Efficient Temperature-Dependent Green’s Function Methods for Realistic Systems: Using Cubic Spline Interpolation to Approximate Matsubara Green’s Functions

Abstract

The popular, stable, robust and computationally inexpensive cubic spline interpolation algorithm is adopted and used for finite temperature Green's function calculations of realistic systems. We demonstrate that with appropriate modifications the temperature dependence can be preserved while the Green's function grid size can be reduced by about two orders of magnitude by replacing the standard Matsubara frequency grid with a sparser grid and a set of interpolation coefficients. We benchmarked the accuracy of our algorithm as a function of a single parameter sensitive to the shape of the Green's function. Through numerous examples, we confirmed that our algorithm can be utilized in a systematically improvable, controlled, and black-box manner and highly accurate one-and two-body energies and one-particle density matrices can be obtained using only around 5% of the original grid points. Additionally, we established that to improve accuracy by an order of magnitude, the number of grid points needs to be doubled, whereas for the Matsubara frequency grid an order of magnitude more grid points must be used. This suggests that realistic calculations with large basis sets that were previously out of reach because they required enormous grid sizes may now become feasible.

Sparse sampling approach to efficient ab initio calculations at finite temperature

“…Self-consistent second order Green's function theory (GF2) is a second order perturbation theory which renormalizes the Green's function [24][25][26][27][28][29][30][31]. Self-consistent GW [15,17,[21][22][23] further renormalizes the interaction.…”

“…Applications to low-energy effective model Hamiltonians include lattice Monte Carlo [5], dynamical mean-field theory [6] with its cluster [7], multi-orbital extensions [8,9], and diagrammatic extensions [10][11][12], and diagrammatic or continuous-time quantum Monte Carlo methods [13,14]. In the context of ab initio calculations of correlated materials, examples include the GW method [15][16][17][18][19][20][21][22][23], the self-consistent second order approximation (GF2) [24][25][26][27][28][29][30][31], variants of the dynamical mean field theory [8,[32][33][34][35][36], and the self-energy embedding theory [37][38][39][40][41][42][43].…”

“…Compact representations of Green's functions are crucial to address this problem. Representations based on power meshes [44,45], Legendre polynomials [27,46], Chebyshev polynomials [47,48], intermediate numerical representations (IR) [49][50][51], quadrature rules [52,53], and spline interpolations [28] have been proposed, as well as high frequency tail expansions [54][55][56].…”

Legendre-spectral Dyson equation solver with super-exponential convergence

“…Amending the representation with a low order high frequency expansion results in polynomial convergence [42][43][44] . In practice, this is problematic, since for systems with a wide range of energy scales, the number of coefficients is controlled by the largest energy scale 14 . Alternatives such as uniform power meshes have had some success 45,46 .…”

Finite temperature quantum embedding theories for correlated systems

“…Here, F weak tot denotes a solution of the entire system using a conserving low-order approximation, for instance self-consistent second order perturbation theory (GF2) [28][29][30][31][32][33] or the GW method [20]. F A denotes all those terms in Φ where all four indices i j k l , , , of v ijkl are contained inside orbital subspace A. F A weak is the approximation to F A within the weak coupling method used for solving the entire system, and F A strong the approximation or exact solution of F A obtained using the higher order method capable of describing 'strong correlation'.…”