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Mathematical analysis of the multiband BCS gap equations in superconductivity. (English) Zbl 1062.82060
Summary: We present a mathematical analysis for the phonon-dominated multiband isotropic and anisotropic BCS gap equations at any finite temperature $$T$$. We establish the existence of a critical temperature $$T_c$$ so that, when $$T<T_c$$, there exists a unique positive gap solution, representing the superconducting phase; when $$T>T_c$$, the only nonnegative gap solution is the zero solution, representing the normal phase. Furthermore, when $$T=T_c$$, we prove that the only gap solution is the zero solution and that the positive gap solution depend on the temperature $$T<T_c$$ monotonically and continuously. In particular, as $$T\to T_c$$, the gap solution tends to zero, which enables us to determine the critical temperature $$T_c$$. In the isotropic case where the entries of the interaction matrix $$K$$ are all constants, we are able to derive an elegant $$T_c$$ equation which says that $$T_c$$ depends only on the largest positive eigenvalue of $$K$$ but does not depend on the other details of $$K$$. In the anisotropic case, we may derive a similar $$T_c$$ equation in the context of the Markowitz-Kadanoff model and we prove that the presence of anisotropic fluctuations enhances $$T_c$$ as in the single-band case. A special consequence of these results is that the half-unity exponent isotope effect may rigorously be proved in the multiband BCS theory, isotropic or anisotropic.

##### MSC:
 82D55 Statistical mechanical studies of superconductors
##### Keywords:
Bardeen-Cooper-Schrieffer gap equations
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##### References:
 [1] Fröhlich, H., Theory of superconductivity state. I. the ground state at the absolute zero of temperature, Phys. rev., 79, 845-856, (1950) · Zbl 0037.43003 [2] Fröhlich, H., Interaction of electrons with lattice vibrations, Proc. R. soc. A, 215, 291-298, (1952) · Zbl 0047.45501 [3] Maxwell, E., Isotope effect in the superconductivity of Mercury, Phys. rev., 78, 477, (1950) [4] Reynolds, C.A.; Serin, B.; Wright, W.H.; Nesbitt, L.B., Superconductivity of isotopes of Mercury, Phys. rev., 78, 487, (1950) [5] Cooper, L.N., Bound electron pairs in degenerate Fermi gas, Phys. rev., 104, 1189-1190, (1956) · Zbl 0074.23705 [6] Bardeen, J.; Cooper, L.N.; Schrieffer, J.R., Theory of superconductivity, Phys. rev., 108, 1175-1204, (1957) · Zbl 0090.45401 [7] Akhiezer, A.I.; Krasil’nikov, V.V.; Peletminskii, S.V.; Yatsenko, A.A., Research on superfluidity and superconductivity on the basis of the Fermi liquid concept, Phys. rep., 245, 1-110, (1994) [8] Balian, R.; Flocard, H.; Veneroni, M., Variational extensions of BCS theory, Phys. rep., 317, 251-358, (1999) [9] Kulic, M.L., Interplay of electron – phonon interaction and strong correlation: the possible way to high-temperature superconductivity, Phys. rep., 338, 1-264, (2000) [10] Yang, Y., On the bardeen – cooper – schrieffer integral equation in the theory of superconductivity, Lett. math. phys., 22, 27-37, (1991) · Zbl 0729.45009 [11] Du, Q.; Yang, Y., The critical temperature and gap solution in the bardeen – cooper – schrieffer theory of superconductivity, Lett. math. phys., 29, 133-150, (1993) · Zbl 0787.65105 [12] McLeod, J.B.; Yang, Y., The uniqueness and approximation of a positive solution of the bardeen – cooper – schrieffer gap equation, J. math. phys., 41, 6007-6025, (2000) · Zbl 1054.82036 [13] Suhl, H.; Matthias, B.T.; Walker, L.R., Bardeen – cooper – schrieffer theory of superconductivity in the case of overlapping bands, Phys. rev. lett., 3, 552-554, (1959) · Zbl 0088.45601 [14] Leggett, A.J., Number-phase fluctuations in two-band superconductors, Prog. theoret. phys., 36, 901-930, (1966) [15] Geilikman, B.T.; Zaitsev, R.O.; Kresin, V.Z., Properties of superconductors having overlapping bands, Sov. phys.: solid state, 9, 642-647, (1967) [16] Ketterson, J.B.; Song, S.N., Superconductivity, (1999), Cambridge University Press Cambridge [17] Kresin, V.Z.; Morawitz, H.; Wolf, S.A., Mechanism of conventional and high $$T_{\text{c}}$$ superconductivity, (1993), Oxford University Press Oxford [18] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press Cambridge · Zbl 0576.15001 [19] Nakajima, S., Paramagnon effect on the BCS transition in he^3, Prog. theoret. phys., 50, 1101-1109, (1973) [20] Markowitz, D.; Kadanoff, L.P., Effect of impurities upon critical temperature of anisotropic superconductors, Phys. rev., 131, 563-575, (1963) [21] Leavens, C.R.; Carbotte, J.P., Gap anisotropy in a weak coupling superconductor, Ann. phys., 70, 338-377, (1972) [22] () [23] Rickayzen, G., Theory of superconductivity, (1965), Interscience New York · Zbl 0138.22703 [24] Allen, P.B.; Mitrovic, B., Theory of superconducting $$T_{\text{c}}$$, Solid state phys., 37, 1-92, (1982) [25] Schrieffer, J.R., Theory of superconductivity, (1988), Addison-Wesley New York · Zbl 0125.24102 [26] Tilley, D.R.; Tilley, J., Superfluidity and superconductivity, (1990), IOP Publication New York [27] Tinkham, M., Introduction to superconductivity, (1996), McGraw-Hill New York [28] Bogoliubov, N.N.; Tolmachev, V.V.; Shirkov, D.V., A new method in the theory of superconductivity, (), 278-355
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