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arXiv:0906.3677 · Physical Review B

Self-consistent T -matrix theory of superconductivity

Břetislav Šopík, P. Lipavský, M. Männel, Klaus Morawetz, Peter Matlock

Published Sep 27, 2011• 12 citations
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Abstract

Using principles of the Fadeev-Lovelace-Watson multiple scattering expansion, a $T$-matrix approximation is derived which coincides with the Galitskii-Feynman $T$ matrix in the normal state and yields the gap in the superconducting state. Unlike other $T$-matrix approaches, the theory satisfies not only the self-consistent Thouless criterion but also the Baym-Kadanoff conditions for a conserving theory in equilibrium. In single-mode approximation it simplifies to the Eliashberg theory.

DOI
10.1103/physrevb.84.094529

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Using principles of the Fadeev-Lovelace-Watson multiple scattering expansion, a TTT-matrix approximation is derived which coincides with the Galitskii-Feynman TTT matrix in the normal state and yields the gap in the superconducting state. Unlike other TTT-matrix approaches, the theory satisfies not only the self-consistent Thouless criterion but also the Baym-Kadanoff conditions for a conserving theory in equilibrium. In single-mode approximation it simplifies to the Eliashberg theory.
We present unified theory of both normal and superconducting states based on the fully self-consistent T\mathcal{T}T-matrix approximation. We argue that the failure of the customary T\mathcal{T}T-matrix approach below the critical temperature is caused by nonphysical repeated collisions. We eliminated the repeated collisions from the Galitskii-Feynman approximation. Obtained corrections are proportional to the inverse volume in the normal state and, thus, vanish in the thermodynamical limit. In the superconducting state these corrections remain finite and describe the Bardeen-Cooper-Schrieffer (BCS) gap. The T\mathcal{T}T-matrix approach goes beyond mean-field BCS theory. It is, thus, well suited for studies of the systems with large fluctuations such as the BCS--Bose-Einstein-condensate crossover or size effects in small metallic grains and the nuclear matter.
The theory of superconducting pairing due to the exchange and kinematical interactions in the t - J model in a paramagnetic state is developed. \nThe Dyson equation for the matrix Green functions in terms of the Hubbard \noperators is obtained in the noncrossing approximation. The linearized self-сonsistent system of Eliashberg equations is proposed to study the temperature and doping dependence of the quasi-particle hole spectrum in \nthe normal state and to calculate the temperature of the superconducting \nphase transition and the symmetry of the gap function.
A general theory of superconductivity is formulated within the thermodynamic Green function method for various types of pairing mediated by phonons, spin fluctuations, and strong Coulomb correlations in the Hubbard and t-J models.A rigorous Dyson equation for matrix Green functions is derived in terms of a self-energy as a many-particle Green function.By applying the noncrossing approximation for the self-energy, a closed selfconsistent system of equations is obtained, similar to the conventional Eliashberg equations.A brief discussion of superconductivity mediated by kinematic interaction with an estimation of a superconducting transition temperature in the Hubbard model is given.
The diagrammatic ttt-matrix approximation has often been adopted to describe a dilute Fermi gas. This approximation, originally considered by V. M. Galitskii (Zh. Eksp. Teor. Fiz. 34, 151 (1958) [Sov. Phys. JETP 7, 104 (1958)]) for a repulsive interparticle interaction, was later widely utilized for an attractive Fermi gas to describe the BCS-BEC crossover from strongly overlapping Cooper pairs in weak coupling to nonoverlapping composite bosons in strong coupling. Several variants of the ttt-matrix approximation have been considered in the literature, which are distinguished by the degree of self-consistency allowed in the building blocks of the diagrammatic structure. Here, we perform a systematic and comparative study of all possible variants on the degree of self-consistency for the ttt-matrix approximation in an attractive Fermi gas, which enables us to confront their outcomes for thermodynamic and dynamical quantities on the same footing in an unbiased way. For definiteness, only the normal phase above the superfluid critical temperature is considered. The dispute that can be raised in this context, about the adequateness of introducing progressive degrees of self-consistency over and above the non-self-consistent ttt-matrix approximation for an attractive Fermi gas, parallels the recent interest in the literature in assessing the importance of various degrees of self-consistency in the context of semiconductors and insulators.
We address conservation laws associated with current, momentum, and energy and show how they can be satisfied within many body theories which focus on pair correlations. Of interest are two well known TTT-matrix theories which represent many body theories which incorporate pairing in the normal state. The first of these is associated with the Nozieres Schmitt-Rink theory, while the second involves the TTT matrix of a BCS-Leggett-like state as identified by Kadanoff and Martin. TTT-matrix theories begin with an ansatz for the single particle self energy and are to be distinguished from $\ensuremath{\Phi}$-derivable theories which introduce an ansatz for a particular contribution to the thermodynamical potential. Conservation laws are equivalent to Ward identities which we address in some detail here. Although $\ensuremath{\Phi}$-derivable theories are often referred to as ``conserving theories,'' a consequence of this work is the demonstration that these two TTT-matrix approaches similarly can be made to obey all conservation laws. When simplifying approximations are made in $\ensuremath{\Phi}$-derivable or other theories, one has to take care that the end results are not incompatible with conservation.
The theory of superconductivity is formulated starting from the configuration-space form of the Bardeen-Cooper-Schrieffer (BCS) wave function. Self-consistent equations for treating nonu- niform systems are obtained. These equations have the advantage over the Bogoliubov-de Gennes equations that an effective condensate wave function and quasiparticle eigenstates can separately be obtained. The intrinsic orbital angular momentum density of the condensate is naturally iden- tified in this formalism due to a manifest two-particle bound-state nature in the one-particle density matrix.