A general Hamiltonian H of electrons in finite concentration, interacting via any two-body coupling inside a crystal of arbitrary dimension, is considered. For simplicity and without loss of generality, a one-band model is used to account for the electron-crystal interaction. The electron motion is described in the Hilbert space ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, spanned by a basis of Slater determinants of one-electron Bloch wave functions. Electron pairs of total momentum K and projected spin \ensuremath{\zeta}=0,\ifmmode\pm\else\textpm\fi{}1 are considered in this work. The Hamiltonian then reads H=
HD+${\ensuremath{\sum}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, where
HD consists of the diagonal part of H in the Slater determinant basis. ${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$ describes the off-diagonal part of the two-electron scattering process which conserves K and \ensuremath{\zeta}. This Hamiltonian operates in a subspace of ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, where the Slater determinants consist of pairs characterized by the same K and \ensuremath{\zeta}. It is shown that the whole set of eigensolutions \ensuremath{\psi},\ensuremath{\epsilon} of the time-independent Schr\"odinger equation (H-\ensuremath{\epsilon})\ensuremath{\psi}=0 divides into two classes, ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ and ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$. The eigensolutions of class 1 are characterized by the property that for each solution ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ there is a single K and \ensuremath{\zeta} such that (
HD+${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$-${\mathrm{\ensuremath{\epsilon}}}_{1}$)${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$=0 where, in general, ${\mathrm{\ensuremath{\psi}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, whereas each solution ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$ of class 2 fulfills (
HD-${\mathrm{\ensuremath{\epsilon}}}_{2}$)${\mathrm{\ensuremath{\psi}}}_{2}$=0. We prove also that the eigenvectors of class 1 have off-diagonal long-range order, whereas those of class 2 do not. Finally, our result shows that off-diagonal long-range order is not a sufficient condition for superconductivity. \textcopyright{} 1996 The American Physical Society.
In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of
N antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order
k has two sets of indices of each
k variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order
N over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete
k-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly
N spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order
M, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
Within the framework of a restricted basis, a bonding and antibonding orbital for the
H2 molecule, a set of boson operators is found which transforms the four configurations of zero total spin (Slater determinants) into the corresponding fully correlated stationary states. In terms of these boson operators the two-body electron Hamiltonian reduces to a one-body diagonal form. This Hamiltonian can be naturally split into two parts, closed-shell and open-shell terms, behaving in very different ways; in the open-shell Hamiltonian the electrons keep their individual character while in the closed-shell Hamiltonian the coupling between electrons is so strong that it is better to treat them in a collective way. In contrast to the usual quasiboson formalism, the bosons considered in this paper commute correctly and consist of linear combinations of four electron operators representing the normal modes of the two-electron pair.
The Fock space Hamiltonian H has a simpler structure than its projection Hn to n-particle Hilbert space. It is therefore recommended to diagonalize H—to the extent that this is possible—before one specifies n. Diagonalization of H is possible if one defines the diagonal and nondiagonal parts of an operator appropriately. It is shown that the diagonalized Fock space Hamiltonian L, called an energy operator, contains all information about the eigenvalues of H in a simply coded form. For a spinfree Hamiltonian, spin can be completely eliminated and all interesting quantities are expressible in terms of spinfree excitation operators (generators of the unitary group). The construction of the wave operator W and the energy operator L is formulated in terms of perturbation theory both in the strictly degenerate and the quasidegenerate version. Three variants are discussed that differ in the normalization of W, namely, a intermediate normalization, b and c unitary normalization with two additional conditions, b: WD=WD+, c:σD=(ln W)D=0. The transformation of operator products to normal product form is achieved by means of a generalization of Wick’s theorem to spinfree quantum chemistry. The contributions to the various orders of L, W, and σ are illustrated by means of diagrams. Only variant c guarantees a connected-diagram expansion of the Fock space energy operator L. None of the three variants satisfies a generalization of the Wigner (2n+1) rule, but with a slight change in the normalization from b to b′ or c to c′ (that does, in the case of c′, not affect the connectedness of the diagram expansion) L(2n) and L(2n+1) are expressible in terms of the W(k), k⩽n. A generalization of the variation principle for the Fock space energy operator is given and its consequences are discussed. Generalized Fock space Brillouin conditions are derived and a variational scheme for the construction of the energy operator is proposed. We finally compare the present Fock space approach with the traditional degenerate perturbation theory in n-particle Hilbert space. If the diagonal part ΩD of an operator Ω is defined in the spirit of the theory of the universal wave and energy operators, then the two approaches lead to the same results (for the same normalization), but the Fock space approach looks simpler and more transparent.
We present stochastic diagonalization results for the ground-state energy and the largest eigenvalue of the two-fermion density matrix of the BCS reduced Hamiltonian, the Hubbard model, and the Hubbard model with correlated hopping. The system-size dependence of this eigenvalue is used to study the existence of Off-Diagonal Long-Range Order in these models. We show that the model with correlated hopping and repulsive on-site interaction can exhibit Off-Diagonal Long-Range Order. Analytical results for some special limiting cases indicate that Off-Diagonal Long-Range Order not always implies superconductivity.