Large-distance Asymptotic Behavior Research Articles

Effective interaction between guest charges immersed in 2D jellium

The model under study is an infinite 2D jellium of pointlike particles with elementary charge e, interacting via the logarithmic potential and in thermal equilibrium at the inverse temperature β. Two cases of the coupling constant Γ≡βe2 are considered: the Debye–Hückel limit Γ→0 and the free-fermion point Γ=2 . In the most general formulation, two guest particles, the one with charge qe (the valence q being an arbitrary integer) and the hard core of radius σ > 0 and the pointlike one with elementary charge e, are immersed in the bulk of the jellium at distance d⩾σ . Two problems are of interest: the asymptotic large-distance behavior of the excess charge density induced in the jellium and the effective interaction between the guest particles. Technically, the induced charge density and the effective interaction are expressed in terms of multi-particle correlations of the pure (translationally invariant) jellium system. It is shown that the separation form of the induced charge density onto its radial and angle parts, observed previously in the limit Γ→0 , is not reproduced at the coupling Γ=2 . Based on an exact expression for the effective interaction between guest particles at Γ=2 , oppositely ( q=0,−1,−2,… ) charged guest particles always attract one another while likely ( q=1,2,… ) charged guest particles repeal one another up to a certain distance d between them and then the mutual attraction takes place up to asymptotically large (finite) distances.

The shape of an axisymmetric meniscus in a static liquid pool: effective implementation of the Euler transformation

Abstract We examine the classical problem of the height of a static liquid interface that forms on the outside of a solid vertical cylinder in an unbounded stagnant pool exposed to air. Gravitational and surface tension forces compete to affect the interface shape as characterized by the Bond number. Here, we provide a convergent power series solution for interface shapes that rise above or fall below the horizontal pool as a function of contact angle and Bond number. We find that the power series solution expressed in terms of the radial distance from the wall is divergent, and thus rewrite the divergent series as a new power series expressed as powers of an Euler transformed variable; this series is modified to match the large distance asymptotic behaviour of the meniscus. The Euler transformation maps non-physical singularities to locations that do not restrict series convergence in the physical domain, while the asymptotic modification increases the rate of convergence of the series overall. We demonstrate that when the divergent series coefficients are used to implement the Euler transformation, finite precision errors are incurred, even for a relatively small number of terms. To avoid such errors, the independent variable in the governing differential equation is changed to that of the Euler transform, and the power series is developed directly without using the divergent series. The resulting power series solution is validated by comparison with a numerical solution of the interface shape and the small Bond number matched asymptotic solution for the height of the interface along the cylinder developed by Lo (1983, J. Fluid Mech., 132, p.65-78). The convergent power series expansion has the ability to exceed the accuracy of the matched asymptotic solution for any Bond number given enough terms, and the recursive nature of the solution makes it straightforward to implement.

Riemann–Hilbert approach to a generalized sine kernel

We derive the large-distance asymptotics of the Fredholm determinant of the so-called generalized sine kernel at the critical point. This kernel corresponds to a generalization of the pure sine kernel arising in the theory of random matrices and has potential applications to the analysis of the large-distance asymptotic behaviour of the so-called emptiness formation probability for various quantum integrable models away from their free fermion point.

Virtual Delbrück scattering and the Lamb shift in light hydrogenlike atoms

We return to the problem of evaluation of the light-by-light contribution to the energy levels of the hydrogen atom. We find an additional contribution directly related to the Delbr\"uck scattering amplitude. The new correction is larger than the previously included light-by-light terms at order~$\alpha^2 (Z\alpha)^6\ln(Z\alpha)\, m_e$. We consider the effective potential in position space using an effective field theory approach and evaluate light-by-light corrections to the energy levels of states with non-zero orbital momentum as well as to the weighted difference of $s$ states. We also determine the large distance asymptotic behaviour of the effective potential induced by the light-by-light scattering in muonic atoms.

