### Abstract

In this paper, 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 → 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.

Original language | English (US) |
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Pages (from-to) | 60-74 |

Number of pages | 15 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 200 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2005 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

**Mathematical analysis of the multiband BCS gap equations in superconductivity.** / Yang, Yisong.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 200, no. 1-2, pp. 60-74. https://doi.org/10.1016/j.physd.2004.09.011

}

TY - JOUR

T1 - Mathematical analysis of the multiband BCS gap equations in superconductivity

AU - Yang, Yisong

PY - 2005/1/1

Y1 - 2005/1/1

N2 - In this paper, 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 Tc so that, when T < Tc, there exists a unique positive gap solution, representing the superconducting phase; when T > Tc, the only nonnegative gap solution is the zero solution, representing the normal phase. Furthermore, when T = Tc, we prove that the only gap solution is the zero solution and that the positive gap solution depend on the temperature T < Tc monotonically and continuously. In particular, as T → Tc, the gap solution tends to zero, which enables us to determine the critical temperature Tc. In the isotropic case where the entries of the interaction matrix K are all constants, we are able to derive an elegant Tc equation which says that Tc 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 Tc equation in the context of the Markowitz-Kadanoff model and we prove that the presence of anisotropic fluctuations enhances Tc 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.

AB - In this paper, 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 Tc so that, when T < Tc, there exists a unique positive gap solution, representing the superconducting phase; when T > Tc, the only nonnegative gap solution is the zero solution, representing the normal phase. Furthermore, when T = Tc, we prove that the only gap solution is the zero solution and that the positive gap solution depend on the temperature T < Tc monotonically and continuously. In particular, as T → Tc, the gap solution tends to zero, which enables us to determine the critical temperature Tc. In the isotropic case where the entries of the interaction matrix K are all constants, we are able to derive an elegant Tc equation which says that Tc 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 Tc equation in the context of the Markowitz-Kadanoff model and we prove that the presence of anisotropic fluctuations enhances Tc 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.

UR - http://www.scopus.com/inward/record.url?scp=10844247872&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10844247872&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2004.09.011

DO - 10.1016/j.physd.2004.09.011

M3 - Article

VL - 200

SP - 60

EP - 74

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -