Full simulation of chiral random matrix theory at nonzero chemical potential by complex Langevin

It is demonstrated that the complex Langevin method can simulate chiral random matrix theory at nonzero chemical potential. The successful match with the analytic prediction for the chiral condensate is established through a shift of matrix integration variables and choosing a polar representation for the new matrix elements before complexification. Furthermore, we test the proposal to work with a Langevin-time-dependent quark mass and find that it allows us to control the fluctuations of the phase of the fermion determinant throughout the Langevin trajectory.

Figure 1

The chiral condensate as a function of the quark mass for one, two and three dynamical flavors. The full lines are the exact analytic predictions, and the points are the results of the complex Langevin dynamics with adaptive step-size. The parameters chosen for the plot are $N=20$, $\mu =1$, $\nu =0$ and $2T=2000$.

Figure 2

The chiral condensate as a function of the quark mass for $\nu =0,1,2$ with $N_f=2$, $N=20$, $\mu =1$ and $\nu =0$. The data points are obtained with the complex Langevin using an adaptive step-size and the full lines are the predictions (5).

Figure 3

The average of the imaginary part of the angular variable ${\theta }_1$ and the average of the imaginary part of the radial variable $r_1$. As a function of the Langevin time $t$, the angular variable flows towards $\mu =1$, marked by the thin vertical line, while the radial variable remains on the real axis within the errors. The error bars are given by the square root of the variance. The parameters in the simulation are $\nu =0$, $N_f=2$, $N=20$, $\mu =1$, $m=1$ and $\nu =0$.

Figure 4

The eigenvalues of the Dirac operator in the complex plane: black for the first 400 time steps and red for the final 400 time steps. Parameters used are $N=20$, $N_f=2$, $\mu =1$, $m=1$, $\nu =0$ and 60000 adaptive steps. Note that the quark mass is initially well inside the eigenvalue distribution.

Figure 5

The argument, $\varphi$, of the phase of the fermion determinant as a function of Langevin time, for $m=1$, $\mu =1$, $N=20$, $N_f=2$ and $\nu =0$. Initially the fermion determinant frequently circles the origin but with the Langevin time the Dirac eigenvalues flow inside the quark mass and the fluctuations of the fermion determinant are damped. The fixed value of the quark mass throughout the run is indicated bu the horizontal red line.

Figure 6

The argument of the phase of the fermion determinant along the Langevin trajectory. With the Langevin-time-dependent quark mass (thin red curve), the fermion determinant does not circle the origin during the Langvin process. Here $\mu =1$, $N=20$, $N_f=2$, $\nu =0$ and the initial value of the quark mass, 3, is outside the cloud of Dirac eigenvalues. As the quark mass reaches the desired value $m=1$, it is kept fixed and the measurement of the chiral condensate can be performed.