Lattice Dirac Operator Research Articles

Equivalence of lattice operators and graph matrices

Abstract We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix called an “antisymmetrized adjacency matrix”, specifically tailored for cycle digraphs (T1 lattice) and simple directed paths (B1 lattice). The nontrivial relationship between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the antisymmetrized adjacency matrix, along with its extensions to higher dimensions, is equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a Cartesian product of cycle digraphs (T1 lattice) and simple directed paths (B1 lattice).

Optimizing shift selection in multilevel Monte Carlo for disconnected diagrams in lattice QCD

The calculation of disconnected diagram contributions to physical signals is a computationally expensive task in Lattice QCD. To extract the physical signal, the trace of the inverse Lattice Dirac operator, a large sparse matrix, must be stochastically estimated. Because the variance of the stochastic estimator is typically large, variance reduction techniques must be employed. Multilevel Monte Carlo (MLMC) methods reduce the variance of the trace estimator by utilizing a telescoping sequence of estimators. Frequency Splitting is one such method that uses a sequence of inverses of shifted operators to estimate the trace of the inverse lattice Dirac operator, however there is no a priori way to select the shifts that minimize the cost of the multilevel trace estimation. In this article, we present a sampling and interpolation scheme that is able to predict the variances associated with Frequency Splitting under displacements of the underlying space time lattice. The interpolation scheme is able to predict the variances to high accuracy and therefore chooses shifts that correspond to an approximate minimum of the cost for the trace estimation. We show that Frequency Splitting with the chosen shifts displays significant speedups over multigrid deflation, and that these shifts can be used for multiple configurations within the same ensemble with no penalty to performance.

Lattice fermions as spectral graphs

We study lattice fermions from the viewpoint of spectral graph theory (SGT). We find that a fermion defined on a certain lattice is identified as a spectral graph. SGT helps us investigate the number of zero eigenvalues of lattice Dirac operators even on the non-torus and non-regular lattice, leading to understanding of the number of fermion species (doublers) on lattices with arbitrary topologies. The procedure of application of SGT to lattice fermions is summarized as follows: (1) One investigates a spectral graph corresponding to a lattice fermion. (2) One obtains a matrix corresponding to the graph. (3) One finds zero eigenvalues of the matrix by use of the discrete Fourier transformation (DFT). (4) By taking an infinite-volume and continuum limits, one finds the number of species. We apply this procedure to the known lattice fermion formulations including Naive fermions, Wilson fermions and Domain-wall fermions, and reproduce the known fact on the number of species. We also apply it to the lattice fermion on the discretized four­dimensional hyperball and discuss the number of fermion species on the bulk. In the end of the paper, we discuss the application of the analysis to lattice fermions on generic lattices with arbitrary topologies, which could lead to constructing a new theorem regarding the number of species.

Role of the axial U(1) anomaly in the chiral susceptibility of QCD at high temperature

Abstract The chiral susceptibility, or the first derivative of the chiral condensate with respect to the quark mass, is often used as a probe for the QCD phase transition since the chiral condensate is an order parameter of SU(2)L × SU(2)R symmetry breaking. However, the chiral condensate also breaks the axial U(1) symmetry, which is usually not studied as it is already broken by the anomaly and apparently has little impact on the transition. We investigate the susceptibilities in the scalar and pseudoscalar channels in order to quantify how much the axial U(1) breaking contributes to the chiral phase transition. Employing a chirally symmetric lattice Dirac operator and its eigenmode decomposition, we separate the axial U(1) breaking effects from others. Our result in two-flavor QCD indicates that both of the connected and disconnected chiral susceptibilities are dominated by axial U(1) breaking at temperatures T ≳ 190 MeV after the quadratically divergent constant is subtracted.

Wigner–Weyl formalism for the lattice models

We discuss application of Wigner–Weyl formalism to the lattice models of condensed matter physics and relativistic quantum field theory. For the noninteracting models our technique relates Wigner transformation of the fermionic Green’s function with the Weyl symbol [Formula: see text] of lattice Dirac operator. We take as an example of the model defined on rectangular lattice the model of Wilson fermions. It represents the regularization of relativistic quantum field theory and, in addition, describes qualitatively certain topological materials in condensed matter physics. In this model we derive expression for [Formula: see text] in the presence of the most general case of external [Formula: see text] gauge field. Next, we solve the Groenewold equation to all orders in powers of the derivatives of [Formula: see text].

