Domain wall fermion

In lattice field theory, domain wall (DW) fermions are a fermion discretization avoiding the fermion doubling problem.[1] They are a realisation of Ginsparg–Wilson fermions in the infinite separation limit ${\displaystyle L_{s}\rightarrow \infty }$ where they become equivalent to overlap fermions.[2] DW fermions have undergone numerous improvements since Kaplan's original formulation[1] such as the reinterpretation by Shamir and the generalisation to Möbius DW fermions by Brower, Neff and Orginos.[3][4]

The original ${\displaystyle d}$-dimensional Euclidean spacetime is lifted into ${\displaystyle d+1}$ dimensions. The additional dimension of length ${\displaystyle L_{s}}$ has open boundary conditions and the so-called domain walls form its boundaries. The physics is now found to ″live″ on the domain walls and the doublers are located on opposite walls, that is at ${\displaystyle L_{s}\rightarrow \infty }$ they completely decouple from the system.

Kaplan's (and equivalently Shamir's) DW Dirac operator is defined by two addends

${\displaystyle D_{\text{DW}}(x,s;y,r)=D(x;y)\delta _{sr}+\delta _{xy}D_{d+1}(s;r)\,}$

with

${\displaystyle D_{d+1}(s;r)=\delta _{sr}-(1-\delta _{s,L_{s}-1})P_{-}\delta _{s+1,r}-(1-\delta _{s0})P_{+}\delta _{s-1,r}+m\left(P_{-}\delta _{s,L_{s}-1}\delta _{0r}+P_{+}\delta _{s0}\delta _{L_{s}-1,r}\right)\,}$

where ${\displaystyle P_{\pm }=(\mathbf {1} \pm \gamma _{5})/2}$ is the chiral projection operator and ${\displaystyle D}$ is the canonical Dirac operator in ${\displaystyle d}$ dimensions. ${\displaystyle x}$ and ${\displaystyle y}$ are (multi-)indices in the physical space whereas ${\displaystyle s}$ and ${\displaystyle r}$ denote the position in the additional dimension.[5]

DW fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry (asymptotically obeying the Ginsparg–Wilson equation).