Form factors of bound states in the XXZ chain

This work focuses on the calculation of the large-volume behaviour of form factors of local operators in the XXZ spin-1/2 chain taken between the ground state and an excited state containing bound states. The analysis is rigorous and builds on various fine properties of the string solutions to the Bethe equations and certain technical hypotheses. These technical hypotheses are satisfied for a generic excited state. The results obtained in this work pave the way for extracting, starting from the first principles, the large-distance and long-time asymptotic behaviour of the XXZ chain’s two-point functions just as the so-called edge singularities of their Fourier transforms.

Thermal form factor approach to the ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime**Dedicated to the memory of Petr Petrovich Kulish.

We use the form factors of the quantum transfer matrix in the zero-temperature limit in order to study the two-point ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime. We obtain novel form factor series representations of the correlation functions which differ from those derived either from the q-vertex-operator approach or from the algebraic Bethe Ansatz approach to the usual transfer matrix. We advocate that our novel representations are numerically more efficient and allow for a straightforward calculation of the large-distance asymptotic behaviour of the two-point functions. Keeping control over the temperature corrections to the two-point functions we see that these are of order in the whole antiferromagnetic massive regime. The isotropic limit of our result yields a novel form factor series representation for the two-point correlation functions of the XXX chain at zero magnetic field.

A novel method for approximating equilibrium single-channel Ca2+ domains

Localized calcium (Ca2+) signals control some of the most fundamental physiological processes, including synaptic transmission as well as its activity-dependent plasticity. Computational and mathematical modeling played a crucial role in the understanding of spatio-temporal Ca2+ dynamics that drives these processes, and showed that Ca2+ concentration around a single Ca2+ channel reaches a quasi-stationary distribution (known as the Ca2+ nanodomain) within tens of microseconds after the opening of the channel, and collapses as rapidly after the closing of the channel. Such localization of Ca2+ in time and space is achieved by its rapid diffusion as well as its binding to its multiple interaction partners collectively called Ca2+ buffers and Ca2+ sensors. One of the successes of mathematical modeling was the development of several analytic approximations describing the equilibrium concentration of Ca2+ as a function of distance from the open Ca2+ channel, such as the Rapid Buffering Approximation (RBA), the Linear Approximation (LA) and the Excess Buffering Approximation (EBA) [1-4]. Each of these approximations has a particular applicability parameter regime created by the interplay between the properties of Ca2+ buffers, in particular their mobility and Ca2+ binding rates, and the strength of the Ca2+ current. Here we present a novel approximation method which does not rely on a specific range of the relevant Ca2+ and buffer parameters, and is based on matching the low-distance and large-distance asymptotic behavior of the concentration function. Even at low orders, the resulting approximation is as accurate as the second-order RBA and EBA approximations [4], but its validity extends far beyond the parameter range of applicability of RBA and EBA. The usefulness of the resulting approximation is two-fold: first, together with the previously developed approximations, the novel method could provide a deeper intuition into the dependence of Ca2+ nanodomain properties on the relevant buffering parameters, and second, it constitutes an efficient numerical approximation tool in the modeling of the Ca2+ signals underlying presynaptic and postsynaptic phenomena.

Correlation functions and momentum distribution of one-dimensional hard-core anyons in optical lattices

We address the problem of calculating the correlation functions in a system of one-dimensional hard-core anyons that can be experimentally realized in optical lattices. Using the summation of form factors we have obtained Fredholm determinant representations for the time-, space-, and temperature-dependent Green's functions which are particularly suited to numerical investigations. In the static case we have also derived the large distance asymptotic behavior of the correlators and computed the momentum distribution function at zero and finite temperature. We present extensive numerical results highlighting the characteristic features of one-dimensional systems with fractional statistics.

Large-distance asymptotic behaviour of multi-point correlation functions in massless quantum models

We provide a microscopic model setting that allows us to readily access the large-distance asymptotic behaviour of multi-point correlation functions in massless, one-dimensional, quantum models. The method of analysis we propose is based on the form factor expansion of the correlation functions and does not build on any field theory reasoning. It constitutes an extension of the restricted sum techniques leading to the large-distance asymptotic behaviour of two-point correlation functions obtained previously.

