Topological Index Theorem on the Lattice through the Spectral Flow of Staggered Fermions

We investigate numerically the spectral flow introduced by Adams for the staggered Dirac operator on realistic (quenched) gauge configurations. We obtain clear numerical evidence that the definition works as expected: there is a clear separation between crossings near and far away from the origin, and the topological charge defined through the crossings near the origin agrees, for most configurations, with the one defined through the near-zero modes of large taste-singlet chirality of the staggered Dirac operator. The crossings are much closer to the origin if we improve the Dirac operator used in the definition, and they move towards the origin as we decrease the lattice spacing.


I. INTRODUCTION
As is well known, smooth SU(N) gauge fields in a 4-dimensional compact differentiable manifold M have associated an integer topological charge where F µν is the gauge potential.
This is not merely a mathematical curiosity; it plays a fundamental role in the understanding of the U A (1) problem through the anomaly [1][2][3], and the Witten-Veneziano formula [4,5]. It is also crucial for the investigation of the θ vacuum in QCD and the strong CP problem [6], and therefore with the current experimental searches for axions [7,8].
The issue of obtaining a theoretically sound and practical definition for the topological charge of lattice gauge fields is an old one. Several definitions exist, each with its own advantages and disadvantages. Some such definitions are purely gluonic, essentially transcribing the continuum definition to the lattice, whereas others take advantage of the index theorem and compute the charge as the index of a conveniently chosen fermionic operator.
In [9], Adams introduced a new definition of the index of a staggered Dirac operator, based on the spectral flow of a related hermitian operator. Some numerical results were obtained there for synthetic configurations in the 2D U(1) model.
The purpose of this paper is to study systematically Adams' definition in 4D (quenched) QCD (preliminary results were presented in [10]).

II. DEFINITION OF THE TOPOLOGICAL CHARGE
The basic observation of Adams in [9] is that when considering the spectral flow definition of the topological charge in the continuum, there is some freedom in the choice of the relevant hermitian operator. By a suitable choice, we can construct an operator which, when implemented in the lattice with a staggered dirac operator has all the required properties.
In the continuum, for a given gauge field, one usually considers the spectral flow of the hermitian operator 2 as a function of m. Because of the key property that if we trace the flow of eigenvalues {λ(m)} of H, the ones corresponding to the zero modes of D, and only those, will change sign at the origin m = 0, each with a slope ±1 which depends on the chirality of the corresponding mode. This gives us the index of D, and through the index theorem, the topological charge Q of the corresponding gauge configuration.
On the lattice, we can substitute in (2) D by the discretized Wilson Dirac operator Now the index can be obtained similarly to the continuum, by counting the number of eigenvalues of H(m) that change sign close to the origin m = 0, taking into account the slope of such crossings. If we try to do the same with the lattice staggered Dirac operator D st , we realize that the procedure does not work anymore, as the corresponding H fails to be hermitian.
The key innovation in Adams [9] is the realization that one can use a different H in the continuum to accomplish the same task, namely This operator is hermitian and verifies (3), and therefore its spectral flow also gives the index of D. But now we can substitute in (5) D by the lattice staggered discretization, where D st is the massless staggered Dirac operator and Γ 5 is the taste-singlet staggered γ 5 [11]. This operator is hermitian, and we can study its spectral flow, λ(m). The would-be zero modes of D st are identified with the eigenmodes for which the corresponding eigenvalue flow λ(m) crosses zero at low values of m, and the chirality of any such mode equals (with our conventions) the sign of the slope of the crossing [9].
Any staggered discretization of the Dirac operator can be used, in principle, to implement 6. We have chosen to work with the unimproved, 1-link staggered Dirac operator [12], and with the highly improved HISQ discretization [13]. In each case we have calculated, using standard numerical algorithms, the smallest (in absolute value) 20 eigenvalues of H st (m) for enough values of m to allow us to determine unambiguously the cuts with the x axis.
To compare with previous work, we have also calculated the low-lying modes of the HISQ Dirac operator at m = 0, and identify the would-be zero modes with the high taste-singlet chirality ones [14,15]. The spectrum of (6) has the exact symmetry λ(m) ↔ −λ(−m), therefore we only need to calculate the flow for, say, m > 0. An equal number of crossings, with identical slopes, will be present for m < 0.
In Fig. 1  for m > 0. For clarity, in most of the figures we plot the variablẽ λ = sgn(λ) |λ| log(|λ|) versus log(m). As we can see in every case we have the expected number of crossings. the HISQ Dirac operators. As before, we calculate the first 20 eigenvalues in absolute value.
The topological charges were also calculated by counting the number of eigenvectors of the HISQ Dirac operator with high chirality [15], and in each case there is agreement between the two definitions and for the spectral flow corresponding to both operators. We can appreciate in the figures that the cuts of the spectral flow with the x axis are closer to the origin m = 0 for the HISQ than for the 1-link operators, and also get closer as we go to smaller lattice spacings. This is consistent with the expectation that in the continuum limit the cuts should move to the origin, and that the improved Dirac operator is closer to the continuum than its unimproved counterpart. In order to make a more quantitative statement, we have computed a histogram (normalized to area one) of the cuts for the three different ensembles and both operators, which is shown in Fig. 5. We can see clearly the large differences between both operators, and how the distribution of cuts moves towards and therefore the topological charge of a configuration is unambiguously defined, even in cases which would be ambiguous using other definitions. For most configurations we have seen that the charge as measured by the number of high taste-singlet chirality modes and by the spectral flow agree. We have also seen the expected differences between the position of the cuts between the 1link and the HISQ operators, as well as the clear move towards zero of the cuts as we decrease the lattice spacing.
For a future work, it would be interesting to repeat this study in full QCD ensembles, including the effect of sea quarks.
Inspired by this construction, it is possible to define an overlap operator starting with a staggered kernel, instead of the usual Wilson one [19], producing a chiral operator representing two tastes of fermions. A similar construction can be carried out to further reduce the degeneracy and produce a one-flavor overlap operator [20]. The question is whether this construction has all the required properties, and is further numerically advantageous as compared with the usual overlap construction. Results are presented in [21][22][23][24].