The principle of compensation of dangerous diagrams (PCDD), used by Bogoliubov to determine the coefficients in the canonical transformation to quasiparticles in boson systems, is obtained by maximizing the overlap between the true ground-state vector and the quasiparticle vacuum state. The zero-momentum state is treated exactly, which implies that the sum of all diagrams leading from the vacuum to a one-quasiparticle state must be zero, in addition to the diagrams leading from the vacuum to the two quasiparticle state. Other criteria, such as the diagonalization of the quasiparticle reaction operator up to terms cubic in the operators, and the absence of one and two quasiparticle contributions to the true ground-state wavefunction, are also shown to lead to the PCDD. A generalization of the Hartree procedure of minimizing the ground-state energy is obtained by replacing the bare quasiparticle interaction with the quasiparticle reaction operator, and is shown to be equivalent to the PCDD. Finally, a perturbation expansion of the PCDD is obtained, and the reducibility of diagrams is discussed.
The principle of compensation of dangerous diagrams (PCDD) postulated by Bogoliubov to determine the coefficients in his canonical transformation in boson systems is obtained from four criteria: (1) the number of quasiparticles in the true ground state is a minimum; (2) the ``best'' approximation to the true density matrix and pair amplitude is made; (3) the expectation value of an arbitrary operator is diagonalized up to terms cubic in the quasiparticle operators; (4) the most convenient starting point for dressing the quasiparticles is used. Quasiparticle Green's functions are introduced to obtain a diagrammatic expansion of the PCDD. The reducibility of the diagrams is also discussed.
The principle of compensation of dangerous diagrams (PCDD) is derived at finite temperature for boson systems by minimizing the average number of Bogoliubov quasiparticles in the system. The conditions obtained state that (a) the amplitude for the creation (or annihilation) of a single quasiparticle is zero and (b) the amplitude for the creation (or annihilation) of a pair of quasiparticles is zero. These conditions are expanded in finite-temperature perturbation theory, using both the density matrix and Green's function methods. In first order the resulting equations are the Hartree-Fock-Bogoliubov equations for a homogeneous boson system at finite temperature which can also be obtained from a free energy variational principle.
The principle of compensation of dangerous diagrams (PCDD) postulated by Bogoliubov to determine the coefficients in the canonical transformation to quasi-particles in superconducting systems is derived from four different criteria (1) the expected number of quasi-particles in the true ground state is a minimum, (2) the one-particle density matrix and the two-particle amplitude determined from the BCS ground state are equated to the true ones, (3) the expectation value of an arbitrary operator is simplified by diagonalizing its quadratic part, and (4) the starting point for the dressing of the quasi-particle is chosen in the most convenient way. The condition obtained for the PCDD is then expressed in terms of quasi-particle Green's functions. The ladder diagrams are eliminated by examining an integral equation for the Green's function describing the creation of two quasi-particles from the vacuum. Finally, the condition obtained here for the PCDD is compared with the condition obtained previously.
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.
During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.