Английская Википедия: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 <math>L_s\rightarrow\infty</math> 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 <math>d</math>-dimensional Euclidean spacetime is lifted into <math>d+1</math> dimensions. The additional dimension of length <math>L_s</math> 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 <math>L_s\rightarrow\infty</math> they completely decouple from the system.
Kaplan's (and equivalently Shamir's) DW Dirac operator is defined by two addends
- <math>
D_\text{DW}(x,s;y,r) = D(x;y)\delta_{sr} + \delta_{xy}D_{d+1}(s;r)\, </math> with
- <math>
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)\, </math> where <math>P_\pm=(\mathbf1\pm\gamma_5)/2</math> is the chiral projection operator and <math>D</math> is the canonical Dirac operator in <math>d</math> dimensions. <math>x</math> and <math>y</math> are (multi-)indices in the physical space whereas <math>s</math> and <math>r</math> 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).
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