A diagrammatical method, based on the second quantization formalism, time-independent Wick's theorem and Feynman-like diagrams, is developed for the calculation of arbitrary matrix elements of n-particle spin-independent operators between the states represented by antisymmetrized products of geminals of an arbitrary kind (i.e., strongly or weakly orthogonal, interacting, nonorthogonal or identical geminals). The method greatly facilitates any algebraic manipulation with geminal type functions, particularly when strong orthogonality is not assumed, and enables one to write down directly the algebraic expressions for the matrix elements of an arbitrary spin-independent operator between the single antisymmetrized products of arbitrary geminals, solely on the basis of simple graphs, which are easily constructed.A diagrammatical method, based on the second quantization formalism, time-independent Wick's theorem and Feynman-like diagrams, is developed for the calculation of arbitrary matrix elements of n-particle spin-independent operators between the states represented by antisymmetrized products of geminals of an arbitrary kind (i.e., strongly or weakly orthogonal, interacting, nonorthogonal or identical geminals). The method greatly facilitates any algebraic manipulation with geminal type functions, particularly when strong orthogonality is not assumed, and enables one to write down directly the algebraic expressions for the matrix elements of an arbitrary spin-independent operator between the single antisymmetrized products of arbitrary geminals, solely on the basis of simple graphs, which are easily constructed.

The Journal of Chemical PhysicsJul 15, 197228 citations

New approach to the one-particle Green's function for finite Fermi systems

J. Schirmer, Lorenz S. Cederbaum, Olaf Walter

A new approach to the one-particle Green's functions $\mathit{G}$ for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part $\mathit{M}$ related to $\mathit{G}$ via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematical way a set of approximation schemes ( $n\mathrm{th}$ -order ADC schemes) that represent infinite partial summations for $\mathit{M}$ and (via the Dyson equation) for $\mathit{G}$ being complete through $n\mathrm{th}$ order of perturbation theory. The corresponding mathematical procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed. While the second- and third-order schemes can be viewed as systematic rederivations of previous approximation schemes, the fourth-order ADC scheme represents a complete fourth-order approximation for the self-energy and the one-particle Green's function which was hitherto not available.A new approach to the one-particle Green's functions $\mathit{G}$ for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part $\mathit{M}$ related to $\mathit{G}$ via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematical way a set of approximation schemes ($n\mathrm{th}$-order ADC schemes) that represent infinite partial summations for $\mathit{M}$ and (via the Dyson equation) for $\mathit{G}$ being complete through $n\mathrm{th}$ order of perturbation theory. The corresponding mathematical procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed. While the second- and third-order schemes can be viewed as systematic rederivations of previous approximation schemes, the fourth-order ADC scheme represents a complete fourth-order approximation for the self-energy and the one-particle Green's function which was hitherto not available.

Physical Review ASep 1, 1983725 citations

Classical Spin Models: An Introduction

Andreas Wipf

Lecture notes in physicsOct 26, 20121 citation

Geometric aspects of quantum spin states

Michael Aizenman, Bruno Nachtergaele

arXiv:cond-mat/9310009Communications in Mathematical PhysicsJul 1, 1994152 citations

Qudit Isotopy

Arthur M. Jaffe, Zhengwei Liu, Alex Wozniakowski

We explore a general diagrammatic framework to understand qudits and their braiding, especially in its relation to entanglement. This involves understanding the role of isotopy in interpreting diagrams that implement entangling gates as well as some standard quantum information protocols. We give qudit Pauli operators $X,Y,Z$ and comment on their structure, both from an algebraic and from a diagrammatic point of view. We explain alternative models for diagrammatic interpretations of qudits and their transformations. We use our diagrammatic approach to define an entanglement-relay protocol for long-distance entanglement. Our approach rests on algebraic and topological relations discovered in the study of planar para algebras. In summary, this work provides bridges between the new theory of planar para algebras and quantum information, especially in questions involving entanglement.We explore a general diagrammatic framework to understand qudits and their braiding, especially in its relation to entanglement. This involves understanding the role of isotopy in interpreting diagrams that implement entangling gates as well as some standard quantum information protocols. We give qudit Pauli operators $X,Y,Z$ and comment on their structure, both from an algebraic and from a diagrammatic point of view. We explain alternative models for diagrammatic interpretations of qudits and their transformations. We use our diagrammatic approach to define an entanglement-relay protocol for long-distance entanglement. Our approach rests on algebraic and topological relations discovered in the study of planar para algebras. In summary, this work provides bridges between the new theory of planar para algebras and quantum information, especially in questions involving entanglement.

arXiv:1602.02671Feb 8, 20164 citations

A new approach to the one-particle Green's functions

\mathit{G}

for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part

\mathit{M}

related to

\mathit{G}

via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematical way a set of approximation schemes (

n\mathrm{th}

-order ADC schemes) that represent infinite partial summations for

\mathit{M}

and (via the Dyson equation) for

\mathit{G}

being complete through

n\mathrm{th}

order of perturbation theory. The corresponding mathematical procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed. While the second- and third-order schemes can be viewed as systematic rederivations of previous approximation schemes, the fourth-order ADC scheme represents a complete fourth-order approximation for the self-energy and the one-particle Green's function which was hitherto not available.

X,Y,Z

and comment on their structure, both from an algebraic and from a diagrammatic point of view. We explain alternative models for diagrammatic interpretations of qudits and their transformations. We use our diagrammatic approach to define an entanglement-relay protocol for long-distance entanglement. Our approach rests on algebraic and topological relations discovered in the study of planar para algebras. In summary, this work provides bridges between the new theory of planar para algebras and quantum information, especially in questions involving entanglement.