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Physics Procedia

Spectral-density polynomial expansion for two-particle Green's function in BdG framework

Yuki Nagai, Y. Ota, Masahiko Machida

Published Jan 1, 2012• 4 citations
Publisher

Abstract

We develop an effcient numerical method on the basis of the spectral-density polynomial expansion to calculate two particle Green's functions in Bogoliubov-de Gennes framework for inhomogeneous superconducting systems. Our method allows systematic evaluation such as spin or charge susceptibility, nuclear magnetic relaxation rate, optical or thermal conductivity even in magnetic fields. In addition, its effciencybecomes remarkable in large-scale parallel performance.

DOI
10.1016/j.phpro.2012.03.413

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We develop an effcient numerical method on the basis of the spectral-density polynomial expansion to calculate two particle Green's functions in Bogoliubov-de Gennes framework for inhomogeneous superconducting systems. Our method allows systematic evaluation such as spin or charge susceptibility, nuclear magnetic relaxation rate, optical or thermal conductivity even in magnetic fields. In addition, its effciencybecomes remarkable in large-scale parallel performance.
We propose a highly efficient numerical method to describe inhomogeneous superconductivity by using the kernel polynomial method in order to calculate the Green's functions of a superconductor. Broken translational invariance of any type (impurities, surfaces, or magnetic fields) can be easily incorporated. We show that limitations due to system size can be easily circumvented and therefore this method opens the way for the study of scenarios and/or geometries that were unaccessible before. The proposed method is highly efficient and amenable to large scale parallel computation. Although we only use it in the context of superconductivity, it is applicable to other inhomogeneous mean-field theories.
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.
We propose the ultra-fast numerical approach to large-scale inhomogeneous superconductors, which we call the Localized Krylov Bogoliubov-de Gennes method (LK-BdG). In the LK-BdG method, the computational complexity of the local Green's function, which is used to calculate the local density of states and the mean-fields, does not depend on the system size NNN. The calculation cost of self-consistent calculations is O(N){\cal O}(N)O(N), which enables us to open a new avenue for treating extremely large systems with millions of lattice sites. To show the power of the LK-BdG method, we demonstrate a self-consistent calculation on the 143806-site Penrose quasicrystal lattice with a vortex and a calculation on 1016064-site two-dimensional nearest-neighbor square-lattice tight-binding model with many vortices. We also demonstrate that it takes less than 30 seconds with one CPU core to calculate the local density of states with whole energy range in 100-millions-site tight-binding model.
An efficient scheme is introduced to calculate the Green function on the basis of polynomials expansion. This scheme has the following advantages: (i) it is possible to calculate both the diagonal and off-diagonal elements of the Green function, (ii) applicable to the system with discrete spectrum, (iii) integrated density of states can also be obtained by the same algorithm, and (iv) it is suitable for vector and parallel processing, and required CPU time and memory size are proportional only to the system size (i.e. ordered N). Applications are made to simple lattice systems for demonstration.
Abstract. In this paper, we propose a spectrally accurate solver for computing the Bogoliubov-de Gennes (BdG) excitations of spin-1 Bose-Einstein condensates (BECs), which are governed by the BdG equations, around the mean-field ground state. The BdG equations are essentially a constrained eigenvalue/eigenfunction system. First, we investigate their analytical properties, including exact eigenpairs, generalized nullspace, and bi-orthogonality of eigenspaces. Second, by combining the standard Fourier spectral method for spatial discretization and a stable Gram-Schmidt bi-orthogonal algorithm, we develop a subspace iterative solver for such a large-scale dense eigenvalue problem, and it proves to be numerically stable, efficient, and accurate. Our solver is matrix-free, and the operator-function evaluation is accelerated by a discrete fast Fourier transform (FFT) with almost optimal efficiency. Therefore, it is memory-friendly and efficient for large-scale problems. Furthermore, we give a rigorous and detailed numerical analysis of the stability and spectral convergence. Finally, we present extensive numerical results to illustrate the spectral accuracy and efficiency, and we investigate the excitation spectrum and Bogoliubov amplitudes around the ground state in 1–3 spatial dimensions.