Recently the complex Langevin method (CLM) has been attracting attention as a solution to the sign problem, which occurs in Monte Carlo calculations when the effective Boltzmann weight is not real positive. An undesirable feature of the method, however, was that it can happen in some parameter regions that the method yields wrong results even if the Langevin process reaches equilibrium without any problem. In our previous work, we proposed a practical criterion for correct convergence based on the probability distribution of the drift term that appears in the complex Langevin equation. Here we demonstrate the usefulness of this criterion in two solvable theories with many dynamical degrees of freedom, i.e., two-dimensional Yang-Mills theory with a complex coupling constant and the chiral Random Matrix Theory for finite density QCD, which were studied by the CLM before. Our criterion can indeed tell the parameter regions in which the CLM gives correct results.

Article funded by SCOAP3

0$ represents the degenerate quark mass. The dynamical variables consist of two general $N\times (N+\nu)$ complex matrices $\Phi_{k} \ (k=1,2)$, where the integer $\nu$ represents the topological index. The action $S_{\rm b}$ in~(\ref{crmt}) is given by \begin{align} S_{\rm b}&=2N \sum_{k=1}^{2}{\rm Tr} (\Psi_k \Phi_k) \,, \label{Sb} \end{align} where $\Psi_k \ (k=1,2)$ are $(N+\nu)\times N$ matrices defined by \begin{align} \Psi_k=(\Phi_k)^\dagger \ . \label{hc-constraint} \end{align} The reason for introducing new matrices representing the Hermitian conjugate of $\Phi_k$ will be clear shortly. The Dirac operator $D$ in~(\ref{crmt}) is given by \begin{align} D&=\left( \begin{matrix} 0 & X \\ Y & 0 \end{matrix} \right) \,, \quad \quad \quad \left\{ \begin{array}{ccc} X & = & e^{\mu} \Phi_1 + e^{-\mu} \Phi_2 \\ Y & = & -e^{-\mu} \Psi_1 - e^\mu \Psi_2 \,, \end{array} \right. \label{Eq:2015Sep01eq2} \end{align} where $\mu$ is the chemical potential. The effective action of this model reads \begin{align} S_{\rm eff}&= S_{\rm b} - N_{\rm f} \ln \det (D+m) \ . \label{Seff} \end{align} When one tries to apply Monte Carlo methods to this model, the sign problem occurs for $\mu \neq 0$ due to the complex fermion determinant $\det (D+m)$. To see that, let us note first that the Dirac operator $D$ satisfies the relation \begin{align} D \gamma_5 & = - \gamma_5 D \,, \quad \quad \quad \gamma_5 =\begin{pmatrix} {\bf 1}_{N} & 0 \\ 0 & -{\bf 1}_{N+\nu} \end{pmatrix} \label{anticommuting} \end{align} for any $\mu$. This implies that all the nonzero eigenvalues of $D$ are paired with the ones with the sign flipped. When $\mu=0$, $D$ is anti-Hermitian and its eigenvalues are purely imaginary, which implies that the determinant $\mathrm{det}(D+m)$ is real semi-positive. On the other hand, when $\mu\neq 0$, $D$ is no longer anti-Hermitian and its eigenvalues can take complex values. In this case, the determinant $\mathrm{det}(D+m)$ is complex in general, which causes the sign problem. Since the model is actually analytically solvable, it serves as a useful toy model for investigating the sign problem that occurs in finite density QCD. Let us apply the CLM to the cRMT with $\mu\neq 0$. First we consider real variables corresponding to the real part and the imaginary part of $(\Phi_k)_{ij}$ and complexify these variables. The action and the observables are extended to holomorphic functions of these complexified variables by analytic continuation. It is easy to convince oneself that this simply amounts to disregarding the constraint~(\ref{hc-constraint}) and extending the action and the observables to holomorphic functions of $\Phi_k$ and $\Psi_k$ ($k=1,2$). A fictitious time evolution of the complex matrices $\Phi_k$ and $\Psi_k$ ($k=1,2$) is given by the complex Langevin equation with gauge cooling as \begin{align} \tilde \Phi_k(t) &= g \, \Phi_k(t) \, h^{-1} \,, \quad \tilde \Psi_k(t)=h \, \Psi_k(t) \, g^{-1} \,, \label{cle-cRMT-gc} \\ \Phi_{k}(t+\epsilon) &= \epsilon\left[-2N\tilde \Phi_{k}(t) - N_{\rm f} e^{(-1)^{k}\mu} \, W^{-1}(\tilde\Phi(t),\tilde\Psi(t)) \, X(\tilde\Phi(t))\right] + \sqrt{\epsilon} \, \eta_{k}(t) \,, \nonumber\\ \Psi_{k}(t+\epsilon) &= \epsilon\left[-2N\tilde \Psi_{k}(t) + N_{\rm f} e^{(-1)^{k+1}\mu} \, Y(\tilde\Psi(t)) \, W^{-1}(\tilde\Phi(t),\tilde\Psi(t)) \right] + \sqrt{\epsilon}\, \eta_{k}^\dagger(t) \,, \label{cle-cRMT} \end{align} where $W=m^2-XY$ is an $N\times N$ matrix. The $N\times (N+\nu)$ matrices $\eta_{k}(t)$ have components taken from complex Gaussian variables normalized by $\langle \eta_{k,ij}(t) \eta_{k',i'j'}^{*} (t') \rangle_\eta = 2 \delta_{kk'} \delta_{ii'} \delta_{jj'} \delta_{tt'}$. Eq.