Physical Review

A Wick Theorem for Spin-One Operators

H. J. Spencer

Published Mar 10, 1968• 35 citations

Abstract

A generalization to all spin magnitudes of the drone-fermion representation (previously known for spin \textonehalf{}), is presented, and the resulting eigenstates investigated. Linear combinations of products of fermion operators acting on a vacuum state can always be found which correspond to spin states which are eigenvectors of ${\mathrm{S}}^{2}$ and ${S}^{z}$. It is then shown that for the case of spin 1 (units of $\ensuremath{\hbar}$) it is possible to construct a Wick theorem which is analogous to the usual temperature form for standard fermion creation and destruction operators.

DOI: 10.1103/physrev.167.430

Related papers

Notes on Wick's theorem in many-body theory

Luca Guido Molinari

In this pedagogical note I present the operator form of Wick's theorem, i.e. a procedure to bring a product of 1-particle creation and destruction operators to normal order, with respect to some reference many-body state. Both the static and the time-ordered cases are presented. For the latter, in particular, I provide a simple proof.

arXiv:1710.09248Oct 25, 20179 citations

Wick's Theorem for Spin Operators and Its Relation to the Coupled-Fermion Representation

Yung‐Li Wang, S. Shtrikman, Herbert B. Callen

It is shown that Wick's theorem and the linked diagram expansion theorem for S = ½ operators (applied elsewhere by the authors to discuss antiferromagnetic spin waves) can be derived by a direct application of conventional theorems to the coupled-fermion representation of spin operators.

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Cumulant Expansion and Wick's Theorem for Spins. Application to the Antiferromagnetic Ground State

Tadashi Arai, Bernard Albert Goodman

Physical ReviewMar 10, 196720 citations

Extension of the Notion of Normal Products and Wick's Theorem to Finite Temperature

Shuichiro Yamasaki

It is shown that normal products of creation and annihilation operators as well as the related theorem of Wick have natural generalizations to finite temperature systems.

Progress of Theoretical PhysicsOct 1, 19831 citation

Extension of Wick’s theorem for many-particle matrix elements

K. Tanabe, K. Enami, Naotaka Yoshinaga

Physical Review CMay 1, 199914 citations

General Wick's theorem for bosonic and fermionic operators

Luca Ferialdi, Lajos Diósi

arXiv:2110.02920Physical Review ANov 15, 20219 citations

Physical Review

A Wick Theorem for Spin-One Operators

H. J. Spencer

Published Mar 10, 1968• 35 citations

Publisher

Abstract

{\mathrm{S}}^{2}

and

{S}^{z}

. It is then shown that for the case of spin 1 (units of

\ensuremath{\hbar}

) it is possible to construct a Wick theorem which is analogous to the usual temperature form for standard fermion creation and destruction operators.

DOI: 10.1103/physrev.167.430

Related papers

Notes on Wick's theorem in many-body theory

Luca Guido Molinari

arXiv:1710.09248Oct 25, 20179 citations

Wick's Theorem for Spin Operators and Its Relation to the Coupled-Fermion Representation

Yung‐Li Wang, S. Shtrikman, Herbert B. Callen

Journal of Applied PhysicsMar 1, 19667 citations

Cumulant Expansion and Wick's Theorem for Spins. Application to the Antiferromagnetic Ground State

Tadashi Arai, Bernard Albert Goodman

A form of Wick's theorem is derived which is applicable to spin operators of arbitrary magnitude in a manner analogous to the decomposition of fermion and boson operator products. Use of the theorem together with Kubo's cumulant rearrangement of perturbation theory leads to a compact prescription for the calculation of the ground-state energy of spin systems. This method is parallel to Goldstone's linked-cluster expansion for fermions, but the expansion for spins, as well as for bosons, contains cumulant correction terms. The Green functions are shown to be expanded similarly in terms of cumulants. The method is illustrated by the calculation of the antiferromagnetic ground state, and circumvents the involved development in the previous work of Davis and Boon. It is found that our procedure gives better convergence than Davis's expansion and that Boon's final formulas as well as his numerical results contain some errors.A form of Wick's theorem is derived which is applicable to spin operators of arbitrary magnitude in a manner analogous to the decomposition of fermion and boson operator products. Use of the theorem together with Kubo's cumulant rearrangement of perturbation theory leads to a compact prescription for the calculation of the ground-state energy of spin systems. This method is parallel to Goldstone's linked-cluster expansion for fermions, but the expansion for spins, as well as for bosons, contains cumulant correction terms. The Green functions are shown to be expanded similarly in terms of cumulants. The method is illustrated by the calculation of the antiferromagnetic ground state, and circumvents the involved development in the previous work of Davis and Boon. It is found that our procedure gives better convergence than Davis's expansion and that Boon's final formulas as well as his numerical results contain some errors.

Physical ReviewMar 10, 196720 citations

Extension of the Notion of Normal Products and Wick's Theorem to Finite Temperature

Shuichiro Yamasaki

It is shown that normal products of creation and annihilation operators as well as the related theorem of Wick have natural generalizations to finite temperature systems.

