Low-Lying Eigenvalues of the Wilson-Dirac Operator

TL;DR

This paper investigates the low-lying eigenvalues and eigenvectors of the Wilson-Dirac operator in lattice QCD, comparing quenched and unquenched simulations and analyzing localization properties in SU(2) and SU(3).

Contribution

It provides new insights into the spectral properties and localization behavior of the Wilson-Dirac operator's eigenmodes in different simulation contexts.

Findings

Eigenvalues differ between quenched and unquenched simulations.

Eigenvectors exhibit localization properties in SU(2) and SU(3).

Results enhance understanding of spectral features in lattice QCD.

Abstract

An exploratory study of the low-lying eigenvalues of the Wilson-Dirac operator and their corresonding eigenvectors is presented. Results for the eigenvalues from quenched and unquenched simulations are discussed. The eigenvectors are studied with respect to their localization properties in the quenched approximation for the cases of SU(2) and SU(3).

Equations

$$Q={\gamma }_{5}M/\left(1+8\kappa \right),$$

$$[M\psi ](x)$$

$$\frac{<{\Psi }_k|Q^2|{\Psi }_k>}{<{\Psi }_k|{\Psi }_k>}.$$

$$P=\frac{1}{V}\frac{({\sum }_{x,\alpha ,a}|{\Psi }_{\alpha a}(x){|}^2{)}^2}{{\sum }_{x,\alpha ,a}|{\Psi }_{\alpha a}(x){|}^4},$$

$$R_V(c)=\frac{1}{V}\#\left\{x:\rho (x)\ge c\cdot {\rho }_{\max }\right\}$$

Full text

1 Introduction

In this contribution we consider the eigenvalues and corresponding eigenvectors of the hermitean operator

[TABLE]

where $M$ is the Wilson-Dirac operator defined through

[TABLE]

The hopping parameter $\kappa$ is related to the bare quark mass $m_{0}$ by $\kappa=(8+2m_{0})^{-1}$. The gauge fields $U_{\mu}(x)$ are mostly taken to be in SU(2) and for some cases in SU(3). The matrix $Q$ is normalized in such a way that the eigenvalues $\lambda$ of $Q$ satisfy $0\leq\lambda^{2}\leq 1$. In the following we will concentrate on those eigenvalues of $Q$ closest to zero, and refer to this part of the spectrum as the “low-lying” eigenvalues.

The eigenvalues of $Q$ are of crucial importance for understanding the performance of algorithms used in dynamical fermion simulations. In particular, the scaling behaviour of the algorithms when approaching the chiral limit can be expressed in terms of the lowest eigenvalue of $Q^{2}$ [1].

Furthermore, the eigenmodes themselves play an interesting physical rôle. In this work we will only address the space-time localization properties of the eigenvectors belonging to low-lying eigenvalues. Additional more challenging questions include their connection to chiral symmetry breaking and topology. Another important aspect is the contribution of the low-lying eigenvalues to physical observables like the pion or $\rho$ meson correlation functions.

We compute the low-lying eigenvalues and the corresponding eigenvectors by using a modified Conjugate Gradient algorithm to minimize the Ritz functional

[TABLE]

For the computation of higher eigenvalues the minimization of (3) is restricted to $\Psi_{k}$ being orthogonal to previously calculated eigenvectors. Convergence is accelerated by intermediate diagonalizations in the space spanned by $\{\Psi_{i}\}$ [2]. This method has the virtue of being numerically stable and of providing rigorous error bounds for the eigenvalues calculated. Moreover the approximate eigenvectors are obtained without additional cost.

2 Low-lying eigenvalues for SU(2)

We first investigate how the spectrum changes with the gauge coupling $g^{2}_{0}$. Recall that for free Wilson fermions and $\kappa<1/8$, the lowest eigenvalue of $Q^{2}$ is 8-fold degenerate. Working in the quenched approximation on an $8^{4}$ lattice at $\kappa=0.12$, we find that up to $g^{2}_{0}=4/\beta$ around $0.8$ the lowest eight eigenvalues are nearly degenerate and well separated from the ninth lowest. Above $g^{2}_{0}\approx 0.8$ the eigenvalues form a dense band. It is interesting to note that this value of $\beta$, where the gap to the ninth eigenvalue closes, is roughly where one might expect the finite temperature phase transition.

The $\kappa$ dependence of the $n$ lowest eigenvalues (with $n=1,3,5,7,10$) at $\beta=2.3$ is shown in Fig. 1 for a lattice size of $12^{4}$. The results in Fig. 1 are averaged over 20 configurations each separated by 500 pure gauge updates. For $\kappa<0.16$ the low-lying eigenvalues are almost degenerate and appear on top of each other in the graph. For $\kappa\geq 0.16$ the eigenvalues drop significantly and their density becomes smaller.

