格林函数的约化位移共轭梯度法：纳米结构超导体的有效数值方法
Reduced-Shifted Conjugate-Gradient Method for a Green's Function: Efficient Numerical Approach in a Nano-structured Superconductor
摘要：
我们提出了减少移位共轭梯度法（RSCG），这是数值有效的计算矩阵元素的格林函数定义为一个哈密顿算子的预解式，通过求解线性方程组所需的精度。这种方法不计算线性方程组的解向量，而是直接计算预解式的矩阵元素。同时得到了不同频率的矩阵元。因此，容易计算表示为这些元素的松原求和的异常值。为了说明我们的方法的力量，我们选择了一个纳米结构的超导系统的平均场Bogoliubov-de Gennes（BdG）的方法。这种方法使我们能够处理系统的制造潜力，其中一个不能有效地使用基于核多项式的方法。我们考虑d波纳米岛超导体的同时解决了大量（~ 50000）松原频率的线性方程组。
英文摘要：
We propose the reduced-shifted Conjugate-Gradient (RSCG) method, which is numerically efficient to calculate a matrix element of a Green's function defined as a resolvent of a Hamiltonian operator, by solving linear equations with a desired accuracy. This method does not calculate solution vectors of linear equations but does directly calculate a matrix element of the resolvent. The matrix elements with different frequencies are simultaneously obtained. Thus, it is easy to calculate the exception value expressed as a Matsubara summation of these elements. To illustrate a power of our method, we choose a nano-structured superconducting system with a mean-field Bogoliubov-de Gennes (BdG) approach. This method allows us to treat with the system with the fabrication potential, where one can not effectively use the kernel-polynomial-based method. We consider the d-wave nano-island superconductor by simultaneously solving the linear equations with a large number (~ 50000) of Matsubara frequencies.
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