Numerical construction of a low-energy effective Hamiltonian in a self-consistent Bogoliubov-de Gennes approach of superconductivity

Abstract

We propose a fast and efficient approach for solving the Bogoliubov-de Gennes (BdG) equations in superconductivity, with a numerical matrix-size reduction procedure proposed by Sakurai and Sugiura [J. Comput. Appl. Math. 159, 119 (2003)]. The resultant small-size Hamiltonian contains the information of the original BdG Hamiltonian in a given energy domain. In other words, the present approach leads to a numerical construction of a low-energy effective theory in superconductivity. The combination with the polynomial expansion method allows a self-consistent calculation of the BdG equations. Through numerical calculations of quasi-particle excitations in a vortex lattice, thermal conductivity, and nuclear magnetic relaxation rate, we show that our approach is suitable for evaluating physical quantities in a large-size superconductor and a nano-scale superconducting device, with the mean-field superconducting theory.

12 pages, 12 figures, to be published in J. Phys. Soc. Jpn

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• Probe-type of superconductivity by impurity in materials with short coherence length: the s-wave and $\eta$-wave phases study
• Reduced-Shifted Conjugate-Gradient Method for a Green's Function: Efficient Numerical Approach in a Nano-structured Superconductor
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