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Physical Review

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

Tadashi Arai, Bernard Albert Goodman

Published Mar 10, 1967• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.155.514

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Physical Review•Jan 10, 1968•6 citations
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.
We propose a spin-free approach to the cumulant decomposition of reduced density matrices of singlet and spin-rotation or SU(2) invariant ensemble of nonsinglet states as in [W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 110, 2800 (1999); 116, 4787 (2002)]. We provide a simple recursive procedure to obtain expressions which relate different spin components of spin-orbital reduced density matrices and cumulants of such states to the spin-free counterparts. These results are used to set up a spin-summation procedure to arrive at a definition of spin-free cumulants of any order. Alternatively, an analytic formula for the spin-free form resulting from a spin summation involving product of two spin-orbital cumulants is derived and its utility in spin-free cumulant decomposition of reduced density matrices is demonstrated. This leads to suitable definitions of spin-free analog of multireference normal ordering and the associated Wicks theorem. The results of this formulation are expected to be useful in investigations of spin-free multireference internally contracted coupled-cluster methods where cumulant approximations to the active reduced density matrices are employed.
The cumulant expansion for spins developed previously is rearranged and extended up to an infinite order in perturbation. A pair of spin-deviation lines is considered as a propagator and, as an additional propagator appears and interacts with the original one, that part of the diagram is defined as a self-energy correction. All cumulant corrections are decomposed and added to the corresponding self-energy corrections, so that self-energy corrections and their cumulant corrections are automatically calculated together exactly. The contribution of a self-energy correction to a diagram depends on the geometry of the incoming and outgoing spin-deviation lines of the self-energy diagram, yielding a new type of perturbation expansion. Numerical results suggest that the nature of the spin-spin correlation in the spin-1/2 case is distinct from that in other cases and that spins behave like fermions rather than bosons in this limit.
We present a compact and simplified proof of a generalized Wick theorem to calculate the Green's function of bosonic and fermionic systems in an arbitrary initial state. It is shown that the decomposition of the noninteracting nnn-particle Green's function is equivalent to solving a boundary problem for the Martin-Schwinger hierarchy; for noncorrelated initial states, a one-line proof of the standard Wick theorem is given. Our result leads to new self-energy diagrams, and an elegant relation with those of the imaginary-time formalism is derived. The theorem is easy to use and can be combined with any ground-state numerical technique to calculate time-dependent properties.
We present in this paper a systematic nonperturbative cluster-cumulant method for deriving thermal averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to ``imaginary time.'' We have generalized the concepts of cumulants in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed ``cluster cumulants'' by us. For a compact treatment, we have derived an alternative thermal field theory in which a time-ordered product is expanded in terms of ``thermal normal products'' of operators and thermal contractions---leading to a ``thermal Wick's theorem.'' The thermal normal products are the finite-temperature analogs of the ordinary normal products and have zero thermal averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This thermal representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch--Balian--de Dominicis theory.
A many-body perturbation theory is developed for the calculation of the antiferromagnetic ground state and lower excited states of one-electron atoms in the insulating state. It is assumed that the interatomic distances are moderately small and that the interactions are well described by exchange coupling, but that distortion of atoms in the lattice is still negligible. The method is essentially the Heitler-London approach: A set of Slater determinants is constructed from nonorthogonal atomic orbitals with all possible spin arrangements. The unperturbed ground state ∣0〉|0〉∣0〉 has an alternating spin configuration, while excited states are generated one by one from ∣0〉|0〉∣0〉 by interchanges of spins and by projecting out the lower excited states. The resulting orthonormal states introduce the concept of quasielectron (or quasihole) states and define creation and destruction operators of those quasiparticles. Although the quasiparticles are correlated with each other through the overlap integrals in the projected wave functions, the energy matrix elements can be decomposed into linked clusters of localized electrons and holes. It is found that the clusters thus generated are cumulants. The expression for the energy matrix is then transformed into the form of an effective Hamiltonian consisting of cumulants and of creation and destruction operators of quasifermions, and the concept of the exchange interaction is defined in terms of those operators. The new formulation not only facilitates the use of the familiar many-body perturbation theory, but also eliminates difficulties in handling spin variables in the Heisenberg theory of magnetism. If only the two-body Coulomb and exchange diagrams are retained and the higher-order diagrams are neglected, however, the effective Hamiltonian takes a form reminiscent of the anisotropic exchange Hamiltonian, ${H}_{\mathrm{spin}}=\ensuremath{-}{\ensuremath{\Sigma}}_{k,h}{J}_{\mathrm{kh}}[{{S}_{k}}^{z}{{S}_{h}}^{z}+\ensuremath{\lambda}({{S}_{k}}^{x}{{S}_{h}}^{x}+{{S}_{k}}^{y}{{S}_{h}}^{y})]$, which is used in the generalized Heisenberg theory and which includes the Ising model in its limit $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. Finally, it is shown that excitations accompanying electron transfers are readily included in the present method as additional diagrams by an extension of the prescriptions for the generation of excited spin states. This yields, in its first approximation, the intra-atomic interaction and the electron-transfer interaction, leading to the Hamiltonian assumed by Hubbard and Anderson.