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Journal of Mathematical Physics

Derivation of an exact quantum-mechanical formula for the second term in the activity expansion of the correlation function S (Q,ω)

R. E. Johnson, Noel K. Pope

Published Jun 1, 1976• 0 citations
Publisher

Abstract

We derive an exact expression for the second term in the activity expansion of S (Q,ω), the Fourier transform of Van Hove’s correlation function, for a quantum system of bosons or fermions (spin neglected). The results for both the self and distinct parts of this function are given in terms of standard two-particle T matrices. The detailed-balance condition and zeroth and first moment relations are confirmed. An expression for the second term in the activity expansion of S (Q), the structure factor, is obtained. The derivation follows directly from the definition of S (Q,ω) using formal operator techniques; in addition, we show that the conserving T approximation for two-particle Green’s functions yields the same result.

DOI
10.1063/1.522989

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We derive an exact expression for the second term in the activity expansion of S (Q,ω), the Fourier transform of Van Hove’s correlation function, for a quantum system of bosons or fermions (spin neglected). The results for both the self and distinct parts of this function are given in terms of standard two-particle T matrices. The detailed-balance condition and zeroth and first moment relations are confirmed. An expression for the second term in the activity expansion of S (Q), the structure factor, is obtained. The derivation follows directly from the definition of S (Q,ω) using formal operator techniques; in addition, we show that the conserving T approximation for two-particle Green’s functions yields the same result.
A many-body theory is presented here for the Van Hove pair (or density) correlation function, in a quantum fluid obeying Boltzmann statistics, of which the Fourier transform $S(q,\ensuremath{\omega})$ determines the differential cross section for inelastic coherent scattering of slow neutrons. The theoretical formalism of two-particle thermodynamic Green's functions is based on Baym and Kadanoff's conserving TTT-matrix approximation, developed for strong short-range (intermolecular) potentials. The only other approximations made are: (i) the smallness of the activity or fugacity $z\ensuremath{\ll}1$, which implies Boltzmann statistics for the single-particle states; and (ii) the TTT matrices are replaced by elastic scattering amplitudes. Our results for $S(q,\ensuremath{\omega})$ developed to order z2{z}^{2}z2 are found to consist of Boltzmann averages of Lorentzian distributions in the energy transfer $\ensuremath{\hbar}\ensuremath{\omega}$ (except for a three-particle scattering term which is developed in the $\ensuremath{\delta}$-function limit of the Lorentzians). Our results are general enough to satisfy detailed balancing at all stages and are found to generalize Van Leeuwen and Yip's results for the classical case. The only unknown in our results is the intermolecular scattering amplitude or phase shifts. Our results seem, however, to break down in the hydrodynamic limit $\ensuremath{\omega}$ and $q\ensuremath{\rightarrow}0$ since each term in our expansion diverges in this limit.
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Abstract We give a descriptive review of the fermionic basis approach to the theory of correlation functions of the XXZ quantum spin chain. The emphasis is on explicit formulae for short-range correlation functions which will be presented in a way that allows for their direct implementation on a computer. Within the fermionic basis approach a huge class of stationary reduced density matrices, compatible with the integrable structure of the model, assumes a factorized form. This means that all expectation values of local operators and all two-point functions, in particular, can be represented as multivariate polynomials in only two functions ρ and ω and their derivatives with coefficients that are rational in the deformation parameter q of the model. These coefficients are of ‘algebraic origin’. They do not depend on the choice of the density matrix, which only impacts the form of ρ and ω . As an example we work out in detail the case of the grand canonical ensemble at temperature T and magnetic field h for q in the critical regime. We compare our exact results for the fourth-neighbour two-point functions with asymptotic formulae for h , T = 0 and for finite h and T .
The quantum dynamics of correlated fermionic or bosonic many-body systems following external excitation can be successfully studied using nonequilibrium Green functions (NEGFs) or reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle self-energy or decoupling of the BBGKY-hierarchy, respectively. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. In a recent paper [E. Schroedter, J.-P. Joost, and M. Bonitz, Condens. Matter Phys. 25, 23401 (2022)] we presented a different approach in which, instead of equations of motion for the many-particle NEGF (or density operators), equations for the correlation functions of fluctuations are analyzed. In particular, we derived the stochastic GWGWGW and polarization approximations that are closely related to the nonequilibrium GWGWGW approximation. Here, we extend this approach to the computation of two-time observables depending on the specific ordering of the underlying operators. In particular, we apply this extension to the calculation of the density correlation function and dynamic structure factor of correlated Hubbard clusters in and out of equilibrium.