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Oxford University Press eBooks

The Bogoliubov Approach

Monique Combescot, Shiue-Yuan Shiau

Published Dec 1, 2015• 0 citations
Publisher

Abstract

Chapter 8 deals with the Bogoliubov approach to the BCS problem. This approach is of fundamental interest because it gives a mathematical support to the BCS ansatz, which was simply proposed by Bardeen, Cooper, and Schrieffer as a trial wave function. This chapter shows that the BCS ansatz corresponds to the ground-state wave function of the one-body Bogoliubov Hamiltonian constructed from the BCS Hamiltonian. This mathematical support is highly welcome because the BCS ansatz contains the questionable idea that composite bosons condense, like elementary bosons, all into the same state, even if the product of N identical electron pair operators substantially differs from that of N single pair operators when N gets large, owing to the moth-eaten effect induced by the Pauli exclusion principle.

DOI
10.1093/acprof:oso/9780198753735.003.0008

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Chapter 8 deals with the Bogoliubov approach to the BCS problem. This approach is of fundamental interest because it gives a mathematical support to the BCS ansatz, which was simply proposed by Bardeen, Cooper, and Schrieffer as a trial wave function. This chapter shows that the BCS ansatz corresponds to the ground-state wave function of the one-body Bogoliubov Hamiltonian constructed from the BCS Hamiltonian. This mathematical support is highly welcome because the BCS ansatz contains the questionable idea that composite bosons condense, like elementary bosons, all into the same state, even if the product of N identical electron pair operators substantially differs from that of N single pair operators when N gets large, owing to the moth-eaten effect induced by the Pauli exclusion principle.
The crucial demonstration that superfluidity was linked to Bose particles and Bose-Einstein condensation came after the discovery of He3 in 1949, which failed to show the characteristic A transition within a reasonable wide temperature interval around the critical temperature for the onset of superfluidity in He4. The idea of the bosonization of electrons due to their pairing, introduced by Ogg (1946) became very attractive as an explanation for superconductivity after Schafroth (1955) showed that a gas of charged bosons with charge 2e would indeed show a Meissner effect, which is one of the most crucial features of superconductors (see Chapter 1). However this phenomenological Bose-gas picture was condemned and later on practically forgotten because it was unable to account quantitatively for the critical parameters of the classical (low-temperature) superconductors. Using for Tc the Bose-Einstein condensation temperature Eq.(l.S) one obtains the utterly meaningless result Tc = lOOOO.R&s; with the atomic density of electron pairs ~ 1022cm~3 and with the effective mass m** = 2me. The failure of the Schafroth picture stems from a very large size £ of pairs in classical superconductors, which can be estimated from the uncertainty principle: The uncertainty of the momentum 8p is estimated from the uncertainty of the kinetic energy 8E ~ vpSp, which should be of the order of Tc, the only characteristic energy of the superconducting state. Therefore and turns out to be more than one micron for simple metals with Tc ~ IK and the Fermi velocity VF — I0scm/s. That means that pairs in classical superconductors overlap strongly and the Shafroth model of real space pairs is wrong. The finally successful theory for classical superconductors was formulated by Bardeen, Cooper and Schrieffer (1957). In this chapter we present the main points of the BCS theory, using the Bogoliubov transformation (1947,1958) for the weak-coupling regime and the Green&s;s function technique for the intermediate coupling in Chapter 3. More details will be found in the excellent books by Schrieffer(1965), De Gennes (1989) and Abrikosov, Gor&s;kov and Dzyaloshinskii (1963).
Abstract One of the important steps leading to the successful formulation of the microscopic theory of superconductivity was the model calculation performed by Cooper (1956) which showed that two electrons, interacting above a filled Fermi sea, had a bound state in the presence of an arbitrarily weak attractive interaction. This chapter first examines Cooper’s calculation. The success of the Cooper model calculation in producing a bound state, with its implication that a proper many-body theory would contain a gap in the excitation spectrum, strongly suggests that the ground state wave function should be constructed from pairs of electrons. However, it is difficult to perform calculations using the many-body pair wave function from Cooper’s model. The chapter then introduces the wave function proposed by Bardeen, Cooper, and Schrieffer (BCS) as an alternative.
The ground-state wave function and some of the excited states of the BCS reduced Hamiltonian are found. In the limit of large volume, the boundary and continuity conditions on the exact wave function lead directly to the equations which Bardeen, Cooper, and Schrieffer found by a variational technique. It is also shown in what sense the BCS trial wave function may be considered asymptotically exact in this limit. Finite-volume corrections are included in an appendix, and explicit calculations are carried out for a one-step model of the kinetic energy which has possible applications to the problem of the finite nucleus.
We review efforts to unify both the Bardeen, Cooper and Schrieffer (BCS) and Bose-Einstein condensation (BEC) pictures of superconductivity. We have finally achieved this in terms of a "\textit{complete} boson-fermion (BF) model" (CBFM) that reduces in special cases to all the main continuum (as opposed to "spin") statistical theories of superconductivity. Our BF model is "complete" in the sense that not only two-electron (2e) but also two-hole (2h) Cooper pairs (CPs) are allowed in arbitrary proportions. In contrast, BCS-Bogoliubov theory--which can also be considered as the theory of a mixture of kinematically independent electrons, 2e- and 2h-CPs--allows only equal, 50%-50%, mixtures of the two kinds of CPs. This is obvious from the perfect symmetry about μμμ, the electron chemical potential, of the well-known Bogoliubov v2(ε)v^{2}(ε)v2(ε) and u2(ε)u^{2}(ε)u2(ε) coefficients, where εεε is the electron energy. The CBFM is then applied to see: a) whether the BCS model interaction for the electron-phonon dynamical mechanism is sufficient to predict the unusually high values of TcT_{c}Tc​ (in units of the Fermi temperature) of ≃0.01−0.1\simeq 0.01-0.1≃0.01−0.1 exhibited by the so-called ``exotic'' superconductors \cite{Brandow} in both 2D and 3D--relative to the low values of ≲10−3\lesssim 10^{-3}≲10−3 more or less correctly predicted by BCS theory for conventional, elemental superconductors; and b) whether it can at least suggest, if not explain, why "hole superconductors" have higher TcT_{c}Tc​'s.
We present a many-body formalism for BCS-BEC crossover, which represents a modification of the Bardeen-Cooper-Schrieffer-Leggett ground state to include four-fermion and higher correlations. In the Bose-Einstein condensate regime, we show how our approach contains the four-fermion behavior of Petrov et al. and associated scattering length add{a}_{dd}add​ at short distances and, second, reduces to composite-boson Bogoliubov physics at long distances. It reproduces the Lee-Yang term, whose numerical value is also fixed by add{a}_{dd}add​. We have also examined the next term beyond the Lee-Yang correction in a phenomenological fashion, building on cloud size data and collective mode experiments, although one has to view this phenomenological analysis with some caution since experiments are in a state of flux and are performed close to unitarity.