Two-flavor QCD simulation with exact chiral symmetry

We perform numerical simulations of lattice QCD with two flavors of dynamical overlap quarks, which have exact chiral symmetry on the lattice. While this fermion discretization is computationally demanding, we demonstrate the feasibility to simulate reasonably large and fine lattices by a careful choice of the lattice action and algorithmic improvements. Our production runs are carried out on a ${16}^3\times 32$ lattice at a single lattice spacing around 0.12 fm. We explore the sea quark mass region down to $m_s/6$, where $m_s$ is the physical strange quark mass, for a good control of the chiral extrapolation in future calculations of physical observables. We describe in detail our setup and algorithmic properties of the production simulations and present results for the static quark potential to fix the lattice scale and the locality of the overlap operator.

Figure 1

Monte Carlo history of the plaquette (left panels) and $N_{\mathrm{inv}}$ (right panels) during first 1000 trajectories with HMC-4D.

Figure 2

Autocorrelation times ${\tau }_{\mathrm{int},\mathrm{plaq}}$ (left panel) and ${\tau }_{\mathrm{int},\mathrm{inv}}$ (right panel) as a function of quark mass $m$. Data for HMC-5D are slightly shifted in the horizontal direction for clarity.

Figure 3

Jackknife error of combined data of the plaquette as a function of bin size. Left and right panels show data at $Q=0$ and at $m=0.050$.

Figure 4

Jackknife error of $N_{\mathrm{inv}}$ in simulations with HMC-4D.

Figure 5

Eigenvalues of $H_{\mathrm{W}}$ smaller than 0.04 at each HMC trajectory and its histogram. Left and right panels show data from test runs at $(\beta ,\mu )=(2.35,0.2)$ and (2.45,0.0) (namely, with and without the determinant $\det [{\Delta }_{\mathrm{W}}]$). We plot data during the first 300 trajectories in each simulation.

Figure 6

Histogram of low-lying eigenvalue $|{\lambda }_{\mathrm{W}}|$ in production runs at $m=0.015$ (thick solid line), 0.050 (thin solid line), and 0.100 (thin dot-dashed line). The small panel shows the same data near ${\lambda }_{\mathrm{W}}=0$.

Figure 7

Lowest eigenvalue of $H_{\mathrm{W}}$ during MD evolution in the first 50 trajectories at $\beta =2.30$ and $m=0.050$.

Figure 8

Measures of reversibility violation ${\Delta }_{U,\mathrm{ave}}$ (left panel) and ${\Delta }_H$ (right panel) for HMC-4D as a function of stopping conditions ${ϵ}_{4\mathrm{D},\mathrm{f}}^{\mathrm{ms}}={ϵ}_{4\mathrm{D},\mathrm{f}}$.

Figure 9

Residual of overlap solvers as a function of the number of the $D_{\mathrm{W}}$ multiplication at $m=0.025$. Thick and thin lines show results for the 5D and 4D solvers. For the 5D solver, we plot the residual for the 4D piece $x$ of the 5D solution vector $x_5$, namely, $r=D(m)x-b$.

Figure 10

Quark mass dependence of the CG iteration count $N_{\mathrm{inv}}$ for the 4D overlap solver. Left and right panels show data for the Hasenbusch preconditioner $D(m^{\prime })$ and the operator with the physical mass $D(m)$. The dashed line shows the fit to Eq. (43).

Figure 11

Projected eigenvalue ${\lambda }_k(0)$ as a function of $m$. We plot data for five lowest-lying modes. The dashed line shows $\lambda =m$.

Figure 12

Quark mass dependence of the CG iteration count $N_{\mathrm{inv}}$ for the 5D solver. Left and right panels show data for $D^{\prime }(m^{\prime })$ and $D^{\prime }(m)$, respectively.

Figure 13

Time history of MD forces $F_2$, $F_1$, $F_g$, and $F_{\mathrm{W}}$ at $m=0.015$ (four leftmost panels) and 0.050 (four rightmost panels) with HMC-4D. Two lines in each panel show the average and maximum values over the degrees of freedom (color, space-time direction, and coordinates).

Figure 14

Time history of MD forces at $m=0.050$ with HMC-5D.

Figure 15

Time history of $\Delta H$ in simulation at $m=0.025$ with HMC-4D. We have a spike at the 2151th trajectory.

Figure 16

Maximum value of $F_{\mathrm{W}}$ and lowest eigenvalue of $H_{\mathrm{W}}$ during MD evolution where a spike in $\Delta H$ is observed.

Figure 17

Acceptance rate $P_{\mathrm{HMC}}$ as a function of $\Delta H$ in simulations with HMC-4D (left panel) and HMC-5D (right panel). The solid line is the expectation of Eq. (45). Data at $(m,Q)=(0.025,0)$ with HMC-4D and at (0.015, 0) with HMC-5D are consistent with the expectation within their huge error and hence are omitted.

Figure 18

Number of $D_{\mathrm{W}}$ multiplications $N_{\mathrm{mult}}$ in HMC-4D (left panel) and HMC-5D (right panel). Open symbols are $N_{\mathrm{mult}}$ in calculations of MD forces and in the Metropolis tests (MTs), whereas the filled symbols are their total.

Figure 19

Comparison of $N_{\mathrm{mult}}$ between HMC-4D and HMC-5D. The left panel shows $N_{\mathrm{mult}}$ only for $T_{P,1}$ and $T_{P,2}$, whereas the right panel is $N_{\mathrm{mult}}$ for all HMC steps.

Figure 20

Effective potential $V_{\mathrm{eff}}(r)$ at $r=2$ and $3\sqrt{3}$. Left and right panels show data at $m=0.015$ and 0.050.

Figure 21

Static quark potential $V(r)$ at $m=0.015$ (left panel) and 0.050 (right panel). The solid line shows the fit of Eq. (50), whereas the figure legend represents the direction of the spatial Wilson line listed in Eq. (47).

Figure 22

Fit range $[r_{\min },r_{\max }]$ dependence of fit parameters $\alpha$ and $\sigma$ in Eq. (50) at $m=0.015$. Our choice $r_{\min }=\sqrt{6}$ and $r_{\max }=8\sqrt{3}$ is shown by dashed lines.

Figure 23

Sommer scale $r_0$ as a function of $r_{\min }$ at $m=0.015$ (top panel) and 0.050 (bottom panel). We set $r_{\max }=8\sqrt{3}$.

Figure 24

Chiral extrapolation of $a(m)$ with the linear form of Eq. (53) (circles). Extrapolated values with other fitting forms in Eqs. (53) (square) and (54) (triangles) are also plotted. Two error bars for these symbols show statistical and total errors.

Figure 25

Lattice spacing determined from $r_0$ in simulations with Iwasaki gauge action. Circles are our estimate Eq. (55) and those in Ref. [57] and our preparatory study. These are compared with results in pure gauge theory [106] (solid line), only with extra fermions [54] (triangles), and with improved Wilson fermions [87] (squares).

Figure 26

Function $f_{\mathrm{W},i}(r)$ at $m=0.025$. The figure legend shows windows in $|{\lambda }_{\mathrm{W}}|$. The dashed lines show the fit Eq. (58).

Figure 27

Inverse of localization length for low-lying modes of $H_{\mathrm{W}}$ as a function of their eigenvalue $|{\lambda }_{\mathrm{W}}|$.

Figure 28

Function $f(r)$ at $m=0.025$ as a function of the distance $r$. The solid line shows the exponential fit $f(r)\propto e^{-r/l}$.