The coarse $\kappa$-scan in Fig. 1 provides only an overall picture of the behaviour of the eigenvalues. On individual configurations, used for Fig. 1, a much finer $\kappa$-scan, guided by a perturbative expansion in small changes in $\kappa$, reveals almost-zero modes with values of $\lambda^{2}$ that can drop to e.g. $O(10^{-18})$, many orders of magnitude smaller than the lowest eigenvalue in Fig. 1. So far, it seems that these almost-zero modes appear above $\kappa_{c}$. If this observation is true for all coupling parameter values of interest, these almost-zero modes would be harmless. Otherwise they would indicate the breakdown of the quenched approximation since any occurance of such a mode could severely distort the configuration sample.

To investigate the situation for dynamical fermions we made unquenched runs for SU(2) at $\beta=2.3$. In Fig. 2 we show the lowest eigenvalue on $8^{4}$ and $16^{4}$ lattices. For the $16^{4}$ lattice, the scan in the sea-$\kappa$ has a high resolution. On some of these configurations we also performed fine valence-$\kappa$ scans. In both cases we did not find any indication of the existence of the almost-zero modes. Fig. 2 also shows that for the $16^{4}$ lattice the lowest eigenvalue in unquenched simulations has a sharp minimum. This suggests that the lowest eigenvalue could be a good quantity to identify the phase transition.

3 Localization of the eigenmodes

A simple quantity to study localization properties of wave functions is the participation ratio [3]

[TABLE]

where $V$ is the lattice volume and $(\alpha,a)$ are Dirac and colour indices. For pointlike localization one obtains $P\rightarrow 0$, whereas $P\rightarrow 1$ for a uniform distribution. The participation ratios which correspond to the eigenvalues of Fig. 1 are shown in Fig. 3. They clearly indicate strongly localized eigenmodes for $\kappa>0.17$, but do not give a clear indication at lower $\kappa$ values.

To investigate this question further we plot the local density $\rho(x)=\|\Psi(x)\|^{2}$ for 3-dimensional cuts through a lattice of size $16^{3}48$ for quenched SU(3) at $\beta=6.0$. Fig. 4 indicates localization also for $\kappa<\kappa_{c}\approx 0.157$; however it is less pronounced than for $\kappa>\kappa_{c}$ (Fig. 5).

For a quantitative analysis of the localization we define

[TABLE]

which measures the relative number of lattice sites with a local density larger than a certain fraction $c$ ($0\leq c\leq 1$) of the maximum value $\rho_{\rm max}$. We find that practically all sites contributing to $R_{V}(c)$ are in a connected region, and that $R_{V}(c)$ scales proportional to the inverse lattice volume.

To characterize the localization we try for $\rho(x)$ a parametrization $\rho(x)\sim\exp\left\{-m_{loc}\|x\|\right\}$, where $\|x\|$ denotes the distance from the site where $\rho=\rho_{\rm max}$. The effective “localization mass” $m_{loc}$ is the local rate with which $\rho(x)$ decays. Taking for $\|x\|$ the effective radius $r_{\rm eff}(c)=\sqrt[4]{2R_{V}(c)/\pi^{2}}$ one obtains $m_{loc}=-\ln(c)/r_{\rm eff}$. A plateau in the c-dependence of $m_{loc}$ would then indicate that $\rho(x)$ decays exponentially according to the above ansatz.

This is indeed observed as Fig. 6 shows for a value of $\kappa$ well below $\kappa_{c}$. One finds a nice stable plateau in $m_{loc}$ completely consistent with an exponential localization.

4 Conclusion

We have presented an exploratory study of the low-lying eigenvalue spectrum of the hermitian Wilson-Dirac operator $Q$ given in eq.(1). In the quenched approximation we find that the lowest eigenvalues drop sharply when the critical hopping parameter is reached. The isolated almost-zero modes we find in the quenched approximation at $\beta=2.3$ have not occurred in our unquenched runs. The eigenmodes corresponding to the low-lying eigenvalues show an exponential localization for $\kappa<\kappa_{c}$. For $\kappa$ above $\kappa_{c}$ they are even stronger (almost pointlike) localized. In general we did not see any qualitative difference between the results from SU(2) and SU(3).

Although the eigenvalues of $Q$ can not be directly related to the complex spectrum of $M$, we expect the qualitative behaviour of the eigenmodes to be similar. Of course, this relation has to be explored. It will also be interesting to relate the low-lying eigenvalues to topological properties of the theory and to quantify the contribution of the low-lying eigenvalues to physical observables. We hope to return to these questions in future publications.

Acknowledgement

We like to thank M. Bäker and M. Lüscher for helpful discussions and suggestions. The numerical work was carried out on APE/Quadrics and T3D systems and we thank DESY-IfH (Zeuthen) and The University of Edinburgh for providing the necessary resources. We acknowledge financial support by PPARC grant GR/J21347 and by the Carnegie Trust for The Universities of Scotland.