Gauge fixing, families index theory, and topological features of the space of lattice gauge fields

The families index theory for the overlap lattice Dirac operator is applied to derive topological features of the space of SU(N) lattice gauge fields on the 4-torus: the topological sectors, specified by the fermionic topological charge, are shown to contain noncontractible even-dimensional spheres when N3, and noncontractible circles in the N=2 case. We describe how certain obstructions to the existence of gauge fixings without the Gribov problem in the continuum setting correspond on the lattice to obstructions to the contractibility of these spheres and circles. We also point out a canonical connection on the space of lattice gauge fields with monopole-like singularities associated with the spheres.     

Investigating the topological structure of quenched lattice QCD with overlap fermions using a multi-probing approximation

The topological charge density and topological susceptibility are determined by a multi-probing approximation using overlap fermions in quenched SU(3) gauge theory. Then we investigate the topological structure of the quenched QCD vacuum, and compare it with results from the all-scale topological density. The results are consistent.Random permuted topological charge density is used to check whether these structures represent underlying ordered properties. The pseudoscalar glueball mass is extracted from the two-point correlation function of the topological charge density. We study 3 ensembles of different lattice spacing a with the same lattice volume 16~3×32. The results are compatible with the results of all-scale topological charge density, and the topological structures revealed by multi-probing are much closer to all-scale topological charge density than those from eigenmode expansion.     

Evaluating the topological charge density with the symmetricmulti-probing method

We evaluate the topological charge density of SU(3) gauge fields on a lattice by calculating the trace of the overlap Dirac matrix employing the symmetric multi-probing(SMP) method in 3 modes. Since the topological charge Q for a given lattice configuration must be an integer number, it is easy to estimate the systematic error(the deviation of Q to the nearest integer). The results demonstrate a high efficiency and accuracy in calculating the trace of the inverse of a large sparse matrix with locality by using the SMP sources when compared to using point sources.We also show the correlation between the errors and probing scheme parameter r_(min), as well as lattice volume N_L and lattice spacing a. It is found that the computational time for calculating the trace by employing the SMP sources is less dependent on N_L than by using point sources. Therefore, the SMP method is very suitable for calculations on large lattices.     

Geometrical Methods in the Study of Topological Effects on the Lattice

We review geometrical definitions of the topological charge for lattice field theories, in particular lattice gauge theories. Some recent Monte Carlo calculations of the topological susceptibility based upon these definitions are described. They are compared with alternative approaches.     

Topology of dynamical lattice configurations including results from dynamical overlap fermions

We investigate how the topological charge density in lattice QCD simulations is affected by violations of chiral symmetry caused by the fermion action. To this end we compare lattice configurations generated with a number of different actions including first configurations generated with exact dynamical overlap quarks. We visualize the topological profiles after mild smearing. In the topological charge correlator we measure the size of the positive core, which is known to shrink to zero extension in the continuum limit. To leading order we find the core size to scale linearly with the lattice spacing with the same coefficient for all actions, even including quenched simulations. In the subleading term the different actions vary over a range of about 10%. Our findings suggest that non-chiral lattice actions at current lattice spacings do not differ much for observables related to topology, both among themselves and compared to overlap fermions.     

Wilson fermions and the topological charge on the lattice

We investigate the realization on the lattice of the relation between the chiral properties of the fermions and the topological properties of the gauge field. A lattice definition of the topological charge density via the U(1) Adler-Bardeen anomaly is analysed with the help of given configurations of nontrivial topology and for 2-dimensional quenched QED.     

Dependence of overlap topological charge density on Wilson mass parameter

In this paper,we analyze the dependence of the topological charge density from the overlap operator on the Wilson mass parameter in the overlap kernel by the symmetric multi-probing source(SMP) method.We observe that non-trivial topological objects are removed as the Wilson mass is increased.A comparison of topological charge density calculated by the SMP method using the fermionic definition with that of the gluonic definition by the Wilson flow method is shown.A matching procedure for these two methods is used.We find that there is a best match for topological charge density between the gluonic definition with varied Wilson flow time and the fermionic definition with varied Wilson mass.By using the matching procedure,the proper flow time of Wilson flow in the calculation of topological charge density can be estimated.As the lattice spacing a decreases,the proper flow time also decreases,as expected.     

Topology and lattice gauge fields

A general discussion of the topology of continuum gauge fields and the problems involved in defining and computing the topology of a lattice gauge field configuration is given. Two definitions of the topological charge for 4-dimensional SU(n) lattice gauge theory are presented. The first of these is geometrically the most straightforward, the second the most useful for efficient numerical calculations.     

Anomalies, gauge field topology, and the lattice

Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I suggest that the fourth power of the naive Dirac operator can provide a natural method to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cooling with respect to the Wilson gauge action, instanton like structures do emerge. As cooling proceeds, these objects tend shrink and finally “fall through the lattice.” Modifying the action can block the shrinking at the expense of a loss of reflection positivity. The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility.     

