Asymptotic Behavior Of Correlation Functions Research Articles

Asymptotic behavior of correlation functions of one-dimensional polar-molecules on optical lattices

We combine a slave-spin approach with a mean-field theory to develop an approximate theoretical scheme to study the density, spin, and, pairing correlation functions of fermionic polar molecules. We model the polar molecules subjected to a one-dimensional periodic optical lattice potential using a generalized t–J model, where the long-range part of the interaction is included through the exchange interaction parameter. For this model, we derive a set of self-consistent equations for the correlation functions, and evaluate them numerically for the long-distance behavior. We find that the pairing correlations are related to spin correlations through the density and the slave-spin correlations. Further, our calculations indicates that the long-range character of the interaction can be probed through these correlation functions. In the absence of exact solutions for the one-dimensional t–J model, our approximate theoretical treatment can be treated as a useful tool to study one dimensional long-range correlated fermions.

Equilibration and GGE in interacting-to-free quantum quenches in dimensions

Ground states ofinteracting QFTs are non-Gaussian states, i.e. their connected n-point correlation functions do not vanish for , in contrast to the free QFT case. We show that, when the ground state of an interacting QFT evolves under a free massive QFT for a long time (a scenario that can be realised by a quantum quench), the connected correlation functions decay and all local physical observables equilibrate to values that are given by a Gaussian density matrix that retains memory only of the two-point initial correlation function. The argument hinges upon the fundamental physical principle of cluster decomposition, which is valid for the ground state of a general QFT. An analogous result was already known to be valid in the case of d = 1 spatial dimensions, where it is a special case of the so-called generalised Gibbs ensemble (GGE) hypothesis, and we now generalise it to higher dimensions. Moreover, in the case of massless free evolution, despite the fact that the evolution may lead not to equilibration but instead to unbounded increase of correlations with time, the GGE gives correctly the leading-order asymptotic behaviour of correlation functions in the thermodynamic and large time limit. The demonstration is performed in the context of a bosonic relativistic QFT, but the arguments apply more generally.

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.

Determination of the ground state energies of the H 2 + , D 2 + and H 2 + molecular ions taking into account relativistic corrections

On the basis of determination of the asymptotic behavior of correlation functions of the corresponding field currents with the corresponding quantum numbers an analytic method for determination of the energy spectrum of three-body Coulomb system is suggested. Our results show that the constituent masses of particles, which we have defined as masses of particles in a bound state, differ from masses of particles in a free-state. The constituent mass to the free state mass relation for the electron is greater than the same mass relation for the proton, deuteron and triton. It was also found that this constituent electron mass has different values in each systems, i.e. in H2+, D2+ and T2+ hydrogen molecular ions. The contributions of exchange and self-energy diagrams were taken into account in the determination of the energy spectrum of the three-body Coulomb system. Our results show that the self-energy diagram contribution is inversely proportional to the square of the constituent mass of particles. This contribution is sufficient for the electron and is negligible for the proton, deuteron and triton. When defining the energy and the wave function (WF), it is necessary to take into account the contributions of both the exchange and self-energy diagrams.

Exact asymptotic behavior of correlation functions for disordered spin-1/2 XXZ chains

We consider an XXZ spin-1/2 chain in the presence of several types of disorder that do not break the XY symmetry of the system. We calculate the complete asymptotic form of the spin-correlation functions at zero temperature at the transition between liquid and disordered phase that occurs for a special value of anisotropy in the limit of small disorder. Apart from a universal power law decay of correlations, we find additional logarithmic corrections due to marginally irrelevant operator of disorder.

A form factor approach to the asymptotic behavior of correlation functions in criticalmodels

We propose a form factor approach for the computation of the large distance asymptoticbehavior of correlation functions in quantum critical (integrable) models. In the largedistance regime we reduce the summation over all excited states to one over theparticle/hole excitations lying on the Fermi surface in the thermodynamic limit. Wecompute these sums, over the so-called critical form factors, exactly. Thus we obtain theleading large distance behavior of each oscillating harmonic of the correlation functionasymptotic expansion, including the corresponding amplitudes. Our method is applicable toa wide variety of integrable models and yields precisely the results stemming from theLuttinger liquid approach, the conformal field theory predictions and our previous analysisof the correlation functions from their multiple-integral representations. We argue that ourscheme applies to a general class of non-integrable quantum critical models aswell.

