Anatomy of a strong residual sign problem on the thimbles

Using a simple Gaussian-like Ansatz for the phase distribution of a theory with a complex action, we show how the thimble integration for the average phase factor can be plagued by a strong residual sign problem when the phase of the complex integration measure conspires with the constant phase of the integrand along the thimble. This strong sign problem prohibits the accurate computation of the average phase factor when it becomes exponentially small, and causes a strong sensitivity to the parameters describing the phase distribution.

Figure 1

Distribution $p(\theta )$ of (2) for $\sigma =4.2$ and $a=0.0095$ (blue) compared to the Gaussian distribution ($a=0$) (red).

Figure 2

Real part $p(\theta )\cos \theta$ of the oscillating integrand in (1) with phase distribution (2) for $\sigma =4.2$ and $a=0.0095$.

Figure 3

Integral $⟨e^{i\theta }⟩$ as a function of $a$ with phase distribution (2) for $\sigma =4.2$. The zero crossing is denoted by $a_0$. The point $a_c$ indicates the transition from the single to double-thimble region, where the residual sign problem sets in.

Figure 4

Constant phase $\varphi$ on the thimbles as a function of $a$ in (2) for $\sigma =4.2$. At $a=a_c$ the system changes from single to double-thimble mode and $\varphi$ becomes nonzero.

Figure 5

Thimble analysis for $\sigma =4.2$ and $a=0.001$ (top) and 0.004 (bottom). As $a<a_c$, the integral is given by a single-thimble. The left pane shows the thimble trajectory in the complex plane, with the saddle point represented by a red bullet. The middle pane shows the magnitude $r(s)$ and the right pane $r(s)\cos \eta (s)$ versus the arc length $s$ on the thimble (we explicitly set $\varphi =0$). The saddle point is located at $s=0$.

Figure 6

Thimble analysis for $\sigma =4.2$ and $a=0.00425$. As $a>a_c$ the integral is given by two mirrored thimbles. Their trajectories in the complex plane are shown in the left pane, with the saddle points represented by red bullets. The middle pane shows the magnitude $r(s)$ and the right pane $r(s)\cos (\varphi +\eta (s))$ versus the arc length $s$ on the thimble ${\mathcal{T}}_{+}$. The saddle point is located at $s=0$.

Figure 7

Thimble analysis for $\sigma =4.2$ and $a=0.009$, 0.0093, 0.0095, and 0.00965 (from top to bottom). As $a>a_c$ the integral is given by two mirrored thimbles. The left pane shows both thimble trajectories in the complex plane, with the saddle points represented by the red bullets. The middle pane shows the magnitude $r(s)$ and the right pane $r(s)\cos (\varphi +\eta (s))$ versus the arc length $s$ on the thimble ${\mathcal{T}}_{+}$. The saddle point is located at $s=0$.

Figure 8

Left: Phase $\eta (s)$ of the Jacobian as a function of the arc length $s$ along the thimble ${\mathcal{T}}_{+}$ for $\sigma =4.2$ and values of $a\in [0.009,a_0]$. The phase remains practically unaltered under such small variations of $a$, as the curves fall on top of one another. Right: Total phase $\varphi +\eta (s)$ of the integrand as a function $s$ for the same parameter values. The variation of the constant phase $\varphi$ with $a$ (see Fig. 4) is responsible for the separation of the total phases when $a$ is varied.