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Register a new userIndex Theorem and Overlap Formalism with Naive and Minimally Doubled Fermions
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We present a theoretical foundation for the Index theorem in naive and minimally doubled lattice fermions by studying the spectral flow of a Hermitean version of Dirac operators. We utilize the point splitting method to implement flavored mass terms, which play an important role in constructing proper Hermitean operators. We show the spectral flow correctly detects the index of the would-be zero modes which is determined by gauge field topology. Using the flavored mass terms, we present new types of overlap fermions from the naive fermion kernels, with a number of flavors that depends on the choice of the mass terms. We succeed to obtain a single-flavor naive overlap fermion which maintains hypercubic symmetry.
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We have performed the first numerical study of minimally doubled fermions of the Karsten-Wilczek class in the quenched approximation. This requires fixing the counterterms, which arise due to hypercubic symmetry breaking induced by the Karsten-Wilczek term. Non-perturbative renormalisation criteria are formulated after a detailed study of the parameter dependence of mesonic observables. Minimisation of the mass anisotropy of the pseudoscalar ground state fixes non-perturbative renormalisation conditions for the counterterm coefficients. These anisotropies are mapped out by probing different euclidean components of the transfer matrix through calculations of the pseudoscalar ground state mass in different directions. The chiral behaviour of the pseudoscalar ground state is studied with the tuned Karsten-Wilczek action for multiple lattice spacings. Light pseudoscalar masses ($ M_{PS} lesssim 250,MeV $) were achieved in the quenched approximation without encountering exceptional configurations. The presence of quenched chiral logarithms is studied under the tentative assumption of Goldstone Boson-like behaviour.
Recently, the interest in local lattice actions for chiral fermions has revived, with the proposition of new local actions in which only the minimal number of doublers appear. The trigger role of graphene having a minimally doubled, chirally invariant, Dirac-like excitation spectrum can not be neglected. The challenge is to construct an action which preserves enough symmetries to be useful in lattice gauge calculations. We present a new approach to obtain local lattice actions for fermions using a reinterpretation of the staggered lattice approach of Kogut and Susskind. This interpretation is based on the similarity with the staggered lattice approach in FDTD simulations of acoustics and electromagnetism. It allows us to construct a local action for chiral fermions which has all discrete symmetries and the minimal number of fermion flavors, but which is non-Hermitian in real space. However, we argue that this will not pose a threat to the usability of the theory.
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 2 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.
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.
HMC histories for light dynamical overlap fermions tend to stay in a fixed topological sector for many trajectories, so that the different sectors are not sampled properly. Therefore the suitable summation of observables, which have been measured in separate sectors, is a major challenge. We explore several techniques for this issue, based on data for the chiral condensate and the (analogue of the) pion mass in the 2-flavour Schwinger model with dynamical overlap-hypercube fermions.
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