Localization and chiral symmetry in three flavor domain wall QCD

We present results for the dependence of the residual mass of domain wall fermions on the size of the fifth dimension and its relation to the density and localization properties of low-lying eigenvectors of the corresponding Hermitian Wilson Dirac operator in three flavor domain wall QCD. Using the DBW2 and Iwasaki gauge actions, we generate ensembles of configurations with a ${16}^3\times 32$ space-time volume and an extent of 8 in the fifth dimension for the sea quarks. We demonstrate the existence of a regime where the degree of locality, the size of chiral symmetry breaking, and the rate of topology change can be acceptable for inverse lattice spacings $a^{-1}\ge 1.6\mathrm{GeV}$, enabling a programme of simulations of $2+1$ flavor QCD to be conducted safely in this region of parameter space.

Figure 1

Schematic diagram of the quenched Aoki phase. A pionic condensate and corresponding nonzero density of near-zero modes is developed throughout most of the negative mass region. In the colored sectors, a nonzero mobility edge is developed and the contours could equally well represent either decreasing low mode density, decreasing pionic condensate, or increasing ${\lambda }_c$ as one moves towards the continuum limit at $g^2=0$.

Figure 2

Spectral flow of DBW2. The horizontal axis is the valence domain wall height, $0\le M_5\le 3.0$, and the vertical axis is the eigenvalue of $H_W$, $-0.3\le {\lambda }_{H_W}\le 0.3$. From the left, $\beta =0.72$, $\beta =0.764$ (first row), $\beta =0.78$, $\beta =0.80$ (second row), and $\beta =0.88$ (third row).

Figure 3

Spectral flow of Iwasaki $\beta =2.13$, $\beta =2.2$ (first row, left to right), and $\beta =2.3$ (second row). Axes are the same as Fig. 2.

Figure 4

Spectral density obtained on a subset of our data with no error analysis, and fixed bin width 0.02. Thus, where the density is low the error could be large and the data should be considered only as a qualitative indication of the nature of our ensembles.

Figure 5

Various measures can be used to define the degree of localization. The inverse participation ratio has been a popular order parameter for delocalization transitions and, while fine for that purpose, counts occupied sites without paying attention to mode shape. This one-dimensional cartoon highlights the advantages of the more robust measure employed in this paper. Given the model of exponential localization, our tactic is to find a localization exponent that forms a tight bound for the eigenvector. This measure will show the influence of satellite peaks in a robust fashion by eliminating all forms of dilution by volume average. Other useful measures that take into account the shape would include appropriately weighted moments of the density function.

Figure 6

Microscopic view of the mobility edge based on a scatter plot of maximal localization lengths of individual low eigenmodes of $H_W$. We overlay the $H_T$ mobility edge ${\lambda }_c$ from fits to the model for $m_{\mathrm{res}}$, Eq. (35), and find good agreement with our diverging localization length for the DBW2 gauge action at $\beta =0.764$.

Figure 7

Microscopic view of the mobility edge based on a scatter plot of maximal localization lengths of individual low eigenmodes of $H_W$. We overlay the $H_T$ mobility edge ${\lambda }_c$ from fits to the model for $m_{\mathrm{res}}$, Eq. (35), and find good agreement with our diverging localization length for the Iwasaki gauge action at $\beta =2.13$.

Figure 8

We display an isosurface through the eigenvector density of a typical near-zero chiral mode of the five-dimensional operator $H_{\mathrm{DWF}}$, bound to a single wall. Here $s$ is the vertical direction, $t$ the left-right axis, and $z$ runs into the page. We show $(x,y)=(3,3)$ the eigenmode exhibits a localized incursion of the eigenmode into the fifth dimension associated with a low mode of $H_W$. This mechanism is responsible for one form of the contributions to chiral symmetry breaking in DWF.

Figure 9

We display an isosurface through the eigenvector density of the same chiral mode displayed in Fig. 8. Here $s$ is the vertical direction, $t$ the left-right axis, and $z$ runs into the page. We show $(x,y)=(11,13)$ the eigenmode exhibits a uniformly well stuck region of this mode. Compared with the previous figure, this degree of binding is the more common case for this typical eigenvector, and one has to search spatially to find any localized incursion.

