Staggered Dirac Operator Research Articles

The chiral condensate of Nf = 2 + 1 QCD from the spectrum of the staggered Dirac operator

We compute the chiral condensate of 2 + 1 QCD from the mode number of the staggered Dirac operator, performing controlled extrapolations to both the continuum and the chiral limit. We consider also alternative strategies, based on the quark mass dependence of the topological susceptibility and of the pion mass, and obtain consistent results within errors. Results are also consistent with phenomenological expectations and with previous numerical determinations obtained with different lattice discretizations.

Deconfinement transition and localization of Dirac modes in finite-temperature Z3 gauge theory on the lattice

We study the localization properties of the eigenmodes of the staggered Dirac operator across the deconfinement transition in finite-temperature $\mathbb{Z}_3$ pure gauge theory on the lattice in 2+1 dimensions. This allows for nontrivial tests of the sea-islands picture of localization, according to which low modes should localize on favorable Polyakov-loop fluctuations in the deconfined phase of a gauge theory. We observe localized low modes in the deconfined phase of the theory, both in the real Polyakov-loop sector, where they are expected, and in the complex Polyakov-loop sectors, where they are not. Our findings expose the limitations of the standard sea-islands picture, and call for its refinement. An improved picture, where spatial hopping terms play a more prominent role, is proposed and found to be in excellent agreement with numerical results.

Topological susceptibility of Nf = 2 + 1 QCD from staggered fermions spectral projectors at high temperatures

We compute the topological susceptibility of Nf = 2 + 1 QCD with physical quark masses in the high-temperature phase, using numerical simulations of the theory discretized on a space-time lattice. More precisely we estimate the topological susceptibility for five temperatures in the range from ∼ 200 MeV up to ∼ 600 MeV, adopting the spectral projectors definition of the topological charge based on the staggered Dirac operator. This strategy turns out to be effective in reducing the large lattice artifacts which affect the standard gluonic definition, making it possible to perform a reliable continuum extrapolation. Our results for the susceptibility in the explored temperature range are found to be partially in tension with previous determinations in the literature.

Localization of Dirac modes in finite-temperature Z2 gauge theory on the lattice

We study the localization properties of the eigenmodes of the staggered Dirac operator in finite-temperature $\mathbb{Z}_2$ pure gauge theory on the lattice in 2+1 dimensions. We find that the low modes turn from delocalized to localized as the system crosses over from the confined to the deconfined phase in the "physical" sector (positive average Polyakov loop) selected by external fermionic probes, while they remain delocalized in the "unphysical" sector (negative average Polyakov loop). This confirms that the close connection between deconfinement and localization of the low Dirac modes in the physical sector, already observed in other models, holds also in the simplest gauge theory displaying a deconfinement transition. We also observe a clear correlation of localized modes with fluctuations of the Polyakov loop away from the ordered value, as expected according to the "sea/islands" picture of localization, and with clusters of negative plaquettes. A novel finding is the presence of localized modes at the high end of the Dirac spectrum in all phases/sectors of the theory.

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 $[{\ensuremath{\gamma}}_{5}\ensuremath{\bigotimes}1]$ and a new shift operator $[1\ensuremath{\bigotimes}{\ensuremath{\xi}}_{5}]$, which respect the same recursion relation as the ${\ensuremath{\gamma}}_{5}$ 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.

A staggered eigensolver based on sparse matrix bidiagonalization

We present a method for calculating eigenvectors of the staggered Dirac operator based on the Golub-Kahan-Lanczos bidiagonalization algorithm. Instead of using orthogonalization during the bidiagonalization procedure to increase stability, we choose to stabilize the method by combining it with an outer iteration that refines the approximate eigenvectors obtained from the inner bidiagonalization procedure. We discuss the performance of the current implementation using QEX and compare with other methods.

An Anderson-like model of the QCD chiral transition

We study the problems of chiral symmetry breaking and eigenmode localisation in finite-temperature QCD by looking at the lattice Dirac operator as a random Hamiltonian. We recast the staggered Dirac operator into an unconventional three-dimensional Anderson Hamiltonian ("Dirac-Anderson Hamiltonian") carrying internal degrees of freedom, with disorder provided by the fluctuations of the gauge links. In this framework, we identify the features relevant to chiral symmetry restoration and localisation of the low-lying Dirac eigenmodes in the ordering of the local Polyakov lines, and in the related correlation between spatial links across time slices, thus tying the two phenomena to the deconfinement transition. We then build a toy model based on QCD and on the Dirac-Anderson approach, replacing the Polyakov lines with spin variables and simplifying the dynamics of the spatial gauge links, but preserving the above-mentioned relevant dynamical features. Our toy model successfully reproduces the main features of the QCD spectrum and of the Dirac eigenmodes concerning chiral symmetry breaking and localisation, both in the ordered (deconfined) and disordered (confined) phases. Moreover, it allows us to study separately the roles played in the two phenomena by the diagonal and the off-diagonal terms of the Dirac-Anderson Hamiltonian. Our results support our expectation that chiral symmetry restoration and localisation of the low modes are closely related, and that both are triggered by the deconfinement transition.

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.

Numerical properties of staggered quarks with a taste-dependent mass term

Abstract The numerical properties of staggered Dirac operators with a taste-dependent mass term proposed by Adams [1, 2] and by Hoelbling [3] are compared with those of ordinary staggered and Wilson Dirac operators. In the free limit and on (quenched) interacting configurations, we consider their topological properties, their spectrum, and the resulting pion mass. Although we also consider the spectral structure, topological properties, locality, and computational cost of an overlap operator with a staggered kernel, we call attention to the possibility of using the Adams and Hoelbling operators without the overlap construction. In particular, the Hoelbling operator could be used to simulate two degenerate flavors without additive mass renormalization, and thus without fine-tuning in the chiral limit.

Theoretical Foundation for the Index Theorem on the Lattice with Staggered Fermions

A way to identify the would-be zero modes of staggered lattice fermions away from the continuum limit is presented. Our approach also identifies the chiralities of these modes, and their index is seen to be determined by gauge field topology in accordance with the index theorem. The key idea is to consider the spectral flow of a certain Hermitian version of the staggered Dirac operator. The staggered fermion index thus obtained can be used as a new way to assign the topological charge of lattice gauge fields. In a numerical study in U(1) backgrounds in two dimensions it is found to perform as well as the Wilson index while being computationally more efficient. It can also be expressed as the index of an overlap Dirac operator with a new staggered fermion kernel.

Polyakov loops and spectral properties of the staggered Dirac operator

We study the spectrum of the staggered Dirac operator in SU(2) gauge fields close to the free limit, for both the fundamental and the adjoint representation. Numerically we find a characteristic cluster structure with spacings of adjacent levels separating into three scales. We derive an analytical formula which explains the emergence of these different spectral scales. The behavior on the two coarser scales is determined by the lattice geometry and the Polyakov loops, respectively. Furthermore, we analyze the spectral statistics on all three scales, comparing to predictions from random matrix theory.

Eigenvalues and eigenvectors of the staggered Dirac operator at finite temperature

We examine the eigenvalues and eigenvectors of the staggered Dirac operator on thermal ensembles created in QCD with two flavors of staggered quarks. We see that across the phase transition a gap opens in the spectrum. For finite volume lattices in the low-temperature phase the eigenvectors are extended, but generic field configurations in the high temperature phase give rise to localized eigenstates. We examine measures of the stability of such localization and find that at finite volumes localization occurs through Mott's mechanism of the formation of mobility edges. However, the band gap between the localized and extended states seem to scale to zero in the limit of large volume.

