Fermion operator

Finally the contraction between two fermion operators can explicitly be computed and one finds that... 

Spectral Representation.—As an application of the invariance properties of quantum electrodynamics we shall now use the results obtained in the last section to deduce a representation of the vacuum expectation value of a product of two fermion operators and of two boson operators. The invariance of the theory under time inversion and more particularly the fact that... 

Normal product of free-field creation and annihilation operators, 606 Normal product operator, 545 operating on Fermion operators, 545 N-particle probability distribution function, 42... 

The only operation used for obtaining this partitioning is the anticommutation rule of the fermion operators. Note, that by adding the F and G terms one falls into the unitarily invariant Absar and Coleman partitioning [32,33] which was obtained by using a Group theoretical approach. 

The basic relations for stndying the properties of the RDMs are the anticommu-tation/commntation relations of gronps of fermion operators since their expectation values give a set of A-representability conditions of the RDMs. Thus,... 

Equations (45) and (49) stress the direct connexion existing between the elements and classes of the Symmetric Group of Permutations and the terms derived by com-muting/anticommuting groups of fermion operators after summing with respect to the spin variables. 

The same expansion can be done for quantum-mechanical problems with halfinteger spin, except that one needs fermion operators, a,1, and a. The bilinear products... 

A -representability conditions [28]. Let us start this description by focusing on the RDM s properties, which may be deduced from their definition as expectation values of density fermion operators. Thus the ROMs are Hermitian, are positive semidefinite, and contract to finite values that depend on the number of electrons, N, and in the case of the HRDMs on the size of the one-electron basis of representation, 2/C. Thus... 

Then one redehnes the annUiilahon operator u, for an occupied spin orbital as the hole creation operator b, and the creation operator a for an occupied spin orbital as the hole annihilation operator bi. The fermion operators for the virtual spin orbitals remain unchanged. 

Valdemoro [28] achieved a close approximation to the 2-RDM by using the anticommutating relation of fermion operators, or what is equivalent, the 7/-representability conditions. This work indicated that the development of 1-RDM functional theories should be couched in terms of explicitly antisymmetric reconstructions of the 2-RDM. 

The standard procedure used to solve Eq. (29) is to transform the spin operators into fermionic operators [61]. Let us define the raising and lowering operators... 

Recalling the basic relation obtained by taking the expectation value of the anticommutator of two fermion operators ... 

Let us consider the equivalent expressions for the 2-RDM blocks, which may be derived by interchanging the fermion operators by following other options than the one used in eq. (7) which gave rise to eq. (10). For the acr blocks one obtains ... 

In the superoperator approach, an abstract linear space is introduced [2], The elements of this space are fermion operators generally expressed as linear combinations of products of creation or annihilation operators,... 

The fermion creation and destruction operators are defined such that apa +a ap = Spq. In analogy to relativistic theory, and more appropriate to the linear response theory to be considered here, the elementary fermion operators ap can be treated as algebraic objects fixed in time, while the orbital functions are solutions of a time-dependent Schrodinger equation... 

It leaves intact the fermion operators related to the /1-th group itself. By virtue of this the two-electron operators WBA result in a renormalization of one-electron terms in the Hamiltonians for each group. <4 = 1,..., M. The expectation values ((b+b ))B are the one-electron densities. The Schrodinger equation eq. (1.193) can be driven close to the standard HFR form. This can be done if one defines generalized Coulomb and exchange operators for group A by their matrix elements in the carrier space of group A ... 

It is known as the three-band Hubbard Hamiltonian [13-15] and can describe the band structure of the parent compounds [16]. Here d (d) and />+ (p) are Fermionic operators for the bands d and p with orbital energy Ed and Ep, respectively. The energy of hopping for an electron (with spin a =t. ) from site i to site j is r. While Up and Ud are Hubbard terms which are present if the occupation number operators rip or rid) are nonzero, Vpd is the intraband Coulomb repulsion. 

An alternative approach is the Slave Boson approximation [21] where the Fermionic operators are defined as the product of Boson and spin operators. There is a constraint with the number of Fermions, n /, and number of Bosons, n n f - - m, = 1. The spin part is treated with a RVB spin model and the charge as a Bose-Einstein condensation problem. These leads to a fractionalization of charges (holons) and spin (spinons), where uncondensed holons exist above the SC domain. The temperature crossover of the spinon pairing and the holon condensation, as a function of doping, is identified as peak in the SC domain. 

Second quantization for composite particles, in the context of quantum chemistry, was elaborated by several authors, e.g. by Girardeau [ 101 ], Kvasnicka [102], Fukutome [103, 92], and Valdemoro [104, 105], to name a few. The present author used creation operators composed of two fermion operators to describe geminals in orthogonal [106, 107, 108, 98] and non-orthogonal [109, 110, 111, 112, 113] basis sets. Second quantization for geminals will be reviewed in Sect. 3.2. 

In the spin-representation the two, three and four electron functions of the basis are simple products of fermion operators. Therefore, the upper limit of their occupation number is one. This upper bound value has also been adopted for the elements in a spin-adapted basis of representation. We know also that the diagonal elements must be positive. Finally, we know the value not only of the trace but also of the partial traces of the spin-adapted matrices (18, 19, 20). 

They use the boson representation of fermion operators to evaluate the leading asymptotic terms in spin correlation functions. For example,... 

As a check, we find the exponent corresponding to (12) from (7) using the functional integral method. Our qp from (13) is actually the phase of the pair operator rather than of the single fermion operator, so (7) loses the factor of two in the exponent. Hence, the result differs from (8) by a factor of four, that is,... 

Yj for spin-flip. The model is that of Luther and Emery [lj, before they introduce new quasi-fermion operators to obtain a gap at Yll — - 3/s. The free energy corresponding to... 

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