P. Contreras

Thanks a lot dear professor i will read this papers ton find the appropriate answer

Best Regards

You should be more specific. For instance, which 'spectral function'? Very generally, in all thermodynamic considerations of systems in grand canonical ensemble one deals with the thermodynamic Hamiltonian K ≡ H - μN, where H is the Hamiltonian operator, N the total-number operator, and μ the chemical potential. In metals in thermal equilibrium, for sufficiently low temperatures the relevant μ to a good approximation coincides with the underlying Fermi energy εF.

P. Contreras : I didn't say as much as you suggest. First, the main question on this page is not very clear, as there are several spectral functions one can think of in the Eliashberg (Eliashberg-McMillan) formalism. Second, my point about the reason for encountering chemical potential / Fermi energy in the microscopic theories of superconductivity was that in these theories the starting point is K and not H. There are also some vital technical reasons for working in the grand canonical ensemble in dealing with the phenomenon of superconductivity (as well as superfluidity of bosons - following the Boguliobov approximation), as in the mean-field theory the ground state is not an eigenstate of the number operator (for superfluidity in boson systems, one has the celebrated Hugenholtz-Pines scheme for surmounting the problem of not having the number conservation - the Boguliobov Hamiltonian does not commute with the total-number operator).

P. Contreras : I do not know in which context you are referring to K(Omega). There is a Pippard kernel denoted by K, which is static but depends on the momentum q, and there is a generalised version of it deduced within the microscopic theory - also denoted by K (more precisely, by Kμν), which is a tensor depending on both q and ω, relating the induced super-current to the external vector potential. For these, consult Sections 49 and 52 of the book by Fetter and Walecka (Quantum Theory of Many-Particle Systems, Dover, 2003).

P. Contreras : You are welcome. The operator K ≡ H - μN in my earlier response on this page is the thermodynamic Hamiltonian (not 'potential') -- a standard term used in the literature (the thermodynamic potential Ω is not an operator, but a real number, deduced from the Helmholtz free energy through a Legendre transformation whereby the number of particles N is replaced by the chemical potential μ as a thermodynamic variable -- you must have mistaken the H in K ≡ H - μN as representing the Helmholtz free energy). In thermodynamic considerations, the operator K is generally not a function of ω if ω represents the angular frequency of some time-dependent external potential coupling to the system; K is generally time-independent (in the Schrödinger picture).

Thanks you professors Behnam Farid and P. Contreras for this interesting involvments; for the spectral fuunction of eliashberg it's described by the equation on the file i want to understand the importance of the fermi level in this equation as you see we have the density of states in fermi level and also if you see in the gamma affiched on fig2 we have the fermi energy in the delta and as i know the difference into delta give to us the information of the infinitesimally passage between the energy of fermi Ef and Ekµ; the other delta also between Ef and Ek+qµ'.

Best Regards Professors

Ilias Serifi : In considering the microscopic theories of superconductivity (as in many other cases), the sums over the underlying k points are replaced by integrals over energy (or frequency) involving the density of electronic states (this is a standard technique, to be found in almost all good textbooks on solid state physics, such as that by Ashcroft and Mermin). In the particular case of superconductors, the latter integrals are subject to sharp cut-offs so that these integrals are over finite intervals around the underlying Fermi energy. Now because in conventional metals, with large bandwidths (in comparison with the relevant boson / phonon energies) the density of the electronic states D(ε) is replaced by the constant D(εF), where εF in the underlying Fermi energy, and the resulting integral is subsequently evaluated analytically. This is an approximate procedure that breaks down when the underlying density of electronic states is not nearly constant in the neighbourhood of ε = εF, but this is how in standard texts, dealing with ideal cases, εF gets all over the place.

Incidentally, I do not know what text you are using that does not provide these technical details. The classic work on the subject is by McMillan [1] (and McMillan and Rowell [2]).

[1] WL McMillan, Phys. Rev. 167, 331 (1968).

Article McMillan, W. L. Transition temperature of strong-coupled sup...

[2] WL McMillan, and JM Rowell, Phys. Rev. Lett. 14, 108 (1965).

Article Lead Phonon Spectrum Calculated from Superconducting Density of States

Dear Ilias,

The Eliashberg model has strong assumptions and one of them is that only depends of a single effective band and a single Fermi energy given a single condensete. On the other hand the Fermi energy is also considered to be much higher than the energy of the phonon, i.e. in the Migdal approximation or adiabatic limit where the Born-Oppenheimer approach can be used because the electrons and the ions are very well separated. Thus this justifies the importance of the Fermi level as the fundamental scale of energies for this model of the superconductivity.

P. Contreras : No problem. As for the Fermi energy, I think I explained in minute detail the way it gets into the relevant expressions. Please consult my previous remarks on this page.

Thanks dear professors Behnam Farid and P. Contreras and also professor Daniel Baldomir for your intersting answer.

Best Regards

Ilias Serifi : My pleasure.