Overcomplete compact representation of two-particle Green's functions

Two-particle Green's functions and the vertex functions play a critical role in theoretical frameworks for describing strongly correlated electron systems. However, numerical calculations at the two-particle level often suffer from large computation time and massive memory consumption. We derive a general expansion formula for the two-particle Green's functions in terms of an overcomplete representation based on the recently proposed “intermediate representation” basis. The expansion formula is obtained by decomposing the spectral representation of the two-particle Green's function. We demonstrate that the expansion coefficients decay exponentially, while all high-frequency and long-tail structures in the Matsubara-frequency domain are retained. This representation therefore enables efficient treatment of two-particle quantities and opens a route to the application of modern many-body theories to realistic strongly correlated electron systems.

Figure 1

Singular values $s_l^{\mathrm{F}}$ for several values of $\mathrm{\Lambda }=\beta {\omega }_{\max }$ [see Eq. (4) for the definition of $s_l^{\mathrm{F}}]$.

Figure 2

(a) Equal-time lines of $G^{3\mathrm{pt}}({\tau }_1,{\tau }_2,{\tau }_3)$ in the ${\tau }_1\text{−}{\tau }_2$ space with ${\tau }_3=0$. (b)–(e) Decomposition of $G^{3\mathrm{pt}}({\tau }_1,{\tau }_2,{\tau }_3)$ into four different functions in Eq. (11). The positions of their discontinuities are represented by broken lines. The squares denoted by the shadows in (b)–(d) are a unit of (anti)periodicity in the ${\tau }_1\text{−}{\tau }_2$ space. The corresponding representations of relative times are shown graphically. The thin (bold) arrow indicates fermionic (bosonic) statistics. The singular term shown in (e) does not depend on any relative time with bosonic statistics.

Figure 3

Equal-time planes of $G^{4\mathrm{pt}}({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4)$ running diagonally through the cubic space: ${\tau }_1={\tau }_2$ (blue), ${\tau }_2={\tau }_3$ (yellow), ${\tau }_1={\tau }_3$ (red). We take ${\tau }_4=0$. These planes meet in the body diagonal ${\tau }_1={\tau }_2={\tau }_3$. Three additional planes ${\tau }_1,{\tau }_2,{\tau }_3=0,\beta$ constitute the bounding box (dashed). Adapted from Ref. [31].

Figure 4

Diagrams for the 16 representations for the four-point Green's function in Table 1. The bold arrows denote imaginary times with the bosonic statistics. The incoming flux at each vertex ${\tau }_i$ equals to $i{\omega }_i$ as demonstrated for #1 and #5. This conservation law can be used for generating entries in Table 1.

Figure 5

$G^{3\mathrm{pt}}({\tau }_1,{\tau }_2,0)$ for the single-site Hubbard model obtained from an exact diagonalization of the system.

Figure 6

Upper three panels: parameters obtained for the three-point Green's function. Lower panel: comparison of exact values and interpolated ones.

Figure 7

$G^{4\mathrm{pt}}({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4)$ in Eq. (29) computed by the exact diagonalization at ${\tau }_3=\beta /2$ and ${\tau }_4=0$.

Figure 8

(a) Comparison between the exact values of $G^{4\mathrm{pt}}({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4)$ and those evaluated by our compact representation at ${\tau }_3=\beta /2$ and ${\tau }_4=0$. (b) Absolute errors between the exact values and the interpolated data. The broken and dotted lines denote the cutoff values for singular values $s_l^{\mathrm{F}}/s_0^{\mathrm{F}}\simeq 2.5\times {10}^{-6}$ and $s_l^{\overline{\mathrm{B}}}/s_0^{\overline{\mathrm{B}}}\simeq 3.5\times {10}^{-5}$, respectively.

Figure 9

Comparison of QMC data and the fit by the expansion formula in Eq. (15). In the 3D plots, the QMC data and the fit are denoted by the blue and red lines, respectively. They are almost on top of each other. In the 2D plots, the QMC data and the fit are denoted by the blue crosses and the red lines, respectively. In all panels, we show the data only for $-40\le n\le 40$ and $-40\le n^{\prime }\le 40$.

Figure 10

Expansion coefficients obtained from fitting the QMC data shown in Fig. 9.

Figure 11

Diagrams representing Eqs. (C7, C8, C9).

Figure 12

Contributions of $G_{\mathrm{normal}}^{3\mathrm{pt}}$ (upper panel) and $G_{\mathrm{singular}}^{3\mathrm{pt}}$ (lower panel) to the data in Fig. 5.