_{-0.05}$ at z = 1.19 and 1.47, respectively, which will become halos with the present-day mass M ∼ 1.5 × 1013 h−1 M⊙. The satellite fraction of the [O ii] emitter sample is found to be fsat ∼ 0.15. All these values are consistent with previous studies of similar samples, but we obtain tighter constraints even in a larger parameter space due to the larger sample size from the HSC. The results obtained for host halos of [O ii] emitters in this paper enable the construction of mock galaxy catalogs and the systematic forecast study of cosmological constraints from upcoming emission line galaxy surveys such as the Subaru Prime Focus Spectrograph survey.

Chaos and Ergodicity in Extended Quantum Systems with Noisy Driving.

We study the time-evolution operator in a family of local quantum circuits with random fields in a fixed direction. We argue that the presence of quantum chaos implies that at large times the time-evolution operator becomes effectively a random matrix in the many-body Hilbert space. To quantify this phenomenon, we compute analytically the squared magnitude of the trace of the evolution operator-the generalized spectral form factor-and compare it with the prediction of random matrix theory. We show that for the systems under consideration, the generalized spectral form factor can be expressed in terms of dynamical correlation functions of local observables in the infinite temperature state, linking chaotic and ergodic properties of the systems. This also provides a connection between the many-body Thouless time τ_{th}-the time at which the generalized spectral form factor starts following the random matrix theory prediction-and the conservation laws of the system. Moreover, we explain different scalings of τ_{th} with the system size observed for systems with and without the conservation laws.

Analyzing clustering of astrophysical gravitational-wave sources: luminosity-distance space distortions

We present a formulation of observed number density fluctuations of gravitational-wave (GW) sources in a three dimensional space. In GW observations, redshift identification for each GW source is a challenging issue, in particular, for high redshift sources. The use of observed luminosity distance as a distance indicator will be a simple yet optimal way for measuring the clustering signal. We derive the density fluctuations of GW sources estimated from observed luminosity distance and sky position of each source. The density fluctuations are distorted as similar to the so-called redshift space distortions in galaxy surveys but with several differences. We then show the two-point correlation function and multipole power spectrum in the presence of the distortion effect. We find that the line-of-sight derivative of the lensing convergence, which does not appear in the redshift-space distortions, leads to significant distortions in the observed correlation function. In addition, the lensing effect affects higher-order multipole power spectra and its signal-to-noise at high redshifts.

Effect of memory on the violation of Leggett–Garg inequality

The Leggett-Garg inequalities impose restrictions on the values taken by some combinations of the two-time correlation functions of observables in order to be explainable by a noninvasive-realist classical model. While in the unitary dynamics, it is straightforward to compute these correlation functions, open system effects bring in subtleties. Specifically, for non-Markovian dynamics, which involves setting up of system-bath correlations, the Leggett-Garg measurements disrupt these correlations, making a full system-bath Hamiltonian approach natural. However, here we point out that the problem can also be dealt with from a reduced dynamics perspective. The key point is that the noise superoperator acting on the system must be suitably updated after measurement interventions. Also considered is the effect of Markovian versus non-Markovian behavior as well as classically non-Markovian processes on the violation of Leggett-Garg inequalities.

Matrix product state of multi-time correlations

For an interacting spatio-temporal lattice system we introduce a formal way of expressing multi-time correlation functions of local observables located at the same spatial point with a time state, i.e. a statistical distribution of configurations observed along a time lattice. Such a time state is defined with respect to a particular equilibrium state that is invariant under space and time translations. The concept is developed within the rule 54 reversible cellular automaton, for which we explicitly construct a matrix product form of the time state, with matrices that act on the three-dimensional auxiliary space. We use the matrix-product state to express equal-space time-dependent density-density correlation function, which, for special maximum-entropy values of equilibrium parameters, agrees with the previous results. Additionally, we obtain an explicit expression for the probabilities of observing all multi-time configurations, which enables us to study distributions of times between consecutive excitations and prove the absence of decoupling of timescales in the rule 54 model.

Anisotropic fluid cosmology: An alternative to dark matter?

