Optimal grouping of arbitrary diagrammatic expansions via analytic pole structure

We present a general method to optimize the evaluation of Feynman diagrammatic expansions which uses an automated symbolic assignment of momentum/energy conserving variables to each diagram. With this symbolic representation, we utilize the pole structure of each diagram to sort the Feynman diagrams into groups that are likely to contain nearly equal or nearly canceling diagrams, and we show that for some model parameters this cancellation is exact. This allows for a potentially massive cancellation during the numerical integration of internal momenta variables, leading to an optimal suppression of the “sign problem” and hence reducing the computational cost. Although we define these groups using a frequency space representation, the equality or cancellation of diagrams within the group remains valid in other representations such as imaginary time used in standard diagrammatic Monte Carlo. As an application of the approach, we apply this method, combined with algorithmic Matsubara integration (AMI) [A. Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 99, 035120 (2019)] and Monte Carlo methods, to the Hubbard model self-energy expansion on a 2D square lattice, which we evaluate and compare with existing benchmarks.

Figure 1

Diagrammatic representation of (left) fermionic line, (middle) interaction line, and (right) two-body interaction $V_{\sigma ,{\sigma }^{\prime }}\left(\mathbf{q}\right)$ between two fermionic lines with spins $\sigma$ and ${\sigma }^{\prime }$. Each line should be assigned with momentum/energy conserving variables.

Figure 2

Diagrammatic illustration of (top) AIL and (bottom) AT procedures in generation of the diagrammatic expansion described in the text. In AIL one interaction (wavy) line is added to the diagram while in AT a tadpole (simple fermionic loop with a wavy tail) is added to the diagram.

Figure 3

Schematic illustration of classes and subclasses of the set $S_m$ of diagrams of order $m$. Diagrams in a class $C_i^m$ have the same pole configuration, which are divided into subclasses of diagrams with similar characters.

Figure 4

Four subclasses of fourth order self-energy diagrams. The diagrams in the top panel belong to class $C_1^4$ with pole configuration (7,0) and the diagrams in the bottom panel belong to class $C_2^4$ with pole configuration (5,1). Collecting pole-IDs for all possible labels, one finds out that the diagrams in each row have the same diagram character, i.e., they belong to the same subclass. Thus, one takes the diagrams in each row as candidates to be either equal or canceling.

Figure 5

Two topologically distinct third order self-energy diagrams which are almost isomorphic. Application of GIT reveals that they are precisely canceling at half-filling.

Figure 6

Schematic illustration of constructing groups of sixth order equal self-energy diagrams at half-filling. There are 515 diagrams after the chemical potential shift at sixth order. We first divide the diagrams into classes according to their pole configurations. Note that $C_1^6$, $C_2^6$, $C_3^6$, and $C_4^6$ are classes of diagrams with pole configurations (11,0,0), (9,1,0), (7,2,0), and (8,0,1), respectively. We then construct the subclasses for each class considering their diagram characters. Finally, the application of GIT within each subclass enables us to discard all the canceling diagrams and find groups of equal diagrams at half-filling.

Figure 7

The contribution at each order to $\mathrm{Im}{\mathrm{\Sigma }}_{\mathbf{k}}\left(i{\nu }_0\right)$ for $U/t=1\to 4$ normalized by the $m=2$ contribution. Data are for parameters $\beta t=5$ and $\mu /t=-1.5$ at $\mathbf{k}=(\pi /8,\pi )$. The AMI results were obtained with $\approx {10}^6$ samples per diagram.

Figure 8

(Top) Real and imaginary parts of the self-energy at second, third and fourth order as well as the result up to fourth order vs real frequency $\nu$, (bottom) Green's function up to fourth order vs real frequency $\nu$. Data are for parameters $U/t=3$, $\mu /t=-1.5$, and $\beta t=5$ at $\mathbf{k}=(\pi /8,\pi )$. We set $\mathrm{\Gamma }/t=0.05$ in the symbolic analytic continuation $i{\nu }_n\to \nu +i\mathrm{\Gamma }$. The AMI results were obtained with $\approx 4\times {10}^7$ samples per diagram.

Figure 9

(Left) Imaginary part of the self-energy on the Matsubara axis at ${\mathbf{k}}_{AN}=(\pi ,0)$ for $\mu =0$, $U/t=3$, and $\beta t=8.33$. Results from DMFT are shown as well as DCA data for 16 and 64-site clusters. (Right) Spectral function $A({\mathbf{k}}_{AN},\nu )$ on the real frequency axis. The DCA results were obtained via maximum entropy inversion [16]. The AMI results assume $\mathrm{\Gamma }/t=0.05$. The AMI results were obtained with $\approx {10}^6$ samples per diagram.

Figure 10

Real and imaginary parts of the self-energy at $i{\nu }_0$ through high-symmetry cuts in the $k_x-k_y$ plane for $\mu =0$, $U/t=4$, and $\beta t=2$. The upper/lower blue squares are the real/imaginary DDMC results from Ref. [32]. The AMI results were obtained with $\approx {10}^7$ samples per diagram.