Legendre-spectral Dyson equation solver with super-exponential convergence

Abstract

Quantum many-body systems in thermal equilibrium can be described by the imaginary time Green's function formalism. However, the treatment of large molecular or solid ab inito problems with a fully realistic Hamiltonian in large basis sets is hampered by the storage of the Green's function and the precision of the solution of the Dyson equation. We present a Legendre-spectral algorithm for solving the Dyson equation that addresses both of these issues. By formulating the algorithm in Legendre coefficient space, our method inherits the known faster-than-exponential convergence of the Green's function's Legendre series expansion. In this basis, the fast recursive method for Legendre polynomial convolution, enables us to develop a Dyson equation solver with quadratic scaling. We present benchmarks of the algorithm by computing the dissociation energy of the helium dimer He 2 within dressed second-order perturbation theory. For this system, the application of the Legendre spectral algorithm allows us to achieve an energy accuracy of 10 −9 E h with only a few hundred expansion coefficients. arXiv:2001.11603v2 [cond-mat.str-el] 5 Apr 2020

Low rank compression in the numerical solution of the nonequilibrium Dyson equation

“…While these do not by themselves change the computational and memory complexity of the solver, they yield a significant reduction in the constant associated with the scaling, and should be used in implementations. Also, in the equilibrium case, spectral methods and specialized basis representations have been used to represent Green's functions with excellent efficiency, and their applicability in the nonequilibrium case has not yet been explored [61,[99][100][101][102].…”

“…(3) and ( 7) for the Matsubara component may be solved independently of the other components. Several efficient numerical methods exist [22,61,62], and we do not consider this topic here.…”

“…It is not our intention here to obtain high accuracy calculations, and we use a low-order discretization for simplicity, but extension to high-order time stepping methods and quadrature rules for history summation is straightforward [22]. G M is computed in advance to near machine precision using a Legendre polynomial-based spectral method with fixed point iteration, and then evaluated on the equispaced τ grid [61,62]. Codes were written in Fortan using OpenBLAS for matrix operations and linear algebra, including the evaluation of history sums in the direct method, and FFTW was used for the FFTs in the fast evaluation of the sums I ,2 m,k [83][84][85].…”

libdlr: Efficient imaginary time calculations using the discrete Lehmann representation

“…As a result, we anticipate the use of the DLR as a basic working tool a variety of equilibrium applications, including continous-time quantum Monte Carlo [17], dynamical mean-field theory [18], self-consistent perturbation theory in both quantum chemistry (GF2) [19][20][21][22] and condensed matter physics [23], as well as Hedin's GW approximation [24,25], including vertex corrections [26,27]. In [28], the DLR was used to discretize imaginary time variables in equilibrium real time contour Green's functions.…”

“…This scaling makes calculations at low temperature and high accuracy prohibitively expensive. Representations of Green's functions by orthogonal polynomials require O √ Λ log(1/ ) degrees of freedom, largely addressing the accuracy issue but remaining suboptimal in the low temperature limit [1][2][3]. This has motivated the development of optimized basis sets in which to expand imaginary time Green's functions; namely the intermediate representation (IR) [4][5][6] with sparse sampling [7], and the more recently introduced discrete Lehmann representation (DLR) [8].…”