Static replica approach to critical correlations in glassy systems

Reconstruction of 2-D porous media using Karhunen–Lóeve expansion

The basic structure of quantum field theory that is used to describe the Standard Model of fundamental interactions of nature is usually formulated for zero temperature. However, the effects of temperature are extremely important for understanding a number of physical processes such as the electro-weak phase transition and quark-gluon plasma. Thermal field theory is the extension of quantum field theory to a non-zero temperature environment and is achieved by modifying the propagators in loop integrations represented by Feynman diagrams. The Python program pySecDec numerically calculates dimensionally-regularized loop integrals in quantum field theory using the sector decomposition approach. It is shown how pySecDec can be applied to thermal field theory numerical calculations using modifications within the Matsubara formalism. Using the formulated algorithm, a 2-point correlation function (such as those occurring in QCD correlation functions) at finite temperature can be numerically calculated for a variety of spacetime dimensions.

Exact three spin correlation function relations for the square and the honeycomb Ising lattices

We consider the AdS/dS CFT correspondence and study the nature of the thermal bath of the de Sitter field theory using holography. Unlike the temperature of a thermal field theory in flat spacetime, the temperature of a superconformal field theory on de Sitter space is an integral part of the theory and leaves intact the conformal symmetry and supersymmetry. In the dual AdS side, there is no black hole. Instead we have cosmological expansion of the de Sitter factor. We consider a number of different observables, such as the entanglement entropy, two point correlation function, Wilson loops corresponding to static and spinning mesons in the field theory, and study their thermal properties using holography. The former two quantities have trivial temperature dependence due to conformal symmetry. We compute the energy of the quark anti-quark bound state for a static meson, as well as the energy and the angular momentum for a spinning meson. We find that there is a maximum distance, as well as a maximum spin for the latter case, beyond which the bound state become unstable. The temperature behavior of the physical quantities in these meson systems are similar to that of the usual thermal field theory with holographic black hole dual. With these examples, we show clearly how the field theory observables get their thermal properties from the bulk despite the absence of a black hole, with the role of the black hole horizon played by the cosmological expansion of the de Sitter factor of the AdS metric.

The Energy Density Functional (EDF) theory is extremely successful within the effective force approach, noticeably the Skyrme or Gogny forces, in reproducing the nuclear binding energies and other nuclear properties along the whole mass table. The EDF is in this case represented formally as the Hartree-Fock (HF) mean field of an effective force, and the associated single particle states can be considered the eigenstates of the corresponding effective mean field. In general, the phenomenological single particle spectrum is not accurately reproduced. To overcome this difficulty one can improve the functional in order to incorporate in the mean field additional correlations in an effective way. In an alternative viable scheme one can introduce explicitly many-body correlations which affect the single particle motion at both dynamic and static levels. In particular, the particle-vibration coupling scheme modifies the positions of the individual single particle energies. In this case one also introduce the fragmentation of the single particle states, a feature that cannot be described at the mean field level. At the same time the so-introduced correlations modify the static single particle potential, and the single particle states that diagonalize the corresponding density matrix are not the orbitals of a mean field with occupation numbers 1 (occupied orbitals) and 0 (unoccupied orbitals) anymore. In this paper we show that both static and dynamic effects on the single particle motion can be introduced within a previously developed scheme, based on the conserving approximations, and that the static part can be identified with the so-called tadpole term introduced by Khodel in the self-consistent theory of finite Fermi systems. The treatment remains at the formal level, but we hope that several aspects of the particle-vibration coupling scheme will be clarified.

The two-dimensional one-component logarithmic Coulomb gas is mapped onto a non-hermitian fermionic field theory. At $\beta=2$, the field theory is free. Correlation functions are calculated and a perturbation theory is discussed for extending to other $\beta$. A phase transition is found at the mean-field level at large $\beta$. Some results are extended to spaces of constant negative curvature.

