Relationship between Hirsch-Fye and weak-coupling diagrammatic quantum Monte Carlo methods

Two weak-coupling continuous time quantum Monte Carlo (CTQMC) methods are shown to be equivalent for Hubbard-type interactions. A relation between these CTQMC methods and the Hirsch-Fye quantum Monte Carlo (HFQMC) method is established, identifying the latter as an approximation within CTQMC and providing a diagrammatic interpretation. Both HFQMC and CTQMC are shown to be equivalent when the number of time slices in HFQMC becomes infinite, implying the same degree of fermion sign problem in this limit.

Figure 1

Example of the diagrams sampled in HFQMC (top) and CTQMC (bottom). The light and dark lines denote propagation of spin-up and spin-down fermions, respectively. The block arrows represent the auxiliary field, associated with an interaction vertex. These vertices are distributed evenly in HFQMC, with interval $\Delta \tau$, while the distribution, the positions, and the number of vertices are arbitrary in the CTQMC. Note that both connected and disconnected diagrams are sampled in both methods. In fact, product of determinants [Eqs. (12, 17)] accounts for all ${\left(k!\right)}^2$ possible diagrams for a particular set of vertices.

Figure 2

The distribution of the contributions $Z_k\left(G_k\right)$ to the partition function (Green’s function) vs the expansion order $k$ for the two-dimensional (2D) Hubbard model with $N_c=16$ sites, $U=W$ (bandwidth), filling $f=0.75$, $\beta =4.5$, and $\alpha =1.01$. Note that the distributions almost perfectly overlap, and their ratio (proportional to $g_k$) varies very little with $k$. Restricting diagrams to one order $k=k_{\text{HF}}$ and restricting vertices to discrete imaginary time grid result in the Hircsh-Fye algorithm with $\Delta \tau t=0.125$.

Figure 3

Comparison of the average sign for the 2D Hubbard model with $N_c=24E$, $U=8t$, $\text{filling}=0.95$, and $\beta t=4.5$. In CTQMC, the average sign depends on parameter $\alpha$ (lower horizontal axis), while in HFQMC, it depends on the size of the time slice $\Delta \tau t$ (upper horizontal axis). Marks on both axes are related by Eq. (20).