Domain wall fermion

{{Short description|Lattice fermion discretisation}}

In lattice field theory, domain wall (DW) fermions are a fermion discretization avoiding the fermion doubling problem.{{cite journal

|title=A method for simulating chiral fermions on the lattice

|volume=288

|issn=0370-2693

|url=http://dx.doi.org/10.1016/0370-2693(92)91112-M

|doi=10.1016/0370-2693(92)91112-m

|number=3–4

|journal=Physics Letters B

|author=Kaplan, David B.

|year=1992

|pages=342–347 |arxiv=hep-lat/9206013

|bibcode=1992PhLB..288..342K

|s2cid=14161004

}} They are a realisation of Ginsparg–Wilson fermions in the infinite separation limit $L\_s\backslash rightarrow\backslash infty$ where they become equivalent to overlap fermions.{{cite journal

|title = Vectorlike gauge theories with almost massless fermions on the lattice

|author = Neuberger, Herbert

|journal = Phys. Rev. D

|volume = 57

|issue = 9

|pages = 5417–5433

|year = 1998

|publisher = American Physical Society

|doi = 10.1103/PhysRevD.57.5417

|arxiv = hep-lat/9710089

|bibcode = 1998PhRvD..57.5417N

|s2cid = 17476701

|url = https://link.aps.org/doi/10.1103/PhysRevD.57.5417}} DW fermions have undergone numerous improvements since Kaplan's original formulation such as the reinterpretation by Shamir and the generalisation to Möbius DW fermions by Brower, Neff and Orginos.{{cite journal

|title = Chiral fermions from lattice boundaries

|journal = Nuclear Physics B

|volume = 406

|number = 1

|pages = 90–106

|year = 1993

|issn = 0550-3213

|doi = 10.1016/0550-3213(93)90162-I

|url = https://dx.doi.org/10.1016/0550-3213%2893%2990162-I

|author = Yigal Shamir|arxiv = hep-lat/9303005

|bibcode = 1993NuPhB.406...90S

|s2cid = 16187316

}}{{cite journal

|title = Möbius Fermions

|journal = Nuclear Physics B - Proceedings Supplements

|volume = 153

|number = 1

|pages = 191–198

|year = 2006

|issn = 0920-5632

|doi = 10.1016/j.nuclphysbps.2006.01.047

|url = https://www.sciencedirect.com/science/article/pii/S0920563206000296

|author = R.C. Brower and H. Neff and K. Orginos|arxiv = hep-lat/0511031

|bibcode = 2006NuPhS.153..191B

|s2cid = 118926750

}}

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 $L\_s\backslash rightarrow\backslash infty$ 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 $P\_\backslash pm=(\backslash mathbf1\backslash pm\backslash gamma\_5)/2$ 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.{{cite book|last1=Gattringer|first1=C.|last2=Lang|first2=C.B.|date=2009|title=Quantum Chromodynamics on the Lattice: An Introductory Presentation|series=Lecture Notes in Physics 788|url=|doi=10.1007/978-3-642-01850-3|location=|publisher=Springer|chapter=10 More about lattice fermions|pages=249–253|isbn=978-3642018497}}

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