Curious behavior of three-dimensional lattice Dirac operators coupled to a monopole background

We investigate numerically the effect of regulating fermions in the presence of singular background fields in three dimensions. For this, we couple free lattice fermions to a background compact U(1) gauge field consisting of a monopole-anti-monopole pair of magnetic charge $\pm Q$ separated by a distance $s$ in a periodic $L^3$ lattice, and study the low-lying eigenvalues of different lattice Dirac operators under a continuum limit defined by taking $L\to\infty$ at fixed $s/L$. As the background gauge field is parity even, we look for a two-fold degeneracy of the Dirac spectrum that is expected of a continuum-like Dirac operator. The naive-Dirac operator exhibits such a parity-doubling, but breaks the degeneracy of the fermion-doubler modes for the $Q$ lowest eigenvalues in the continuum limit. The Wilson-Dirac operator lifts the fermion-doublers but breaks the parity-doubling in the $Q$ lowest modes even in the continuum limit. The overlap-Dirac operator shows parity-doubling of all the modes even at finite $L$ that is devoid of fermion-doubling, and singles out as a properly regulated continuum Dirac operator in the presence of singular gauge field configurations albeit with a peculiar algorithmic issue.

New Nf=2 pseudofermion action for Monte Carlo simulations of a lattice field theory with domain-wall fermions

We construct a novel $ N_f = 2 $ pseudofermion action for Monte-Carlo simulation of lattice gauge theory with domain-wall fermions (DWF), of which the effective four-dimensional lattice Dirac operator is equal to the overlap-Dirac operator with the argument of the sign function equal to $ H = c \gamma_5 D_w (1 + d D_w)^{-1} $, where $ c $ and $ d $ are parameters, and $D_w$ is the standard Wilson-Dirac operator plus a negative parameter $-m_0 \; (0 < m_0 < 2)$. This new action is particularly useful for the challenging simulations of lattice gauge theories with large $N_f = 2n $ DWF, on the large lattices, and in the strong-coupling regime.

Chiral-spin symmetry emergence in baryons and eigenmodes of the Dirac operator

Truncating the low-lying modes of the lattice Dirac operator results in an emergence of the chiral-spin symmetry $SU(2)_{CS}$ and its flavor extension $SU(2N_F)$ in hadrons. These are symmetries of the quark - chromo-electric interaction and include chiral symmetries as subgroups. Hence the quark - chromo-magnetic interaction, which breaks both symmetries, is located at least predominantly in the near - zero modes. Using as a tool the expansion of propagators into eigenmodes of the Dirac operator we here analytically study effects of a gap in the eigenmode spectrum on baryon correlators. We find that both $U(1)_A$ and $SU(2)_L \times SU(2)_R$ emerge automatically if there is a gap around zero. Emergence of larger $SU(2)_{CS}$ and $SU(4)$ symmetries requires in addition a microscopical dynamical input about the higher-lying modes and their symmetry structure.

Shift of symmetries of naive and staggered fermions in QCD-like lattice theories

We study the global symmetries of naive lattices Dirac operators in QCD-like theories in any dimension larger than two. In particular we investigate how the chosen number of lattice sites in each direction affects the global symmetries of the Dirac operator. These symmetries are important since they do not only determine the infra-red spectrum of the Dirac operator but also the symmetry breaking pattern and, thus, the lightest pseudo-scalar mesons. We perform the symmetry analysis and discuss the possible zero modes and the degree of degeneracy of the lattice Dirac operators. Moreover we explicitly identify a "reduced" lattice Dirac operator which is the naive Dirac operator apart from the degeneracy. We verify our predictions by comparing Monte Carlo simulations of QCD-like theories in the strong coupling limit with the corresponding random matrix theories.

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.

