Chiral symmetry and taste symmetry from the eigenvalue spectrum of staggered Dirac operators

We investigate general properties of the eigenvalue spectrum for improved staggered quarks. We introduce a new chirality operator $[{\gamma }_5\otimes 1]$ and a new shift operator $[1\otimes {\xi }_5]$, which respect the same recursion relation as the ${\gamma }_5$ operator in the continuum. Then we show that matrix elements of the chirality operator sandwiched between two eigenstates of the staggered Dirac operator are related to those of the shift operator by the Ward identity of the conserved $U(1{)}_A$ symmetry of staggered fermion actions. We perform a numerical study in quenched QCD using HYP staggered quarks to demonstrate the Ward identity. We introduce a new concept of leakage patterns which collectively represent the matrix elements of the chirality operator and the shift operator sandwiched between two eigenstates of the staggered Dirac operator. The leakage pattern provides a new method to identify zero modes and nonzero modes in the Dirac eigenvalue spectrum. This method is as robust as the spectral flow method but requires much less computing power. Analysis using a machine learning technique confirms that the leakage pattern is universal, since the staggered Dirac eigenmodes on normal gauge configurations respect it. In addition, the leakage pattern can be used to determine a ratio of renormalization factors as a by-product. We conclude that it might be possible and realistic to measure the topological charge $Q$ using the Atiya-Singer index theorem and the leakage pattern of the chirality operator in the staggered fermion formalism.

Figure 1

Eigenvalue spectrum of staggered Dirac operator on a $Q=0$ gauge configuration. (a) ${\lambda }_i^2$ and (b) ${\lambda }_i$.

Figure 2

Eigenvalue spectrum of staggered Dirac operator on a $Q=-1$ gauge configuration. (a) ${\lambda }_i^2$ and (b) ${\lambda }_i$.

Figure 3

The phase $\theta$ as a function of $\lambda$. Red circles represent numerical results for $\theta$. The blue line represents the prediction from the theory. Here we use a gauge configuration with $Q=-1$ for the measurement.

Figure 4

Leakage pattern for would-be zero modes at $Q=-1$. Here, the red bar represents leakage to ${\lambda }_{i=2n-1}>0$ with $i$ odd, and the blue bar represents leakage to its parity partner ${\lambda }_{i=2n}=-{\lambda }_{2n-1}$ with $i$ even.

Figure 5

Leakage pattern for nonzero modes at $Q=-1$.

Figure 6

$[{\gamma }_5\otimes 1]$ leakage pattern for nonzero modes at $Q=-1$.

Figure 7

$[1\otimes {\xi }_5]$ leakage pattern for nonzero modes at $Q=-1$.

Figure 8

$S_5$ as a function of $|\ell -j|$. Numerical values are given in Table 7.

Figure 9

Matrix elements of $|{\mathrm{\Gamma }}_5|$ for 200 and 32 of the lowest eigenmodes on a gauge configuration with $Q=2$. Here, indices on both axes are the eigenvalue index. The color of each square represents the magnitude of the corresponding matrix element. Black lines indicate borders of nonzero mode quartets, and red lines are those of zero mode quartets.

Figure 10

Examples for our samples. Every sample contains only one nonzero mode quartet. There are eight kinds of classes according to the location of the borders of the quartet.

Figure 11

The same as Fig. 1 except for $Q=-2$.

Figure 12

The same as Fig. 1 except for $Q=-3$.

Figure 13

$[{\gamma }_5\otimes 1]$ leakage pattern for the first quartet of would-be zero modes at $Q=-2$.

Figure 14

$[1\otimes {\xi }_5]$ leakage pattern for the first quartet of would-be zero modes at $Q=-2$.

Figure 15

$[{\gamma }_5\otimes 1]$ leakage pattern for the second quartet of would-be zero modes at $Q=-2$.

Figure 16

$[1\otimes {\xi }_5]$ leakage pattern for the second quartet of would-be zero modes at $Q=-2$.

Figure 17

$[{\gamma }_5\otimes 1]$ leakage pattern for the third quartet of would-be zero modes at $Q=-3$.

Figure 18

$[1\otimes {\xi }_5]$ leakage pattern for the third quartet of would-be zero modes at $Q=-3$.

Figure 19

$[{\gamma }_5\otimes 1]$ leakage pattern for the first quartet of nonzero modes at $Q=0$.

Figure 20

$[1\otimes {\xi }_5]$ leakage pattern for the first quartet of nonzero modes at $Q=0$.

Figure 21

$[{\gamma }_5\otimes 1]$ leakage pattern for the second quartet of nonzero modes at $Q=0$.

Figure 22

$[1\otimes {\xi }_5]$ leakage pattern for the second quartet of nonzero modes at $Q=0$.

Figure 23

$[{\gamma }_5\otimes 1]$ leakage pattern for the first quartet of nonzero modes at $Q=-2$.

Figure 24

$[1\otimes {\xi }_5]$ leakage pattern for the first quartet of nonzero modes at $Q=-2$.

Figure 25

$[{\gamma }_5\otimes 1]$ leakage pattern for the first quartet of nonzero modes at $Q=-3$.

Figure 26

$[1\otimes {\xi }_5]$ leakage pattern for the first quartet of nonzero modes at $Q=-3$.