We present a general scheme based on nonlinear response theory to calculate the expansion of correlation functions such as the pair-correlation function or the exchange-correlation hole of an inhomogeneous many-particle system in terms of density derivatives of arbitrary order. We further derive a consistency condition that is necessary for the existence of the gradient expansion. This condition is used to carry out an infinite summation of terms involving response functions up to infinite order from which it follows that the coefficient functions of the gradient expansion can be expressed in terms of the local density profile rather than the background density around which the expansion is carried out. We apply the method to the calculation of the gradient expansion of the one-particle density matrix to second order in the density gradients and recover in an alternative manner the result of Gross and Dreizler [Gross and Dreizler, Z. Phys. A 302, 103 (1981)], which was derived using the Kirzhnits method. The nonlinear response method is more general and avoids the turning point problem of the Kirzhnits expansion. We further give a description of the exchange hole in momentum space and confirm the wave vector analysis of Langreth and Perdew [Langreth and Perdew, Phys. Rev. B 21, 5469 (1980)] for this case. This is used to derive that the second-order gradient expansion of the system averaged exchange hole satisfies the hole sum rule and to calculate the gradient coefficient of the exchange energy without the need to regularize divergent integrals.
Abstract We identify a nontrivial yet tractable quantum field theory model with space/time anisotropic scale invariance, for which one can exactly compute certain four-point correlation functions and their decompositions via the operator-product expansion(OPE). The model is the Calogero model, non-relativistic particles interacting with a pair potential
∣x−y∣2g g x − y 2 in one dimension, considered as a quantum field theory in one space and one time dimension via the second quantisation. This model has the anisotropic scale symmetry with the anisotropy exponent z = 2. The symmetry is also enhanced to the Schrödinger symmetry. The model has one coupling constant g and thus provides an example of a fixed line in the renormalisation group flow of anisotropic theories. We exactly compute a nontrivial four-point function of the fundamental fields of the theory. We decompose the four-point function via OPE in two different ways, thereby explicitly verifying the associativity of OPE for the first time for an interacting quantum field theory with anisotropic scale invariance. From the decompositions, one can read off the OPE coefficients and the scaling dimensions of the operators appearing in the intermediate channels. One of the decompositions is given by a convergent series, and only one primary operator and its descendants appear in the OPE. The scaling dimension of the primary operator we computed depends on the coupling constant. The dimension correctly reproduces the value expected from the well-known spectrum of the Calogero model combined with the so-called state-operator map which is valid for theories with the Schrödinger symmetry. The other decomposition is given by an asymptotic series. The asymptotic series comes with exponentially small correction terms, which also have a natural interpretation in terms of OPE.
The authors analyse the nonlinear Schrodinger model (the one-dimensional Bose gas) by means of conformal field theory in conjunction with a newly developed cutoff procedure. This makes it possible to obtain exact expressions for current correlators in the model. A number of new results are presented, including the expression for the two-point current correlator on a finite-size strip and the general n-point current correlator on the plane. All these results have the full time dependence and are presented to order 1/g, where g is the strength of the repulsive delta -function interaction.