Progress of Theoretical PhysicsOct 1, 19831 citation

Extension of Wick’s theorem for many-particle matrix elements

K. Tanabe, K. Enami, Naotaka Yoshinaga

Wick's theorem for the expectation value of an operator between many-particle states is generalized to the case of a homogeneous linear transformation operator $\^O$ , which leaves invariant the $2N$ -dimensional linear space composed of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the group $\mathrm{SO}(2N,C)$ for the case of fermion, or the group $\mathrm{Sp}(2N,C)$ for the case of boson. This theorem is applied to derive a reduction theorem for the matrix element of the type $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\alpha}}_{m}\^O{\ensuremath{\beta}}_{m+1}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\beta}}_{m+2}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\beta}}_{2n}^{\ifmmode\dagger\else\textdagger\fi{}}|\ensuremath{\beta}〉,$ whose many-quasiparticle states and vacua appearing in both sides are represented in different quasiparticle pictures as indicated by $\ensuremath{\alpha}$ and $\ensuremath{\beta}.$ The reduction formula is useful in calculating any nuclear transition matrix element for the quantum-number projection calculation associated with the constrained Hartree-Fock-Bogoliubov solutions, and also the generator coordinate method with quantum number projection.Wick's theorem for the expectation value of an operator between many-particle states is generalized to the case of a homogeneous linear transformation operator $\^O$, which leaves invariant the $2N$-dimensional linear space composed of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the group $\mathrm{SO}(2N,C)$ for the case of fermion, or the group $\mathrm{Sp}(2N,C)$ for the case of boson. This theorem is applied to derive a reduction theorem for the matrix element of the type $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\alpha}}_{m}\^O{\ensuremath{\beta}}_{m+1}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\beta}}_{m+2}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\beta}}_{2n}^{\ifmmode\dagger\else\textdagger\fi{}}|\ensuremath{\beta}〉,$ whose many-quasiparticle states and vacua appearing in both sides are represented in different quasiparticle pictures as indicated by $\ensuremath{\alpha}$ and $\ensuremath{\beta}.$ The reduction formula is useful in calculating any nuclear transition matrix element for the quantum-number projection calculation associated with the constrained Hartree-Fock-Bogoliubov solutions, and also the generator coordinate method with quantum number projection.

Physical Review CMay 1, 199914 citations

General Wick's theorem for bosonic and fermionic operators

Luca Ferialdi, Lajos Diósi

Wick's theorem provides a connection between time ordered products of bosonic or fermionic fields, and their normal ordered counterparts. We consider a generic pair of operator orderings and we prove, by induction, the theorem that relates them. We name this the general Wick's theorem (GWT) because it carries Wick's theorem as special instance, when one applies the GWT to time and normal orderings. We establish the GWT both for bosonic and fermionic operators, i.e., operators that satisfy c-number commutation and anticommutation relations respectively. We remarkably show that the GWT is the same, independent of the type of operator involved. By means of a few examples, we show how the GWT helps treat demanding problems by reducing the amount of calculations required.Wick's theorem provides a connection between time ordered products of bosonic or fermionic fields, and their normal ordered counterparts. We consider a generic pair of operator orderings and we prove, by induction, the theorem that relates them. We name this the general Wick's theorem (GWT) because it carries Wick's theorem as special instance, when one applies the GWT to time and normal orderings. We establish the GWT both for bosonic and fermionic operators, i.e., operators that satisfy c-number commutation and anticommutation relations respectively. We remarkably show that the GWT is the same, independent of the type of operator involved. By means of a few examples, we show how the GWT helps treat demanding problems by reducing the amount of calculations required.

arXiv:2110.02920Physical Review ANov 15, 20219 citations

{\mathrm{S}}^{2}

and

{S}^{z}

. It is then shown that for the case of spin 1 (units of

\ensuremath{\hbar}

) it is possible to construct a Wick theorem which is analogous to the usual temperature form for standard fermion creation and destruction operators.

Wick's theorem for the expectation value of an operator between many-particle states is generalized to the case of a homogeneous linear transformation operator

\^O

, which leaves invariant the

2N

-dimensional linear space composed of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the group

\mathrm{SO}(2N,C)

for the case of fermion, or the group

\mathrm{Sp}(2N,C)

for the case of boson. This theorem is applied to derive a reduction theorem for the matrix element of the type $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\alpha}}_{m}\^O{\ensuremath{\beta}}_{m+1}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\beta}}_{m+2}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\beta}}_{2n}^{\ifmmode\dagger\else\textdagger\fi{}}|\ensuremath{\beta}〉,$ whose many-quasiparticle states and vacua appearing in both sides are represented in different quasiparticle pictures as indicated by $\ensuremath{\alpha}$ and $\ensuremath{\beta}.$ The reduction formula is useful in calculating any nuclear transition matrix element for the quantum-number projection calculation associated with the constrained Hartree-Fock-Bogoliubov solutions, and also the generator coordinate method with quantum number projection.