~(\ref{cle-cRMT-gc}) represents the gauge cooling with $g\in \mathrm{GL}(N,\mathbb{C})$ and $h\in \mathrm{GL}(N+\nu,\mathbb{C})$, which are obtained by complexifying the $\mathrm{U}(N)\times \mathrm{U}(N+\nu)$ symmetry of the original model~(\ref{crmt}). In ref.~\cite{Mollgaard-ml-2013qra}, the same model was studied by the CLM without gauge cooling, and it was found that one obtains wrong results for small quark mass or large chemical potential. The reason for this failure is the singular-drift problem~\cite{Nishimura-ml-2015pba}, which occurs due to eigenvalues of $(D+m)$ close to zero. In ref.~\cite{Nagata-ml-2016alq}, we proposed to use the gauge cooling to avoid the singular drift problem as much as possible.\footnote{While the gauge cooling is found to be useful in avoiding the singular drift problem in the present model, it is found to be ineffective in a similar model~\cite{Stephanov-ml-1996ki} according to a recent study~\cite{Bloch-ml-2017sex}.} There, three different types of ``norm'' were considered so that the gauge transformations $g$ and $h$ in~(\ref{cle-cRMT-gc}) can be determined by minimizing them. A counterpart of the unitarity norm~(\ref{n}) in gauge theory, which is called the Hermiticity norm, can be defined as \begin{align} \mathcal N_{\rm H} &= \frac{1}{N}\, \sum_{k=1,2} {\rm Tr} [ (\Psi_k-\Phi_k^\dagger)^\dagger (\Psi_k-\Phi_k^\dagger) ] \,, \end{align} which measures the violation of the relation~(\ref{hc-constraint}). It turned out, however, that the gauge cooling with this norm does not reduce the singular drift problem. This led us to consider a norm that is related directly to the eigenvalue distribution of the Dirac operator. A simple choice is given by \begin{align} \mathcal N_1 = \frac{1}{N}{\rm Tr} \left[(X+Y^\dagger)(X+Y^\dagger)^\dagger\right] \,, \label{normtype1} \end{align} which measures the deviation of the Dirac operator~(\ref{Eq:2015Sep01eq2}) from an anti-Hermitian matrix. The gauge cooling with this norm has an effect of making the eigenvalue distribution of $D+m$ narrower in the real direction. Another choice is given by \begin{align} \mathcal N_{2} = \sum_{a=1}^{n_{\rm ev}} e^{-\xi \alpha_a} \,, \label{normtype2} \end{align} where $\xi$ is a real parameter and $\alpha_a$ are the real semi-positive eigenvalues of $M^\dagger M$ with $M=D+m$. In eq.~(\ref{normtype2}), we take a sum over the $n_{\rm ev}$ smallest eigenvalues of $M^\dagger M$. The gauge cooling with this norm has an effect of achieving $\alpha_a \gtrsim 1/\xi$ and suppressing the appearance of small $\alpha_a$. Since $\alpha_a \gtrsim 1/\xi$ implies $|\lambda_a|^2 \gtrsim 1/\xi$, where $\lambda_a$ are the eigenvalues of $M$, it is also expected to suppress the appearance of $\lambda_a$ close to zero. In some cases, the use of the norm~(\ref{normtype1}) or~(\ref{normtype2}) causes the excursion problem. In order to avoid this, we consider a combined norm \begin{align} \hat{\mathcal N}_{i}(s) = s \mathcal N_{\rm H} + (1-s) \mathcal N_i \quad \quad \quad \mbox{for~}i=1,2 \,, \label{Ntot} \end{align} where $s$ $(0 \le s \le 1)$ is a tunable parameter. Let us discuss how we define the probability distribution of the drift term. Here we set the topological index $\nu=0$ for simplicity so that the dynamical variables $\Phi_k$ and $\Psi_k$ are $N \times N$ square matrices. We denote the drift terms of $\Phi_k$ and $\Psi_k$ by $F_k$ and $G_k$ ($k=1,2$), and represent the eigenvalues of $(F_k^\dagger F_k)^{1/2}$ and $(G_k^\dagger G_k)^{1/2}$ by $v_k^{(a)}$ and $w_k^{(a)}$ ($a=1, \cdots , N$), respectively. Then we define the probability distribution of the drift term as \begin{align} p(u)=\frac{1}{2N}\int \prod_{k=1,2}d\Phi_k d\Psi_k \sum_{k=1}^{2}\sum_{a=1}^{N} \left(\delta(u-v^{(a)}_k(\Phi,\Psi)) +\delta(u- w^{(a)}_k(\Phi,\Psi))\right) P(\Phi,\Psi) \,, \label{p of crmt} \end{align} where $P(\Phi,\Psi)$ is the probability distribution of $\Phi_k(t)$ and $\Psi_k(t)$ in the $t\rightarrow \infty$ limit. This definition of $p(u)$ respects the $\U(N)\times \U(N)$ symmetry of the original theory. \begin{figure} \centerline{ \includegraphics[width=7cm]{cond_r_mut2n30nf2_ncdt5d-5_compare_histogram-eps-converted-to}\quad \includegraphics[width=7cm]{nocooling_FdaggerF_mono-eps-converted-to} } \centerline{ \includegraphics[width=7cm]{cond_r_mut2n30nf2_coolingABr1d-2_compare_histogram-eps-converted-to}\quad \includegraphics[width=7cm]{coolingN1_FdaggerF_mono-eps-converted-to} } \centerline{ \includegraphics[width=7cm]{cond_r_mut2n30nf2_nev2xi300r1d-2_compare_histogram-eps-converted-to}\quad \includegraphics[width=7cm]{coolingN2_FdaggerF_mono-eps-converted-to} } ]]>