Recently, the complex Langevin method has been applied successfully to finite density QCD either in the deconfinement phase or in the heavy dense limit with the aid of a new technique called the gauge cooling. In the confinement phase with light quarks, however, convergence to wrong limits occurs due to the singularity in the drift term caused by small eigenvalues of the Dirac operator including the mass term. We propose that this singular-drift problem should also be overcome by the gauge cooling with different criteria for choosing the complexified gauge transformation. The idea is tested in chiral Random Matrix Theory for finite density QCD, where exact results are reproduced at zero temperature with light quarks. It is shown that the gauge cooling indeed changes drastically the eigenvalue distribution of the Dirac operator measured during the Langevin process. Despite its non-holomorphic nature, this eigenvalue distribution has a universal diverging behavior at the origin in the chiral limit due to a generalized Banks-Casher relation as we confirm explicitly.

Phase Diagram of QCD Lattice QCD

Article funded by SCOAP3

Introduction

The method and the model

The complex Langevin method (CLM)

Chiral Random Matrix Theory (cRMT)

0$at zero temperature and finite chemical potential$\mu$~\cite{Bloch-ml-2012bh,Osborn-ml-2004rf}. The partition function is defined by~\cite{Bloch-ml-2012bh} \begin{equation} Z = \int d\Phi_1d\Phi_2 \, [\det (D+m)]^{N_{\rm f}} e^{-S_{\rm b}} \,, \label{crmt} \end{equation} where$\Phi_{k} \ (k=1,2)$are general$N\times (N+\nu)$complex matrices.\footnote{The model~(\ref{crmt}) is equivalent to the one investigated in ref.~\cite{Mollgaard-ml-2013qra} as one can show by a simple change of variables~\cite{Bloch-ml-2012bh}. } The integer$\nu$represents the topological index, which gives the number of exact zero eigenvalues of the Dirac operator$Dgiven by \begin{align} D&=\left( \begin{matrix} 0 & X \\ Y & 0 \end{matrix} \right) \,, \label{eq:Dirac} \\ X&= e^{\mu} \Phi_1 + e^{-\mu} \Phi_2 \,,\vphantom{\left( \begin{matrix} 0 & X \\ Y & 0 \end{matrix} \right)} \label{Eq:2015Sep01eq1}\\ Y&=-e^{-\mu} \Psi_1 - e^\mu \Psi_2 \,,\vphantom{\left( \begin{matrix} 0 & X \\ Y & 0 \end{matrix} \right)} \label{Eq:2015Sep01eq2} \end{align} where we have defined(N+\nu)\times N$matrices$\Psi_k \ (k=1,2)$by \begin{equation} \Psi_k=(\Phi_k)^\dagger \label{hc-constraint} \end{equation} for later convenience. The bosonic action$S_{\rm b}$in~(\ref{crmt}) is given by \begin{equation} S_{\rm b}=2N \sum_{k=1}^{2}{\rm Tr} (\Psi_k \Phi_k) \ . \label{Sb} \end{equation} The effective action of the system can be written as \begin{equation} S_{\rm eff}= S_{\rm b} - N_{\rm f} \ln \det (D+m) \ . \label{Seff} \end{equation} Strictly speaking, the logarithm in~(\ref{Seff}) has an ambiguity due to the branch cut~\cite{Mollgaard-ml-2013qra,Mollgaard-ml-2014mga}. This is not an issue, however, since the CLM can be formulated in terms of the weight$w=[\det (D+m)]^{N_{\rm f}} e^{-S_{\rm b}}$without ever having to refer to the effective action~(\ref{Seff}) as was pointed out in ref.~\cite{Nishimura-ml-2015pba}. With that in mind, we keep on using the effective action~(\ref{Seff}) just for simplicity of terminology. As observables in this model, one can consider the chiral condensate$\Sigma$and the baryon number density$n_B$defined by \begin{equation} \Sigma = \frac{1}{N} \frac{\partial}{\partial m} \log Z \,, \quad \quad n_B = \frac{1}{N} \frac{\partial}{\partial \mu} \log Z \ . \label{chiral} \end{equation} The partition function of the cRMT can be calculated analytically using the orthogonal polynomial method~\cite{Osborn-ml-2004rf}. It turns out that the partition function is independent of the chemical potential$\mu$, and hence the baryon number density is exactly zero and the chiral condensate has no$\mu$dependence. (See, e.g., ref.~\cite{Mollgaard-ml-2013qra} for an explicit expression of the chiral condensate$\Sigma$suitable for numerical evaluation.) In the gauge cooling, symmetries of the system play a crucial role. Let us consider a transformation \begin{equation} \Phi_k \to \Phi_k' = g \Phi_k h^{-1},\quad \Psi_k \to \Psi_k' = h \Psi_k g^{-1}, \quad (k=1,2) \,, \label{Eq:2015Apr14eq1} \end{equation} which leaves the bosonic action invariant. In order for~(\ref{hc-constraint}) to be satisfied for the transformed configuration,\footnote{Upon complexifying the variables in the CLM, the constraint~(\ref{hc-constraint}) is disregarded, and hence the symmetry enhances to~(\ref{Eq:2015Apr14eq1}) with$g\in \mathrm{GL}(N)$and$h\in \mathrm{GL}(N+\nu)$. In view of this, we use$g^{-1}$and$h^{-1}$instead of$g^{\dagger}$and$h^{\dagger}$. } we need to have$g\in \mathrm{U}(N)$and$h\in \mathrm{U}(N+\nu)$. The matrices$X$and$Y$in~(\ref{Eq:2015Sep01eq1}) and~(\ref{Eq:2015Sep01eq2}) transform in the same way as$\Phi_k$and$\Psi_k$, respectively, and hence the Dirac operator$D$defined by~(\ref{eq:Dirac}) transforms as \begin{equation} \left( \begin{matrix} 0 & X \\ Y & 0 \end{matrix} \right) \to \left( \begin{matrix} g & ~ \\ ~ & h \end{matrix} \right) \left( \begin{matrix} 0 & X \\ Y & 0 \end{matrix} \right) \left( \begin{matrix} g^{-1} & ~ \\ ~ & h^{-1} \end{matrix} \right) \ . \label{D-transform} \end{equation} Therefore, both the effective action~(\ref{Seff}) and the observables~(\ref{chiral}) are invariant under the transformation~(\ref{Eq:2015Apr14eq1}). Note that the Dirac operator$Dsatisfies \begin{align} D \gamma_5 & = - \gamma_5 D \,, \label{anticommuting} \\ \gamma_5 &=\begin{pmatrix} {\bf 1}_{N} & 0 \\ 0 & -{\bf 1}_{N+\nu} \end{pmatrix} \label{gamma5-def} \end{align} for any\mu$, which implies that the nonzero eigenvalues of$D$appear in pairs with opposite signs. Furthermore, when$\mu=0$, the Dirac operator$D$is anti-Hermitian$D^\dagger = - D$and hence its eigenvalues are purely imaginary. Thus, one can show that$\mathrm{det}(D+m)$is real positive in this case. On the other hand, when$\mu\neq 0$,$D$is no longer anti-Hermitian, and its eigenvalues take general complex values. The determinant$\mathrm{det}(D+m)$becomes complex in general, which causes the sign problem. ]]> Application of the CLM to the cRMT Gauge cooling for the singular-drift problem New types of norm for the gauge cooling 1/\xi$. Since $\alpha_a > 1/\xi$ implies $|\lambda_a|^2 > 1/\xi$, where $\lambda_a$ are the eigenvalues of $M$, the appearance of $\lambda_a$ close to zero is also suppressed. In actual simulations, the excursion problem and the singular-drift problem may occur at the same time. In that case, we take a linear combination of the Hermiticity norm and one of the new types of norm as \begin{equation} \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{equation} where $s$ $(0 \le s \le 1)$ is a tunable parameter. Note that the eigenvalue distribution of the Dirac operator $D$ is invariant under $\mathrm{GL}(N,\mathbb{C})\times \mathrm{GL}(N+\nu,\mathbb{C})$. However, the Langevin dynamics is modified nontrivially\footnote{The symmetry enhancement from ${\rm U}(N)\times {\rm U}(N+\nu)$ to $\mathrm{GL}(N,\mathbb{C})\times \mathrm{GL}(N+\nu,\mathbb{C})$ occurs for the action and the observables but not for the Langevin process itself. Despite this fact, the use of gauge cooling in the CLM can be justified. See ref.~\cite{Nagata-ml-2015uga} for explicit demonstration based on the Fokker-Planck equation.} since the noise term respects only ${\rm U}(N)\times {\rm U}(N+\nu)$. As a result, the eigenvalue distribution for the configuration obtained in the next Langevin step is affected nontrivially by the gauge cooling even if one averages over the Gaussian noise. ]]>

Details of the gauge cooling procedure

Results of the CLM with or without gauge cooling

The generalized Banks-Casher relation

Summary and discussions

Acknowledgments

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