Physics: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]\displaystyle{ 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]\displaystyle{ d }[/math]-dimensional Euclidean spacetime is lifted into [math]\displaystyle{ d+1 }[/math] dimensions. The additional dimension of length [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ D_\text{DW}(x,s;y,r) = D(x;y)\delta_{sr} + \delta_{xy}D_{d+1}(s;r)\, }[/math]
with
- [math]\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)\, }[/math]
where [math]\displaystyle{ P_\pm=(\mathbf1\pm\gamma_5)/2 }[/math] is the chiral projection operator and [math]\displaystyle{ D }[/math] is the canonical Dirac operator in [math]\displaystyle{ d }[/math] dimensions. [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are (multi-)indices in the physical space whereas [math]\displaystyle{ s }[/math] and [math]\displaystyle{ 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).
References
- ↑ 1.0 1.1 Kaplan, David B. (1992). "A method for simulating chiral fermions on the lattice". Physics Letters B 288 (3–4): 342–347. doi:10.1016/0370-2693(92)91112-m. ISSN 0370-2693. Bibcode: 1992PhLB..288..342K. http://dx.doi.org/10.1016/0370-2693(92)91112-M.
- ↑ Neuberger, Herbert (1998). "Vectorlike gauge theories with almost massless fermions on the lattice". Phys. Rev. D (American Physical Society) 57 (9): 5417–5433. doi:10.1103/PhysRevD.57.5417. Bibcode: 1998PhRvD..57.5417N. https://link.aps.org/doi/10.1103/PhysRevD.57.5417.
- ↑ Yigal Shamir (1993). "Chiral fermions from lattice boundaries". Nuclear Physics B 406 (1): 90–106. doi:10.1016/0550-3213(93)90162-I. ISSN 0550-3213. Bibcode: 1993NuPhB.406...90S. https://dx.doi.org/10.1016/0550-3213%2893%2990162-I.
- ↑ R.C. Brower and H. Neff and K. Orginos (2006). "Möbius Fermions". Nuclear Physics B - Proceedings Supplements 153 (1): 191–198. doi:10.1016/j.nuclphysbps.2006.01.047. ISSN 0920-5632. Bibcode: 2006NuPhS.153..191B. https://www.sciencedirect.com/science/article/pii/S0920563206000296.
- ↑ 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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