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).

References

1. Kaplan, David B. (1992). "A method for simulating chiral fermions on the lattice". Physics Letters B. 288 (3–4): 342–347. arXiv:hep-lat/9206013. Bibcode:1992PhLB..288..342K. doi:10.1016/0370-2693(92)91112-m. ISSN 0370-2693. S2CID 14161004.
2. Neuberger, Herbert (1998). "Vectorlike gauge theories with almost massless fermions on the lattice". Phys. Rev. D. 57 (9). American Physical Society: 5417–5433. arXiv:hep-lat/9710089. Bibcode:1998PhRvD..57.5417N. doi:10.1103/PhysRevD.57.5417. S2CID 17476701.
3. Yigal Shamir (1993). "Chiral fermions from lattice boundaries". Nuclear Physics B. 406 (1): 90–106. arXiv:hep-lat/9303005. Bibcode:1993NuPhB.406...90S. doi:10.1016/0550-3213(93)90162-I. ISSN 0550-3213. S2CID 16187316.
4. R.C. Brower and H. Neff and K. Orginos (2006). "Möbius Fermions". Nuclear Physics B - Proceedings Supplements. 153 (1): 191–198. arXiv:hep-lat/0511031. Bibcode:2006NuPhS.153..191B. doi:10.1016/j.nuclphysbps.2006.01.047. ISSN 0920-5632. S2CID 118926750.
5. Gattringer, C.; Lang, C.B. (2009). "10 More about lattice fermions". Quantum Chromodynamics on the Lattice: An Introductory Presentation. Lecture Notes in Physics 788. Springer. pp. 249–253. doi:10.1007/978-3-642-01850-3. ISBN 978-3642018497.