Determination of the mass spectrum of the bound state in the framework of the relativistic hamiltonian approach

We propose one a variant of calculation of the energy spectrum of bound state systems with relativistic corrections. In the framework of quantum field theory, an expression that takes into account relativistic corrections to the mass of the bound state with a known nonrelativistic pair interaction potential is proposed on the basis of calculating the asymptotic behavior of correlation functions of the corresponding field currents with the necessary quantum numbers. Excluding the time variables allows one to determine nonperturbative corrections to the interaction potential. The following results have been obtained in the framework of this approach. The nonperturbative corrections arising due to the relativistic nature of a system to the interaction Hamiltonian are determined. The dependence of the constituent mass of bound-state forming particles on the free state mass and on the orbital and radial quantum numbers is analytically derived. The energy level shift of muonic hydrogen taking into account relativistic corrections is calculated. The energy spectrum of a wide class of potentials that describe the Coulomb bound state is analytically derived with relativistic corrections. The mass spectrum of the glueballs and the constituent masses of the gluons are analytically calculated taking into account spin-orbit, spin—spin, and tensor interactions. Our numerical results have shown very good agreement with the lattice data. Taking into account nonperturbative and nonlocality characters of interactions, the mass spectrum of the mesons consisting of light-light and light-heavy quarks with orbital and radial excitations is determined. Our results show that good agreement with the experimental data for the slope and the intercept of the Regge trajectory can be obtained only taking into account the nonperturbative and the nonlocal characters of interactions. The dependences of the constituent masses of constituent particles on the masses of a free state are certain. When quarks are light, then the difference between current and valent masses of quarks is greater than valent masses of quarks, and when quarks are heavy, then the difference between these masses is insignificant. One of the alternative variants of taking nonlocality into account has been suggested for the definition of properties of hadrons at large distances. The dependence of the constituent masses of constituent particles on the radius of confinement is determined.

Fine structure of the asymptotic expansion of cyclic integrals

The asymptotic expansion of $n$-dimensional cyclic integrals was expressed as a series of functionals acting on the symmetric function involved in the cyclic integral. In this article, we give an explicit formula for the action of these functionals on a specific class of symmetric functions. These results are necessary for the computation of the O(1) part in the long-distance asymptotic behavior of correlation functions in integrable models.

Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions

We describe a method to derive, from first principles, the long-distance asymptotic behavior ofcorrelation functions of integrable models in the framework of the algebraic Bethe ansatz. Weapply this approach to the longitudinal spin–spin correlation function of the XXZ Heisenberg spin-1/2 chain (with magnetic field) in the disordered regime as well as to the density–densitycorrelation function of the interacting one-dimensional Bose gas. At leading order, theresults confirm the Luttinger liquid and conformal field theory predictions.

Integrable models of strongly correlated particles with correlated hopping

The exact solution is obtained for the eigenvalues and eigenvectors for two models of strongly correlated particles with single-particle correlated and uncorrelated pair hoppings. The asymptotic behavior of correlation functions are analysed in different regions, where the models exhibit different physical behavior.

Temperature corrections to conformal field theory

We consider finite temperature dynamical correlation functions in the interacting delta-function Bose gas. In the low-temperature limit the asymptotic behaviour of correlation functions can be determined from conformal field theory. In the present work we determine the deviations from conformal behaviour at low temperatures.

New Exactly Solvable One-Parameter Model of Strongly Correlated Electrons

We present a two band fermion model consisting of a subband of conduction electrons and a subband of local electron pairs interacting via interband interactions. The model is solved in one dimension by means of the Bethe ansatz for an empty subband of local electron pairs. The ground state energy, the chemical potential, the critical exponents, describing the asymptotic behavior of correlation functions at long distances, the effective electron transport mass have been calculated numerically for an arbitrary density of conduction electrons.

Thermodynamics of an integrable model for electrons with correlated hopping

A new supersymmetric model for electrons with generalized hopping terms and Hubbard interaction on a one-dimensional lattice is solved by means of the Bethe Ansatz. We investigate the phase diagram of this model by studying the ground state and excitations of the model as a function of the interaction parameter, electronic density and magnetization. Using arguments from conformal field theory we can study the critical exponents describing the asymptotic behaviour of correlation functions at long distances.

