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International Journal of Quantum Chemistry

A density-matrix formulation of the hartree-fock-bogoliubov equations for space-dependent superconductivity

William L. Clinton

Published Jun 18, 2009• 5 citations
Publisher

Abstract

The Hartree-Fock-Bogoliubov equations that govern the quasi-particle states of space-dependent superconductivity are formulated in terms of the pure state density matrices ρn(′, )  Φn() Φ (). If is the Hartree-Fock-Bogoliubov Hamiltonian where HH is the normal metal part and Δ() the pairing potential, then the density operator equations are The pairing potential Δ() = V Σnun() V() tanh (βEE/2) is a mixed state quantity and, just as in the normal metal, the equations must be solved self-consistently. We use a steepest descent (optimum gradient) technique, developed by McWeeny for Hartree-Fock equations, to derive an iterative algorithm for solving the above equations. The formulism is developed in general for nonzero temperatures. The method is applied to the T ≈︁ TT region.

DOI
10.1002/qua.560070755

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The Hartree-Fock-Bogoliubov equations that govern the quasi-particle states of space-dependent superconductivity are formulated in terms of the pure state density matrices ρn(′, )  Φn() Φ (). If is the Hartree-Fock-Bogoliubov Hamiltonian where HH is the normal metal part and Δ() the pairing potential, then the density operator equations are The pairing potential Δ() = V Σnun() V() tanh (βEE/2) is a mixed state quantity and, just as in the normal metal, the equations must be solved self-consistently. We use a steepest descent (optimum gradient) technique, developed by McWeeny for Hartree-Fock equations, to derive an iterative algorithm for solving the above equations. The formulism is developed in general for nonzero temperatures. The method is applied to the T ≈︁ TT region.
To describe comprehensively the superconducting phase together with the other phases realized in the copper oxcides, the normal phase, the antiferro-magnetic and the spin-glass phases, an application of the constrained thermal Hartree-Fock-Bogoliubov (CTHFB) approximation is proposed. A set of the electron number- and the magnetization-constrained formalism is exemplified by employing the one-band Hubbard model. The CTHFB equation is applicable to the microscopic interaction whether it includes the phonon-mediated pairing interaction, or it provides a self-sustaining mechanism of the Cooper pair formation. A new type of the order parameter is introduced by extending the generalized Bogoliubov transformation to mix electron spin projections onto the quantization axis (i.e. S^z\hat{S}_{z}S^z​). This yields the lowered minimum of the grand potential, which corresponds to the physical solution violating time-reversal invariance. The gradient method is proposed to solve the CTHFB equation, the non-linear eigenvalue broblem with constraints.
Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we rely on a single variational principle which optimizes the characteristic function. The Bloch equation is used as a constraint to define the equilibrium state, and the trial quantities are a density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity are respectively a Schrödinger-like equation and a Heisenberg-- like equation with an imaginary time running backwards. General conditions for the trial spaces are stated that warrant the preservation of thermodynamic relations. When the trial spaces are chosen to be of the independent-quasi-particle type, the ensuing coupled equations provide an extension of the Hartree-Fock-Bogoliubov approximation, which optimizes the characteristic function. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature TTT, the level density and the strength of the pairing force, are investigated analytically and numerically.
The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics, Bose–Einstein condensation and quantum chemistry. We reformulate the Dirac–Frenkel approximation principle in terms of reduced density matrices and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov–de Gennes and Hartree–Fock–Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov–de Gennes equations in energy space and discuss conserved quantities.
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.
Hartree-Fock-Bogoliubov (HFB) theory is generalized by eliminating the restriction to quasiparticle vacuum states. However, the fundamental HFB transformation to quasiparticles is retained. The theory is formulated in terms of the quasiparticle density matrix RRR, but, in contrast to conventional HFB, there is no restriction to vacuum densities R2=R{R}^{2}=RR2=R.