Abstract
The equivalence of domain wall and overlap fermion formulations is demonstrated for lattice gauge theories in 2+1 spacetime dimensions with parity-invariant mass terms. Even though the domain wall approach distinguishes propagation along a third direction with projectors ${1\over2}(1\pm\gamma_3)$, the truncated overlap operator obtained for finite wall separation $L_s$ is invariant under interchange of $\gamma_3$ and $\gamma_5$. In the limit $L_s\to\infty$ the resulting Ginsparg-Wilson relations recover the expected U($2N_f$) global symmetry up to O($a$) corrections. Finally it is shown that finite-$L_s$ corrections to bilinear condensates associated with dynamical mass generation are characterised by whether even powers of the symmetry-breaking mass are present; such terms are absent for antihermitian bilinears such as $i\bar\psi\gamma_3\psi$, markedly improving the approach to the large-$L_s$ limit.
Highlights
Relativistic fermions moving in 2 spatial dimensions are the focus of much attention, in part due to the stability of Dirac points in graphene and surface states of topological band insulators when the underlying Hamiltonian is symmetric under time reversal and spatial inversion
Hands / Physics Letters B 754 (2016) 264–269 the direction separating the walls is governed by γ3, can maintain the equivalence between γ3 and γ5 rotations for finite Ls; the reason for O (a) violations of U(2) symmetries even in the overlap limit Ls → ∞; and a better understanding of why finite-Ls corrections are minimised by choosing i ψγ3ψ, rather than ψψ, as the bilinear condensate to focus on
First we review the passage from the domain wall formulation of lattice fermions to the overlap operator
Summary
Relativistic fermions moving in 2 spatial dimensions are the focus of much attention, in part due to the stability of Dirac points in graphene and surface states of topological band insulators when the underlying Hamiltonian is symmetric under time reversal and spatial inversion (see, e.g. [1]). Hands / Physics Letters B 754 (2016) 264–269 the direction separating the walls is governed by γ3, can maintain the equivalence between γ3 and γ5 rotations for finite Ls; the reason for O (a) violations of U(2) symmetries even in the overlap limit Ls → ∞; and a better understanding of why finite-Ls corrections are minimised by choosing i ψγ3ψ , rather than ψψ , as the bilinear condensate to focus on In this brief technical Letter I outline how the overlap operator is recovered in the Ls → ∞ limit of the domain wall formulation using a familiar sequence of matrix algebra operations. As well as providing a firm conceptual foundation for domain wall fermions and their symmetry properties in 2 + 1d, the proof sheds light on each of these outstanding issues
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