We use anisotropic fluid cosmology to describe the present, dark energy-dominated, universe. Similarly to what has been proposed for galactic dynamics, the anisotropic fluid gives an effective description of baryonic matter, dark energy and their possible interaction, without assuming the presence of dark matter. The resulting anisotropic fluid spacetime naturally generates inhomogeneities at small scales, triggered by an anisotropic stress, and could therefore be responsible for structure formation at these scales. Solving the cosmological equations, we show that the dynamics of the scale factor $a$ is described by usual FLRW cosmology and decouples completely from that describing inhomogeneities. We assume that the cosmological anisotropic fluid inherits the equation of state from that used to describe galaxy rotation curves. We show that, in the large scale regime, the fluid can be described as a generalized Chaplygin gas and fits well the distance modulus experimental data of type Ia supernovae, thus correctly modelling the observed accelerated expansion of the universe. Conversely, in the small scale regime, we use cosmological perturbation theory to derive the power spectrum $P(k)$ for mass density distribution. At short wavelengths, we find a $1/k^4$ behaviour, in good accordance with the observed correlation function for matter distribution at small scales.

Two-Dimensional Correlation Function of Binary Black Hole Coalescences

We compute the two-dimensional correlation functions of the binary black hole coalescence detections in LIGO-Virgo’s first and second observation runs. The sky distribution of binary black hole coalescence events is tested for correlations at different angular scales by comparing the observed correlation function to two reference functions that are obtained from mock datasets of localization error regions uniformly distributed in the sky. No excess correlation at any angular scale is found. The power-law slope of the correlation function is estimated to be γ = 2.24 ± 0.33 at the three- σ confidence level, a value consistent with the measured distribution of galaxies.

Space-like dynamics in a reversible cellular automaton

In this paper we study the space evolution in the Rule 54 reversible cellular automaton, which is a paradigmatic example of a deterministic interacting lattice gas. We show that the spatial translation of time configurations of the automaton is given in terms of local deterministic maps with the support that is small but bigger than that of the time evolution. The model is thus an example of space-time dual reversible cellular automaton, i.e. its dual is also (in general different) reversible cellular automaton. We provide two equivalent interpretations of the result; the first one relies on the dynamics of quasi-particles and follows from an exhaustive check of all the relevant time configurations, while the second one relies on purely algebraic considerations based on the circuit representation of the dynamics. Additionally, we use the properties of the local space evolution maps to provide an alternative derivation of the matrix product representation of multi-time correlation functions of local observables positioned at the same spatial coordinate.

Corrfunc – a suite of blazing fast correlation functions on the CPU

ABSTRACT The two-point correlation function (2PCF) is the most widely used tool for quantifying the spatial distribution of galaxies. Since the distribution of galaxies is determined by galaxy formation physics as well as the underlying cosmology, fitting an observed correlation function yields valuable insights into both. The calculation for a 2PCF involves computing pair-wise separations and consequently, the computing time-scales quadratically with the number of galaxies. The next-generation galaxy surveys are slated to observe many millions of galaxies, and computing the 2PCF for such surveys would be prohibitively time-consuming. Additionally, modern modelling techniques require the 2PCF to be calculated thousands of times on simulated galaxy catalogues of at least equal size to the data and would be completely unfeasible for the next-generation surveys. Thus, calculating the 2PCF forms a substantial bottleneck in improving our understanding of the fundamental physics of the Universe, and we need high-performance software to compute the correlation function. In this paper, we present corrfunc – a suite of highly optimized, openmp parallel clustering codes. The improved performance of corrfunc arises from both efficient algorithms as well as software design that suits the underlying hardware of modern CPUs. corrfunc can compute a wide range of 2D and 3D correlation functions in either simulation (Cartesian) space or on-sky coordinates. corrfunc runs efficiently in both single- and multithreaded modes and can compute a typical two-point projected correlation function [wp(rp)] for ∼1 million galaxies within a few seconds on a single thread. corrfunc is designed to be both user-friendly and fast and is publicly available at https://github.com/manodeep/Corrfunc.

Renormalization and BRST Symmetry in Donaldson–Witten Theory

The presence of a BRST symmetry in topologically twisted gauge theories makes a precise analysis of these theories feasible. While the global BRST symmetry suggests that correlation functions of BRST exact observables vanish, this decoupling might be obstructed due to a contribution from the boundary of field space. Motivated by divergent BRST exact observables on the Coulomb branch of Donaldson-Witten theory, we put forward a new prescription for the renormalization of correlation functions on the Coulomb branch. This renormalization is based on the relation between Coulomb branch integrals and integrals over a modular fundamental domain, and establishes that BRST exact observables indeed decouple in Donaldson-Witten theory.

