Domain wall fermion information

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 $L_{s}\rightarrow \infty$ 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 $d$ -dimensional Euclidean spacetime is lifted into $d+1$ dimensions. The additional dimension of length $L_{s}$ 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 $L_{s}\rightarrow \infty$ they completely decouple from the system.

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

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

with

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

^ ^a ^b 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.
^ 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.
^ 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.
^ 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.
^ 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.

[domain_walls_1992-1] 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.

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