In this work the basic formalism of non-equilibrium Green’s functions is presented and then applied to study a Ward identity in linear response theory, namely the frequency sum-rule. It can be proven that the frequency sum-rule is satisfied when the quantities involved are calculated using perturbation theory within a conserving approximation for the self-energy. To illustrate this equality along with other properties of the response function, a numerical application that solves the Kadanoff-Baym equations for systems of Hubbard chains was used. The results showed that the frequency sum-rule was satisfied to the same extent by all the conserving approximations used as by the exact diagonalization numerical results. The density response function was analyzed diagrammatically for a series of conserving approximations for the self-energy and this demonstrated that even for a first order in perturbation theory approximation for the self-energy, the response function has a corresponding complex, third order in the perturbation diagrammatic structure.In this work the basic formalism of non-equilibrium Green’s functions is presented and then applied to study a Ward identity in linear response theory, namely the frequency sum-rule. It can be proven that the frequency sum-rule is satisfied when the quantities involved are calculated using perturbation theory within a conserving approximation for the self-energy. To illustrate this equality along with other properties of the response function, a numerical application that solves the Kadanoff-Baym equations for systems of Hubbard chains was used. The results showed that the frequency sum-rule was satisfied to the same extent by all the conserving approximations used as by the exact diagonalization numerical results. The density response function was analyzed diagrammatically for a series of conserving approximations for the self-energy and this demonstrated that even for a first order in perturbation theory approximation for the self-energy, the response function has a corresponding complex, third order in the perturbation diagrammatic structure.

Jan 1, 20100 citations

Collisional frequency shifts of absorption lines in an atomic hydrogen gas

C. J. Pethick, Henk T. C. Stoof

We consider the effect of interactions on the line shape of the two-photon $1s\ensuremath{-}2s$ transition in a (doubly) spin-polarized atomic hydrogen gas in terms of the interatomic interaction potentials. We show that the frequency-weighted sum rule for the intensity of the line is not given simply in terms of the pseudopotentials that describe the interactions between low-energy atoms. The origin of the departures from the simple pseudopotential result for the frequency-weighted sum rule is traced to what we refer to as incoherent contributions to the spectral weight. These arise from more complicated final states of the many-body systems than the ones usually considered. In particular, we show how the relevant response function may be treated in a manner similar to the density-density response function for Fermi liquids, and express it as a coherent part coming from single particle-hole pairs, and an incoherent part coming from other excitations. We argue that in experiments only the coherent part of the response of the system is observed, and its contribution to the frequency-weighted sum rule is shown to be given correctly by the pseudopotential approximation. Finally we calculate the width of the coherent part of the line due to collisional damping.We consider the effect of interactions on the line shape of the two-photon $1s\ensuremath{-}2s$ transition in a (doubly) spin-polarized atomic hydrogen gas in terms of the interatomic interaction potentials. We show that the frequency-weighted sum rule for the intensity of the line is not given simply in terms of the pseudopotentials that describe the interactions between low-energy atoms. The origin of the departures from the simple pseudopotential result for the frequency-weighted sum rule is traced to what we refer to as incoherent contributions to the spectral weight. These arise from more complicated final states of the many-body systems than the ones usually considered. In particular, we show how the relevant response function may be treated in a manner similar to the density-density response function for Fermi liquids, and express it as a coherent part coming from single particle-hole pairs, and an incoherent part coming from other excitations. We argue that in experiments only the coherent part of the response of the system is observed, and its contribution to the frequency-weighted sum rule is shown to be given correctly by the pseudopotential approximation. Finally we calculate the width of the coherent part of the line due to collisional damping.

arXiv:cond-mat/0102105Physical Review AJun 8, 200128 citations

Spectral density and sum rules for second-order response functions

Barry Bradlyn, Peter M. Abbamonte

Sum rules for response functions represent some of the few exact results in many-body physics, relating the dynamical response functions to equilibrium ground state properties. To date, however, sum rules have been examined primarily for linear response functions. In this work, the authors introduce a spectral density representation for second-order response functions, using it to systematically derive families of sum rules, including a nonlinear extension of the well-known $f$ -sum rule. This work places important constraints on future nonlinear spectroscopic experiments.Sum rules for response functions represent some of the few exact results in many-body physics, relating the dynamical response functions to equilibrium ground state properties. To date, however, sum rules have been examined primarily for linear response functions. In this work, the authors introduce a spectral density representation for second-order response functions, using it to systematically derive families of sum rules, including a nonlinear extension of the well-known $f$-sum rule. This work places important constraints on future nonlinear spectroscopic experiments.

arXiv:2404.16144Physical review. B./Physical review. BDec 18, 202414 citations

f

-sum rule. This work places important constraints on future nonlinear spectroscopic experiments.