Figure 10

Normalized overlap of different $s$ slices of a chiral five-dimensional eigenmode of $H_{\mathrm{DWF}}$ with the first 256 four-dimensional eigenmodes of $H_W$. We take the valence operator with $L_s=16$, and demonstrate numerically that only a few low-lying four-dimensional eigenmodes dominate the large $L_s$ contributions to chirality mixing operators such as $m_{\mathrm{res}}$ by plotting the decomposition of the lowest five-dimensional mode $|{\alpha }_{0j}(s){|}^2$ as a function of the fifth coordinate $s$, and four-dimensional mode number $j$.

Figure 11

Residual mass fits for valence quarks with $m=0.04$ and $L_s=8$, 12, 16, 24, 32 on the DBW2 $\beta =0.72$ ensemble with sea quark masses $m_{ud}=m_s=0.04$ and for the sea quark $L_s=8$.

Figure 12

Residual mass fits for valence quarks with $m=0.04$ and $L_s=8$, 12, 16, 24, 32 on the DBW2 $\beta =0.764$ ensemble with sea quark masses $m_{ud}=m_s=0.04$ and for the sea quark $L_s=8$.

Figure 13

Residual mass fits for valence quarks with $m=0.04$ and $L_s=8$, 12, 16, 24, 32 on the DBW2 $\beta =0.78$ ensemble with sea quark masses $m_{ud}=m_s=0.04$ and for the sea quark $L_s=8$.

Figure 14

Residual mass fits for valence quarks with $m=0.04$ and $L_s=8$, 12, 16, 24, 32 on the DBW2 $\beta =0.80$ ensemble with sea quark masses $m_{ud}=m_s=0.04$ and for the sea quark $L_s=8$.

Figure 15

Residual mass fits for valence quarks with $m=0.04$ and $L_s=8$, 12, 16, 24, 32 on the DBW2 $\beta =0.88$ ensemble with sea quark masses $m_{ud}=m_s=0.04$ and for the sea quark $L_s=8$.

Figure 16

Residual mass fits for valence quarks with $m=0.04$ and $L_s=8$, 12, 16, 24, 32 on the Iwasaki $\beta =2.13$ ensemble with sea quark masses $m_{ud}=m_s=0.04$ and for the sea quark $L_s=8$.

Figure 17

Residual mass fits for valence quarks with $m=0.04$ and $L_s=8$, 12, 16, 24, 32 on the Iwasaki $\beta =2.2$ ensemble with sea quark masses $m_{ud}=m_s=0.04$ and for the sea quark $L_s=8$.

Figure 18

Residual mass fits for valence quarks with $m=0.04$ and $L_s=8$, 12, 16, 24, 32 on the Iwasaki $\beta =2.3$ ensemble with sea quark masses $m_{ud}=m_s=0.04$ and for the sea quark $L_s=8$.

Figure 19

Valence $m_{\mathrm{res}}(L_s)$ dependence of partially quenched DWF with the DBW2 gauge action as a function of $L_s$, using $L_s=8$ and the indicated $\beta$ values on a ${16}^3\times 32$ volume for the ensembles. The sea and valence quark masses are 0.04 throughout.

Figure 20

Valence $m_{\mathrm{res}}(L_s)$ dependence of partially quenched DWF with the Iwasaki gauge action as a function of $L_s$, using $L_s=8$ and the indicated $\beta$ values on a ${16}^3\times 32$ volume for the ensembles. The sea and valence quark masses are 0.04 throughout.

Figure 21

Dependence of the plaquette as a function of the dynamical $L_s$, obtained from low statistics runs. The dependence of this observable shows the expected $C+e^{-\alpha L_s}$ for an exponential approach to $L_s=\infty$, and thus demonstrates nicely the connection between finite $L_s$ DWF simulations and those with exactly chiral fermions.

Figure 22

The Monte Carlo trajectory history for global topological charge with the DBW2 gauge action at couplings $\beta =0.72$, 0.764, 0.78, 0.8. The $\beta =0.88$ history is offset by 3000 and displays a clear loss of topology tunneling. With any reasonable computational effort, a simulation with the current algorithm and these parameters would be practically nonergodic.

Figure 23

The Monte Carlo trajectory history for global topological charge with the Iwasaki gauge action at couplings $\beta =2.13$, 2.2, 2.3. The tunneling rate remains acceptable and leaves headroom for taking a continuum limit with acceptable topological sampling without any further algorithmic improvement.