Polyakov loop fluctuations in the Dirac eigenmode expansion

We investigate correlations of the Polyakov loop fluctuations with eigenmodes of the lattice Dirac operator. Their analytic relations are derived on the temporally odd-number size lattice with the normal non-twisted periodic boundary condition for the link-variables. We find that the low-lying Dirac modes yield negligible contributions to the Polyakov loop fluctuations. This property is confirmed to be valid in confined and deconfined phase by numerical simulations in SU(3) quenched QCD. These results indicate that there is no direct, one-to-one correspondence between confinement and chiral symmetry breaking in QCD in the context of different properties of the Polyakov loop fluctuation ratios.

The eigSUMR inverter for overlap fermions

We discuss the usage and applicability of deflation methods for the overlap lattice Dirac operator, focusing on calculating the eigenvalues using a method similar to the eigCG algorithm used for other Dirac operators. The overlap operator, which contains several theoretical advantages over other formulations of lattice Quantum Chromodynamics, is more computationally expensive because it requires the computation of the matrix sign function. The principal change made compared to deflation methods for other formulations of lattice QCD is that it is necessary for best performance to tune the accuracy of the matrix sign function as the computation proceeds. We present two possible relaxation strategies, one which provides a rigorous bound for the eigenvalues but seems to be too conservative in practice, and a second which is less conservative but, while its stability is not guaranteed, seems to work well in practice.We adapt the original eigCG algorithm for two of the preferred inversion algorithms for overlap fermions, GMRESR(relCG) and GMRESR(relSUMR). Before deflation, the rate of convergence of these routines in terms of iterations is similar, but, since the Shifted Unitary Minimal Residual (SUMR) algorithm only requires one call to the matrix sign function compared to the two calls required for Conjugate Gradient (CG), SUMR is usually preferred for single inversions of the Dirac operator. We construct bounds for the required accuracy of the matrix sign function during the eigenvalue calculation. For the SUMR algorithm, we use a variant of the Galerkin projection to perform the deflation; while for the CG algorithm, we are able to use a considerably superior spectral pre-conditioner. The superior performance of the spectral pre-conditioner, and its need for less accurate eigenvalues, almost erodes SUMR’s advantage over CG as an inversion algorithm.We see factor of three gains for the inversion algorithm from the deflation on our small test lattices; we expect larger gains over the undeflated algorithms in realistic simulations on larger lattices and with smaller masses. There is, however, a significant cost in the eigenvalue calculation because we cannot relax the accuracy of the matrix sign function as aggressively when calculating the eigenvalues as we do while performing the inversions. This set-up cost is, however, more than compensated for the gain in the deflation if enough right hand sides are required.

Definition of the covariant lattice Dirac operator

In the continuum the definitions of the covariant Dirac operator and of the gauge covariant derivative operator are tightly intertwined. We point out that the naive discretization of the gauge covariant derivative operator is related to the existence of local unitary operators which allow the definition of a natural lattice gauge covariant derivative. The associated lattice Dirac operator has all the properties of the classical continuum Dirac operator, in particular antihermiticy and chiral invariance in the massless limit, but is of course non-local in accordance to the Nielsen-Ninomiya theorem. We show that this lattice Dirac operator coincides in the limit of an infinite lattice volume with the naive gauge covariant generalization of the SLAC derivative, but contains non-trivial boundary terms for finite-size lattices. Its numerical complexity compares pretty well on finite lattices with smeared lattice Dirac operators.

Colorful SU(2) center vortices in the continuum and on the lattice

The spherical vortex as introduced in [G. Jordan et al., Phys. Rev. D 77, 014515 (2008)] is generalized. A continuum form of the spherical vortex is derived and investigated in detail. The discrepancy between the gluonic lattice topological charge and the index of the lattice Dirac operator described in previous papers is identified as a discretization effect. The importance of the investigations for Monte Carlo configurations is discussed.

Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature

We reconsider constraints on the eigenvalue density of the Dirac operator in the chiral symmetric phase of 2 flavor QCD at finite temperature. To avoid possible ultra-violet(UV) divergences, we work on a lattice, employing the overlap Dirac operator, which ensures the exact "chiral" symmetry at finite lattice spacings. Studying multi-point correlation functions in various channels and taking their thermodynamical limit (and then taking the chiral limit), we obtain stronger constraints than those found in the previous studies: both the eigenvalue density at the origin and its first and second derivatives vanish in the chiral limit of 2 flavor QCD. In addition we show that the axial U(1) anomaly becomes invisible in susceptibilities of scalar and pseudo scalar mesons, suggesting that the 2nd order chiral phase transition with the O(4) scaling is not realized in 2 flavor QCD. Possible lattice artifacts when non-chiral lattice Dirac operator is employed are briefly discussed.

Lattice QCD with open boundary conditions and twisted-mass reweighting

Lattice QCD simulations at small lattice spacings and quark masses close to their physical values are technically challenging. In particular, the simulations can get trapped in the topological charge sectors of field space or may run into instabilities triggered by accidental near-zero modes of the lattice Dirac operator. As already noted in ref. [1], the first problem is bypassed if open boundary conditions are imposed in the time direction, while the second can potentially be overcome through twisted-mass determinant reweighting [2]. In this paper, we show that twisted-mass reweighting works out as expected in QCD with open boundary conditions and 2+1 flavours of O(a) improved Wilson quarks. Further algorithmic improvements are tested as well and a few physical quantities are computed for illustration.

The chiral magnetic effect and chiral symmetry breaking in SU(3) quenched lattice gauge theory

We study some properties of the non-Abelian vacuum induced by strong external magnetic field. We perform calculations in the quenched SU(3) lattice gauge theory with tadpole-improved Luscher-Weisz action and chirally invariant lattice Dirac operator. The following results are obtained: The chiral symmetry breaking is enhanced by the magnetic field. The chiral condensate depends on the strength of the applied field as a power function with exponent n = 1.6 +/- 0.2. There is a paramagnetic polarization of the vacuum. The corresponding susceptibility and other magnetic properties are calculated and compared with the theoretical estimations. There are non-zero local fluctuations of the chirality and electromagnetic current, which grow with the magnetic field strength. These fluctuations can be a manifestation of the Chiral Magnetic Effect (CME).

Determining low-energy constants in partially quenched Wilson chiral perturbation theory

In the low energy effective theory describing the partially quenched extension of two light Wilson fermions, three low energy constants (LECs) appear in terms proportional to a^2 (a being the lattice spacing). We propose methods to separately calculate these LECs, typically called W'_6, W'_7 and W'_8. While only one linear combination of these constants enters into physical quantities, different combinations enter into the description of the spectral density and eigenvalue distributions of the lattice Dirac operator and its Hermitian counterpart. Thus it is useful to be able to determine the LECs separately. Our methods require studying certain correlation functions for either two or three pion scattering, which are accessible only in the partially quenched extension of the theory.

New one-flavor hybrid Monte Carlo simulation method for lattice fermions with [formula omitted] hermiticity

We propose a new method for Hybrid Monte Carlo (HMC) simulations with odd numbers of dynamical fermions on the lattice. It employs a different approach from polynomial or rational HMC. In this method, γ5 hermiticity of the lattice Dirac operators is crucial and it can be applied to Wilson, domain-wall, and overlap fermions. We compare HMC simulations with two degenerate flavors and (1+1) degenerate flavors using optimal domain-wall fermions. The ratio of the efficiency, (number of accepted trajectories)/(simulation time), is about 3:2. The relation between pseudofermion action of chirally symmetric lattice fermions in four-dimensional (overlap) and five-dimensional (domain-wall) representation are also analyzed.

Brillouin improvement for Wilson fermions

We present a parameter-free Wilson-type lattice Dirac operator with an 81-point stencil for the covariant derivative and the Laplacian which attempts to minimize the breaking of rotational symmetry near the boundary of the Brillouin zone. The usefulness of this "Brillouin operator" in practical applications is explored by studying the scaling of pseudoscalar decay constants in quenched QCD, with rather good results in the physical charm region. We also investigate the suitability of this operator as a kernel to the overlap procedure. Here, the resulting overlap operator is found to be cheaper to construct and significantly better localized than the variety with the standard Wilson kernel.