The flavour projection of staggered fermions and the quarter-root trick

It is shown that the flavour projection of staggered fermions can be written as a projection between the fields on four separate, but parallel, lattices, where the fields on each are modified forms of the standard staggered fermion field. Because the staggered Dirac operator acts equally on each lattice, it respects this flavour projection. We show that the system can be gauged in the usual fashion and that this does not interfere with flavour projection. We also consider the path integral, showing that, prior to flavour projection, it evaluates to the same form on each lattice and that this form is equal to that used in the quarter-root trick. The flavour projection leaves a path integral for a single flavour of field on each lattice.

Rotational Symmetry and a Light Mode in Two-Dimensional Staggered Fermions

To obtain a light mode in two-dimensional staggered fermions, we introduce four new local operators that preserve the rotational invariance for a staggered Dirac operator. To split the masses of tastes, three cases are considered. The mass matrix and the propagator for free theories are analyzed. We find that one of the three cases may allow the identification of a single mode through the mass splitting. In this case, it is possible that a heavy mode acquires an infinite mass on even sites or odd sites.

A comparative study of overlap and staggered fermions in QCD

We perform a comparative study of the infrared properties of overlap and staggered fermions in QCD. We observe that the infrared spectrum of the APE/HYP improved staggered Dirac operator develops a four-fold near-degeneracy and is in quantitative agreement with the infrared spectrum of the overlap operator. The near-degeneracy allows us to identify the zero modes of the staggered operator and we find that the number of zero modes is in line with the topological index of the overlap operator.

Staggered eigenvalue mimicry

We study the infrared part of the spectrum for UV-filtered staggered Dirac operators and compare them to the overlap counterpart. With sufficient filtering and at small enough lattice spacing the staggered spectra manage to ``mimic'' the overlap version. They show a 4-fold near-degeneracy, and a clear separation between would-be zero modes and non-zero modes. This suggests an approximate index theorem for filtered staggered fermions and a correct sensitivity to the topology of QCD. Moreover, it supports square-rooting the staggered determinant to obtain dynamical ensembles with $N_f=2$.

Locality of the square-root method for improved staggered quarks

We study the effects of improvement on the locality of square-rooted staggered Dirac operators in lattice QCD simulations. We find the localization lengths of the improved operators (fat7tad and asqtad) to be very similar to that of the one-link operator studied by Bunk et al. being at least the Compton wavelength of the lightest particle in the theory, even in the continuum limit. We conclude that improvement has no effect. We discuss the implications of this result for the locality of the ${n}^{\mathrm{t}\mathrm{h}}$-rooted fermion determinant used to reduce the number of sea quark flavors, and for possible staggered valence quark formulations.

Comparing the R algorithm and RHMC for staggered fermions

The R algorithm is widely used for simulating two flavours of dynamical staggered fermions. We give a simple proof that the algorithm converges to the desired probability distribution to within O(δπ 2) errors, but show that the relevant expansion parameter is ( δτ/m) 2 , m being the quark mass. The Rational Hybrid Monte Carlo (RHMC) algorithm provides an exact (i.e., has no step size errors) alternative for simulating the square root of the staggered Dirac operator. We propose using it to test the validity of the R algorithm for simulations carried out with δτ ≈ m.

Lattice QCD at finite temperature: Evidence for calorons from the eigenvectors of the Dirac operator

We analyze the eigenvalues and eigenvectors of the staggered Dirac operator in quenched lattice QCD in the vicinity of the deconfinement phase transition using the Lüscher-Weisz gauge action. The spectral and localization properties of the low-lying eigenmodes show characteristic differences between the Z 3 sectors above the critical temperature T c . These findings can be interpreted in terms of calorons.

Zero-modes of the QED Neuberger Dirac operator

We consider 4 d compact lattice QED in the quenched approximation. First, we briefly summarize the spectrum of the staggered Dirac operator and its connection with random matrix theory. Afterwards we present results for the low-lying eigenmodes of the Neuberger overlap-Dirac operator. In the strong coupling phase we find exact zero-modes. Subsequently we discuss possibly related topological excitations of the U(1) lattice gauge theory.