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arXiv:1003.2860 · Physical Review B

Density-matrix functionals for pairing in mesoscopic superconductors

Denis Lacroix A, Guillaume Hupin

Published Oct 13, 2010• 10 citations
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Abstract

A functional theory based on single-particle occupation numbers is developed for pairing. This functional, that generalizes the Bardeen-Cooper-Schrieffer approach, directly incorporates corrections due to particle number conservation. The functional is benchmarked with the pairing Hamiltonian and reproduces perfectly the energy for any particle number and coupling.

DOI
10.1103/physrevb.82.144509

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A functional theory based on single-particle occupation numbers is developed for pairing. This functional, that generalizes the Bardeen-Cooper-Schrieffer approach, directly incorporates corrections due to particle number conservation. The functional is benchmarked with the pairing Hamiltonian and reproduces perfectly the energy for any particle number and coupling.
We improve the Bardeen-Cooper-Schrieffer wave function with a fixed particle number so as to incorporate many-body correlations beyond the mean-field treatment. It is shown that the correlations lower the ground-state energy far more than Cooper-pair condensation in the weak-coupling region. Moreover, they naturally bring a superposition over the number of condensed particles. Thus, Cooper-pair condensation is special among the various bound-state formations of quantum mechanics in that number fluctuations are necessarily present in the condensate through the dynamical exchange of particles with the non-condensate reservoir. On the basis of this result, we propose ΔNcon⋅Δϕ≳1\varDelta N_{\rm con}\cdot \varDelta ϕ\gtrsim 1ΔNcon​⋅Δϕ≳1 as the uncertainty relation relevant to the number-phase fluctuations in superconductors and superfluids, where the number of condensed particles NconN_{\rm con}Ncon​ is used instead of the total particle number NNN. The formula implies that a macroscopic phase ϕϕϕ can be established even in number-fixed superconductors and superfluids since ΔNcon≫1\varDelta N_{\rm con}\gg 1ΔNcon​≫1.
A new microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. Instead of the variation principle, we start from the Heisenberg equations of motion for density matrix operators ${K}_{12}\ensuremath{\equiv}{a}_{2}{a}_{1}$. The theory generalizes the well-known particle-particle random-phase approximation. Its analytical equations are as simple as that of the BCS theory and can be solved within a similar computer time if the model space is not too large. The current theory conserves the exact particle number and is valid at arbitrary pairing strength (including those below the BCS critical strength). It is also of interest to other mesoscopic systems such as ultrasmall metallic grains.
Different steps leading to the new functional for pairing based on natural orbitals and occupancies proposed earlier [D. Lacroix and G. Hupin, Phys. Rev. B 82, 144509 (2010)] are carefully analyzed. Properties of quasiparticle states projected onto good particle numbers are first reviewed. These properties are used to (i) prove the existence of such a functional, (ii) provide an explicit functional through a 1/N1/N1/N expansion starting from the BCS approach, and (iii) give a compact form of the functional summing up all orders in the expansion. The functional is benchmarked in the case of the picket-fence pairing Hamiltonian where even and odd systems are studied, using the blocking technique, at various particle numbers and coupling strengths, with uniform and random single-particle level spacing. In all cases, very good agreement is found, with a deviation of $<$1% compared to the exact energy.
In this work, a new functional is introduced to treat pairing correlations in finite many-body systems. Guided by the projected BCS framework, the energy is written as a functional of occupation numbers. It is shown to generalize the BCS approach and to provide an alternative to Variation After Projection framework. Illustrations of the new approach are given for the pairing Hamiltonian for various particle numbers and coupling strengths. In all case, a very good agreement with the exact solution is found.
The dynamics of BCS (Bardeen-Cooper-Schrieffer) superconductors is fairly well understood due to the availability of a mean-field solution for the pairing Hamiltonian, a solution which gives the quantum state of a superconductor as a state of almost-free fermions interacting only with a condensate. As a result, transition probabilities may be computed, and expressed in terms of matrix elements of electron creation and annihilation operators between approximate eigenstates. These matrix elements are also called ``coherence factors.'' Mean-field theory is however not sufficient to describe all eigenstates of a superconductor, a deficiency which is hardly important in (or very close to) equilibrium, but one that becomes relevant in certain out-of-equilibrium situations. We report here on a computation of matrix elements (coherence factors) for the pairing Hamiltonian between any ``two-arc'' eigenstates in the thermodynamic limit.