A construction of the Schr\"odinger Functional for M\"obius Domain Wall Fermions

TL;DR

This paper develops a Schr"odinger Functional framework for M"obius domain wall fermions, extending previous methods and addressing symmetry constraints to ensure proper chiral properties at the boundary.

Contribution

It introduces a parity-symmetric parameter set for MDWF with SF boundary conditions, enabling consistent chiral symmetry breaking and boundary behavior.

Findings

Eigenvalue analysis confirms proper boundary conditions.

Ginsparg-Wilson relation violation is controlled at tree-level.

Computational cost is comparable to standard DWF with SF.

Abstract

We construct the Schr\"odinger Functional (SF) setup for the M\"obius domain wall fermions (MDWF). The method is an extension of the method proposed by Takeda for the standard domain wall fermion. In order to fulfill the requirement that the lattice Dirac operator with the SF boundary obeys the L\"uscher's universality argument: the lattice chiral fermion with the SF boundary condition breaks the chiral symmetry at the temporal boundary, we impose the parity symmetry with respect to the fifth-direction on the MDWF operator. This additional symmetry restricts the choice of the parameter of the MDWF so that the optimal parameter from the Zolotarev optimal approximation cannot be applied. We introduce a modified parameter set having the fifth-dimensional parity symmetry. We investigate the MDWF with the SF boundary by observing eigenvalues of the Hermitian operator and the Ginsparg-Wilson relation violation at the tree-level. We compare the computational cost with that of the standard DWF with the SF scheme.