We present an efficient basis for imaginary time Green's functions based on a low rank decomposition of the spectral Lehmann representation. The basis functions are simply a set of well-chosen exponentials, so the corresponding expansion may be thought of as a discrete form of the Lehmann representation using an effective spectral density which is a sum of $\delta$ functions. The basis is determined only by an upper bound on the product $\beta \omega_{\max}$, with $\beta$ the inverse temperature and $\omega_{\max}$ an energy cutoff, and a user-defined error tolerance $\epsilon$. The number $r$ of basis functions scales as $\mathcal{O}\left(\log(\beta \omega_{\max}) \log (1/\epsilon)\right)$. The discrete Lehmann representation of a particular imaginary time Green's function can be recovered by interpolation at a set of $r$ imaginary time nodes. Both the basis functions and the interpolation nodes can be obtained rapidly using standard numerical linear algebra routines. Due to the simple form of the basis, the discrete Lehmann representation of a Green's function can be explicitly transformed to the Matsubara frequency domain, or obtained directly by interpolation on a Matsubara frequency grid. We benchmark the efficiency of the representation on simple cases, and with a high precision solution of the Sachdev-Ye-Kitaev equation at low temperature. We compare our approach with the related intermediate representation method, and introduce an improved algorithm to build the intermediate representation basis and a corresponding sampling grid.
翻译:Green 的功能基于光谱 Lehmann 代表的低级别分解。 基函数只是一套精选的指数, 因此相应的扩展可以被看作使用有效的光谱密度( 美元=delta$ ) 函数的Lehmann 代表的离散形式。 基值只能由产品 $\beta\\ omega ⁇ max} 的上限来确定。 美元=beta$ 反温, 美元=oomega{max} 的离散代表, 一种能量截断, 以及一个用户定义的错差的中值正值代表 $\ epselderSaloldal 。 基函数的美元值是 $\\\\\\\\ left\\\\\\\\\\\\\\\\\\\\\ omga\ max\\ 美元=====xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 。 直数函数和直数的直数=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx