Nigel Cundy

Topological tunnelling with dynamical overlap fermions

Tunnelling between different topological sectors with dynamical chiral fermions is difficult because of a poor mass scaling of the pseudo-fermion estimate of the determinant. For small fermion masses it is virtually impossible using standard methods. However, by projecting out the small Wilson eigenvectors from the overlap operator, and treating the correction determinant exactly, we can significantly increase the rate of topological sector tunnelling and reduce substantially the auto-correlation time. We present and compare a number of different approaches, and advocate a method which allows topological tunnelling even at low mass with little addition to the computational cost.

Topology of dynamical lattice configurations including results from dynamical overlap fermions

Abstract 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.

Low-lying Wilson Dirac operator eigenvector mixing in dynamical overlap hybrid Monte Carlo

Current dynamical overlap fermion hybrid Monte Carlo simulations encounter large fermionic forces when there is mixing between eigenvectors of the kernel operator with near zero-eigenvalues. This leads to low acceptance rates when there is a large density of near zero eigenvalues. I present a method where these large forces are eliminated and the large action jumps seen when two eigenvalues approach zero are significantly reduced. This significantly increases the stability of the algorithm, and allows the use of larger integration time steps.

A renormalisation group derivation of the overlap formulation

Abstract Starting from the continuum Dirac operator, I construct a renormalisation group blocking which transforms the continuum action into a lattice action, and I specifically consider the Wilson and overlap formalisms. For Wilson fermions the inverse blocking is non-local and thus invalid. However, I proceed to demonstrate that it is possible to construct a valid, local, blocking which, though dependent on the lattice spacing, generates the lattice overlap fermion action from the continuum action. Using this renormalisation group blocking for overlap fermions, I re-derive the Ginsparg–Wilson equations and the lattice chiral symmetry, and show that the standard Ginsparg–Wilson relation is not the most general way of expressing chiral symmetry on the lattice, nor, for overlap fermions, the most natural. I suggest how this reformulation of the Ginsparg–Wilson relation combined with the renormalisation group formulation of overlap fermions could allow the construction of a CP -invariant lattice chiral gauge theory.

New solutions to the Ginsparg–Wilson equation

Abstract The overlap operator is just the simplest of a class of Dirac operators with an exact chiral symmetry. I demonstrate how a general class of chiral Dirac operators can be constructed, show that they have no fermion doublers and that they are all exponentially local, and test my conclusions numerically for a few examples. However, since these operators are more expensive than the overlap operator, it is unlikely that they will be useful in practical simulations.

A Non-perturbative operator product expansion

Nucleon structure functions can be observed in Deep Inelastic Scattering experiments, but it is an outstanding challenge to confront them with fully non-perturbative QCD results. For this purpose we investigate the product of electromagnetic currents (with large photon momenta) between quark states (of low momenta). By means of an Operator Product Expansion the structure function can be decomposed into matrix elements of local operators, and Wilson coefficients. For consistency both have to be computed non-perturbatively. Here we present precision results for a set of Wilson coefficients. They are evaluated from propagators for numerous quark momenta on the lattice, where the use of chiral fermions suppresses undesired operator mixing. This over-determines the Wilson coefficients, but reliable results can be extracted by means of a Singular Value Decomposition.

The Operator Product Expansion on the Lattice

We investigate the Operator Product Expansion (OPE) on the lattice by directly measuring the product (where J is the vector current) and comparing it with the expectation values of bilinear operators. This will determine the Wilson coefficients in the OPE from lattice data, and so give an alternative to the conventional methods of renormalising lattice structure function calculations. It could also give us access to higher twist quantities such as the longitudinal structure function F_L = F_2 - 2 x F_1. We use overlap fermions because of their improved chiral properties, which reduces the number of possible operator mixing coefficients.

Confinement From The Gauge Invariant Abelian Decomposition

A common approach while considering confinement is to study the dominance of an Abelian subgroup of the SU(3) gauge Links. A good way to find the Abelian component of the field is through the Cho-Guan-De gauge invariant Abelian Decomposition, which uses a carefully chosen direction vector

A lattice Dirac operator for QCD with light dynamical quarks

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Hadron structure in terms of OPE with non-perturbative Wilson coefficients

to split the gauge field into an Abelian restricted field and a remnant coloured field. The restricted field can be further subdivided into topological and non-topological terms. We show that there is a choice of