A one-dimensional integrable model of fermions with multi-particle hopping

A model with both single-particle and multi-particle hopping of electrons on a one-dimensional 'triangular' lattice is formulated and solved exactly by Bethe ansatz. On the basis of the exact calculation of the asymptotic behaviour of correlation functions we find a transition between the normal state and a state with a tendency to 'superconductivity'. The latter state is confined to small densities of electrons up to a critical density rho c.

Calculation and Interpretation of Hadron Correlation Functions in Lattice QCD

Several new developments in the calculation and interpretation of hadron density-density correlation functions are presented. The asymptotic behavior of correlation functions is determined from a tree diagram path integral. A method is developed to use this behavior to correct for leading image contributions on a finite periodic spatial lattice and to correct for the finite temporal extent of the lattice. Equal time correlation functions are shown to determine a sum of the ground state rms radius plus a polarization correction, and it is shown how to extract the hadron polarizability from unequal time correlation functions. Image-corrected correlation functions calculated in quenched lattice QCD are presented and used to determine the size of the pion and nucleon.

Competition between singlet formation and magnetic ordering in one-dimensional spin systems.

We study the behavior of a model of two XXZ spin chains coupled by an isotropic exchange interaction ${\mathit{J}}_{\mathit{K}}$ as the anisotropies and spin velocities in the individual chains are varied. In the limit of large antiferromagnetic ${\mathit{J}}_{\mathit{K}}$ we map the model's low-energy sector onto an Ising model in a transverse field; in the limit of large ferromagnetic ${\mathit{J}}_{\mathit{K}}$ we map the model's low-energy sector onto that of a single spin-one chain. For small ${\mathit{J}}_{\mathit{K}}$, we study the model perturbatively via Abelian and non-Abelian bosonization. We determine the possible phases, approximate locations of the phase boundaries, universality classes of the transitions, and asymptotic behavior of correlation functions. We believe our model captures the essential physics of the interplay between the compensation of local moments and magnetic ordering, which may be important for the heavy-fermion materials.

CRITICAL EXPONENTS IN THE ONE-DIMENSIONAL HUBBARD MODEL

Exact Bethe Ansatz results on the spectrum of large but finite Hubbard chains in conjunction with methods from conformal quantum field theory can be used to obtain exact results for the asymptotic behaviour of correlation functions. We review this method and discuss some interesting consequences of the results.

Critical exponents of the degenerate Hubbard model

The authors study the critical behaviour of the SU(N) generalization of the one-dimensional Hubbard model with arbitrary degeneracy N. Using the integrability of this model by Bethe ansatz they are able to compute the spectrum of the low-lying excitations in a large but finite box for arbitrary values of the electron density and of the Coulomb interaction. This information is used to determine the asymptotic behaviour of correlation functions at zero temperature in the presence of external fields lifting the degeneracy. The critical exponents depend on the system parameters through an N*N dressed charge matrix implying the relevance of the interaction of charge- and spin-density waves.

Logarithmic corrections in the one-dimensional Hubbard model with attraction

The free energy in the vicinity of a half-filled band for the one-dimensional Hubbard model with attraction is calculated as a function of density (chemical potential). The expansion of the elementary excitation energies for a lattice of finite length N is accomplished with an accuracy of the order (N2 ln N)−1. An explicit expression for the free energy and the elementary excitation spectrum as functions of external field or of volume N is necessary for studying the asymptotic behavior of correlation functions.

Asymptotic behavior of correlation functions for electric potential and field fluctuations in a classical one- component plasma

The correlations of the electric potential fluctuations in a classical one-component plasma are studied for large distances between the observation points. The two-point correlation function for these fluctuations is known to decay slowly for large distances, even if exponential clustering holds for the charge correlation functions. In this paper the asymptotic behavior of the generalk-point electric potential correlation functions is analyzed. Each of these correlation functions can be split into a reducible part, which is given by a sum of products of lower-order correlation functions, and a remaining irreducible part. It is shown, on the basis of an exponential clustering hypothesis for the charge correlation functions, that for allk⩾3 the irreducible parts of the electric potential correlation functions decay faster than any inverse power of the distance, if one or more of the observation points move far away from the others. Hence, the two-point electric potential correlation function is the only one with a slow algebraic decay. The same statement holds for the correlation functions of the electric field fluctuations.