For many purposes the dynamics of a many body system are adequately described by the familiar autocorrelation function \( S\left( {\vec q,w} \right) \). \( S\left( {\vec q,w} \right) \) is the Fourier transform of \(\left\langle {u\left( {\vec r,t} \right)u\left( {\vec r't'} \right)} \right\rangle\), where \(u\left( {\vec r,t} \right)\) represents some characteristic dynamic variable such as spin, density, etc. Scattering experiments using light, neutrons, electrons or x-rays, are routinely directed toward the measurement of \( S\left( {\vec q,w} \right) \). However in recent years it has become increasingly appreciated that such scattering probes may also couple strongly to higher powers of the dynamic variable and may thus explore more complex dynamic correlation functions than \( S\left( {\vec q,w} \right) \). Indeed the operation of certain selection rules, particularly for light scattering, often permits rather direct and unambiguous measurement of the four point correlation function \( \left\langle {u\left( {{{\vec r}_1}{t_1}} \right)u\left( {{{\vec r}_2}{t_2}} \right)u\left( {{{\vec r}_3}{t_3}} \right)u\left( {{{\vec r}_4}{t_4}} \right)} \right\rangle \). This type of function contains information on the interactions among those elementary excitations, which are described to lowest order by \( S\left( {\vec q,w} \right) \). In addition, various limits of this function determine such quantities as specific heat, acoustic attenuation, and other transport properties.

We present a field theory of Jain's composite fermion model as generalised to the bilayer quantum Hall systems. We define operators which create composite fermions and write the Hamiltonian exactly in terms of these operators. This is seen to be a complexified version of the familiar Chern Simons theory. In the mean-field approximation, the composite fermions feel a modified effective magnetic field exactly as happens in usual Chern Simons theories, and plateaus are predicted at the same values of filling factors as Lopez and Fradkin and Halperin . But unlike normal Chern Simons theories, we obtain all features of the first-quantised wavefunctions including its phase, modulus and correct gaussian factors at the mean field level. The familiar Jain relations for monolayers and the Halperin wavefunction for bilayers come out as special cases.

We have investigated the redshift space distortions in the optically selected Durham/UKST Galaxy Redshift Survey using the 2-point galaxy correlation function perpendicular and parallel to the observer’s line of sight, �(�,�). We present results for the real space 2-point correlation function, �(r), by inverting the optimally estimated projected correlation function, which is obtained by integration of �(�,�), and find good agreement with other real space estimates. On small, non-linear scales we observe an elongation of the constant �(�,�) contours in the line of sight direction. This is due to the galaxy velocity dispersion and is the common “Finger of God” effect seen in redshift surveys. Our result for the one-dimensional pairwise rms velocity dispersion is 1/2 = 416 ± 36kms 1 which is consistent with those from recent redshift surveys and canonical values, but inconsistent with SCDM or LCDM models. On larger, linear scales we observe a compression of the �(�,�) contours in the line of sight direction. This is due to the infall of galaxies into overdense regions and the Durham/UKST data favours a value of ( 0.6 /b)�0.5, where is the mean mass density of the Universe and b is the linear bias factor which relates the galaxy and mass distributions. Comparison with other optical estimates yield consistent results, with the conclusion that the data does not favour an unbiased critical-density universe.

We compute exactly various 4−point correlation functions of shortest scalar operators in bi-scalar planar four-dimensional “fishnet” CFT. We apply the OPE to extract from these functions the exact expressions for the scaling dimensions and the structure constants of all exchanged operators with an arbitrary Lorentz spin. In particular, we determine the conformal data of the simplest unprotected two-magnon operator analogous to the Konishi operator, as well as of the one-magnon operator. We show that at weak coupling 4−point correlation functions can be systematically expanded in terms of harmonic polylogarithm functions and verify our results by explicit calculation of Feynman graphs at a few orders in the coupling. At strong coupling we obtain that the correlation functions exhibit the scaling behaviour typical for semiclassical description hinting at the existence of the holographic dual.