Large-Distance and Long-Time Asymptotic Behavior of the Reduced Density Matrix in the Non-Linear Schrödinger Model

Starting from the form factor expansion in finite volume, we derive the multidimensional generalization of the so-called Natte series for the time- and distance-dependent reduced density matrix at zero temperature in the non-linear Schrödinger model. This representation allows one to read-off straightforwardly the long-time/large-distance asymptotic behaviour of this correlator. This method of analysis reduces the complexity of the computation of the asymptotic behaviour of correlation functions in the so-called interacting integrable models, to the one appearing in free-fermion equivalent models. We compute explicitly the first few terms appearing in the asymptotic expansion. Part of these terms stems from excitations lying away from the Fermi boundary, and hence go beyond what can be obtained using the CFT/Luttinger liquid-based predictions.

Extended Grimus–Stockinger theorem and inverse-square law violation in quantum field theory

We study corrections to the Grimus–Stockinger theorem dealing with the large-distance asymptotic behavior of the external wave-packet modified neutrino propagator within the framework of a field-theoretical description of the neutrino-oscillation phenomenon. The possibility is discussed that these corrections, responsible for breakdown of the classical inverse-square law (ISL), can lead to measurable effects at small but macroscopic distances accessible in the SBL (anti)neutrino experiments and in particular can provide an explanation of the well-known reactor antineutrino anomaly.

Correlation lengths of the repulsive one-dimensional Bose gas

We investigate the large-distance asymptotic behavior of the static density-density and field-field correlation functions in the one-dimensional Bose gas at finite temperature. The asymptotic expansions of the Bose gas correlators are obtained performing a specific continuum limit in the similar low-temperature expansions of the longitudinal and transversal correlation functions of the $XXZ$ spin chain. In the lattice system the correlation lengths are computed as ratios of the largest and next-largest eigenvalues of the $XXZ$ spin chain quantum transfer matrix. In both cases, lattice and continuum, the correlation lengths are expressed in terms of solutions of Yang-Yang type [C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1151 (1969)] nonlinear integral equations which are easily implementable numerically.

Long-time and large-distance asymptotic behavior of the current–current correlators in thenon-linear Schrödinger model

We present a new method allowing one to derive the long-time and large-distanceasymptotic behavior of the correlation functions of quantum integrable modelsfrom their exact representations. Starting from the form factor expansion of thecorrelation functions in finite volume, we explain how to reduce the complexity of thecomputation in the so-called interacting integrable models to the one appearing infree fermion equivalent models. We apply our method to the time-dependentzero-temperature current–current correlation function in the non-linear Schrödinger modeland compute the first few terms in its asymptotic expansion. Our result goes beyond theconformal field theory based predictions: in the time-dependent case, excitationsother than the ones on the Fermi surface contribute to the leading orders of theasymptotics.

Black Holes in 5D Hořava Lifshitz Theory of Gravity

In this talk we discuss black hole solutions in five dimensions in the context of Hořava–Lifshtz gravity. There are several classes of solutions which are characterized by an AdS5, dS5 or flat large distance asymptotic behaviour, plus the standard 1/r2 tail of the usual five-dimensional Schwarzschild black holes. In addition there are solutions with an unconventional short or large distance behaviour, and for a special range of the coupling parameters solutions which coincide with black hole solutions of conventional relativistic five-dimensional Gauss–Bonnet gravity.

Black hole solutions in 5D Horava-Lifshitz gravity

We study the full spectrum of spherically symmetric solutions in the five dimensional non-projectable Horava-Lifshitz type gravity theories. For appropriate ranges of the coupling parameters, we have found several classes of solutions which are characterized by an AdS_5, dS_5 or flat large distance asymptotic behaviour, plus the standard 1/r^2 tail of the usual five-dimensional Schwarzschild black holes. In addition we have found solutions with an unconventional short or large distance behaviour, and for a special range of the coupling parameters solutions which coincide with black hole solutions of conventional relativistic five-dimensional Gauss-Bonnet gravity.

