[Paper Review] Spectral Flow and Index Theorem for Staggered Fermions

E. Follana, V. Azcoiti+2 more|arXiv (Cornell University)|Nov 15, 2011

Quantum Chromodynamics and Particle Interactions4 references3 citations

TL;DR

This paper proposes a novel numerical method to compute the topological charge in lattice QCD using spectral flow of a hermitian operator derived from the staggered Dirac operator. It demonstrates that the index from spectral flow—defined by zero-mode crossings—agrees precisely with the index from counting high-chirality low-lying modes, confirming Adams' definition as robust and unambiguous even at finite lattice spacing.

ABSTRACT

We investigate numerically the spectral flow introduced by Adams for the staggered Dirac operator on realistic gauge configurations. We study both the unimproved and the HISQ Dirac operators. We compare the spectral flow index with the index obtained by identifying low-lying modes of large chirality.

Motivation & Objective

• To test the viability of Adams' spectral flow definition of topological charge in realistic 4D quenched SU(3) gauge configurations.
• To compare the spectral flow index with the conventional index obtained from high-chirality low-lying modes of the Dirac operator.
• To assess the robustness of the spectral flow method in cases where the chirality-based method becomes ambiguous.
• To evaluate the performance of both the unimproved (1-link) and highly improved (HISQ) staggered Dirac operators in capturing topological structure.
• To investigate the separation between low- and high-mass crossings in spectral flow, ensuring unambiguous topological charge assignment.

Proposed method

• Define a hermitian operator $ H_{st}(m) = iD_{st} - m\Gamma_5 $, where $ D_{st} $ is the staggered Dirac operator and $ \Gamma_5 $ is the taste-singlet $ \gamma_5 $.
• Compute the spectral flow $ \lambda(m) $ of $ H_{st}(m) $ by numerically diagonalizing the operator across a range of mass parameters $ m $.
• Identify topological charge as the number of eigenvalue crossings through zero at low $ m $, with chirality determined by the sign of the slope.
• Compare this spectral flow index with the index obtained by counting low-lying modes with high taste-singlet chirality.
• Use configurations from a tree-level Symanzik and tadpole-improved quenched QCD ensemble with $ a \approx 0.077 $ fm.
• Analyze both HISQ and 1-link Dirac operators to assess the impact of improved fermion actions on topological charge resolution.

Experimental results

Research questions

• RQ1 Does Adams' spectral flow definition yield a consistent and unambiguous topological charge in realistic 4D SU(3) gauge configurations?
• RQ2 How well does the spectral flow index agree with the conventional index from high-chirality mode counting?
• RQ3 Are low-lying and high-lying spectral crossings sufficiently separated to allow reliable topological charge assignment?
• RQ4 How does the choice of Dirac operator (HISQ vs. 1-link) affect the location and clarity of spectral crossings?
• RQ5 Can the spectral flow method resolve topological ambiguities that arise in the chirality-based method at finite lattice spacing?

Key findings

• The spectral flow method correctly identifies the topological charge $ Q $, with exactly $ 4Q $ eigenvalue crossings at low $ m $, matching the expected index.
• For configurations with $ Q = 0, -1, +2 $, the spectral flow shows clear, isolated crossings at low $ m $, confirming unambiguous charge assignment.
• The spectral flow for the HISQ Dirac operator produces crossings at significantly smaller $ m $ values than the 1-link operator, indicating better continuum behavior.
• In configurations where chirality-based classification fails due to near-degenerate high-chirality modes, the spectral flow method remains unambiguous and assigns $ Q = 0 $.
• No additional crossings are observed at large $ m $ (up to $ \mathcal{O}(200) $), confirming good separation between low- and high-mass crossings.
• The spectral flow method remains well-defined even when the chirality-based method fails, due to its reliance on crossing behavior rather than mode classification.