No sign problem in one-dimensional path integral Monte Carlo simulation of fermions: A topological proof

This paper shows that, in one dimension, due to its topology, a closed-loop product of short-time propagators is always positive, despite the fact that each antisymmetric free fermion propagator can be of either sign.

Figure 1

Case A illustrates the sign change of the antisymmetric two-particle propagator when the line connecting its initial and final particle positions crosses the line $x_2=x_1$. The propagator is not traced over. Cases B and C illustrate the tracing, or closed-loop products of three and seven antsymmetric propagators, respectively. They show that any closed-loop product of propagators must cross the line $x_2=x_1$ an even number of times and, therefore, must remains positive.

Figure 2

Intensity plot of $G_0(x_1^{\prime },x_2^{\prime };x_1,x_2;\tau )$ as a function of $x_1$ and $x_2$ for $x_1^{\prime }=0.6, x_2^{\prime }=0.3$. Left: $\tau =0.05$. Right: $\tau =0.2$. The nodal lines no longer change at $\tau \ge 0.2$.

Figure 3

Intensity plot of $G_0(x_1^{\prime },x_2^{\prime };x_1,x_2;0.2)$ as a function of $x_1$ and $x_2$ for $x_1^{\prime }=0.1, x_2^{\prime }=0.4$.