Test for a universal behavior of Dirac eigenvalues in the complex Langevin method

We apply the complex Langevin (CL) method to a chiral random matrix theory (ChRMT) at nonzero chemical potential and study the nearest neighbor spacing (NNS) distribution of the Dirac eigenvalues. The NNS distribution is extracted using an unfolding procedure for the Dirac eigenvalues obtained in the CL method. For large quark mass, we find that the NNS distribution obeys the Ginibre ensemble as expected. For small quark mass, the NNS distribution follows the Wigner surmise for the correct convergence case, while it follows the Ginibre ensemble for the wrong convergence case. The Wigner surmise is physically reasonable from the chemical potential independence of the ChRMT. The Ginibre ensemble is known to be favored in a phase-quenched QCD at finite chemical potential. Our result suggests a possibility that the originally universal behavior of the NNS distribution is preserved even in the CL method for the correct convergence case.

Figure 1

The cumulative spectral function $N(x,y)$ (histogram) and its average part $N_{\text{ave}}(x,y)$ (solid line) as a function of $x$ at $N=20$, $\tilde{m}=2$, and $\tilde{\mu }=3$ in the Exp representation. They are obtained from eigenvalues included in a strip around $y=0$. The top and bottom panels are obtained with a cubic function and a quintic function, respectively.

Figure 2

Dirac eigenvalues at $N=20$, $\tilde{m}=2$, and $\tilde{\mu }=3$ before the unfolding procedure (left panel) and after the unfolding procedure (right panel). They are obtained from configurations in late Langevin time.

Figure 3

The chiral condensate at $\tilde{\mu }=1$ as a function of $\tilde{m}$. The solid line shows the exact value in the ChRMT [25] and the dotted line shows the exact value in the phase-quenched (PQ) ChRMT [29]. The results of the Exp and Hyp representations are expressed as squares and circles. At large $\tilde{m}$, both representations produce exact results well. Results of the Hyp representation slightly deviate from exact results at small $\tilde{m}$. Dotted vertical lines show the values of $\tilde{m}$ at which the NNS distribution is obtained.

Figure 4

The NNS distribution of the Hyp notation at $N=20$, $\tilde{m}=2$, and $\tilde{\mu }=1$.

Figure 5

The NNS distribution of the Hyp notation at $N=20$, $\tilde{m}=30$, and $\tilde{\mu }=1$.

Figure 6

The NNS distribution of the Exp notation at $N=20$, $\tilde{m}=2$, and $\tilde{\mu }=1$.

Figure 7

The NNS distribution of the Exp notation at $N=20$, $\tilde{m}=30$, and $\tilde{\mu }=1$.

Figure 8

Chiral condensate at $\tilde{\mu }=3$ as a function of $\tilde{m}$. The solid line shows the exact value in the ChRMT [25] and the dotted line shows the exact value in the phase-quenched (PQ) ChRMT [29]. The results of the Exp and Hyp representations are expressed as squares and circles. At large $\tilde{m}$, both representations produce the exact result well. The result of the Hyp representation at small $\tilde{m}$ deviates from the exact result. Dotted vertical lines show the values of $\tilde{m}$ at which the NNS distribution is obtained.

Figure 9

The NNS distribution of the Hyp notation at $N=20$, $\tilde{m}=2$, and $\tilde{\mu }=3$.

Figure 10

The NNS distribution of the Hyp notation at $N=20$, $\tilde{m}=30$, and $\tilde{\mu }=3$.

Figure 11

The NNS distribution of the Exp notation at $N=20$, $\tilde{m}=2$, and $\tilde{\mu }=3$.

Figure 12

The NNS distribution of the Exp notation at $N=20$, $\tilde{m}=30$, and $\tilde{\mu }=3$.

Figure 13

The NNS distribution at $N=20$, $\tilde{m}=2$, and $\tilde{\mu }=3$ in Exp with a cubic fitting function in the top panel and that with a quintic fitting function in the bottom panel.

Figure 14

The NNS distribution at $N=20$, $\tilde{m}=2$, and $\tilde{\mu }=3$ in Hyp with a cubic fitting function in the top panel and that with a quintic fitting function in the bottom panel.