Overcoming the sign problem in one-dimensional QCD by new integration rules with polynomial exactness

In this paper we describe a new integration method for the groups $U(N)$ and $SU(N)$, for which we verified numerically that it is polynomially exact for $N\le 3$. The method is applied to the example of one-dimensional QCD with a chemical potential. We explore, in particular, regions of the parameter space in which the sign problem appears due the presence of the chemical potential. While Markov chain Monte Carlo fails in this region, our new integration method still provides results for the chiral condensate on arbitrary precision, demonstrating clearly that it overcomes the sign problem. Furthermore, we demonstrate that also in other regions of parameter space our new method leads to errors which are reduced by orders of magnitude.

Figure 1

Failure of MC-MC methods. Comparison of the relative error of the chiral condensate $\chi$ using polynomially exact (bottom) and Monte Carlo (top) quadrature rules for $SU(2)$. The polynomially exact rule used $n=8$ integration points, $m=0.25$, $\mu =1.0$, and the error bars have been computed from 20 independent repetitions.

Figure 2

Order of the quadrature-rule point evaluation of the partition function integrand, ${(2^{-n}{e}^{n\mu })}^3$, see (6), compared to the analytic values of the partition functions for $U(3)$ and $SU(3)$ (see Theorem 2), using $n=20$, $\mu =1$. As discussed in the paper, the ratio $Z_{\text{analytic}}/2^{-3n}{e}^{3n\mu }$ determines the relative errors of the partition function and the chiral condensate to a large extent. In particular, we identify three regions (I, II, and III) in which the relative error exhibits qualitatively different behavior. (These computations were performed with 1024-bit floating point arithmetic.)

Figure 3

Comparison of the relative error of the used methods, namely the polynomially exact and Monte Carlo quadrature rules to calculate the partition function $Z$ for $SU(2)$ and $SU(3)$ (left column, top to bottom), and $U(1)$, $U(2)$, and $U(3)$ (right column, top to bottom) with $n=20$, $\mu =1$, $m\in [0.001,100]$. Averages and standard deviations (error bars) have been computed from 50 independent computations. Here we used double precision to carry through the numerical calculations. The different behaviors of the relative error regarding different values of $m$ are divided into regions I, II and III, corresponding to Fig. 2.

Figure 4

Comparison of the relative error as shown in Fig. 3 but here using 1024-bit extended floats. Averages and standard deviations (error bars) have been computed from 50 independent computations for $U(1)$, $U(2)$, and $SU(2)$, and from 10 independent computations for $SU(3)$ and $U(3)$. Again $n=20$, $\mu =1$, and $m\in [0.001,100]$ are used.

Figure 5

Comparison of the relative error of the chiral condensate $\chi ={\partial }_m\ln Z$ using polynomially exact and Monte Carlo quadrature rules for $SU(2)$ and $SU(3)$ (left column, top to bottom), and $U(1)$, $U(2)$, and $U(3)$ (right column, top to bottom) with $n=8$, $\mu =1.0$, and $m\in [{10}^{-11},10^3]$. We use 1024-bit extended floats. Averages and standard deviations (error bars) have been computed from 50 independent computations for $SU(2)$, $U(1)$, and $U(2)$ and from 5 for $SU(3)$ and $U(3)$. The different behaviors of the error regarding different values of $m$ are divided into regions I, II, and III.