^{*}

^{3}.

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.

In lattice simulations of quantum chromodynamics (QCD) at nonzero chemical potential the action in the partition function is generically complex such that standard importance sampling Monte Carlo methods are unusable. One way to circumvent this problem is to apply the well known density of states method to this particular setting by splitting the complex action in its real and imaginary parts, and considering the density of the phase of the complex weights in the partition function generated by their magnitudes

To apply the density of states method, the phase density

Herein we present a simple example illustrating that the average phase factor obtained from such a fit is neither necessarily stable under slight variations of the fit parameters, nor can it be computed accurately when the sign problem becomes too strong. To compute this oscillatory integral we opted to use the thimble integration, which is often believed to reduce the sign problem and make it manageable. One of our motivations was to investigate the mechanism by which the thimble integration yields exponentially small values for

When integrating along a thimble, the magnitude of the complex integrand falls off in a Gaussian like manner at either side of the saddle point. Nevertheless, there is a potential sign problem due to the

Although we applied the thimble integration to the one-dimensional oscillatory integral

In Sec.

In lattice simulations of QCD at nonzero chemical potential the weights in the partition function are generically complex due to the fermion determinant. The distribution

As

To improve upon this, it was suggested to consider an

The Gaussian Ansatz was however questioned by the observation that higher order corrections in the cumulant expansion, which involve delicate volume cancellations, could invalidate the Gaussian value of

The distribution

This particular functional form for

Although the analysis performed in this paper is valid for any value of

Note that similar actions were considered in a one-dimensional model

The phase distribution

Distribution

Real part

Although the main focus of this paper will be the computation of

Integral

In the thimble formulation

As thimbles are trajectories of constant phase

Let us first briefly consider a pure Gaussian distribution in

A salient feature of the Gaussian distribution is that the average phase factor

To determine the saddle points and thimble trajectories we rewrite the integrand as

Although the solutions

Note that the saddle points are explicit functions of the parameters

Once the saddle points are known, a further analysis is performed to determine which thimbles are relevant to the thimble integration.

For

On the other hand, for

The aim is to understand how exponentially small integral values arise in the thimble framework, and to investigate the impact of small variations in

Summary of results for the thimble analysis of the integral

We first look at the constant phase along the relevant thimbles. For this, we substitute the value of the saddle point on the relevant thimble in the action

For

Constant phase

When studying the thimble integration for various values of

the path(s) of the relevant thimble(s) in the complex plane with their saddle point(s);

the

the real part of

Thimble analysis for

Thimble analysis for

Thimble analysis for

We now analyze the thimble integration as function of the parameter

When

We now further increase

Left: Phase

When integrating along the thimble the magnitude

At first sight, this may look like a rather artificial problem, considering that it is merely caused by the zero crossing of

The average phase factor

Although, the Gaussian distribution is a simple candidate for a fit function that allows for an accurate computation of

We extended the Gaussian distribution with a quartic term in the exponential, which was suggested by simulations of random matrix theory at nonzero chemical potential. Higher order corrections were not considered as their contributions were consistent with zero, due to the large statistical errors on the higher moments of the phase in the RMT simulations. However, as the simple quartic extension considered here already produces a strong residual sign problem, there is little reason to expect that higher order corrections, which are presumably present and may be computed more accurately using the LLR method, will mend this deficiency. Such a systematic extension to higher order is currently being investigated.

We performed a detailed analysis of the thimble integration to compute

In Monte Carlo simulations of physical systems with a complex action,

As an endnote, we observe that, although we have only presented results for

I dedicate this paper to the memory of Mike Pennington. Moreover, I would like to thank Falk Bruckmann, Robert Lohmayer and Simon Reiser for useful discussions. This work was supported by the DFG collaborative research center SFB/TRR-55.