Discrete Lehmann representation of three-point functions

TL;DR

The paper tackles the high cost of representing and manipulating three-point functions in quantum many-body physics by generalizing the discrete Lehmann representation (DLR) to two-variable (i.e., two Matsubara frequencies) functions. It develops a two-dimensional, explicit basis consisting of exponentials in imaginary time and simple poles in Matsubara frequency, with a low-rank size $r=\mathcal{O}(\log^2\Lambda\ \log^2\epsilon^{-1})$, and introduces efficient algorithms for grid construction, coefficient recovery, and fast Matsubara summation with $\mathcal{O}(r^3)$ scaling. The approach is validated on the Hubbard atom and a multi-level Anderson impurity model, achieving machine-precision accuracy while dramatically reducing memory and computational cost. This framework enables robust implementation of three-point object methods (e.g., TRILEX, vertex corrections in GW) with controlled accuracy and reduced resource requirements. The work also outlines directions for extending to four-point functions and for refining the underlying spectral representations.

Abstract

We present a generalization of the discrete Lehmann representation (DLR) to three-point correlation and vertex functions in imaginary time and Matsubara frequency. The representation takes the form of a linear combination of judiciously chosen exponentials in imaginary time, and products of simple poles in Matsubara frequency, which are universal for a given temperature and energy cutoff. We present a systematic algorithm to generate compact sampling grids, from which the coefficients of such an expansion can be obtained by solving a linear system. We show that the explicit form of the representation can be used to evaluate diagrammatic expressions involving infinite Matsubara sums, such as polarization functions or self-energies, with controllable, high-order accuracy. This collection of techniques establishes a framework through which methods involving three-point objects can be implemented robustly, with a substantially reduced computational cost and memory footprint.

Figures (5)

• Figure 1: $\epsilon$-rank $R$ of $\Phi^f$ (solid lines), which determines the number of DLR grid points, or degrees of freedom in our representation, as a function of $\Lambda$ and $\epsilon$. The quantity $3r^2 + r$ (dash-dotted lines), the maximum possible rank, is also plotted. $R$ scales as $\mathcal{O}(\log^2(\Lambda))$, and is less than the maximum rank, indicating that further compression of the representation \ref{['eq:dlr2d']} is possible.
• Figure 2: The $R = 1180$ DLR nodes $(i\nu_{m_j}, i\nu_{n_j})$ for $\Lambda = 1024$ and $\epsilon = 10^{-8}$, in dimensionless units. The inset is a zoom-in to the low-frequency region.
• Figure 3: $l^2(i \nu_n)$ error, compared with an exact reference, of the DLR expansion as a function of $\beta$ and the truncation tolerance $\epsilon$, for the correlation, vertex, and polarization functions (computed using the method of Sec. \ref{['sec:matsum']}) of the Hubbard atom example.
• Figure 4: $l^2(i \nu_n)$ error, compared with a reference computed by exact diagonalization, of the DLR expansion as a function of the truncation tolerance $\epsilon$ for the correlation, vertex, and polarization functions (computed using the method of Sec. \ref{['sec:matsum']}) of the single impurity Anderson model example. Error $\epsilon$ is also indicated as a dashed line, and we observe errors close to this specified tolerance.
• Figure 5: $l^2(i \nu_n)$ error of the polarization functions computed using the residue calculus algorithm of Appendix \ref{['app:matsumres']}, shown as a function of $\beta$ and the truncation tolerance $\epsilon$ for the Hubbard atom example (left), and as a function of the truncation tolerance for the single impurity Anderson model example (right). These can be compared with the third panels of Figs. \ref{['fig:hub_errvbeta']} and \ref{['fig:siam_errveps']}, respectively, which use the Matsubara summation algorithm described in Sec. \ref{['sec:matsum']}.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4