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

Partial self-consistency and analyticity in many-body perturbation theory: Particle number conservation and a generalized sum rule

Daniel Karlsson, Robert van Leeuwen

We consider a general class of approximations which guarantees the conservation of particle number in many-body perturbation theory. To do this we extend the concept of $\mathrm{\ensuremath{\Phi}}$ derivability for the self-energy $\mathrm{\ensuremath{\Sigma}}$ to a larger class of diagrammatic terms in which only some of the Green's function lines contain the fully dressed Green's function $G$ . We call the corresponding approximations for $\mathrm{\ensuremath{\Sigma}}$ partially $\mathrm{\ensuremath{\Phi}}$ derivable. A special subclass of such approximations, which are gauge invariant, is obtained by dressing loops in the diagrammatic expansion of $\mathrm{\ensuremath{\Phi}}$ consistently with $G$ . These approximations are number conserving but do not have to fulfill other conservation laws, such as the conservation of energy and momentum. From our formalism we can easily deduce whether commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial $\mathrm{\ensuremath{\Phi}}$ derivability plays an important role in the derivation of a generalized sum rule for the particle number, which reduces to the Luttinger-Ward theorem in the case of a homogeneous electron gas, and the Friedel sum rule in the case of the Anderson model. To do this we need to ensure that the Green's function has certain complex analytic properties, which can be guaranteed if the spectral function is positive-semidefinite. The latter property can be ensured for a subset of partially $\mathrm{\ensuremath{\Phi}}$-derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half diagrams. For the case in which the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure, and multivaluedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.We consider a general class of approximations which guarantees the conservation of particle number in many-body perturbation theory. To do this we extend the concept of $\mathrm{\ensuremath{\Phi}}$ derivability for the self-energy $\mathrm{\ensuremath{\Sigma}}$ to a larger class of diagrammatic terms in which only some of the Green's function lines contain the fully dressed Green's function $G$. We call the corresponding approximations for $\mathrm{\ensuremath{\Sigma}}$ partially $\mathrm{\ensuremath{\Phi}}$ derivable. A special subclass of such approximations, which are gauge invariant, is obtained by dressing loops in the diagrammatic expansion of $\mathrm{\ensuremath{\Phi}}$ consistently with $G$. These approximations are number conserving but do not have to fulfill other conservation laws, such as the conservation of energy and momentum. From our formalism we can easily deduce whether commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial $\mathrm{\ensuremath{\Phi}}$ derivability plays an important role in the derivation of a generalized sum rule for the particle number, which reduces to the Luttinger-Ward theorem in the case of a homogeneous electron gas, and the Friedel sum rule in the case of the Anderson model. To do this we need to ensure that the Green's function has certain complex analytic properties, which can be guaranteed if the spectral function is positive-semidefinite. The latter property can be ensured for a subset of partially $\mathrm{\ensuremath{\Phi}}$-derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half diagrams. For the case in which the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure, and multivaluedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.

arXiv:1606.07486Physical review. B./Physical review. BSep 13, 201614 citations

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. We call the corresponding approximations for $\mathrm{\ensuremath{\Sigma}}$ partially $\mathrm{\ensuremath{\Phi}}$ derivable. A special subclass of such approximations, which are gauge invariant, is obtained by dressing loops in the diagrammatic expansion of $\mathrm{\ensuremath{\Phi}}$ consistently with

G

. These approximations are number conserving but do not have to fulfill other conservation laws, such as the conservation of energy and momentum. From our formalism we can easily deduce whether commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial $\mathrm{\ensuremath{\Phi}}$ derivability plays an important role in the derivation of a generalized sum rule for the particle number, which reduces to the Luttinger-Ward theorem in the case of a homogeneous electron gas, and the Friedel sum rule in the case of the Anderson model. To do this we need to ensure that the Green's function has certain complex analytic properties, which can be guaranteed if the spectral function is positive-semidefinite. The latter property can be ensured for a subset of partially $\mathrm{\ensuremath{\Phi}}$-derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half diagrams. For the case in which the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure, and multivaluedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.