Index theorem and universality properties of the low-lying eigenvalues of improved staggered quarks

We study various improved staggered quark Dirac operators on quenched gluon backgrounds in lattice QCD generated using a Symanzik-improved gluon action. We find a clear separation of the spectrum into would-be zero modes and others. The number of would-be zero modes depends on the topological charge as expected from the index theorem, and their chirality expectation value is large ( approximately 0.7). The remaining modes have low chirality and show clear signs of clustering into quartets and approaching the random matrix theory predictions for all topological charge sectors. We conclude that improvement of the fermionic and gauge actions moves the staggered quarks closer to the continuum limit where they respond correctly to QCD topology.     

Irrational charge from topological order

Topological or deconfined phases of matter exhibit emergent gauge fields and quasiparticles that carry a corresponding gauge charge. In systems with an intrinsic conserved U(1) charge, such as all electronic systems where the Coulombic charge plays this role, these quasiparticles are also characterized by their intrinsic charge. We show that one can take advantage of the topological order fairly generally to produce periodic Hamiltonians which endow the quasiparticles with continuously variable, generically irrational, intrinsic charges. Examples include various topologically ordered lattice models, the three-dimensional resonating valence bond liquid on bipartite lattices as well as water and spin ice. By contrast, the gauge charges of the quasiparticles retain their quantized values.     

Monopoles, instantons and chiral condensate in lattice QCD

We analyze the interplay of topological objects in four-dimensional QCD at finite temperature on the lattice. The distributions of color magnetic monopoles in the maximum abelian gauge are computed around instantons. Studies are performed in both pure and full QCD and in both the confinement and deconfinement phase. We find an enhanced probability for monopoles inside the core of an instanton on gauge field average. This is independent of the topological charge definition used. For specific gauge field configurations we visualize the situation graphically. Moreover the correlation of monopole loops and instantons with the chiral condensate is investigated. Strong evidence is found that clusters of the quark condensate and topological objects coexist locally on individual configurations.     

Analytic study of θ vacua on the lattice

A new definition of the topological charge density for four-dimensional lattice gauge theory is given. Using a systematic expansion we find a cusp in the vacuum energy at θ = π signaling the spontaneous breaking of CP there. Unlike its two-dimensional analogue (QED₂), QCD confines at θ = π. As a by-product an expression for the topological mass term for (2+1)-dimensional lattice gauge theory is obtained.     

Quantitative comparison of filtering methods in lattice QCD

We systematically compare filtering methods used to extract topological excitations (like instantons, calorons, monopoles and vortices) from lattice gauge configurations, namely APE-smearing and spectral decompositions based on lattice Dirac and Laplace operators. Each of these techniques introduces ambiguities, which can invalidate the interpretation of the results. We show, however, that all these methods, when handled with care, reveal very similar topological structures. Hence, these common structures are free of ambiguities and faithfully represent infrared degrees of freedom in the QCD vacuum. As an application we discuss an interesting power law for the clusters of filtered topological charge.     

Axial Anomaly and Topological Charge in Lattice Gauge Theory with Overlap Dirac Operator

An explicit, detailed evaluation of the classical continuum limit of the axial anomaly and index density of the overlap Dirac operator is carried out in the infinite volume setting and in a certain finite volume setting where the continuum limit involves an infinite volume limit. Our approach is based on a novel power series expansion of the overlap Dirac operator. The correct continuum expression is reproduced when the parameter m₀ is in the physical region 0<m₀<2. This is established for a broad range of continuum gauge fields. An analogous result for the fermionic topological charge, given by the index of the overlap Dirac operator, is then established for a class of topologically nontrivial fields in the aforementioned finite volume setting. Problematic issues concerning the index in the infinite volume setting are also discussed.     

Axial anomaly in lattice abelian gauge theory in arbitrary dimensions

The axial anomaly of lattice abelian gauge theory on a hyper-cubic regular lattice in arbitrary even dimensions is investigated by applying the method of exterior differential calculus. The topological invariance, gauge invariance and locality of the axial anomaly determine the explicit form of the topological part. The anomaly is obtained up to a multiplicative constant for finite lattice spacing and can be interpreted as the Chern character of the abelian lattice gauge theory.     

Lattice θ vacua

We study some aspects of θ vacua by Monte-Carlo simulations of the SU(2) lattice gauge theory using a definition proposed recently for the total lattice topological charge. We find no phase transition up to θ = 0.8π. Beyond this point, limited statistical accuracy prevents a definite conclusion. Our results are in surprisingly good agreement with the dilute gas picture.     

Topological charge of (lattice) gauge fields

Using recently derived explicit formulae for the 2- and 3-cochains in SU(2) gauge theory, we are able to integrate the Chern-Simons density analytically. We arrive — in SU(2) — at a local algebraic expression for the topological charge, which is the sum of local winding numbers associated with the corners (lattice points) of the cells covering the manifold plus contributions from possible isolated gauge singularities which manifest themselves as “vortices” in the 1-, 2- or 3-cochains. Among others we consider hypercubic geometry — i.e. covering the manifold by hypercubes — which is of particular interest to lattice Monte Carlo applications. Finally, we extend our results to SU(3) gauge theory.