General approach to the sign problem: Factorization method with multiple observables

The sign problem is a notorious problem, which occurs in Monte Carlo simulations of a system with the partition function whose integrand is not real positive. The basic idea of the factorization method applied on such a system is to control some observables in order to determine and sample efficiently the region of configuration space which gives important contribution to the partition function. We argue that it is crucial to choose appropriately the set of the observables to be controlled in order for the method to work successfully in a general system. This is demonstrated by an explicit example, in which it turns out to be necessary to control more than one observable. Extrapolation to large system size is possible due to the nice scaling properties of the factorized functions, and known results obtained by an analytic method are shown to be consistently reproduced.

Figure 1

The function $\frac{1}{N^2}\log w(x)$ is plotted for $N=8$, 12, 16. The solid line represents the asymptotic behavior (25) with the coefficients extrapolated to $N=\infty$.

Figure 2

The function $\frac{1}{N^2}\frac{d}{dx}\log {\rho }^{(0)}(x)$ is plotted for $N=16$, 32, 64. The dashed line is drawn to guide the eye. We also plot $-\frac{d}{dx}\underset{N\to \infty }{lim}\{\frac{1}{N^2}\log w(x)\}$ obtained from Fig. 1, where the two solid lines show the margin of error. The value of $x$ at the intersecting point is consistent with the GEM result $⟨{\tilde{\lambda }}_2⟩\simeq 1.4$.

Figure 3

The function $\frac{1}{N^2}\frac{d}{dx}\log {\rho }_{\mathcal{O}}^{(0)}(x)$ is plotted for $N=8$, 16, 32. The dashed line is drawn to guide the eye. We also plot $-\frac{d}{dx}\underset{N\to \infty }{lim}\{\frac{1}{N^2}\log w_{\mathcal{O}}(x)\}$, where the two solid lines show the margin of error. From this figure, we find that the position of the peak shifts from $x=0.92$ to $x=0.85(3)$ due to the effect of the phase.