Determining a local Hamiltonian from a single eigenstate

We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is ``yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the ``k-correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.

Some statistical properties of almost Anosov diffeomorphisms

Abstract For a kind of almost Anosov diffeomorphisms, we study the relationship among the existence of Sinai-Ruelle-Bowen (SRB) measures, the local differentiability near the indifferent fixed points, and space dimension, where the almost Anosov diffeomorphisms are hyperbolic everywhere except for the indifferent fixed points. As a consequence, there are C2 almost Anosov diffeomorphisms that admit σ-finite (infinite) SRB measures in spaces with dimensions bigger than one; there exist C2 almost Anosov diffeomorphisms with finite SRB measures in spaces with dimensions bigger than three. Further, we obtain the lower and upper polynomial bounds for the decay rates of the correlation functions of the Holder observables for the maps admitting finite SRB measures.

Spin correlation function theory of spin-charge conversion effects

Theory of spin-charge conversion effects in spintronics are presented in terms of correlation functions of physical observables, spin and electric current. Direct and inverse spin Hall effects and spin pumping effect are studied considering metallic systems with random spin-orbit interaction and spatially nonuniform Rashba interaction. The theory is free from ambiguity associated with spin current, and provides clear physical picture of the spin-charge conversion effects. In the present approach, so-called the spin current transmission efficiency is essentially the nonuniform component of magnetic susceptibility.

Black holes from large N singlet models

The emergent nature of spacetime geometry and black holes can be directly probed in simple holographic duals of higher spin gravity and tensionless string theory. To this end, we study time dependent thermal correlation functions of gauge invariant observables in suitably chosen free large N gauge theories. At low temperature and on short time scales the correlation functions encode propagation through an approximate AdS spacetime while interesting departures emerge at high temperature and on longer time scales. This includes the existence of evanescent modes and the exponential decay of time dependent boundary correlations, both of which are well known indicators of bulk black holes in AdS/CFT. In addition, a new time scale emerges after which the correlation functions return to a bulk thermal AdS form up to an overall temperature dependent normalization. A corresponding length scale was seen in equal time correlation functions in the same models in our earlier work.

Exponential decay of correlations functions in MIXMAX generator of pseudorandom numbers

We are developing further our earlier suggestion to use high entropy Anosov C-systems for the Monte-Carlo simulations. The hyperbolic Anosov C-systems have exponential instability of their trajectories and as such have mixing of all orders and nonzero Kolmogorov entropy. Of special interest are C-systems that are defined on a high dimensional torus. The C-systems on a torus are perfect candidates to be used for Monte-Carlo simulations. The correlation functions of the physical observables which are defined on a torus phase space are tend to zero and become uncorrelated exponentially fast. It is important to specify the parameters of a dynamical C-system which quantify the exponential decay. We have found that the upper bound on the rate of the exponential decay of the correlation functions universally depends on the value of the system entropy. This result allows to define decorrelation and relaxation times in terms of entropy and characterise the statistical properties of the MIXMAX generator.

Semiclassical asymptotics of $$\mathbf {GL}_N({\mathbb {C}})$$GLN(C) tensor products and quantum random matrices

The Littlewood--Richardson process is a discrete random point process arising from the isotypic decomposition of tensor products of irreducible representations of $\operatorname{GL}_N(\mathbb{C})$. Biane--Perelomov--Popov matrices are quantum random matrices obtained as the geometric quantization of random Hermitian matrices with deterministic eigenvalues and uniformly random eigenvectors. As first observed by Biane, correlation functions of certain global observables of the LR process coincide with correlation functions of linear statistics of sums of classically independent BPP matrices, thereby enabling a random matrix approach to the statistical study of $\operatorname{GL}_N(\mathbb{C})$ tensor products. In this paper, we prove an optimal result: classically independent BPP matrices become freely independent in any semiclassical/large-dimension limit. This proves and generalizes a conjecture of Bufetov and Gorin, and leads to a Law of Large Numbers for the BPP observables of the LR process which holds in any and all semiclassical scalings.