One-dimensional impenetrable anyons in thermal equilibrium: III. Large distance asymptotics of the space correlations

Using the determinant representation for the field–field correlation functions of impenetrable anyons at finite temperature obtained in [1], we derive a system of nonlinear partial differential equations completely characterizing the correlators. The system is the same as the one for impenetrable bosons but with different initial conditions. The large-distance asymptotic behavior of the correlation functions is obtained from the analysis of the Riemann–Hilbert problem associated with the system of differential equations. We calculate both the exponential and pre-exponential factors in the asymptotics of the field–field correlators. The asymptotics derived in this way agree with those of the free fermions and impenetrable bosons in the appropriate limits, κ → 1 and κ → 0, of the statistics parameter κ, and coincide with the predictions of the conformal field theory at low temperatures.

A Trickiness of the High-Temperature Limit for Number Density Correlation Functions in Classical Coulomb Fluids

The Debye-Hückel theory describes rigorously the thermal equilibrium of classical Coulomb fluids in the high-temperature β→ 0 regime (β denotes the inverse temperature). It is generally believed that the Debye-Hückel theory and the systematic high-temperature expansion provide an adequate description also in the region of small strictly positive values of β > 0. This hypothesis is tested in the present paper on a two-dimensional Coulomb gas of pointlike +/− unit charges interacting via a logarithmic potential which is equivalent to an integrable sine-Gordon field model. In particular, we apply a form factor method to obtain the exact asymptotic large-distance behavior of particle correlation functions, considered in the charge and number density combinations. We first determine the general forms of the leading and subleading asymptotic terms at strictly positive β > 0 and then evaluate their high-temperature β→ 0 forms. In the case of the charge correlation function, the leading asymptotic term at a strictly positive β > 0 is also the leading one in the high-temperature β→ 0 regime. On the contrary, the β→ 0 behavior of the number density correlation function is accompanied by an interference between the first two asymptotic terms. Consequently, the large-distance behavior of this function exhibits a discontinuity when going from strictly positive values of β > 0 to the Debye-Hückel limit β→ 0. This is the crucial conclusion of the paper: the large-distance asymptotics and the high-temperature limit do not commute for the density correlation function of the two-dimensional Coulomb gas.

Anomalous Effects of “Guest” Charges Immersed in Electrolyte: Exact 2D Results

We study physical situations when one or two "guest" arbitrarily-charged particles are immersed in the bulk of a classical electrolyte modelled by a Coulomb gas of positive/negative unit point-like charges, the whole system being in thermal equilibrium. The models are treated as two-dimensional with logarithmic pairwise interactions among charged constituents; the (dimensionless) inverse temperature $\beta$ is considered to be smaller than 2 in order to ensure the stability of the electrolyte against the collapse of positive-negative pairs of charges. Based on recent progress in the integrable (1+1)-dimensional sine-Gordon theory, exact formulas are derived for the chemical potential of one guest charge and for the asymptotic large-distance behavior of the effective interaction between two guest charges. The exact results imply, under certain circumstances, anomalous effects such as an effective attraction (repulsion) between like-charged (oppositely-charged) guest particles and the charge inversion in the electrolyte vicinity of a highly-charged guest particle. The adequacy of the concept of renormalized charge is confirmed in the whole stability region of inverse temperatures and the related saturation phenomenon is revised.

Sine-Gordon/Coulomb Gas Soliton Correlation Functions and an Exact Evaluation of the Kosterlitz--Thouless Critical Exponent

We present an exact derivation for the asymptotic large distance behavior of the spin two-point correlation function in the XY-model. This allows for the exact obtainment of the critical exponent $\eta=1/4$ at the Kosterlitz-Thouless transition that occurs in this model and in the 2D neutral Coulomb gas and which has been previously obtained by scaling arguments. In order to do that, we use the language of sine-Gordon theory to obtain a Coulomb Gas description of the XY-model spin correlation function, which becomes identified with the soliton correlator of that theory. Using a representation in terms of bipolar coordinates we obtain an exact expression for the asymptotic large distance behavior of the relevant correlator at $\beta^2=8\pi$, which corresponds to the Kosterlitz-Thouless transition. The result is obtained by approaching this point from the plasma (high-temperature) phase of the gas. The vortex correlator of the XY-model is also obtained using the same procedure.

CORRELATION FUNCTIONS OF THE XXZ SPIN-½ HEISENBERG CHAIN: RECENT ADVANCES (REVIEW)

We review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.