Domain wall fermion explained
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
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
-dimensional
Euclidean spacetime is lifted into
dimensions. The additional dimension of length
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
they completely decouple from the system.
Kaplan's (and equivalently Shamir's) DW Dirac operator is defined by two addends
DDW(x,s;y,r)=D(x;y)\deltasr+\deltaxyDd+1(s;r)
with
Dd+1(s;r)=\deltasr-
)P-\deltas+1,r-(1-\deltas0)P+\deltas-1,r+m\left(P-\delta
\delta0r+P+\deltas0
\right)
where
is the
chiral projection operator and
is the canonical Dirac operator in
dimensions.
and
are (multi-)
indices in the physical space whereas
and
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).
Notes and References
- A method for simulating chiral fermions on the lattice . 288 . 0370-2693 . 10.1016/0370-2693(92)91112-m . 3–4 . Physics Letters B . Kaplan, David B. . 1992 . 342–347 . hep-lat/9206013 . 1992PhLB..288..342K . 14161004 .
- Vectorlike gauge theories with almost massless fermions on the lattice . Neuberger, Herbert . Phys. Rev. D . 57 . 9 . 5417–5433 . 1998 . American Physical Society . 10.1103/PhysRevD.57.5417 . hep-lat/9710089 . 1998PhRvD..57.5417N . 17476701 .
- Chiral fermions from lattice boundaries . Nuclear Physics B . 406 . 1 . 90–106 . 1993 . 0550-3213 . 10.1016/0550-3213(93)90162-I . Yigal Shamir. hep-lat/9303005 . 1993NuPhB.406...90S . 16187316 .
- Möbius Fermions . Nuclear Physics B - Proceedings Supplements . 153 . 1 . 191–198 . 2006 . 0920-5632 . 10.1016/j.nuclphysbps.2006.01.047 . R.C. Brower and H. Neff and K. Orginos. hep-lat/0511031 . 2006NuPhS.153..191B . 118926750 .
- Book: Gattringer. C.. Lang. C.B.. 2009. Quantum Chromodynamics on the Lattice: An Introductory Presentation. Lecture Notes in Physics 788. 10.1007/978-3-642-01850-3. Springer. 10 More about lattice fermions. 249–253. 978-3642018497.
