Particle-hole

The matrices A, B, Q are of infinite dimension since there are an infinite number (2N+1, N — oo) of k-values and thus an infinite number of k-states in each band. Moreover, there is an equation for each triplet formed by a k-value and two band indices. This triplet represents a particle-hole excitation that is vertical in order to preserve the momentum. As is the case in many polymeric techniques, the infinite sum over k is transformed into an integration in the first Brillouin zone ... 

Again using the completeness of the particle-hole states (eq. 10), we find that the Bethe sum rule (eq. 6) is fulfilled. 

The above observation suggests an intriguing relationship between a bulk property of infinite nuclear matter and a surface property of finite systems. Here we want to point out that this correlation can be understood naturally in terms of the Landau-Migdal approach. To this end we consider a simple mean-field model (see, e.g., ref.[16]) with the Hamiltonian consisting of the single-particle mean field part Hq and the residual particle-hole interaction Hph-... 

In the Landau-Migdal approach the effective isovector particle-hole interaction Hph is given by... 

In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle-hole propagators, which have been at the heart of Jens electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9-15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions ... 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

M. V. Mihailovic and M. Rosina, The particle-hole states in some light nuclei calculated with the two-body density matrix of the ground state. Nucl. Phys. A237, 229-234 (1975). 

M. Rosina and C. Garrod, The particle-hole matrix its connection with the symmetries and collective features of the ground state. J. Math. Phys. 10, 1855 (1969). 

M. Rosina and M. V. Mihailovic, The determination of the particle—hole excited states by using the variational approach to the ground state two-body density matrix, in International Conference on Properties of Nuclear States, Montreal 1969, Les Presses de I Universite de Montreal, 1969. 

M. Bouten, P. Van Leuven, M. V. Mihailovic, and M. Rosina, A new particle-hole approach to collective states. Nucl. Phys. A202, 127—144 (1973). 

M. V. Mihailovic and M. Rosina, Particle-hole states in light nuclei, in Proceedings of the International Corferenee on Nuclear Self-Consistent Fields, Trieste 1975 (G. Ripka and M. Porneuf, eds.), North-HoUand, Amsterdam, 1975, p. 37. 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

By particle-hole duality, the same block structure appears in the spin-adapted two-electron RDM. The four blocks of the 2-RDM have the following traces [57] ... 

M. B. Ruskai, iV-representability problem particle-hole equivalence. J. Math. Phys. 11, 3218 (1970). 

This generalized particles-holes separating approach generates an algorithm (GP-H) that emphasizes the role of the 2-RDM— the variable of the 2-CSE— and it is computationally more economical [54]. 

The CSE allows us to recast A-representability as a reconstruction problem. If we knew how to build from the 2-RDM to the 4-RDM, the CSE in Eq. (12) furnishes us with enough equations to solve iteratively for the 2-RDM. Two approaches for reconstruction have been explored in previous work on the CSE (i) the explicit representation of the 3- and 4-RDMs as functionals of the 2-RDM [17, 18, 20, 21, 29], and (ii) the construction of a family of higher 4-RDMs from the 2-RDM by imposing ensemble representability conditions [20]. After justifying reconstmction from the 2-RDM by Rosina s theorem, we develop in Sections III.B and III.C the functional approach to the CSE from two different perspectives—the particle-hole duality and the theory of cumulants. 

Many-body problems in quantum mechanics are usually described by the number of particles N in the system and the probabilities of finding those particles at different locations in space. If the rank of the one-particle basis is a finite number r, an equally valid description of the system may be given by specifying the number of holes r N in the system and the probabilities of finding these holes at different locations in space. This possibility for an equivalent representation of the system by particles or holes is known as the particle-hole duality. By using the fermion anticommutation relation... 

Valdemoro and co-workers [14] realized that these particle-hole relations could be written in the following form ... 

The reconstruction functionals, derived in the previous section through the particle-hole duality, may also be produced through the theory of cumulants [21,22,24,26,39,55-57]. We begin by constructing a functional whose derivatives with respect to probe variables generate the reduced density matrices in second quantization. Because we require that additional derivatives increase the number of second quantization operators, we are led to the following exponential form ... 

In the Af-representability literature these positivity conditions are known as the D- and the g-conditions [5, 7, 63]. The two-particle RDM and the two-hole RDM are linearly related via the particle-hole duality,... 

That is, both the 2-CM and the 2-G matrix have common elements, but a given element occupies different positions in each matrix. In other words, while the labels of the row/column of the 2-CM refer, as in the 2-RDM, to two particlesitwo holes, the labels of the row/column of the 2-G matrix refer to particle-hole/hole-particle. Thus, although both the 2-CM and the 2-G matrices describe similar types of correlation effects, only the 2-CM describes pure two-body correlation effects. This is because the 2-CM natural tensorial contractions vanish, and thus there is no contribution to the natural contraction of the 2-RDM into the one-body space whereas the 2-G natural tensorial contractions are functionals of the 1-RDM. 

Particularly noteworthy is the particle—hole symmetry. Let us define (one- and two-) hole density matrices [17]... 

One can further define a particle-hole density matrix, which also has the same two-particle cumulant... 

Similar inequalities hold for the two-hole and particle-hole density matrices [21,22] ... 

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