It is shown that certain sum rule identities exist which relate correlation functions for $n$ Potts spins on the boundary of a planar lattice for $n\geq 4$. Explicit expressions of the identities are obtained for $n=4,5$. It is also shown that the identities provide the missing link needed for a complete determination of the duality relation for the $n$-point correlation function. The $n=4$ duality relation is obtained explicitly. More generally we deduce the number of correlation identities for any $n$ as well as an inversion relation and a conjecture on the general form of the duality relation.

The spatial two point galaxy correlation function, ξ(r), is, at present, the most useful statistic for comparing theoretical models to observational data. We have derived an expression for the dynamical evolution of ξ for structures arising from Gaussian initial conditions under the assumption that non-linear evolution may be described by the Zel’dovich approximation. The observed angular correlation function of galaxies, ω(θ), may place useful constraints on the spectrum of initial fluctuations on large scales. The angular correlation function due to the growth of structure from Gaussian fluctuations with the Cold Dark Matter spectrum is calculated. It is shown that ω g for the Ω = 1 biassed CDM model exhibits a gentle falloff near the position of the break found by Groth and Peebles in the Shane-Wirtanen catalogue.

In recent years, there have been two independent but related developments in the study of irrelevant deformations in two dimensional quantum field theories (QFTs). The first development is the deformation of a two dimensional QFT by the determinant of the energy momentum stress tensor, commonly referred to as $T{\bar T}$ deformation. The second development is in two dimensional holographic field theories which are dual to string theory in asymptotically Anti-de Sitter (AdS) spacetimes. In this latter development, the deformation is commonly referred to as single-trace $T{\bar T}$ deformation. The single-trace $T{\bar T}$ deformation corresponds in the bulk to a string background that interpolates between AdS spacetime in the infrared (IR) and a linear dilaton spacetime (vacuum of little string theory (LST)) in the ultraviolet (UV). It serves as a useful tool and guide to better understand and explore holography in asymptotically AdS and non-AdS spacetimes in a controlled setting. In particular, it is useful to gain insights into holography in flat spacetimes. The dissertation is devoted to the study of single-trace $T{\bar T}$ deformation and its single-trace generalizations in theories with $U(1)$ currents, namely $J\bar T$ and $T\bar J$ deformations, in the context of gauge/gravity duality. In the dissertation I present new results in the study of holography in asymptotically non-AdS spacetimes. I discuss two point correlation functions in single-trace $T{\bar T}$ deformation, and entanglement entropy and entropic $c$-function in single-trace $T{\bar T}$, $J\bar T$ and $T\bar J$ deformations.

The goal of this chapter is to review recent analytical results about the growth of a (static) correlation length in glassy systems, and the connection that can be made between this length scale and the equilibrium correlation time of its dynamics. The definition of such a length scale is first given in a generic setting, including finite‐dimensional models, along with rigorous bounds linking it to the correlation time. We then present some particular cases (finite connectivity mean‐field models, and Kac limit of finite‐dimensional systems) where this length can be actually computed.

We discuss the stability of semiclassical gravity solutions with respect to small quantum corrections by considering the quantum fluctuations of the metric perturbations around the semiclassical solution. We call the attention to the role played by the symmetrized 2-point quantum correlation function for the metric perturbations, which can be naturally decomposed into two separate contributions: intrinsic and induced fluctuations. We show that traditional criteria on the stability of semiclassical gravity are incomplete because these criteria based on the linearized semiclassical Einstein equation can only provide information on the expectation value and the intrinsic fluctuations of the metric perturbations. By contrast, the framework of stochastic semiclassical gravity provides a more complete and accurate criterion because it contains information on the induced fluctuations as well. The Einstein–Langevin equation therein contains a stochastic source characterized by the noise kernel (the symmetrized 2-point quantum correlation function of the stress tensor operator) and yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. These points are illustrated with the example of Minkowski space-time as a solution to the semiclassical Einstein equation, which is found to be stable under both intrinsic and induced fluctuations.