Fermion bag solutions to some sign problems in four-fermion field theories

Lattice four-fermion models containing $N$ flavors of staggered fermions, which are invariant under $Z_2$ and $U(1)$ chiral symmetries, are known to suffer from sign problems when formulated using the auxiliary field approach. Although these problems have been ignored in previous studies, they can be severe. Here we show that the sign problems disappear when the models are formulated in the fermion bag approach, allowing us to solve them rigorously for the first time.

Figure 1

Nearest neighbor lattice sites (open circles) of a fixed dual site (filled circle) $\tilde{x}$ is represented by $[x,\tilde{x}]$ (left), while the nearest neighbor dual sites of a fixed lattice site is denoted by $⟨\tilde{x},x⟩$ (right).

Figure 2

Distributions of positive (left) and negative (right) weight configurations as a function of $\log |\mathrm{Det}(D[\overline{\varphi }])|$. One million configurations and 5000 configurations were generated at $6^3$ and ${12}^3$ lattices, respectively. The distribution of positive configurations is almost identical to the one with negative configurations suggesting a severe sign problem.

Figure 3

An illustration of the four types of four-fermion couplings (or bonds) generated through the auxiliary field integration. From left to right we have an $S$, $L$, $F$, and $B$ bond, respectively.