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Journal of physics. A, Proceedings of the Physical Society. General

Projection in the occupation-number space and the canonical transformation

V. N. Fomenko

Published Jan 1, 1970• 66 citations
Publisher

Abstract

A method for projecting the Bardeen-Cooper-Schrieffer states with respect to the particle number is suggested. This method does not involve the Fowler-Darwin integrals which are usually applied for this purpose. Energy matrix elements of an arbitrary single-particle operator and overlap integrals are calculation in the quasi- particle representation and are expressed analytically in terms of the u and v parameters. The volume of calculations according to the method suggested increases linearly with the size of the system in the case of factorizable pairing matrix elements (the weaker the particle number fluctuations, the smaller the quantity of calculations). The method is illustrated by some model systems. In the paper the limiting case of weak correlations is considered and comparison with perturbation theory is carried out.

DOI
10.1088/0305-4470/3/1/002

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Progress of Theoretical Physics•May 1, 1969•1 citation
A method for projecting the Bardeen-Cooper-Schrieffer states with respect to the particle number is suggested. This method does not involve the Fowler-Darwin integrals which are usually applied for this purpose. Energy matrix elements of an arbitrary single-particle operator and overlap integrals are calculation in the quasi- particle representation and are expressed analytically in terms of the u and v parameters. The volume of calculations according to the method suggested increases linearly with the size of the system in the case of factorizable pairing matrix elements (the weaker the particle number fluctuations, the smaller the quantity of calculations). The method is illustrated by some model systems. In the paper the limiting case of weak correlations is considered and comparison with perturbation theory is carried out.
A variational principle is formulated to determine the single-particle states, their pairing, and the occupation number distribution for a trial state vector of the Bardeen, Cooper, Schrieffer type. The equations which are derived generalize those of the Hartree-Fock method obtained with a Slater determinant trial wave function. It is pointed out that in a suitable representation the vacuum state of a general quasi-particle transformation has such a trial form which exhibits directly the pairing of single-particle states. Another variational principle determines the excitation energies.Two coupling cases are distinguished: the commutative case in which the self-consistent densities and energies are related to quantities which all commute, and the more general noncommutative case. The latter is of importance in critical-field phenomena. The equations for the commutative case can be written in a matrix form which retains its validity in the more general noncommutative case. The simple matrix commutator equations appear as direct generalizations of the density matrix form of the Hartree-Fock equations.The equations for small oscillations have an equally simple form. Their connection with a diagonal representation of the quasi-particle energies is exhibited in a way which remains valid in the general coupling case. The "unphysical" solutions are excluded by the supplementary condition. The contact with the Green's function approach is established. The generalized matrix form of the Green's function equations shows especially clearly the symmetry properties of the method.
The canonical operator exp [−β(H − μN)] associated with the Bardeen-Cooper-Schrieffer (BCS) model Hamiltonian of superconductivity is represented as a functional integral by the use of Feynman's ordering parameter. General properties of the partition function in this representation are discussed. Taking the inverse volume of the system as an expansion parameter, it is possible to calculate the thermodynamic potential including terms independent of the volume. This yields a new proof that the BCS variational value is asymptotically exact. The behavior of the canonical operator for large volume is described and related to the state of free quasiparticles. A study of the terms of the thermodynamic potential which are of smaller order in the volume in the low-temperature limit, shows that the ground state energy is nondegenerate and belongs to a number eigenstate.
Wick's theorem for the expectation value of an operator between many-particle states is generalized to the case of a homogeneous linear transformation operator Oˆ\^OOˆ, which leaves invariant the 2N2N2N-dimensional linear space composed of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the group SO(2N,C)\mathrm{SO}(2N,C)SO(2N,C) for the case of fermion, or the group Sp(2N,C)\mathrm{Sp}(2N,C)Sp(2N,C) for the case of boson. This theorem is applied to derive a reduction theorem for the matrix element of the type $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\alpha}}_{m}\^O{\ensuremath{\beta}}_{m+1}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\beta}}_{m+2}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\beta}}_{2n}^{\ifmmode\dagger\else\textdagger\fi{}}|\ensuremath{\beta}〉,$ whose many-quasiparticle states and vacua appearing in both sides are represented in different quasiparticle pictures as indicated by $\ensuremath{\alpha}$ and $\ensuremath{\beta}.$ The reduction formula is useful in calculating any nuclear transition matrix element for the quantum-number projection calculation associated with the constrained Hartree-Fock-Bogoliubov solutions, and also the generator coordinate method with quantum number projection.
The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two-, and four-body operators within a basis of particle-number-projected BCS states are analytically expressed in terms of a single function Q(N)Q(N)Q(N) depending on the number of particles, NNN. The function Q(N)Q(N)Q(N) is calculated using a recursion in NNN in which the shells and the BCS angles are kept the same for any iteration step. An illustrative example is numerically considered in a restricted single-particle space. Some specific features of the standard BCS, the projection after variation approach, and the variation after projection formalism are pointed out.
The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two- and four-body operators within a basis of particle number projected BCS states, are analytically expressed in terms of a single function Q(N) depending on the number of particles, NNN. The function Q(N) is calculated using a recursion in NNN in which the shells and the BCS angles are kept the same for any step of iteration. An illustrative example is numerically considered in a restricted single particle space. Some specific features for the standard BCS, the projection after variation approach as well as for the variation after projection formalism, are pointed out.